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--- abstract: 'In this paper is proposed a geometric solution to the dark energy, assuming that the space can be divided into regions of size $\sim L_{p}$ and energy $\sim E_{p}$. Significantly this assumption generate a energy density similar to the energy density observed for the vaccum energy, the correct solution for the coincidence problem and the state equation characteristic of quintessence in the comoving coordinates. Similarly is studied the ultraviolet and infrarred limits and the amount of dark energy in the Universe.' author: - 'Miguel A. García-Aspeitia' title: 'About the Geometric Solution to the Problems of Dark Energy.' --- Introduction. ============= One of the most intriguing problem of the modern cosmology is the accelerated expansion of the Universe. The best explanation for this problem, is the existence of an unknown kind of energy not predicted by the standard model of particles nor by the general theory of relativity. This dark component is called dark energy (DE) with the property of accelerate the rate expansion of the Universe. Currently exist differents kind of models trying to explain the nature and the behavior of DE. For example the best models for DE are quintessence, phantom energy, cosmological constant and higher dimensional theories, each one of them try to mimic the behavior of DE. However the discordance between the theoretical predictions and the observations puts into question the validity of the models at large and Planck scales. To understand the detail of DE we will enumerate the main problems in the following way 1. *The fine tuning problem.* Observational evidence show that the energy density today must be $\vert\rho^{obs}\vert\leqslant2\times10^{-10}erg/cm^{3}$ [@Carroll], this imply that the theoretical predictions must relate two quantities appear to be unrelated [@Bousso1] $$t_{DE}\sim t_{obs} \label{time}$$ where $t_{DE}\sim\rho_{DE}^{-1/2}$ is the domination time of the DE and $t_{obs}$ is the time at which observer exist. 2. *The coincidence problem.* Another problem caused by DE is the coincidence where this problem can be summarized in the following question: Why the Universe starts the acceleration today ($\sim13.7\times10^{9} yrs$)? Any good model should address this question. 3. *The “central” Problems.* At this point we refer to the main features of DE as: the amount of DE in the Universe, the state equation and the convergence of $\langle\rho\rangle$ in the infrarred and ultraviolet limit. In the following sections we will focuse on addressing one by one the above points with the aim of found some physical explanation to the problem of DE. In the following, is used the CGS units, unless explicitly written. The three main problems of dark energy. ======================================= The fine tuning problem. ------------------------ Before to mention the proposal, it is important to stress that we work in a *physical coordinates* [@Liddle]. *The proposal.* The idea is assume that the Universe is full of minimal regions of equal size [@Miguel]. It is assumed that a region smaller that the one given by the equation will collapse into a Planckian black hole where all the physical information is lost. Then it is possible to write this minimal regions as $$L_{p}=\frac{1}{\sqrt{2}}\sqrt{\frac{\hbar G}{c^3}}\approx\sqrt{\frac{\hbar G}{c^3}}, \label{Lp}$$ where $G$ is the gravitational Newton constant, $c$ is the light velocity and $\hbar$ is the reduced Planck constant. It is important to remark that the last expression in the equation is the well known Planckian longitude. Then, intuitively we assume that inside of compact region of the Universe with size $L_{0}$ exist $n_{0}$ minimal regions of size $L_{p}$ immersed in the following way $$n_{0}=\left(\frac{L_{0}}{L_{p}}\right).$$ It is possible to observe that the we made a count of the number of Planck regions in a preferred direction and not in the three spatial directions of the space (see Figure \[fig:1\]). The answer to why this is done can have deeper implications, which will be discussed in the section IV. ![Sketch of the grid hypothesis. In the figure it is observed the minimal regions of the “bricks” $L_{p}$ and the minimal energy $E_{p}$ as well as the size of the Universe $L_{0}$ and the preferential direction of the counting $n_{0}$.[]{data-label="fig:1"}](Grid.pdf) Returning to the idea, it is possible to assume that each region have a energy [@Miguel] written as $$E_{p}=\frac{1}{2\sqrt{2}}\sqrt{\frac{\hbar c^{5}}{G}}\approx\sqrt{\frac{\hbar c^{5}}{G}}, \label{Ep}$$ where the last expression of the equation is the Planck energy. Then it is possible to obtain the total energy provided by all the regions in the Universe as $$\langle E_{T_{0}}\rangle\approx\sum_{i=1}^{n_{0}}E_{p_{i}}=n_{0}E_{p}.$$ On the other hand, observational evidence show that the Universe at large scales behaves as homogeneous and isotropic space expanding in time with a flat geometry. Flatness implies that the geometry of the hypersurface is Euclidean $\mathbb{R}^3$ then, it is possible to define the volumen of the Universe as [@Bousso] $$V_{0}\approx\frac{4}{3}\pi L^{3}_{0}.$$ Defining the energy density as $\langle\rho_{Y}\rangle=\langle E_{T_{0}}\rangle/V_{0}$ it is possible to write in the following way $$\langle\rho_{Y}\rangle_{L_{0}}\approx\frac{3}{4\pi L_{0}^{3}}\sum_{i=1}^{L_{0}/L_{p}}E_{pi}\approx\left(\frac{3c^4}{8\pi G}\right)L_{0}^{-2}. \label{enrgdens}$$ Now, using the evidence that the Universe is finite it is possible to compactified in a hypersurface and assign an approximate size of the Universe actually as $L_{0}\approx ct_{0}$, at a fixed moment of time, where $t_{0}$ is the actual age of the Universe [@Holo]. Then the previous expression can be written as $$\langle\rho_{Y}\rangle_{t_{0}}\approx\left(\frac{3c^2}{8\pi G}\right)t_{0}^{-2}, \label{t}$$ where it is shown the comparison between the present cosmological time and the energy density of the DE as $t_{0}\sim\rho_{Y}^{-1/2}$. Helping to solve the fine tuning problem where $\langle\rho_{Y}\rangle_{t_{0}}\approx8.62\times10^{-9}erg/cm^{3}$. The Recent Acceleration. ------------------------ Another problem for the dark energy is the coincidence of the recent acceleration ($\sim4.32\times10^{17} s$). In this model, we explore a possible solution to the problem in the following way: In fact, the Universe has a specific quantity of barions, radiation, neutrinos and dark matter with the possibility of colapse the Universe with its gravitational interactions. Then, it is possible to obtain with observations the total mass of the Universe as $M_{u}\sim10^{56}g$. With this amount of matter it is necessary a minimum of energy to accelerate the Universe which can be written as $$E_{min}=n_{acc}E_{p}\gtrsim G\frac{M_{u}^2}{L_{acc}},$$ where $n_{acc}=(L_{acc}/L_{p})$. Assuming that we know the Newtonian potential, it is straightforward to demonstrate the following equation $$L_{acc}\gtrsim\sqrt{\frac{GL_{p}}{E_{p}}}M_{u}=\frac{\sqrt{2}GM_{u}}{c^{2}}\approx R_{s}, \label{accel}$$ where $L_{acc}$ is the size of the Universe in the moment of acceleration. It is possible to observe that the minimal longitude must be approximately the Schwarzschild radius $R_{s}$. Adding numbers to the last equation is obtained $L_{acc}\gtrsim1.0482\times10^{28}$ $cm$ $\approx L_{0}$ this imply that $t_{acc}\approx3.49\times10^{17} s$ which coincides in a good way with the moment of acceleration. The Universe start the acceleration at this age and never before while the mass is of $M_{u}\sim10^{56}g$. The Amount of Dark Energy in the Universe. ------------------------------------------ In cosmology, the critical energy density relate the content in the Universe with its geometry and is defined as $$\rho_{crit}(t_{0})=\frac{3H^2_{0}c^{2}}{8\pi G}, \label{crit}$$ where $H_{0}$ is the Hubble rate today. If $\rho>\rho_{crit}$ the geometry is $\mathbb{S}^{3}$, $\rho\sim\rho_{crit}$ the geometry is $\mathbb{R}^{3}$ and $\rho<\rho_{crit}$ the geometry is $\mathbb{H}^{3}$. On the other hand, the equation can be written in terms of the Hubble parameter $H(t_{0})=H_{0}=t_{0}^{-1}$ due to the dimensionality between the Hubble parameter and the time $[H_{0}]=[seg^{-1}]=[t_{0}^{-1}]$. Therefore, it is possible to write $$\langle\rho_{Y}\rangle_{t_{0}}\approx\frac{3H^{2}_{0}}{8\pi G}c^2, \label{DE}$$ using dimensional analysis. Defining the density parameter as $\Omega_{Y}\equiv\langle\rho_{Y}\rangle/\rho_{crit}$ it is possible to obtain $$\Omega_{Y}\approx1.$$ For this reason we assume that this component is the dominant component in the actual Universe, coinciding with the observations about the dark energy which is more than $70\%$ of the known Universe. The Limits and the State Equation. ================================== The Ultraviolet and Infrarred Limit. ------------------------------------ Now, the questions is: What is the two limits of the equation ? What is the behavior of the energy density in the singularity? and What is the behavior of the energy density in infinity? 1. *Ultraviolet Limit ($L\to\infty$).* When the Universe tends to infinity, the equation can be written as $$\lim_{L\to\infty}\langle\rho_{Y}\rangle_{L}\approx\lim_{L\to\infty}\frac{3}{4\pi L^{3}}\sum_{i=1}^{L/L_{p}}E_{pi}\to0, \label{limsup}$$ this imply that the equation of the energy density for the DE is convergent to zero in infinity reducing the energy density as the Universe grows. In comparison, the integral energy density is divergent in infinity, for this reason must be calculated with a wave number cutoff [@Weinberg] as $$\langle\rho\rangle=\int_{0}^{\Lambda}\frac{4\pi k^{2}c^{2}dk}{(2\pi\hbar)^{3}}\frac{1}{2}\sqrt{k^{2}+m^{2}}\approx\frac{\Lambda^{4}}{16\pi^{2}},$$ where $\Lambda>>m$. 2. *Infrarred Limit ($L\to L_{p}$).* On the other when the Universe decreases it is assumed that the minimal length will be $L_{p}$, then from the equation it is obtained the following expression $$\begin{aligned} \lim_{L\to L_{p}}\langle\rho_{Y}\rangle_{L}&&\approx\lim_{L\to L_{p}}\frac{3}{4\pi L^{3}}\sum_{i=1}^{L/L_{p}}E_{pi}\nonumber\\&&\approx\frac{3c^{7}}{4\pi G^{2}\hbar}, \label{liminf}\end{aligned}$$ the overwhelming Planck energy density shown in the equation could plunge us into a Universe always dominated by the dark sector, however inflation could help us to solve the problem because the exponential growth of the e-foldings permit the entry of a greater amount of space controlling the growth of the energy provided by the Planck regions (see equation ). The State Equation. ------------------- Now we focuse in obtain the state equation for this characteristic density $\rho_{Y}$. Then from the equation it is possible to write $$\langle\rho_{Y}\rangle\approx\rho_{0}a^{-2}, \label{III1}$$ where $$\rho_{0}=\frac{3c^{2}}{8\pi Gd^{2}},$$ where we fix at a comoving coordinates system $\vec{L}=a(t)\vec{d}$ and $a(t)$ is the scale factor [@Liddle]. On the other hand the state equation can be written as $\omega_{Y}\equiv\langle p_{Y}\rangle/\langle\rho_{Y}\rangle$ and the conservation law in a comoving system reads as $$\frac{\partial}{\partial t}\langle\rho_{Y}\rangle+3\frac{\dot{a}}{a}\left[\langle\rho_{Y}\rangle+\langle p_{Y}\rangle\right]=0, \label{III3}$$ with the equations and the pressure can be written as $\langle p_{Y}\rangle\approx-(1/3)\rho_{0}a^{-2}$, this implies that the state equation reads as $$\omega_{Y}\approx-\frac{1}{3},$$ where the last equation is clearly similar to the state equation of quintessence $-1<\omega<-1/3$ where the energy density decreases with the scale factor as $\rho_{Q}\varpropto a(t)^{-3(1+\omega)}$. Discussion. =========== It is important to remark that the proposed hypothesis generates a good results to the problems of dark energy. This may mean that the Universe is made of minimum regions $\sim L_{p}$ and minimum energy $\sim E_{p}$ such as a quantum grid. An important feature is the count in a preferential direction of the number of regions $n_{0}$ in the grid, this has important consequences in the understand of the microscopic structure of the space time. For example, if we do not consider a preferential direction and do the count in the three spatial directions $n_{0}^{3}$ (just as intuition tell us) we obtain the following expression for the energy density $$\langle\rho^{3D}\rangle\approx\frac{3}{4\pi}\left(\frac{E_{p}}{L^{3}_{p}}\right)=\rho_{0}^{3D}=cte,$$ where the numerical value of the last expression is $\langle\rho^{3D}\rangle\approx1.3866\times10^{113}erg/cm^{3}$ regaining the problem of vaccum energy implemented by the energy density integral calculated by @Weinberg. Even more, using the state equation and the conservation law it is possible to obtain $$\langle p^{3D}\rangle\approx-\langle\rho^{3D}\rangle,$$ where the last equation is associated with the cosmological constant $\omega\approx-1$. Summarizing, if we perform the count in a preferred direction, we get the values needed to adjust to the observations of dark energy being quintessence the model associated with that hypothesis. On the other hand if we perform the count in the three dimensional directions we regaining the problem of the vaccum energy and the associated model is the cosmological constant. This leads us to believe that there is a preferential direction which the counting is done, violating Lorentz symmetry. However an interesting alternative *ad hoc* to this model (instead of considering a preferential direction) is consider that the Universe can be one dimensional at short scales and at large scales behave as a four dimensional Universe as the proposition of @Dimensions avoiding the preferential direction. It should be noted that this model can be considered as a toy model, where the results should be formalized, however can help us to clarify questions about dark energy currently unsolved. Further investigation may be made to study inflation since not only could help us to the problem of cosmic acceleration, but it also could provide clues about inflation. Acknowledgement. ================ I thank Yasmín Alcántara for encouragement to publish this result. In the same way I would like to acknowledge the help and the enlightening conversation of Abdel Pérez-Lorenzana and Juan Magaña. This work was partially supported by CONACyT México, under grant 49865-F, 216536/219673, Instituto Avanzado de Cosmologia (IAC) collaboration. [99]{} A. Liddle. Wiley, Second Edition. R. Bousso, B. Freivogel, S. Leichenauer and V. Rosenhaus. arXiv:1012.2869v1 \[hep-th\] (2010) S. M. Carroll, Liv. Rev. in Rel. arXiv:astro-ph/0004075v2 S. Lloyd, Phys. Rev. Lett., **23**, Vol 88 (2002) E. Lattman, Eur. J. Phys. **30** (2009) R. Bousso, Rev. Mod. Phys. **74**: 825-874. arXiv:hep-th/0203101v2 (2002) M. A. García-Aspeitia. Submitted to Classical and Quantum Gravity. S. Weinberg, Rev. Mod. Phys. Vol. **61**, No. 1 (1989) Anchordoqui L., Dai D., Fairbairn M., Landsberg G and Stojkovic D., arXiv:1002.5914v1 (2010)

--- abstract: 'The concepts of Feynman integrals in white noise analysis are used to realize the Feynman integrand for a charged particle in a constant magnetic field as a Hida distribution. For this purpose we identify the velocity dependent potential as a so called generalized Gauss kernel.' address: - | Functional Analysis and Stochastic Analysis Group,\ Department of Mathematics,\ University of Kaiserslautern, 67653 Kaiserslautern, Germany - | Functional Analysis and Stochastic Analysis Group,\ Department of Mathematics,\ University of Kaiserslautern, 67653 Kaiserslautern, Germany - | Functional Analysis and Stochastic Analysis Group,\ Department of Mathematics,\ University of Kaiserslautern, 67653 Kaiserslautern, Germany author: - Wolfgang Bock - Martin Grothaus - Sebastian Jung title: The Feynman integrand for the Charged Particle in a Constant Magnetic field as White Noise Distribution --- Introduction ============ As an alternative approach to quantum mechanics Feynman introduced the concept of path integrals ([@F48; @Fe51; @FeHi65]), which was developed into an extremely useful tool in many branches of theoretical physics. In this article we use concepts for realizing Feynman integrals in the framework of white noise analysis. The Feynman integral for a particle moving from $0$ at time $0$ to $\mathbf{y} \in {\mathbb{R}}^d$ at time $t$ under the potential $V$ is given by $$\label{eqnfey} {\rm N} \int_{\mathbf{x}(0)=0, \mathbf{x}(t)=\mathbf{y}} \int \exp\left(\frac{i}{\hbar} \int_0^t \frac{1}{2}m\dot{\mathbf{x}}^2 -V(\mathbf{x},\dot{\mathbf{x}}) \, d\tau \right) \prod_{0<\tau<t} d\mathbf{x}(\tau),\quad \hbar = \frac{h}{2\pi}.$$ Here $h$ is Planck’s constant, and the integral is thought of being over all paths with $\mathbf{x}(0)=0$ and $\mathbf{x}(t)=\mathbf{y}$.\ In the last fifty years there have been many approaches for giving a mathematically rigorous meaning to the Feynman integral by using e.g.  analytic continuation,limits of finite dimensional approximations or Fresnel integrals. Instead of giving a complete list of publications concerning Feynman integrals we refer to [@AHKM08] and the references therein. Here we choose a white noise approach. white noise analysis is a mathematical framework which offers generalizations of concepts from finite-dimensional analysis, like differential operators and Fourier transform to an infinite-dimensional setting. We give a brief introduction to white noise analysis in Section 2, for more details see [@Hid80; @HKPS93; @Ob94; @BK95; @Kuo96]. Of special importance in white noise analysis are spaces of generalized functions and their characterizations. In this article we choose the space of Hida distributions, see Section 2.\ The idea of realizing Feynman integrals within the white noise framework goes back to [@HS83]. There the authors used exponentials of quadratic (generalized) functions in order to give meaning to the Feynman integral in configuration space representation $${\rm N}\int_{\mathbf{x}(0) =0, \mathbf{x}(t)=y} \exp\left(\frac{i}{\hbar} S(\mathbf{x}) \right) \, \prod_{0<\tau<t} \, d\mathbf{x}(\tau) ,\quad \hbar = \frac{h}{2\pi},$$ with the classical action $S(\mathbf{x})= \int_0^t \frac{1}{2} m \dot{\mathbf{x}}^2 -V(\mathbf{x})\, d\tau$. We use these concepts of quadratic actions in white noise analysis, which were further developed in [@GS98a] and [@BG10] to give a rigorous meaning to the Feynman integrand $$\begin{gathered} \label{integrandpot} I_V = {\rm Nexp}\left( \frac{i}{\hbar}\int_0^t \frac{m}{2} \dot{\mathbf{x}}(\tau)^2 d\tau +\frac{1}{2}\int_0^t \dot{\mathbf{x}}(\tau)^2 d\tau\right)\\ \times \exp\left(-\frac{i}{\hbar} \int_0^t V(\mathbf{x}(\tau),\dot{\mathbf{x}}(\tau),\tau) \, d\tau\right) \cdot \delta_0(\mathbf{x}(t)-y)\end{gathered}$$ as a Hida distribution. In this expression the sum of the first and the third integral in the exponential is the action $S(\mathbf{x},\mathbf{\dot{x}})$, and the delta function (Donsker’s delta) serves to pin trajectories to $\mathbf{y}$ at time $t$. The second integral is introduced to simulate the locally Lebesgue integral by a local compensation of the fall-off of the Gaussian reference measure $\mu$. Furthermore we use a two-dimensional Brownian motion starting in $0$ as the path i.e. $$\label{varchoice} \mathbf{x}(\tau)=\sqrt{\frac{\hbar}{m}}\mathbf{B}(\tau).$$ The construction is done in terms of the $T$-transform (infinite-dimensional version of the Fourier transform w.r.t a Gaussian measure), which characterizes Hida distributions, see Theorem \[charthm\]. At the same time, the $T$-transform of the constructed Feynman integrands provides us with their generating functional. Finally using the generating functional, we can show that the generalized expectation (generating functional at zero) gives the Greens function to the corresponding Schrödinger equation.\ In this article we consider the potential given by the action of a constant magnetic field to a moving particle. From classical physics it is well-known, that a magnetic field is influencing the so-called Lorentz force on a charged particle moving through this field. The corresponding potential term of a charged particle moving in the $(1,2)$-plane is given by $$({\mathbf{x}},\dot{{\mathbf{x}}}) \mapsto V_{\rm{mag}}({\mathbf{x}},\dot{{\mathbf{x}}})= -\frac{q H_3}{c} \left(x_1\dot{x_2}-\dot{x_1}x_2\right),$$ where $q$ is the charge, $H_3$ the strength of the magnetic field vector orthogonal to the $(1,2)$-plane and $c$ the speed of light. These are the core results of this article: - The concepts of generalized Gauss kernels from [@GS98a] and [@BG10] are used to construct the Feynman integrand for a charged particle in a constant magnetic field as a Hida distribution, see Theorem \[magnetictheorem\]. - The results in Theorem \[magnetictheorem\] provide us with the generating functional for a charged particle in a constant magnetic field. - The generalized expectations (generating functional at zero) yields the Greens functions to the corresponding Schrödinger equation. White Noise Analysis ==================== Gel’fand Triples ---------------- Starting point is the Gel’fand triple $S_d({\mathbb{R}}) \subset L^2_d({\mathbb{R}},dx) \subset S'_d({\mathbb{R}})$ of the ${\mathbb{R}}^d$-valued, $d \in {\mathbb{N}}$, Schwartz test functions and tempered distributions with the Hilbert space of (equivalence classes of) ${\mathbb{R}}^d$-valued square integrable functions w.r.t. the Lebesgue measure as central space (equipped with its canonical inner product $(\cdot, \cdot)$ and norm $\|\cdot\|$), see e.g.  [@W95 Exam. 11]. Since $S_d({\mathbb{R}})$ is a nuclear space, represented as projective limit of a decreasing chain of Hilbert spaces $(H_p)_{p\in {\mathbb{N}}}$, see e.g. [@RS75a Chap. 2] and [@GV68], i.e.  $$S_d({\mathbb{R}}) = \bigcap_{p \in {\mathbb{N}}} H_p,$$ we have that $S_d({\mathbb{R}})$ is a countably Hilbert space in the sense of Gel’fand and Vilenkin [@GV68]. We denote the inner product and the corresponding norm on $H_p$ by $(\cdot,\cdot)_p$ and $\|\cdot\|_p$, respectively, with the convention $H_0 = L^2_d({\mathbb{R}}, dx)$. Let $H_{-p}$ be the dual space of $H_p$ and let $\langle \cdot , \cdot \rangle$ denote the dual pairing on $H_{p} \times H_{-p}$. $H_{p}$ is continuously embedded into $L^2_d({\mathbb{R}},dx)$. By identifying $L_d^2({\mathbb{R}},dx)$ with its dual $L_d^2({\mathbb{R}},dx)'$, via the Riesz isomorphism, we obtain the chain $H_p \subset L_d^2({\mathbb{R}}, dx) \subset H_{-p}$. Note that $\displaystyle S'_d({\mathbb{R}})= \bigcup_{p\in {\mathbb{N}}} H_{-p}$, i.e. $S'_d({\mathbb{R}})$ is the inductive limit of the increasing chain of Hilbert spaces $(H_{-p})_{p\in {\mathbb{N}}}$, see e.g. [@GV68]. We denote the dual pairing of $S_d({\mathbb{R}})$ and $S'_d({\mathbb{R}})$ also by $\langle \cdot , \cdot \rangle$. Note that its restriction on $S_d({\mathbb{R}}) \times L_d^2({\mathbb{R}}, dx)$ is given by $(\cdot, \cdot )$. We also use the complexifications of these spaces denoted with the subindex ${\mathbb{C}}$ (as well as their inner products and norms). The dual pairing we extend in a bilinear way. Hence we have the relation $$\langle g,f \rangle = (\mathbf{g},\overline{\mathbf{f}}), \quad \mathbf{f},\mathbf{g} \in L_d^2({\mathbb{R}})_{{\mathbb{C}}},$$ where the overline denotes the complex conjugation. White Noise Spaces ------------------ We consider on $S_d' ({\mathbb{R}})$ the $\sigma$-algebra ${\mathcal{C}}_{\sigma}(S_d' ({\mathbb{R}}))$ generated by the cylinder sets $\{ \omega \in S_d' ({\mathbb{R}}) | \langle \xi_1, \omega \rangle \in F_1, \dots ,\langle \xi_n, \omega \rangle \in F_n\} $, $\xi_i \in S_d({\mathbb{R}})$, $ F_i \in {\mathcal{B}}({\mathbb{R}}),\, 1\leq i \leq n,\, n\in {\mathbb{N}}$, where ${\mathcal{B}}({\mathbb{R}})$ denotes the Borel $\sigma$-algebra on ${\mathbb{R}}$.\ The canonical Gaussian measure $\mu$ on $C_{\sigma}(S_d'({\mathbb{R}}))$ is given via its characteristic function $$\begin{aligned} \int_{S_d' ({\mathbb{R}})} \exp(i \langle {\bf f}, \boldsymbol{\omega} \rangle ) d\mu(\boldsymbol{\omega}) = \exp(- \tfrac{1}{2} \| {\bf f}\|^2 ), \;\;\; {\bf f} \in S_d({\mathbb{R}}),\end{aligned}$$ by the theorem of Bochner and Minlos, see e.g. [@Mi63], [@BK95 Chap. 2 Theo. 1. 11]. The space $(S_d'({\mathbb{R}}),{\mathcal{C}}_{\sigma}(S_d'({\mathbb{R}})), \mu)$ is the basic probability space in our setup. The central Gaussian spaces in our framework are the Hilbert spaces $(L^2):= L^2(S_d'({\mathbb{R}}),$ ${\mathcal{C}}_{\sigma}(S_d' ({\mathbb{R}})),\mu)$ of complex-valued square integrable functions w.r.t. the Gaussian measure $\mu$.\ Within this formalism a representation of a d-dimensional Brownian motion is given by $$\label{BrownianMotion} {\bf B}_t ({\boldsymbol \omega}) :=(B_t(\omega_1), \dots, B_t(\omega_d)):= ( \langle {\mathbf{1}}_{[0,t)},\omega_1 \rangle, \dots \langle {\mathbf{1}}_{[0,t)},\omega_d \rangle),$$ with ${\boldsymbol \omega}=(\omega_1,\dots, \omega_d) \in S'_d({\mathbb{R}}),\quad t \geq 0,$ in the sense of an $(L^2)$-limit. Here ${\mathbf{1}}_A$ denotes the indicator function of a set $A$. The Hida triple --------------- Let us now consider the Hilbert space $(L^2)$ and the corresponding Gel’fand triple $$(S) \subset (L^2) \subset (S)'.$$ Here $(S)$ denotes the space of Hida test functions and $(S)'$ the space of Hida distributions. In the following we denote the dual pairing between elements of $(S)$ and $(S)'$ by $\langle \! \langle \cdot , \cdot \rangle \!\rangle$. Instead of reproducing the construction of $(S)'$ here we give its characterization in terms of the $T$-transform.\ We define the $T$-transform of $\Phi \in (S)'$ by $$T\Phi({\bf f}) := \langle\!\langle \exp(i \langle {\bf f}, \cdot \rangle),\Phi \rangle\!\rangle, \quad {\bf f}:= ({ f_1}, \dots ,{ f_d }) \in S_{d}({\mathbb{R}}).$$ - Since $\exp(i \langle {\bf f},\cdot \rangle) \in (S)$ for all $f \in S_d({\mathbb{R}})$, the $T$-transform of a Hida distribution is well-defined. - For ${\bf f} = 0$ the above expression yields $\langle\!\langle \Phi, 1 \rangle\!\rangle$, therefore $T\Phi(0)$ is called the generalized expectation of $\Phi \in (S)'$. In order to characterize the space $(S)'$ by the $T$-transform we need the following definition. A mapping $F:S_{d}({\mathbb{R}}) \to {\mathbb{C}}$ is called a [*U*-functional]{} if it satisfies the following conditions: - For all ${\bf{f, g}} \in S_{d}({\mathbb{R}})$ the mapping ${\mathbb{R}}\ni \lambda \mapsto F(\lambda {\bf f} +{\bf g} ) \in {\mathbb{C}}$ has an analytic continuation to $\lambda \in {\mathbb{C}}$ ([**[ray analyticity]{}**]{}). - There exist constants $0<C,D<\infty$ and a $p \in {\mathbb{N}}_0$ such that $$|F(z{\bf f})|\leq C\exp(D|z|^2 \|{\bf f} \|_p^2),$$ for all $z \in {\mathbb{C}}$ and ${\bf f} \in S_{d}({\mathbb{R}})$ ([**[growth condition]{}**]{}). This is the basis of the following characterization theorem. For the proof we refer to [@PS91; @Kon80; @HKPS93; @KLPSW96]. \[charthm\] A mapping $F:S_{d}({\mathbb{R}}) \to {\mathbb{C}}$ is the $T$-transform of an element in $(S)'$ if and only if it is a U-functional. Theorem \[charthm\] enables us to discuss convergence of sequences of Hida distributions by considering the corresponding $T$-transforms, i.e.  by considering convergence on the level of U-functionals. The following corollary is proved in [@PS91; @HKPS93; @KLPSW96]. \[seqcor\] Let $(\Phi_n)_{n\in {\mathbb{N}}}$ denote a sequence in $(S)'$ such that: - For all ${\bf f} \in S_{d}({\mathbb{R}})$, $((T\Phi_n)({\bf f}))_{n\in {\mathbb{N}}}$ is a Cauchy sequence in ${\mathbb{C}}$. - There exist constants $0<C,D<\infty$ such that for some $p \in {\mathbb{N}}_0$ one has $$|(T\Phi_n)(z{\bf f })|\leq C\exp(D|z|^2\|{\bf f}\|_p^2)$$ for all ${\bf f} \in S_{d}({\mathbb{R}}),\, z \in {\mathbb{C}}$, $n \in {\mathbb{N}}$. Then $(\Phi_n)_{n\in {\mathbb{N}}}$ converges strongly in $(S)'$ to a unique Hida distribution. Let $\,{\bf{B}}(t)$, $t\geq 0$, be the $d$-dimensional Brownian motion as in . Consider $$\frac{{\bf{B}}(t+h,\boldsymbol{\omega}) - {\bf{B}}(t,\boldsymbol{\omega})}{h} = (\langle \frac{{\mathbf{1}}_{[t,t+h)}}{h} , \omega_1 \rangle , \dots (\langle \frac{{\mathbf{1}}_{[t,t+h)}}{h} , \omega_d \rangle),\quad h>0.$$ Then in the sense of Corollary \[seqcor\] it exists $$\begin{aligned} \langle {\boldsymbol\delta_t}, {\boldsymbol \omega} \rangle := (\langle \delta_t,\omega_1 \rangle, \dots ,\langle \delta_t,\omega_d \rangle):= \lim_{h\searrow 0} \frac{{\bf{B}}(t+h,\boldsymbol{\omega}) - {\bf{B}}(t,\boldsymbol{\omega})}{h}.\end{aligned}$$ Of course for the left derivative we get the same limit. Hence it is natural to call the generalized process $\langle {\boldsymbol\delta_t}, {\boldsymbol \omega} \rangle$, $t\geq0$ in $(S)'$ vector valued white noise. One also uses the notation ${\boldsymbol \omega}(t) =\langle{\boldsymbol\delta_t}, {\boldsymbol \omega} \rangle$, $t\geq 0$. Another useful corollary of Theorem \[charthm\] concerns integration of a family of generalized functions, see [@PS91; @HKPS93; @KLPSW96]. \[intcor\] Let $(\Lambda, {\mathcal{A}}, \nu)$ be a measure space and $\Lambda \ni\lambda \mapsto \Phi(\lambda) \in (S)'$ a mapping. We assume that its $T$–transform $T \Phi$ satisfies the following conditions: 1. The mapping $\Lambda \ni \lambda \mapsto T(\Phi(\lambda))({\bf f})\in {\mathbb{C}}$ is measurable for all ${\bf f} \in S_d({\mathbb{R}})$. 2. There exists a $p \in {\mathbb{N}}_0$ and functions $D \in L^{\infty}(\Lambda, \nu)$ and $C \in L^1(\Lambda, \nu)$ such that $${\left|T(\Phi(\lambda))(z{\bf f})\right|} \leq C(\lambda)\exp(D(\lambda) {\left|z\right|}^2 {\left\|{\bf f}\right\|}^2),$$ for a.e. $ \lambda \in \Lambda$ and for all ${\bf f} \in S_d({\mathbb{R}})$, $z\in {\mathbb{C}}$. Then, in the sense of Bochner integration in $H_{-q} \subset (S)'$ for a suitable $q\in {\mathbb{N}}_0$, the integral of the family of Hida distributions is itself a Hida distribution, i.e. $\!\displaystyle \int_{\Lambda} \Phi(\lambda) \, d\nu(\lambda) \in (S)'$ and the $T$–transform interchanges with integration, i.e.  $$T\left( \int_{\Lambda} \Phi(\lambda) \, d\nu(\lambda) \right)(\mathbf{f}) = \int_{\Lambda} T(\Phi(\lambda))(\mathbf{f}) \, d\nu(\lambda), \quad \mathbf{f} \in S_d({\mathbb{R}}).$$ Based on the above theorem, we introduce the following Hida distribution. \[D:Donsker\] We define Donsker’s delta at $x \in {\mathbb{R}}$ corresponding to $0 \neq {\boldsymbol\eta} \in L_{d}^2({\mathbb{R}})$ by $$\delta_0(\langle {\boldsymbol\eta},\cdot \rangle-x) := \frac{1}{2\pi} \int_{{\mathbb{R}}} \exp(i \lambda (\langle {\boldsymbol\eta},\cdot \rangle -x)) \, d \lambda$$ in the sense of Bochner integration, see e.g. [@HKPS93; @LLSW94; @W95]. Its $T$–transform in ${\bf f} \in S_d({\mathbb{R}})$ is given by $$T(\delta_0(\langle {\boldsymbol\eta},\cdot \rangle-x)({\bf f}) = \frac{1}{\sqrt{2\pi \langle {\boldsymbol\eta}, {\boldsymbol\eta}\rangle}} \exp\left( -\frac{1}{2\langle {\boldsymbol\eta},{\boldsymbol\eta} \rangle}(i\langle {\boldsymbol\eta},{\bf f} \rangle - x)^2 -\frac{1}{2}\langle {\bf f},{\bf f}\rangle \right), \, \, \mathbf{f} \in S_d({\mathbb{R}}).$$ Generalized Gauss Kernels ------------------------- Here we review a special class of Hida distributions which are defined by their T-transform, see e.g. [@HS83],[@HKPS93],[@GS98a]. Proofs and more details for can be found in [@BG10]. Let $\mathcal{B}$ be the set of all continuous bilinear mappings $B:S_{d}({\mathbb{R}}) \times S_{d}({\mathbb{R}}) \to {\mathbb{C}}$. Then the functions $$S_d({\mathbb{R}})\ni \mathbf{f} \mapsto \exp\left(-\frac{1}{2} B({\bf f},{\bf f})\right) \in {\mathbb{C}}$$ for all $B\in \mathcal{B}$ are U-functionals. Therefore, by using the characterization of Hida distributions in Theorem \[charthm\], the inverse T-transform of these functions $$\Phi_B:=T^{-1} \exp\left(-\frac{1}{2} B\right)$$ are elements of $(S)'$. \[GGK\] The set of [**[generalized Gauss kernels]{}**]{} is defined by $$GGK:= \{ \Phi_B,\; B\in \mathcal{B} \}.$$ [[@GS98a]]{} \[Grotex\] We consider a symmetric trace class operator $\mathbf{K}$ on $L^2_{d}({\mathbb{R}})$ such that $-\frac{1}{2}<\mathbf{K}\leq 0$, then $$\begin{aligned} \int_{S'_{d}({\mathbb{R}})} \exp\left(- \langle \omega,\mathbf{K} \omega\rangle \right) \, d\mu(\boldsymbol{\omega}) = \left( \det(\mathbf{Id +2K})\right)^{-\frac{1}{2}} < \infty.\end{aligned}$$ For the definition of $\langle \cdot,\mathbf{K} \cdot \rangle$ see the remark below. Here $\mathbf{Id}$ denotes the identity operator on the Hilbert space $L^2_{d}({\mathbb{R}})$, and $\det(\mathbf{A})$ of a symmetric trace class operator $\mathbf{A}$ on $L^2_{d}({\mathbb{R}})$ denotes the infinite product of its eigenvalues, if it exists. In the present situation we have $\det(\mathbf{Id +2K})\neq 0$. Therefore we obtain that the exponential $g= \exp(-\frac{1}{2} \langle \cdot,\mathbf{K} \cdot \rangle)$ is square-integrable and its T-transform is given by $$Tg({\bf f}) = \left( \det(\mathbf{Id+K}) \right)^{-\frac{1}{2}} \exp\left(-\frac{1}{2} ({\bf f}, \mathbf{(Id+K)^{-1}} {\bf f})\right), \quad {\bf f} \in S_{d}({\mathbb{R}}).$$ Therefore $\left( \det(\mathbf{Id+K}) \right)^{\frac{1}{2}}g$ is a generalized Gauss kernel. - \[traceL2\] Since a trace class operator is compact, see e.g. [@RS75a], we have that $\mathbf{K}$ in the above example is diagonalizable, i.e.  $$\mathbf{K}\mathbf{f} = \sum_{k=1}^{\infty} k_n (\mathbf{f},\mathbf{e}_n)\mathbf{e}_n, \quad \mathbf{f} \in L_d^2({\mathbb{R}},dx),$$ where $(\mathbf{e}_n)_{n\in {\mathbb{N}}}$ denotes an eigenbasis of the corresponding eigenvalues $(k_n)_{n\in {\mathbb{N}}}$ with $k_n \in (-\frac{1}{2}, 0 ]$, for all $n \in {\mathbb{N}}$. Since $K$ is compact, we have that $\lim\limits_{n\to \infty} k_n =0$ and since $\mathbf{K}$ is trace class we also have $\sum_{n=1}^{\infty} (\mathbf{e}_n, -\mathbf{K} \mathbf{e}_n)< \infty$. We define for ${\boldsymbol \omega }\in S_d'({\mathbb{R}})$ $$\begin{aligned} - \langle {\boldsymbol \omega }, \mathbf{K} {\boldsymbol \omega } \rangle := \lim_{N \to \infty} \sum_{n=1}^N \langle \mathbf{e}_n, {\boldsymbol \omega }\rangle (-k_n)\langle \mathbf{e}_n,{\boldsymbol \omega } \rangle. \end{aligned}$$ Then as a limit of measurable functions ${\boldsymbol \omega } \mapsto -\langle {\boldsymbol \omega }, \mathbf{K} {\boldsymbol \omega } \rangle$ is measurable and hence $$\begin{aligned} \int\limits_{S_d'({\mathbb{R}})} \exp(- \langle {\boldsymbol \omega }, \mathbf{K} {\boldsymbol \omega }\rangle ) \, d\mu({\boldsymbol \omega }) \in [0, \infty].\end{aligned}$$ The explicit formula for the $T$-transform and expectation then follow by a straightforward calculation with help of the above limit procedure. - In the following, if we apply operators or bilinear forms defined on $L^2_d({\mathbb{R}})$ to generalized functions from $S'_d({\mathbb{R}})$, we are always having in mind the interpretation as in \[traceL2\]. \[D:Nexp\][@BG10]$\;$ Let $\mathbf{K}: L^2_d({\mathbb{R}}, dx)_{{\mathbb{C}}} \to L^2_d({\mathbb{R}}, dx)_{{\mathbb{C}}}$ be linear and continuous such that: - $\mathbf{Id+K}$ is injective. - There exists $p \in {\mathbb{N}}_0$ such that $(\mathbf{Id+K})(L^2_{d}({\mathbb{R}},\,dx)_{{\mathbb{C}}}) \subset H_{p,{\mathbb{C}}}$ is dense. - There exist $q \in{\mathbb{N}}_0$ such that $\mathbf{(Id+K)^{-1}} :H_{p,{\mathbb{C}}} \to H_{-q,{\mathbb{C}}}$ is continuous with $p$ as in (ii). Then we define the normalized exponential $$\label{Nexp} {\rm{Nexp}}(- \frac{1}{2} \langle \cdot ,\mathbf{K} \cdot \rangle)$$ by $$T({\rm{Nexp}}(- \frac{1}{2} \langle \cdot ,\mathbf{K} \cdot \rangle))({\bf f}) := \exp(-\frac{1}{2} \langle {\bf f}, \mathbf{(Id+K)^{-1}} {\bf f} \rangle),\quad {\bf f} \in S_d({\mathbb{R}}).$$ The “normalization” of the exponential in the above definition can be regarded as a division of a divergent factor. In an informal way one can write $$\begin{gathered} T({\rm{Nexp}}(- \frac{1}{2} \langle \cdot ,\mathbf{K} \cdot \rangle))({\mathbf f})=\frac{T(\exp(- \frac{1}{2} \langle \cdot ,\mathbf{K} \cdot \rangle))(\mathbf{f})}{T(\exp(- \frac{1}{2} \langle \cdot ,\mathbf{K} \cdot \rangle))(0)}\\ =\frac{T(\exp(- \frac{1}{2} \langle \cdot ,\mathbf{K} \cdot \rangle))(\mathbf{f})}{\sqrt{\det(\mathbf{Id+K})}} , \quad {\bf f} \in S_d({\mathbb{R}}), \end{gathered}$$ i.e.  if the determinant in the Example \[Grotex\] above is not defined, we can still define the normalized exponential by the T-transform without the diverging prefactor. The assumptions in the above definition then guarantee the existence of the generalized Gauss kernel in . \[pointprod\] For sufficiently “nice” operators $\mathbf{K}$ and $\mathbf{L}$ on $L^2_{d}({\mathbb{R}})_{{\mathbb{C}}}$ we can define the product $${\rm{Nexp}}\big( - \frac{1}{2} \langle \cdot,\mathbf{K} \cdot \rangle \big) \cdot \exp\big(-\frac{1}{2} \langle \cdot,\mathbf{L}\cdot \rangle \big)$$ of two square-integrable functions. Its $T$-transform is then given by $$\begin{gathered} T\Big({\rm{Nexp}}( - \frac{1}{2} \langle \cdot,\mathbf{K} \cdot\rangle ) \cdot \exp( - \frac{1}{2} \langle \cdot,\mathbf{L} \cdot\rangle )\Big)({\bf f})\\ =\sqrt{\frac{1}{\det(\mathbf{Id+L(Id+K)^{-1}})}} \exp(-\frac{1}{2} \langle {\bf f}, \mathbf{(Id+K+L)^{-1}} {\bf f} \rangle ),\quad {\bf f} \in S_{d}({\mathbb{R}}), \end{gathered}$$ in the case the right hand side indeed is a U-funcional. \[prodnexp\] Let $\mathbf{K}: L^2_{d}({\mathbb{R}}, dx)_{{\mathbb{C}}} \to L^2_{d}({\mathbb{R}}, dx)_{{\mathbb{C}}}$ be as in Definition \[D:Nexp\], i.e. ${\rm{Nexp}}(- \frac{1}{2} \langle \cdot ,\mathbf{K} \cdot \rangle)$ exists. Furthermore let $\mathbf{L}: L^2_d({\mathbb{R}}, dx)_{{\mathbb{C}}} \to L^2_d({\mathbb{R}}, dx)_{{\mathbb{C}}}$ be trace class. Then we define $${\rm{Nexp}}( - \frac{1}{2} \langle \cdot,\mathbf{K} \cdot\rangle ) \cdot \exp( - \frac{1}{2} \langle \cdot,\mathbf{L} \cdot\rangle )$$ via its $T$-transform, whenever $$\begin{gathered} T\Big({\rm{Nexp}}( - \frac{1}{2} \langle \cdot,\mathbf{K} \cdot\rangle ) \cdot \exp( - \frac{1}{2} \langle \cdot,\mathbf{L} \cdot\rangle )\Big)({\bf f})\\ =\sqrt{\frac{1}{\det(\mathbf{Id+L(Id+K)^{-1}})}} \exp(-\frac{1}{2} \langle {\bf f}, \mathbf{(Id+K+L)^{-1}} {\bf f} \rangle ),\quad {\bf f} \in S_{d}({\mathbb{R}}), \end{gathered}$$ is a U-functional. In the case $\mathbf{g} \in S_d({\mathbb{R}})$, $c\in{\mathbb{C}}$ the product between the Hida distribution $\Phi$ and the Hida test function $\exp(i \langle \mathbf{g},. \rangle + c)$ can be defined because $(S)$ is a continuous algebra under pointwise multiplication. The next definition is an extension of this product. \[linexp\] The pointwise product of a Hida distribution $\Phi \in (S)'$ with an exponential of a linear term, i.e.  $$\Phi \cdot \exp(i \langle {\bf g}, \cdot \rangle +c), \quad {\bf g} \in L^2_{d}({\mathbb{R}})_{{\mathbb{C}}}, \, c \in {\mathbb{C}},$$ is defined by $$T(\Phi \cdot \exp(i\langle {\bf g}, \cdot \rangle + c))({\bf f}):= T\Phi({\bf f}+{\bf g})\exp(c),\quad {\bf f} \in S_d({\mathbb{R}}),$$ if $T\Phi$ has a continuous extension to $L^2_d({\mathbb{R}})_{{\mathbb{C}}}$ and the term on the right-hand side is a U-functional in ${\bf f} \in S_d({\mathbb{R}})$. \[donsker\] Let $D \subset {\mathbb{R}}$ with $0 \in \overline{D}$. Under the assumption that $T\Phi$ has a continuous extension to $L^2_d({\mathbb{R}})_{{\mathbb{C}}}$, ${\boldsymbol\eta}\in L^2_d({\mathbb{R}})_{{\mathbb{C}}}$, $y \in {\mathbb{R}}$, $\lambda \in \gamma_{\alpha}:=\{\exp(-i\alpha)s|\, s \in {\mathbb{R}}\}$ and that the integrand $$\gamma_{\alpha} \ni \lambda \mapsto \exp(-i\lambda y)T\Phi({\bf f}+\lambda {\boldsymbol\eta}) \in {\mathbb{C}}$$ fulfills the conditions of Corollary \[intcor\] for all $\alpha \in D$. Then one can define the product $$\Phi \cdot \delta_0(\langle {\boldsymbol\eta}, \cdot \rangle-y),$$ by $$T(\Phi \cdot \delta_0(\langle {\boldsymbol\eta}, \cdot \rangle-y))({\bf f}) := \lim_{\alpha \to 0} \int_{\gamma_{\alpha}} \exp(-i \lambda y) T\Phi({\bf f}+\lambda {\boldsymbol\eta}) \, d \lambda.$$ Of course under the assumption that the right-hand side converges in the sense of Corollary \[seqcor\], see e.g. [@GS98a]. This definition is motivated by the definition of Donsker’s delta, see Definition \[D:Donsker\]. [[@BG10]]{}\[thelemma\] Let $\mathbf{L}$ be a $d\times d$ block operator matrix on $L^2_{d}({\mathbb{R}})_{{\mathbb{C}}}$ acting componentwise such that all entries are bounded operators on $L^2({\mathbb{R}})_{{\mathbb{C}}}$. Let $\mathbf{K}$ be a d $\times d$ block operator matrix on $L^2_{d}({\mathbb{R}})_{{\mathbb{C}}}$, such that $\mathbf{Id+K}$ and $\mathbf{N}=\mathbf{Id}+\mathbf{K}+\mathbf{L}$ are bounded with bounded inverse. Furthermore assume that $\det(\mathbf{Id}+\mathbf{L}(\mathbf{Id}+\mathbf{K})^{-1})$ exists and is different from zero (this is e.g. the case if $\mathbf{L}$ is trace class and -1 in the resolvent set of $\mathbf{L}(\mathbf{Id}+\mathbf{K})^{-1}$). Let $M_{\mathbf{N}^{-1}}$ be the matrix given by an orthogonal system $({\boldsymbol\eta}_k)_{k=1,\dots J}$ of non–zero functions from $L^2_d({\mathbb{R}})$, $J\in {\mathbb{N}}$, under the bilinear form $\left( \cdot ,\mathbf{N}^{-1} \cdot \right)$, i.e.  $(M_{\mathbf{N}^{-1}})_{i,j} = \left( {\boldsymbol\eta}_i ,\mathbf{N}^{-1} {\boldsymbol\eta}_j \right)$, $1\leq i,j \leq J$. Under the assumption that either $$\begin{aligned} \Re(M_{\mathbf{N}^{-1}}) >0 \quad \text{ or }\quad \Re(M_{\mathbf{N}^{-1}})=0 \,\text{ and } \,\Im(M_{\mathbf{N}^{-1}}) \neq 0,\end{aligned}$$ where $M_{\mathbf{N}^{-1}}=\Re(M_{\mathbf{N}^{-1}}) + i \Im(M_{\mathbf{N}^{-1}})$ with real matrices $\Re(M_{\mathbf{N}^{-1}})$ and $\Im(M_{\mathbf{N}^{-1}})$,\ then $$\Phi_{\mathbf{K},\mathbf{L}}:={\rm Nexp}\big(-\frac{1}{2} \langle \cdot, \mathbf{K} \cdot \rangle \big) \cdot \exp\big(-\frac{1}{2} \langle \cdot, \mathbf{L} \cdot \rangle \big) \cdot \exp(i \langle \cdot, {\bf g} \rangle) \cdot \prod_{i=1}^J \delta_0 (\langle \cdot, {\boldsymbol\eta}_k \rangle-y_k),$$ for ${\bf g} \in L^2_{d}({\mathbb{R}},{\mathbb{C}}),\, t>0,\, y_k \in {\mathbb{R}},\, k =1\dots,J$, exists as a Hida distribution.\ Moreover for ${\bf f} \in S_d({\mathbb{R}})$ $$\begin{gathered} \label{magicformula} T\Phi_{\mathbf{K},\mathbf{L}}({\bf f})=\frac{1}{\sqrt{(2\pi)^J \det((M_{\mathbf{N}^{-1}}))}} \sqrt{\frac{1}{\det(\mathbf{Id}+\mathbf{L}(\mathbf{Id}+\mathbf{K})^{-1})}}\\ \times \exp\bigg(-\frac{1}{2} \big(({\bf f}+{\bf g}), \mathbf{N}^{-1} ({\bf f}+{\bf g})\big) \bigg) \exp\bigg(-\frac{1}{2} (u,(M_{\mathbf{N}^{-1}})^{-1} u)\bigg),\end{gathered}$$ where $$u= \left( \big(iy_1 +({\boldsymbol\eta}_1,\mathbf{N}^{-1}({\bf f}+{\bf g})) \big), \dots, \big(iy_J +({\boldsymbol\eta}_J,\mathbf{N}^{-1}({\bf f}+{\bf g})) \big) \right).$$ The Feynman integrand for a Charged Particle in a Constant Magnetic Field ========================================================================= In classical mechanics a charged particle moving through a magnetic field $\mathbf{H}=(0,0,H_3)$ has the Lagrangian $$L({\mathbf{x}}, \dot{{\mathbf{x}}}) = \frac{1}{2}m (\dot{{\mathbf{x}}_1}^2 +\dot{{\mathbf{x}}_2}^2 +\dot{{\mathbf{x}}_3}^2) + \frac{q H_3}{c} \left(x_1\dot{x_2}-\dot{x_1}x_2\right),$$ where $m$ is the mass of the particle. We denote the constant in front of the potential term by $k:=\frac{q H_3}{c}$. We see that, beneath the dependence on the spatial coordinates, the potential term depends explicitly on the velocities.\ Since the above three dimensional system can be separated to the free motion parallel to the magnetic field vector and a motion in the plane orthogonal to the magnetic field vector, we restrict ourselves to the two-dimensional system.\ In the following we realize rigorously the ansatz $$\begin{gathered} \label{eqfey} I_{\rm{mag}} = {\rm Nexp}\left( \frac{i}{\hbar}\int_{0}^t \frac{\dot{{\mathbf{x}}}(\tau)^2}{2m} d\tau +\frac{1}{2}\int_{t_0}^t \dot{{\mathbf{x}}}(\tau)^2\right) \\ \times \exp\left(-\frac{ik}{\hbar} \int_{0}^t \left(x_1(\tau) \dot{x_2}(\tau)-\dot{x_1}(\tau)x_2(\tau)\right) \, d\tau\right) \cdot \delta_0({\mathbf{x}}(t)-{\mathbf{y}}),\end{gathered}$$ for the Feynman integrand of a charged particle in a constant magnetic field, with the help of Lemma . See the introduction for a physical motivation. In the path ${\mathbf{x}}$ is realized by a two-dimensional Brownian motion starting in $0$ at time $t_0=0$. Then the first term in can be written as an exponential of quadratic type and gives a generalized Gauss kernel, see Definition \[GGK\]. Indeed with $\hbar = m =1,$ $$\label{Nexpquad} {\rm Nexp}\left( i \int_0^t \frac{\dot{{\mathbf{x}}}(\tau)^2}{2} d\tau +\frac{1}{2}\int_0^t \dot{{\mathbf{x}}}(\tau)^2\right) ={\rm Nexp}\left(-\frac{1}{2} \langle (\omega_1 , \omega_2),\mathbf{K} (\omega_1 , \omega_2) \rangle \right),$$ with $\mathbf{K} := -(i+1)\mathbf{P}_{[0,t)} := -(i+1) \left(\begin{array}{l l } P_{[0,t)} & 0\\ 0 & P_{[0,t)} \end{array}\right)$, where $P_{[0,t)}$ denotes the orthogonal projection in $L^2({\mathbb{R}})_{{\mathbb{C}}}$ given by the multiplication with ${\mathbf{1}}_{[0,t)}$.\ In the following we derive the desired properties for applying Lemma \[thelemma\]. First we write also the potential term in in a quadratic way. \[magneticL\] The operator matrix $$\begin{aligned} {\mathbf{L}}=\mathbf{P}_{[0,t)}\left(\begin{matrix} 0 & ik\left(A-A^*\right)\\ik\left(A^* - A\right) & 0\end{matrix}\right)\mathbf{P}_{[0,t)},\label{magnetiLformula}\end{aligned}$$ fulfills $$\begin{aligned} \frac12\langle\mathbf{f},{\mathbf{L}}\mathbf{f}\rangle &=-ik\int\limits_{0}^t \left(\int_0^{\tau} f_1(s) \, ds f_2(\tau)-f_1(\tau) \int_0^{\tau}f_2(s) \, ds \,\right) d\tau, \quad 0\leq t<\infty,\end{aligned}$$ where $\mathbf{f} = (f_1,f_2) \in L_2^2({\mathbb{R}})$ and operator $A$ is defined by $$A f(\tau) = {\mathbf{1}}_{[0,t)}(\tau) \int_{[0,\tau)} f(s) \, ds,\quad f \in L^2({\mathbb{R}}), \tau\in {\mathbb{R}}.$$ $A^*$ denotes its adjoint w.r.t. the bilinear dual pairing $ \langle \cdot,\cdot \rangle$. Moreover ${\mathbf{L}}$ is symmetric w.r.t. $\langle \cdot , \cdot \rangle$. [**Proof:**]{} With ${\mathbf{L}}$ as above we have by the symmetry of the dual pairing $$\begin{aligned} \langle\mathbf{f},{\mathbf{L}}\mathbf{f}\rangle&=\left\langle\left(\begin{matrix}f_1\\ f_2\end{matrix}\right), \left(\begin{matrix} 0 & ikP_{[0,t)}\left(A-A^*\right)\\ikP_{[0,t)}\left(A^*-A\right) & 0\end{matrix} \right)\left(\begin{matrix}f_1\\\ f_2\end{matrix}\right) \right\rangle\\ &=\left\langle f_1,ikP_{[0,t)}Af_2\right\rangle-\left\langle f_1 , ikP_{[0,t)}A^*f_2\right\rangle\\ &\hspace{35mm}+\left\langle f_2,ikP_{[0,t)}A^*f_1\right\rangle-\left\langle f_2, ikP_{[0,t)}A f_1\right\rangle\\ &=2\left\langle f_1,ikP_{[0,t)} A f_2\right\rangle-2\left\langle f_2,ikP_{[0,t)}A f_1\right\rangle\\ &=2ik\int\limits_{0}^t \left(\int_0^{\tau} f_1(s) \, ds f_2(\tau)-f_1(\tau) \int_0^{\tau}f_2(s) \, ds\right),\end{aligned}$$ since $P_{[0,t)}$ and $A$ commute. $\blacksquare$ If we extend $\langle \cdot, {\mathbf{L}}\cdot \rangle$ informally to an element $\boldsymbol{\omega}\in S'_2({\mathbb{R}})$ we have $$\label{80} \frac{1}{2} \langle \boldsymbol{\omega}, {\mathbf{L}}\boldsymbol{\omega} \rangle = -ik\int\limits_{0}^t B_\tau(\omega_1)\omega_2(\tau)- B_\tau(\omega_2)\omega_1(\tau) \,d\tau,$$ where $B_\tau$ is the representation of a one-dimensional Brownian motion as in (4). The term in corresponds in this case with the potential term in the Feynman integrand . \[magneticinvertability\] The operator ${\mathbf{N}}:L^2_{2}({\mathbb{R}},dx)_{{\mathbb{C}}}{\rightarrow}L^2_{2}({\mathbb{R}},dx)_{{\mathbb{C}}}$ given by $$\begin{aligned} {\mathbf{N}}:=\mathbf{P}_{[0,t)}\left(\begin{matrix} -i Id &ik\left(A-A^*\right)\\ ik\left(A^*-A\right)&-i Id \end{matrix}\right)\mathbf{P}_{[0,t)}+\mathbf{P}_{[0,t)^c},\end{aligned}$$ is bijective. Here $A$ and $A^*$ are as in Proposition \[magneticeigenbladeterminant\], $0< t< \infty$. [**Proof:**]{} We have $$\begin{gathered} \label{magneticNdefinition} {\mathbf{N}}={\mathbf{Id}}+{\mathbf{K}}+{\mathbf{L}}\\ \hspace{5 mm}=\mathbf{P}_{[0,t)}\left(\begin{matrix} -i&ik\left(A-A^*\right)\\ ik\left(A^*-A\right)&-i\\ \end{matrix}\right)\mathbf{P}_{[0,t)}+\mathbf{P}_{[0,t)^c} =-i\mathbf{P}_{[0,t)}{\mathbf{N}}_1 \mathbf{P}_{[0,t)}+\mathbf{P}_{[0,t)^c}.\\\end{gathered}$$ Denote the restriction of $\mathbf{P}_{[0,t)} {\mathbf{N}}_1$ to $L^2_2([0,t),{\mathbb{R}})_{{\mathbb{C}}}$ by ${\mathbf{N}}_2$. Then bijectivity of ${\mathbf{N}}_2$ implies bijectivity of ${\mathbf{N}}$ and $${\mathbf{N}}^{-1} = i {\mathbf{N}}_2^{-1}\mathbf{P}_{[0,t)} + \mathbf{P}_{[0,t)^c}.$$ For this we show that ${\mathbf{N}}_2$ is Fredholm with $\ker({\mathbf{N}}_2)=\{0\}$. First we show that $$\left(\begin{array}{l l} 0 & -k (A-A^* ) \\ -k( A^* -A ) & 0 \end{array}\right)=:\left(\begin{array}{l l} 0 & M \\ M^* & 0 \end{array}\right)$$ is compact on $L^2_2 ([0,t),{\mathbb{R}})_{{\mathbb{C}}}$. Then we have $${\mathbf{N}}_2 = \left(\begin{array}{l l} Id & M \\ M^* & Id \end{array} \right),$$ is a compact perturbation of the identity on $L^2_2 ([0,t),{\mathbb{R}})_{{\mathbb{C}}}$. We have $$\begin{gathered} \left(Af\right)(\tau)=\int\limits_{0}^{\tau} f(s)\,ds=\int\limits_{0}^t{\mathbf{1}}_{[0,\tau)}(s)f(s)\,ds\, \\ \text{and }\,\left(A^*f\right)(\tau)=\int\limits_\tau^t f(s)\,ds=\int\limits_{0}^t{\mathbf{1}}_{[\tau,t)}(s)f(s)\,ds.\end{gathered}$$ If ${\mathbf{1}}_{[0,\tau)}$ and ${\mathbf{1}}_{[\tau,t)}$ are Hilbert-Schmidt-kernels, the above integral operators $A$ and $A^*$ are compact operators on $L^2([0,t),dx)_{{\mathbb{C}}}$ and so are $M$ and $M^*$. Indeed $$\begin{aligned} &\int\limits_{0}^t\int\limits_{0}^t ({\mathbf{1}}_{[0,s)}(\tau))^2 \, d\tau ds=\int\limits_{0}^t\int\limits_{0}^t ({\mathbf{1}}_{[s,t)}(\tau))^2 \, d\tau ds=\frac12 t^2<\infty.\end{aligned}$$ $M$ as well as $M^*$can be written as the limit of a sequence of finite rank operators $(M_n)_{n\in {\mathbb{N}}}$ and $(M_n^*)_{n\in {\mathbb{N}}}$, respectively, in operator norm. Then: $$\begin{gathered} \sup_{\|(f_1 , f_2)\|\leq 1} \left\| \Bigg(\left(\begin{array}{l l} 0 & M \\ M^* & 0 \end{array}\right)- \!\left(\begin{array}{l l} 0 & M_n \\ M_n^* & 0 \end{array}\right)\Bigg)\left(\begin{array}{l} f_1 \\ f_2 \end{array} \right)\right\|\\ \leq\!\sup\limits_{\|f_1\| \leq1}\|(M-M_n)f_1\|+\!\sup\limits_{\|f_2\| \leq1}\!\|(M^*-M^*_n)f_2\|,\end{gathered}$$ where the right hand side tends to zero as $n$ goes to $\infty$. Hence, $\left(\begin{array}{l l} 0 & M \\ M^* & 0 \end{array}\right)$ as the limit of finite rank operators is compact.\ It is left to show that $\ker\left({\mathbf{N}}_2\right)=\{0\}$. Let $$\begin{aligned} \left(\begin{matrix} Id&M\\ M^*&Id \end{matrix}\right) \left(\begin{matrix} f_1\vphantom{\left(A-A^*\right)}\\f_2\vphantom{\left(A-A^*\right)} \end{matrix} \right)= \left(\begin{matrix} 0\vphantom{\left(A-A^*\right)}\\0\vphantom{\left(A-A^*\right)} \end{matrix} \right).\end{aligned}$$ This leads to the system $$\begin{aligned} f_1(s)&=k\int\limits_{0}^sf_2(\tau)d\tau-k\int\limits_s^tf_2(\tau)d\tau,\quad s \in [0,t),\\ f_2(s)&=k\int\limits_s^tf_1(\tau)d\tau-k\int\limits_{0}^sf_1(\tau)d\tau,\quad s \in [0,t).\end{aligned}$$ An analogue calculation as in the proof of Proposition \[magneticeigenbladeterminant\], below, yields $f_1\equiv f_2\equiv0$, which gives $\ker\left({\mathbf{N}}_2\right)=\{0\}$. $\blacksquare$ Now we want to determine the prefactor in Equation . Recall that the determinant of a diagonalizable operator is defined as the product of its eigenvalues, if it exists. We have the following proposition. \[magneticeigenbladeterminant\] Let ${\mathbf{K}}$ be as in , ${\mathbf{L}}$ as in Proposition \[magneticL\]. Then - For ${\mathbf{L}}({\mathbf{Id}}+{\mathbf{K}})^{-1}:L^2_{2}({\mathbb{R}},dx)_{{\mathbb{C}}}{\rightarrow}L^2_{2}({\mathbb{R}},dx)_{{\mathbb{C}}}$, the non-vanishing eigenvalues and their corresponding eigenvectors are $$\begin{aligned} \lambda_n&=\frac{2k}{(2n-1)\pi}t,\\ e_n(\cdot)&=c_1\left(\begin{matrix} {\mathbf{1}}_{[0,t)}(\cdot)\cos\left(\frac{2k}{\lambda_n}\cdot\right)\\ {\mathbf{1}}_{[0,t)}(\cdot)\sin\left(\frac{2k}{\lambda_n}\cdot\right) \end{matrix}\right)+c_2\left(\begin{matrix} {\mathbf{1}}_{[0,t)}(\cdot)\sin\left(\frac{2k}{\lambda_n}\cdot\right)\\ -{\mathbf{1}}_{[0,t)}(\cdot)\cos\left(\frac{2k}{\lambda_n}\cdot\right) \end{matrix}\right),\end{aligned}$$ $n\in{\mathbb{Z}},c_1,c_2\in{\mathbb{C}}$, where the multiplicity of the eigenvalues is 2. - We have for the determinant $$\begin{aligned} \det\left({\mathbf{Id}}+{\mathbf{L}}({\mathbf{Id}}+{\mathbf{K}})^{-1}\right)=\cos^2\!\left(kt\right).\end{aligned}$$ [**Proof:**]{} (i): We want to calculate the eigenvalues of $$\begin{aligned} &{\mathbf{L}}({\mathbf{Id}}+{\mathbf{K}})^{-1}\\[3mm] =&\left(\begin{matrix} 0 & ikP_{[0,t)}\left(A-A^*\right)P_{[0,t)}\\ ikP_{[0,t)}\left(A^*-A\right)P_{[0,t)} & 0\end{matrix} \right)\\ &\hspace{55mm}\times \left(\begin{matrix} iP_{[0,t)}+P_{[0,t)^c}&0\vphantom{ikP_{[0,t)}\left(A-A^*\right)}\\ 0\vphantom{ikP_{[0,t)}\left(A-A^*\right)}&iP_{[0,t)}+P_{[0,t)^c} \end{matrix}\right)\\[5mm] =&\left(\begin{matrix} 0&kP_{[0,t)}\left(A^*-A\right)P_{[0,t)}\\ kP_{[0,t)}\left(A-A^*\right)P_{[0,t)}&0 \end{matrix}\right) = \left( \begin{matrix} 0&M\\ M^*&0 \end{matrix}\right)\mathbf{P}_{[0,t)},\end{aligned}$$ with $M$ and $M^*$, as in Lemma \[magneticinvertability\], respectively. Hence we interpret the operator matrix as an operator from $L^2_{2}([0,t),dx)_{\mathbb{C}}$ into itself and restrict the desired eigenfunctions from now on to this interval. Assume $$\begin{aligned} \left(\begin{matrix} 0&M\\ M^*&0 \end{matrix}\right) \left(\begin{matrix} e_{n,1}\vphantom{kP_{[0,t)}\left(A-A^*\right)}\\ e_{n,2}\vphantom{kP_{[0,t)}\left(A-A^*\right)} \end{matrix}\right)=\lambda_n\left(\begin{matrix} e_{n,1}\vphantom{kP_{[0,t)}\left(A-A^*\right)}\\ e_{n,2}\vphantom{kP_{[0,t)}\left(A-A^*\right)} \end{matrix}\right), \quad \left(\begin{matrix} e_{n,1} \\e_{n,2} \end{matrix} \right)\in L^2_2([0,t), dx)_{{\mathbb{C}}}, n \in {\mathbb{Z}}.\end{aligned}$$ Then $$\begin{aligned} &k\left(A^*-A\right)e_{n,2}=\lambda_ne_{n,1}(\cdot)\label{41}\\ \text{and}\hspace{3mm}&k\left(A-A^*\right)e_{n,1}=\lambda_ne_{n,2}(\cdot)\label{42}.\end{aligned}$$ Differentiation yields $$-2ke_{n,2}=\lambda_ne'_{n,1} \quad \text{ and }\quad 2ke_{n,1}=\lambda_ne'_{n,2}.$$ Hence $$\begin{aligned} e''_{n,1}+\frac{4k^2}{\lambda_n^2}e_{n,1}=0 \quad \text{ and }\quad e''_{n,2}+\frac{4k^2}{\lambda_n^2}e_{n,2}=0.\end{aligned}$$ Due to general theory on ordinary differential equations, the solutions read $$\begin{aligned} e_{n,1}(\cdot)&=c_1\cos\left(\frac{2k}{\lambda_n}\cdot\right)+c_2\sin\left(\frac{2k}{\lambda_n}\cdot\right)\label{43}\\ \text{and}\ \ e_{n,2}(\cdot)&=d_1\cos\left(\frac{2k}{\lambda_n}\cdot\right)+d_2\sin\left(\frac{2k}{\lambda_n}\cdot\right)\label{44},\end{aligned}$$ $c_1,c_2,d_1,d_2\in{\mathbb{C}}$. Inserting this into the integral Equation (\[42\]) we get $$\begin{aligned} e_{n,2}(s)&=\frac k{\lambda_{n}}\int\limits_{0}^sc_1\cos\left(\frac{2k}{\lambda_n}\tau\right)+c_2\sin\left(\frac{2k}{\lambda_n}\tau\right)d\tau\\* &\hspace{10mm}-\frac k{\lambda_n}\int\limits_{s}^tc_1\cos\left(\frac{2k}{\lambda_n}\tau\right)+c_2\sin\left(\frac{2k}{\lambda_n}\tau\right)d\tau,\quad s \in [0,t),\end{aligned}$$ which leads to $$\begin{aligned} e_{n,2}(s)&=c_1\sin\left(\frac{2k}{\lambda_n}s\right)-c_2\cos\left(\frac{2k}{\lambda_n}s\right)\\* &\hspace{4mm}+c_2\left(\cos\left(\frac{2k}{\lambda_n}t\right)+1\right)-c_1\sin\left(\frac{2k}{\lambda_n}t\right),\quad s \in [0,t).\end{aligned}$$ Since $e_{n,2}$ is of the form (\[44\]) we have $d_1=-c_2$, $d_2=c_1$ and $$\begin{aligned} c_2\left(\cos\left(\frac{2k}{\lambda_n}t\right)+1\right)-c_1\sin\left(\frac{2k}{\lambda_n}t\right)=0\label{45}.\end{aligned}$$ \ Now of course (\[41\]) must also hold for $s=0$, thus $$\lambda_nc_1=\lambda_ne_{n,1}(0)=k\int\limits_{0}^tc_1\sin\left(\frac{2k}{\lambda_n}\tau\right)-c_2\cos\left(\frac{2k}{\lambda_n}\tau\right)d\tau,$$ which implies $$\begin{aligned} c_1&\left(\cos\left(\frac{2k}{\lambda_n}t\right)+1\right)=-c_2\sin\left(\frac{2k}{\lambda_n}t\right)\label{46}.\end{aligned}$$ First assume $c_1=0$, then we have with $$c_2\left(\cos\left(\frac{2k}{\lambda_n}t\right)+1\right)=0,$$ and with $$-c_2\sin\left(\frac{2k}{\lambda_n}t\right)=0.$$ But as we assume $(e_{n,1},e_{n,2})^T$ to be an eigenvector, the functions $e_{n,1}$ and $e_{n,2}$ may not both be the zero function, i.e. $c_2\neq 0$.\ Hence we have $$\sin\left(\frac{2k}{\lambda_n}t\right)=0 \text{ and }\cos\left(\frac{2k}{\lambda_n}t\right) = -1,$$ which is equivalent to $$\begin{aligned} \frac{2k}{\lambda_n}t=(2n-1)\pi,\end{aligned}$$ for some $n\in\mathbb{Z}$, i.e.  $$\begin{aligned} \lambda_n=\frac{2k}{(2n-1)\pi}t,\ n\in{\mathbb{Z}}.\end{aligned}$$ If we assume $c_2=0$, then $$c_1\sin\left(\frac{2k}{\lambda_n}t\right)=0=c_1\left(\cos\left(\frac{2k}{\lambda_n}t\right)+1\right).$$ This again is equivalent to $$\begin{aligned} \frac{2k}{\lambda_n}t=(2n-1)\pi,\end{aligned}$$ for some $n\in\mathbb{Z}$, i.e. , $$\begin{aligned} \lambda_n=\frac{2k}{(2n-1)\pi}t,\ n\in{\mathbb{Z}}.\end{aligned}$$ Assume $c_1\neq0\neq c_2$, then we multiply on both sides with and obtain: $$c_1 c_2 \left(\cos^2\left(\frac{2k}{\lambda_n}t\right) + 2 \cos\left(\frac{2k}{\lambda_n}t\right) +1 \right) = - c_1 c_2 \sin^2\left(\frac{2k}{\lambda_n}t\right)$$ which gives $$2 \cos\left(\frac{2k}{\lambda_n}t\right) = - \left(\sin^2\left(\frac{2k}{\lambda_n}t\right) +\cos^2\left(\frac{2k}{\lambda_n}t\right)\right)-1.$$ Thus again $$\cos\left(\frac{2k}{\lambda_n}t\right) = -1.$$ Inserted in , we also obtain $$\sin\left(\frac{2k}{\lambda_n}t\right)=0.$$ At first sight (\[45\]) and (\[46\]) give restrictions to the choice of $c_1$ and $c_2$. But naturally, if we have an eigenvector consisting of the two functions $e_{n,1}$ and $e_{n,2}$, corresponding to a certain $\lambda_n$, the factors of $c_1$ and $c_2$ in (\[45\]) and (\[46\]) become zero and the aforementioned can be choosen arbitrary. So an eigenfunction to the eigenvalue $\lambda_n$ is always of the form $$\begin{aligned} s \mapsto e_n(s)=\left(\begin{matrix} e_{n,1}(s)\\ e_{n,2}(s) \end{matrix}\right)& =c_1\left(\begin{matrix} \cos\left(\frac{2k}{\lambda_n}(s)\right)\\ \sin\left(\frac{2k}{\lambda_n}(s)\right) \end{matrix}\right)+c_2\left(\begin{matrix} \sin\left(\frac{2k}{\lambda_n}(s)\right)\\ -\cos\left(\frac{2k}{\lambda_n}(s)\right) \end{matrix}\right),\\\end{aligned}$$ where $c_1,c_2\in{\mathbb{C}}$ are arbitrary and the involved vectors are clearly linearly independent. Thus the dimension of the eigenspace corresponding to $\lambda_n$ and therewith its multiplicity is 2.\ (ii): In (i) we calculated the eigenvalues and eigenfunctions of $\mathbf{L}({\mathbf{Id}}+{\mathbf{K}})^{-1}$ considered as an operator from $L^2_2([0,t),dx)_{{\mathbb{C}}}$ to itself. The eigenfunctions form a basis of $L^2_2([0,t),dx)_{{\mathbb{C}}}$, but surely not of $L^2_2({\mathbb{R}},dx)_{{\mathbb{C}}}$. However, we can extend the set of eigenfunctions to a basis of $L^2_2({\mathbb{R}},dx)_{{\mathbb{C}}}$ by adding an arbitrary basis $L^2_2([0,t)^c,dx)_{{\mathbb{C}}}$. Note that because of the projection on $[0,t)$ in $\mathbf{L}({\mathbf{Id}}+{\mathbf{K}})^{-1}$ all basis functions of $L^2_2([0,t)^c,dx)_{{\mathbb{C}}}$ are eigenvectors to the eigenvalue $0$. Since the spectrum of ${\mathbf{Id}}+\mathbf{L}({\mathbf{Id}}+{\mathbf{K}})^{-1}$ is just shifted by $1$. This part does not give a contibution to the determinant (multiplication by $1$).\ Note for the nonvanishing eigenvalues of $\mathbf{L}({\mathbf{Id}}+{\mathbf{K}})^{-1}$ we have $$\lambda_n=-\lambda_{-n+1}, \quad \text{for all } n\in{\mathbb{Z}},$$ thus $$\begin{aligned} (1+\lambda_n)(1+\lambda_{-n+1})=1-\lambda_n^2.\end{aligned}$$\ Finally $$\begin{gathered} \det\left({\mathbf{Id}}+{\mathbf{L}}({\mathbf{Id}}+{\mathbf{K}})^{-1}\right)=\prod\limits_{n\in{\mathbb{Z}}}(1+\lambda_n)^2\\ = \prod\limits_{n\in{\mathbb{N}}} (1+\lambda_n)(1+\lambda_{-n+1})=\prod\limits_{n\in{\mathbb{N}}}\left(1-\frac{4k^2}{(2n-1)^2\pi^2}t^2\right)^2=\cos^2\!\left(kt\right).\end{gathered}$$ $\blacksquare$\ In the following we calculate the preimages of ${\boldsymbol \eta}_1=\left(\begin{matrix} {\mathbf{1}}_{[0,t)}\\ 0 \end{matrix}\right)$ and ${\boldsymbol \eta}_2=\left(\begin{matrix} 0\\ {\mathbf{1}}_{[0,t)} \end{matrix}\right)$ under $N$. With the help of this we can obtain $\left({\boldsymbol \eta_i}, \mathbf{N^{-1}} {\boldsymbol \eta_j}\right)$, $1\leq i,j\leq2$, and hence the matrix $M_{\mathbf{N^{-1}}}$ used in Equation . \[P:Preimage\] Let ${\mathbf{N}}$ as in equation (\[magneticNdefinition\]). Then $$\begin{aligned} \text{(i)}\quad{\mathbf{N}^{-1}}\left(\begin{matrix} {\mathbf{1}}_{[0,t)}\\ 0 \end{matrix}\right)&=\left(\begin{matrix} f_1\\ f_2\end{matrix}\right)={\mathbf{f}}\in L^2_{2}([0,t),dx)_{{\mathbb{C}}},\\ \text{(ii)}\quad{\mathbf{N}^{-1}}\left(\begin{matrix} 0\\ {\mathbf{1}}_{[0,t)} \end{matrix}\right)&=\left(\begin{matrix} g_1\\ g_2\end{matrix}\right)={\mathbf{g}}\in L^2_{2}([0,t),dx)_{{\mathbb{C}}},\end{aligned}$$ with $$\begin{aligned} f_1(s)&:=i\cos\left(2ks\right)+i\frac{\sin\left(2kt\right)}{\cos\left(2kt\right)+1}\sin\left(2ks\right)=:g_2(s)\label{f1},\quad s \in [0,t)\\ f_2(s)&:=i\frac{\sin\left(2kt\right)}{\cos\left(2kt\right)+1}\cos\left(2ks\right)-i\sin\left(2ks\right)=-:g_1(s)\label{f2},\quad s \in [0,t).\end{aligned}$$ [**Proof:**]{} We have to check that $$\begin{aligned} -i\left(\begin{matrix} Id&M\\ M^*&Id \end{matrix}\right) \left(\begin{matrix} f_1\vphantom{\left(A-A^*\right)}\\f_2\vphantom{\left(A-A^*\right)} \end{matrix} \right)= \left(\begin{matrix} {\mathbf{1}}_{[0.t)}\vphantom{\left(A-A^*\right)}\\0\vphantom{\left(A-A^*\right)} \end{matrix} \right),\end{aligned}$$ see the proof of Lemma \[magneticinvertability\]. The corresponding system of equations reads $$\begin{aligned} -if_1+ik\left(Af_2 -A^*f_2\right)&=1\label{48}\\ ik\left(A^*f_1-Af_1\right)-if_2&=0\label{49}.\end{aligned}$$ Let $s \in [0,t)$, then $$\left((A- A^*)\sin(2k\cdot)\right)(s) = \int_0^s \sin(2k\tau) \, d\tau -\int_s^t \sin(2k\tau) \, d\tau =-\frac{\cos(2ks)}{k} + \frac{1+ \cos(2kt)}{2k},$$ and $$\left((A- A^*)\cos(2ks)\right)(s) = \int_0^s \cos(2k\tau) \, d\tau - \int_s^t \cos(2k\tau) \, d\tau = \frac{\sin(2ks)}{k} - \frac{\sin(2kt)}{2k}.$$ Thus $$\begin{gathered} \left((A- A^*) f_2\right)(s) \\ = i\frac{\sin\left(2kt\right)}{\cos\left(2kt\right)+1} \left((A- A^*)\cos\left(2k\cdot\right)\right)(s) -i\left((A- A^*)\sin\left(2k\cdot\right)\right)(s) \\ = i(\frac{\sin\left(2kt\right)}{\cos\left(2kt\right)+1} (\frac{\sin(2ks)}{k} - \frac{\sin(2kt)}{2k}) +\frac{\cos(2ks)}{k} - \frac{1+ \cos(2kt)}{2k}).\\\end{gathered}$$ So we get $$\begin{gathered} -if_1(s)+ik\left(Af_2(s) -A^*f_2(s)\right)\\ = \cos(2ks) + \frac{\sin\left(2kt\right)}{\cos\left(2kt\right)+1}\sin\left(2ks\right) -\frac{\sin\left(2kt\right)}{\cos\left(2kt\right)+1}\sin(2ks) \\ + \frac{\sin^2(2kt)}{2\cos\left(2kt\right)+1} - \cos(2ks) + \frac{1+ \cos(2kt)}{2} = 1.$$\end{gathered}$$ Furthermore $$\begin{gathered} \left((A^*-A) f_1\right)(s)\\ =i\left((A^*- A)\cos\left(2k\cdot\right)\right)(s)+i\frac{\sin\left(2kt\right)}{\cos\left(2kt\right)+1}\left((A^*- A)\sin\left(2k\cdot\right)\right)(s)\\ = \frac{i}{k}\left( \frac{\sin(2kt)}{\cos(2kt)+1} \cos(2ks) -\sin(2ks) \right).\end{gathered}$$ And hence we obtain $$\begin{gathered} -ik\left(Af_1(s)-A^*f_1(s)\right)-if_2(s) \\ = -\frac{\sin(2kt)}{\cos(2kt)+1} \cos(2ks) +\sin(2ks) +\frac{\sin(2kt)}{\cos(2kt)+1} \cos(2ks) -\sin(2ks)=0.\end{gathered}$$ Thus (i) is shown. (ii) can be shown analogously. An analogue computation also yields for $\mathbf{g}$. $\blacksquare$ Now all conditions of Lemma \[thelemma\] are fulfilled. Hence we have the following theorem. \[magnetictheorem\] Let $0\leq 0< t<\infty$ with $$\begin{aligned} \frac{kt}{\pi},\frac{kt}{\pi}+\frac{1}{2} \notin {\mathbb{Z}}.\end{aligned}$$ Then the Feynman integrand $I_{\rm{mag}}$ for a charged particle in a constant the magnetic field exists as a Hida Distribution. Moreover the integrand can be written as $$I_{\rm{mag}}={\text{Nexp}}\left(-\frac{1}{2} \langle {\boldsymbol \omega},\mathbf{K} {\boldsymbol \omega} \rangle\right)\cdot\exp\left(-\frac{1}{2} \langle {\boldsymbol \omega},{\mathbf{L}}{\boldsymbol \omega} \rangle\right)\cdot\delta_0\left(\mathbf{B}_{t} -{{\mathbf{y}}}\right),$$ where ${\mathbf{y}}=(y_1,y_2)^T\in{\mathbb{R}}^2$ and the operators $\mathbf{K}$ as in and ${\mathbf{L}}$ as in Proposition \[magneticL\].Its $T$-transform in $\boldsymbol{\varphi} \in S_2({\mathbb{R}})$ is given by $$\begin{gathered} \label{magneticTtransform} TI_{mag}(\boldsymbol{\varphi})\notag=\frac{k}{2\pi i}\frac{1}{\cos(\left(kt\right)}\exp\left(-\frac12\left(\boldsymbol{\varphi},{\mathbf{N}^{-1}}\boldsymbol{\varphi}\right)\right)\\ \times\exp\left(-\frac{ik}{2}\cot\left(kt\right)\Bigg(\left(iy_1+\frac12\left({\mathbf{1}}_{[0,t)},({\mathbf{N}^{-1}}\boldsymbol{\varphi})_1\right)+\frac12\left(\boldsymbol{\varphi},\mathbf{f}\right)\right)^2\right.\\ \hspace{45mm}\left.\left.+\left(iy_2+\frac12\left({\mathbf{1}}_{[0,t)},({\mathbf{N}^{-1}}\boldsymbol{\varphi})_2\right)+\frac12\left(\boldsymbol{\varphi},\mathbf{g}\right)\right)^2\Bigg)\right)\right.,\end{gathered}$$ for all $\boldsymbol{\varphi}\in S_2({\mathbb{R}})$. Here ${\mathbf{f}}=({\mathbf{1}}_{[0,t)} f_1,{\mathbf{1}}_{[0,t)} f_2)^T,{\mathbf{g}}=({\mathbf{1}}_{[0,t)} g_1,{\mathbf{1}}_{[0,t)} g_2)^T$ with $$\begin{aligned} f_1(s)&=i\cos\left(2kt\right)+i\frac{\sin\left(2kt\right)}{\cos\left(2kt\right)+1}\sin\left(2ks\right)=g_2(s), \quad s \in [0,t)\\ f_2(s)&=i\frac{\sin\left(2kt\right)}{\cos\left(2kt\right)+1}\cos\left(2ks\right)-i\sin\left(2ks\right)=-g_1(s),\quad s \in [0,t).\end{aligned}$$ The generalized expectation ($T$-transform in $\boldsymbol{\varphi}=0$) gives $$\label{magneticpropagator} TI_{mag}(0) =\frac{k}{2\pi i}\frac{1}{\cos\left(kt\right)}\exp\left(\frac{ik}{2}\cot\left(kt\right)\left(y_1^2+y_2^2 \right)\right),$$ which coincides with the Greens function for a charged particle in a magnetic field see e.g. [@KL85], [@G96]. [**Proof:**]{} By Proposition \[P:Preimage\] we have that $M_{{\mathbf{N}}^{-1}}$ is completely imaginary and thus fulfills the conditions of Lemma \[thelemma\]. The prefactor in the exponential function in exists whenever the $\cot(kt) \neq \infty$, which is for $kt \neq n \pi , n \in {\mathbb{Z}}.$ Furthermore ${\mathbf{N}}$ is invertible by Theorem \[magneticinvertability\]. By Proposition \[magneticeigenbladeterminant\] we have the the determinant of ${\mathbf{Id}}+{\mathbf{L}}({\mathbf{Id}}+{\mathbf{K}})^{-1}$ exists and the prefactor in is finite whenever the $\cos{kt} \neq 0$, i.e. $kt\neq n + \frac{1}{2}$, for $n \in {\mathbb{N}}$. Hence we have that the conditions of Lemma \[thelemma\] are fulfilled and $$I_{\rm{mag}}={\text{Nexp}}\left(-\frac{1}{2} \langle {\boldsymbol \omega},\mathbf{K} {\boldsymbol \omega} \rangle\right)\cdot\exp\left(-\frac{1}{2} \langle {\boldsymbol \omega},{\mathbf{L}}{\boldsymbol \omega} \rangle\right)\cdot \delta_0\left(\mathbf{B}_{t} -{{\mathbf{y}}}\right),$$ is a Hida disribution. The $T$-transform is provided by Lemma \[thelemma\] $\blacksquare$ \[magneticjusitfication\] At the critical time $t$, with $\frac{kt}{\pi}\in{\mathbb{Z}}$ or $\frac{kt}{\pi}\in{\mathbb{Z}}+\frac{1}{2}{\mathbb{Z}}$ the Feynman propagator again is the Dirac delta function at 0. In the theory of Maslov (Morse) correction this singularity is called caustics, see e.g. [@GS98a] Remark 5.2 and [@S81]. Another typical example for caustics beneath a charged particle in a constant magnetic field is the harmonic oscillator, [@GS98a]. Note that the Greens function for small times always exists. In this article we considered the charged particle in a magnetic field without any electric induced force. 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--- abstract: 'A detailed analytical inspection of light scattering by a particle with high refractive index $m+i\kappa$ and small dissipative constant $\kappa$ is presented. We have shown that there is a dramatic difference in the behavior of the electromagnetic field within the particle (inner problem) and the scattered field outside it (outer problem). With an increase in $m$ at fix values of the other parameters, the field within the particle asymptotically converges to a periodic function of $m$. The electric and magnetic type Mie resonances of different orders overlap substantially. It may lead to a giant concentration of the electromagnetic energy within the particle. At the same time, we demonstrate that identical transformations of the solution for the outer problem allow to present each partial scattered wave as a sum of two partitions. One of them corresponds to the $m$-independent wave, scattered by a perfectly reflecting particle and plays the role of a background, while the other is associated with the excitation of a sharply-$m$-dependent resonant Mie mode. The interference of the partitions brings about a typical asymmetric Fano profile. The explicit expressions for the parameters of the Fano profile have been obtained “from the first principles” without any additional assumptions and/or fitting. In contrast to the inner problem, at an increase in m the resonant modes of the outer problem die out, and the scattered field converges to the universal, $m$-independent profile of the perfectly reflecting sphere. Numerical estimates of the discussed effects for a gallium phosphide particle are presented.' author: - - 'Andrey E. Miroshnichenko' bibliography: - 'Mie\_Fano.bib' title: 'Giant In-Particle Field Concentration and Fano Resonances at Light Scattering by High-Refractive Index Particles' --- Introduction ============ Presently the resonant light scattering by particles related to excitation of different eigenmodes attracts a great deal of attention of researchers all around the world [@Zhao:MT:2009; @Rybin:PRL:2009; @Evlyukhin:PRB:2011; @Staude:ACSN:2013; @Hancu:NL:2014; @Kuznetsov:NC:2014]. In addition to purely academic interest there is a broad spectrum of applications of the phenomenon in physics, chemistry, biology, medicine, data storage and processing, telecommunications, micro- and nanotechnologies, etc., see, e.g., [@Novotny:Book:2006; @Rybin:PRB:2013]. In particular, plenty of hopes were pinned on the resonant excitation of localized and/or bulk plasmons in metal nanoparticles [@Klimov:Book:2014]. Unfortunately, plasmonic resonances in such nanoparticles are usually accompanied with rather large dissipative losses, which in many cases diminish the advantages of the resonances. For this reason recently the frontier of the corresponding study has been shifted to light scattering by dielectric particles with low losses and high refractive index (HRI) $m$. In contrast to the plasmonic resonances, they exhibit the high $Q$-factor Mie resonances of both electric and magnetic types  which bring more opportunities for wider applications in sensing, spontaneous emission enhancement, and unidirectional scattering. Despite the fact that the exact Mie solution, describing light scattering by a sphere with an arbitrary size and material properties, is known for more than a hundred years, and the case of a sphere with HRI has been repeatedly discussed in textbooks and monographs, see, e.g. [@Landau:T8:1984; @Hulst::1981], some important peculiarities of this problem have not been disclosed yet. Meanwhile, the quantitative feature of HRI brings about qualitatively new effects and paradoxes, which merely do not exist at moderate values of the refractive index, see, e.g. [@Evlyukhin:PRB:2010; @Tribelsky:EPL:2012; @Geffrin:NC:2012]. In the present paper we produce a detailed, systematic study of light scattering by a HRI particle. Specifically, we show that the scattering may be accompanied with a *giant concentration* of the electromagnetic field within the particle and reveal the nature of the Fano resonances exhibited by partial scattered waves. It is well known that at the limit $m \rightarrow \infty$ a small (relative to the incident light wavelength) dielectric sphere scatters light as a perfectly reflecting one (PRS) “into which neither the electric, nor the magnetic field penetrates" [@Landau:T8:1984]. Then, it may be concluded that the electromagnetic field within the scattering particle should vanish at $m \rightarrow \infty$. Seemingly, the conclusion is supported by the argument that a HRI implies a high polarizability of the sphere. Then, at $m \rightarrow \infty$ from the point of view of the polarizability by an external electric field a dielectric sphere becomes equivalent to a perfectly conducting one [@Landau:T8:1984], and the electric field induced within the particle owing to its polarization by the incident light should compensate the field inducing the polarization. That is to say, the field within the particle should vanish. In fact, the question is much more subtle, and the actual situation is far from this simple picture. The point is that the wavelength inside the particle vanishes at $m \rightarrow \infty$. Then, no matter how small the particle is, at large enough $m$ the wavelength within the particle becomes smaller than the particle size. In this case the incident wave may resonantly excite the Mie electromagnetic eigenmodes in the particle. Moreover, an unlimited growth in $m$ results in infinite cascades of these resonances. The interference of the resonant eigenmodes with the incident wave and, what is the most important, with each other gives rise to dramatic changes in the aforementioned simple scattering process. To reveal these changes is the goal of our study. To this end the full Mie theory is employed. We show that while at $m \rightarrow \infty$ the scattered field for the outer problem does converge to the one for the PRS (no matter, whether the sphere is small, or large), the field within the particle, though it sounds paradoxical, does not have any limit at all. Such a difference between the outer and inner problems is related to the different lineshapes of the Mie resonances in the two problems. For the former an increase in $m$ makes the resonances less pronounced. For the latter in the non-dissipative limit the amplitude of the resonances increases with an increase in $m$. In the case of a finite dissipation rate (regardless how small it is), the growth of the amplitudes eventually saturates and the resonance lines become periodic functions of $m$. In both the cases the field within the particle does not tend to any fixed limit at $m \rightarrow \infty$. It is important to stress, that the resonance lines of different orders and different origin (i.e., electric and magnetic) may overlap substantially. The mentioned peculiarities of the inner field in the vicinity of the resonances may result in a *giant concentration* of the electromagnetic energy inside the particle. At realistic values of the refractive index and the proper selection of the particle radius the field inside the particle may exceed the one in the incident wave in several orders of magnitude. Such a huge field may give rise to numerous nonlinear effects. For this reason the discussed results may appear extremely important in the design and fabrication of highly-nonlinear nanostructures. Regarding the outer problem, it is known that the scattering coefficients in this problem have the well-pronounced asymmetric Fano resonance lines. Recent publications of Rybin et al, Ref. [@Rybin:OE:2013; @Rybin:FTT:2014; @Rybin:SciRep:2014] should be mentioned in this context. Based on the analysis of the exact Lorenz-Mie solution for a cylinder the authors of these publications have revealed that the resonant Mie scattering can be presented through infinite cascades of the Fano resonances between the narrow-line resonant Mie scattering and the non-resonant (background) scattering from the object. The analytical expressions for both the partitions have been obtained through the Maxwell boundary conditions. The numerical fit of the lineshape, resulting from the exact solution in the vicinity of the resonances, to the conventional Fano profile [@Fano:PR:1961] has allowed the authors to obtain the dependence of the Fano asymmetry parameter $q$ [@Fano:PR:1961] on the ratio of the radius of the cylinder R to the wavelength of the incident light $x =2\pi R/\lambda$ (size parameter) in rather a broad range of its variations. They also have shown that in the inspected cases $q(x) \sim −\cot x$. This dependence agrees with their previous results for disordered photonic crystals [@Rybin:NatCom:2012], as well as with the general expression for $q$ in terms of the phase shift of the background partition [@Connerade:RPP:1988]. Despite the study of these authors is a big step forward to understanding the essence of the Fano resonances at light scattering by a particle, they have not disclosed the physical nature of the background partition. Regarding the results obtained by the numerical fit, the great advantage of this procedure is the possibility to fit any curve with any set of the basic functions. However, precisely because of that, based on the fitting solely, one never can answer the question whether the studied profile is the Fano profile indeed, or it is *just fitted* to that profile. It also remains unclear how far beyond the inspected numerical domain the obtained results could be extended, e.g., what happens with the modes with the multipolarity higher than that, examined by the authors, etc. For this reason, a self-consistent analytical examination of the problem, connecting the parameters of the Fano profiles with the fundamental parameters of the light scattering $x$ and $m$ would be highly desirable. Such a study is produced in the present paper. Specifically, we show that the Fano profile could be obtained by identical transformations of the exact Mie solution. As a result, the exact analytical expressions for the parameters of the profile are obtained automatically. Regarding the background partition, we reveal that it is just the corresponding partial wave scattered by a PRS. We should emphasize that the resonances discussed in the present paper have nothing to do with the well known *whispering gallery modes*. These modes are associated with waves propagating along the surface of the particle, confined there by “continuous total internal reflection" [@Vahala:Nature:2003] and cannot be excited by a plane incident wave. The closest to the topic of our study is another well known phenomenon: the ripple structure in the spectrum at light scattering by droplets with $m \approx 1.5$ and related problems, see, e.g., [@Chylek:JOSA:1976; @Hulst::1981; @Chylek:ApplPhys:1985]. The resonances we discuss and the ripple structure, both have the same nature. However, the large value of the refractive index makes the manifestation of our resonances quite different from what is known for the ripple structure. In addition, the specific characteristics of the resonances examining in our paper, usually are not discussed in connection with the ripple structure. Note also the very close effects discussed in Ref. [@Arruda:JOSA:2010] for a magnetic particle. However, up to now materials with large magnetic permeability at optical frequencies remain hypothetical objects, while the ones with HRI may be easily found among the common semiconductors, see below. It is necessary to stress in this context that all the effects discussed in the present paper occur to *optically thick* particles, i.e., $mx$ should be of the order of unity, or larger than that. However, an important peculiarity of the HRI particles is that this condition may hold for *geometrically small* particles with $x \ll 1$, cf. [@Tribelsky:EPL:2012]. The paper is arranged as follows. In Section II we consider general properties of the Mie solution at the limit of large, purely real $m$. In Section III the lineshapes and linewidths of partial resonances for the scattered field and the field, concentrated within the particle are inspected. In Section IV the generic nature of the Fano resonances for partial modes of the scattered field is disclosed, and explicit expressions for the parameters of the Fano profile are obtained from the first principles by means of identical transformations of the exact Mie solution. In we show that in the non-dissipative limit the entire set of the infinite number of the cascades of the resonances possesses a certain scaling and may be reduced to a universal set of lines by simple scale transformations. In Section VI, effects of finite dissipation are inspected. In Section VII the resonances at a fixed $m$ and varying size parameter are discussed and the manifestation of the resonances in a particle made of gallium phosphide is presented as an example. In Conclusion a brief summary of the obtained results is presented. In Appendix certain cumbersome but important calculations are performed. Last, but not least. The only thing we do below is a detailed analysis of the well known Mie solution at the range of high refractive index of the scattering sphere. It would have been nothing but a mathematical exercise, if it did not reveal new unusual features of the phenomenon. It did. Let us proceed discussing these features in detail. Large refractive index limit {#sec:2} ============================ The subsequent analysis is based upon the exact Mie solution describing light scattering by a spatially homogeneous sphere with an arbitrary radius $R$ and a given permittivity $\varepsilon$. The case of a cylinder and core-shell structures may be inspected in the same manner. Since we are interested in the optical properties of a particle with low dissipation, we first focus on a discussion of the non-dissipative limit, i.e., a purely real positive refractive index $m \equiv \sqrt{\varepsilon}$. The general case of a complex refractive index along with the applicability condition for the non-dissipative limit will be produced later on. According to the Mie solution, the scattered field is presented as an infinite series of partial multipolar contributions (dipolar, quadrupolar, etc.) of the two types: TE and TM — the so-called electric and magnetic modes. For the sake of briefness the electric responses only are discussed here in detail. The behavior of the magnetic submodes is alike. Therefore, the corresponding discussion of these modes will be rather brief. The key quantities of the Mie solution are the properly normalized complex amplitudes of the scattered $a_n,\;b_n$ and internal $c_n,\;d_n$ field components (scattering coefficients). These coefficients should satisfy the boundary conditions, following from the continuity of the tangential components of the electric and magnetic fields at the surface of the particle. The conditions are split into two independent pairs for the electric and magnetic modes, respectively [@Bohren::1998] $$\begin{aligned} & & \xi_n(x)a_n + \psi_n(mx)d_n= \psi_n(x), \label{Eq:BC1electr} \\ & & m\xi'_n(x)a_n +\psi'_n(mx)d_n = m \psi'_n(x), \label{Eq:BC2electr}\\ & & m\xi_n(x)b_n + \psi_n(mx)c_n = m\psi_n(x), \label{Eq:BC1magn}\\ & & \xi'_n(x)b_n + \psi'_n(mx)c_n = \psi'_n(x) \label{Eq:BC2magn}.\end{aligned}$$ Here $x=2\pi R/\lambda$ is the size parameter; $\psi_n(z)$, , $\psi_n(z)=zj_n(z)$ and $\chi_n(z)=-zy_n(z)$ are the Riccati-Bessel functions; $j_n(z),\;y_n(x)$ stand for the spherical Bessel functions [@Bohren::1998]; ${}'=\partial/\partial z$ designates derivative with respect to the entire argument. Integer $n$ indicates the multipolarity of the mode, so that $n=1,\;2...$ correspond to the dipole mode, quadrupole mode, etc. It should be stressed that functions $\psi_n(z)$ and $\chi_n(z)$ are real for Solving Eqs. , with respect to $a_n$, $d_n$, we obtain [@Bohren::1998] $$\begin{aligned} a_n &=& \frac{m\psi_n(mx)\psi_n'(x) - \psi_n(x)\psi_n'(mx)}{m\psi_n(mx)\xi_n'(x) - \xi_n(x)\psi_n'(mx)} \label{Eq:a_n} \\ d_n &=& \frac{im}{m\psi_n(mx)\xi_n'(x) - \xi_n(x)\psi_n'(mx)},\label{Eq:d_n}\end{aligned}$$ where the identity $$\label{Eq:identity_1} \psi_n(x)\xi'_n(x) - \psi'_n(x)\xi_n(x) \equiv i,$$ following from the expression for the Wronskian of the Riccati-Bessel functions [@Abramowitz::1965], has been employed. The corresponding treatment of the magnetic modes yields $$\begin{aligned} b_n &=& \frac{m\psi_n(x)\psi'_n(mx)-\psi_n(mx)\psi'_n(x)}{m\xi_n(x)\psi'_n(mx)-\psi_n(mx)\xi'_n(x)} \label{Eq:b_n} \\ c_n &=& -\frac{im}{m\xi_n(x)\psi'_n(mx)-\psi_n(mx)\xi'_n(x)} \label{Eq:c_n}\end{aligned}$$ Note, that according to Eqs. , , $d_n \equiv c_n \equiv 1$ at $m=1$, i.e., when the optical properties of the particle are identical to those of the embedding medium. Thus, $|d_n|,\;|c_n|$ may be regarded as the enhancement parameters for the field within the particle. Eqs. , , , may be written in the equivalent, identical form: $$\begin{aligned} a_n & = & \frac{F_n^{(a)}}{F_n^{(a)}+i G_n^{(a)}}\;,\;\;d_n=\frac{i\;m}{F_n^{(a)}+i G_n^{(a)}},\label{Eq:an_dn_FG}\\ b_n & = & \frac{F_n^{(b)}}{F_n^{(b)}+i G_n^{(b)}}\;,\;\;c_n=-\frac{i\;m}{F_n^{(b)}+i G_n^{(b)}},\end{aligned}$$ with $$\begin{aligned} F_n^{(a)}&=&m\psi'_n(x)\psi_n(m x)-\psi_n(x)\psi'_n(m x)\;,\label{Eq:Fna}\\ G_n^{(a)}&=&-m\chi'_n(x)\psi_n(m x)+\chi_n(x)\psi'_n(m x)\;,\label{Eq:Gna}\\ F_n^{(b)}&=&m\psi_n(x)\psi'_n(mx)-\psi_n(mx)\psi'_n(x)\;,\label{Eq:Fnb}\\ G_n^{(b)}&=&-m\chi_n(x)\psi'_n(mx) + \chi'_n(x)\psi_n(mx). \label{Eq:Gnb}\end{aligned}$$ Let us focus on the electric modes. At the limit of the large refractive index we can keep just the first terms in Eqs.–: $$\begin{aligned} F_n^{(a)}\xrightarrow[{m \gg 1}]{}m\psi'_n(x)\psi_n(m x)\;,\label{Eq:Fna_m->infty}\\ G_n^{(a)}\xrightarrow[{m \gg 1}]{}-m\chi'_n(x)\psi_n(m x)\;. \label{Eq:Gna_m->infty}\end{aligned}$$ Eqs. – lead to two important conclusions. First, the partial scattering coefficients $a_n$ converge to the $m$-[*independent*]{} form valid for the PRS: $$a_n\xrightarrow[{m \gg 1}]{}a_n^{\rm (PRS)}\!=\!\frac{F_n^{(a,{\rm PRS})}}{F_n^{(a,{\rm PRS})}+i G_n^{(a,{\rm PRS})}}\!=\!\frac{\psi'_n(x)}{\xi'_n(x)}, \label{Eq:a_n-PRS}$$ with $$\label{Eq:Fna_PRS-Gna_PRS} F_n^{(a,{\rm PRS})}=\psi'_n(x),\;\;G_n^{(a,{\rm PRS})}=-\chi'_n(x).$$ Note that expression, Eq.  can be also obtained from the boundary condition, Eq. , if we suppose there $d_n = 0$, as it should be for a PRS, or by direct solution of the light scattering problem for a PRS [^1]. Second, the internal coefficients $d_n$ in this limit still keep the dependence on $m$, converging to $$\begin{aligned} \label{Eq:d_n_at_m->infty} d_n\xrightarrow[m \gg 1]{}d_n^{(\lim)}=\frac{i}{\psi_n(mx)\xi'_n(x)}.\end{aligned}$$ It is relevant to mention that while expressions, Eqs. , satisfy boundary condition, Eq.  identically, boundary condition, Eq.  for these expressions is satisfied only asymptotically at It is interesting to note also, that the limit, Eqs. , corresponds to the scattering coefficient for a sphere made of a perfect electric conductor  [@Balanis:Book:2012]. However, in the problem in question a sphere made of *a perfect insulator* with zero conductivity (Im$\,\varepsilon = 0$) is considered. The coincidence of the two limits occurs owing to the fact, that despite the conductivity current in our case is zero, the displacement current plays its role. If the field [**E**]{} inside the particle does not vanish, the displacement current diverges at Re$\,\varepsilon \rightarrow \infty$. It brings about exactly the same result as that, following from the divergence of the conductivity current in a perfect electric conductor. It should be stressed that limit, Eq. (\[Eq:a\_n-PRS\]) is not valid in the vicinities of the points, where $\psi_n(mx)=0$. Moreover, exactly at these points the internal coefficient diverges $d_n^{(\lim)}\rightarrow\infty$, see Eq. (\[Eq:d\_n\_at\_m-&gt;infty\]). It allows us to conclude that the condition $$\label{Eq:psi=0} \psi_n(mx)=0$$ corresponds to the resonances excited in a HRI dielectric sphere at large enough $m$. This resonance condition can be further simplified taking into account that in the range $m x>n^2$ (known as the Fraunhofer regime) the Riccati-Bessel functions are reduced to simple trigonometrical ones, so that the condition $\psi_n(mx)=0$ reads $$\begin{aligned} \psi_n(m x)\cong\sin\left(m x-\frac{n\pi}{2}\right)=0 \label{Eq:sin=0}\;.\end{aligned}$$ An analogous treatment of the magnetic coefficients $b_n$ (for the outer problem) and $c_n$ (for the inner) brings about the following expressions: $$\begin{aligned} F_n^{(b)}\xrightarrow[{m \gg 1}]{}m\psi_n(x)\psi'_n(m x)\;,\label{Eq:Fnb_m->infty}\\ G_n^{(b)}\xrightarrow[{m \gg 1}]{}-m\chi_n(x)\psi'_n(m x)\;. \label{Eq:Gnb_m->infty}\end{aligned}$$ $$b_n\xrightarrow[{m \gg 1}]{}b_n^{\rm (PRS)}\!=\!\frac{F_n^{(b,{\rm PRS})}}{F_n^{(b,{\rm PRS})}+i G_n^{(b,{\rm PRS})}}\!=\!\frac{\psi_n(x)}{\xi_n(x)}, \label{Eq:b_n-PRS}$$ $$\label{Eq:Fnb_PRS-Gnb_PRS} F_n^{(b,{\rm PRS})}=\psi_n(x),\;\;G_n^{(b,{\rm PRS})}=-\chi_n(x),$$ $$\begin{aligned} \label{Eq:c_n_at_m->infty} c_n\xrightarrow[m \gg 1]{}c_n^{(\lim)}=-\frac{i}{\psi'_n(mx)\xi_n(x)},\end{aligned}$$ with the following condition for the magnetic modes resonances: $$\begin{aligned} \psi'_n(m x)=0\;, \label{Eq:psi_prime=0}\end{aligned}$$ leading in the Fraunhofer regime to the equation $$\label{Eq:cos=0} \psi'_n(m x)\cong \cos\left(m x-\frac{n\pi}{2}\right)=0.$$ The two resonance conditions may be rewritten in a unified form: $$\begin{aligned} & & m^{({\rm res},E)}_{n,p}\cong(n+2p)\frac{\pi}{2x}, \label{Eq:m_res_E}\\ & & m^{({\rm res},H)}_{n,p}\cong(n+2p+1)\frac{\pi}{2x} \label{Eq:m_res_H}.\end{aligned}$$ where $p$ is a non-negative integer number. The meaning of resonance conditions, Eqs. , becomes absolutely clear, if we recall that $x = 2\pi R/\lambda$ and $\lambda/m$ is the wavelength inside the scattering particle. Then, Eqs. , may be rewritten as follows: $$\begin{aligned} & & \frac{\lambda}{m^{({\rm res},E)}_{n,p}}\left(\frac{n}{2}+p\right) \cong 2R, \\ & & \frac{\lambda}{m^{({\rm res},H)}_{n,p}}\left(\frac{n+1}{2}+p\right) \cong 2R,\end{aligned}$$ i.e., the resonances occur when an integer number of the half-waves equals the diameter of the sphere, cf. [@Kuznetsov:SR:2012]. These resonance conditions lead to a number of interesting conclusions, cf. [@Hulst::1981]. First, each multipole has, in general, infinite number of resonances, associated with $p=0,1,2,\ldots$. Second, owing to the additional degree of freedom related to variations of $p$, there is the *multiple degeneracy* of the resonances. Specifically, the resonances with different multipolarity $n$ occur at one and the same value of $m$, provided for these resonances the variation of $n$ is compensated by the corresponding variation of $p$, i.e., $n+2p$ for the electric resonant modes remains the same and equals $n+1+2p$ for magnetic. Next, at a give $n$ the points of the $n$th electric resonances correspond to those of the $(n+1)th$ magnetic and vice versa. Note also that at a fixed $n$ the points of resonances of one type (i.e., either electric, or magnetic) are situated in the $m$-axis just in the middle of the spacing between the points of the other type, so that the maxima for the electric modes correspond to the minima of magnetic and the other way around, see Eqs. , . In Fig. \[fig:fig2\] we plot the dependence of the first two electric $d_{1,2}$ and magnetic $c_{1,2}$ internal coefficients vs. size parameter $x$ and refractive index $m$, calculated according to the exact Mie solution. In this figure we also plot asymptotic resonance conditions, Eqs. (\[Eq:m\_res\_E\]), (\[Eq:m\_res\_H\]). The agreement between the position of the resonances according to the exact solution and approximate conditions, Eqs. (\[Eq:m\_res\_E\]), (\[Eq:m\_res\_H\]) is surprisingly good. It should be stressed, however, that the results obtained do not mean a strong overlap of the resonances yet. The point is that the right-hand-sides of Eqs. , are just the first terms of the corresponding asymptotic expansions in powers of small $1/(mx)$. Higher order terms, dropped in these equations bring about the mismatches between the points of the resonances for different modes. To claim the strong overlap of the resonances we must make sure that the mismatches are smaller than the corresponding linewidths. Let us proceed to a discussion of these issues. ![(color online) Dependence of electric $d_{1,2}$ (a,c) and magnetic $c_{1,2}$ (b,d) internal coefficients vs size parameter $x$ and refractive index $m$, calculated according to the exact Mie solution. Dashed lines indicate the asymptotic resonance conditions, obtained in the limit $m\rightarrow\infty$ , see Eqs. (\[Eq:m\_res\_E\]), (\[Eq:m\_res\_H\]) at $p = 0,\,1,\,2,\,3.$ Despite the employed values of $m$ are not so large, the agreement between the dashed lines and the positions of the actual resonances is very good. []{data-label="fig:fig2"}](Fig2 "fig:"){width="\columnwidth"}\ Linewidth and lineshape {#sec:3} ======================= To understand the meaning of the existence of the limit at $m \rightarrow \infty$ for scattering coefficients $a_n$, and its absence for internal ones, $d_n$, we have to inspect Eqs. – more carefully. First of all note, direct calculations of $a_n$ based upon Eqs. , , , , brings about the uncertainty of the type 0/0. To resolve the lineshape, let us consider a small departure of the refractive index from a certain resonant value: $\delta m = m - m^{\rm (res)}$, where $m^{\rm (res)}$ satisfies Eq. . Here and in what follows for simplicity of notations we put $m^{\rm (res)}$ for $m^{({\rm res}, E)}_{n,p}$. Then, bearing in mind that in the vicinity of the resonance $\psi_n\left((m^{{\rm (res)}}+\delta m)x\right) \cong \psi'_n(m^{{\rm (res)}}x)x\delta m$, we readily obtain the following formulas for $F^{(a)}_n\!\!,\; G^{(a)}_n$: $$\begin{aligned} \!\!\!\!\!\!\!\!F^{(a)}_n &\cong& \psi_n'(m^{\rm (res)}x)\left\{m^{\rm (res)}\psi_n'(x)x\delta m - \psi_n(x) \right\}\!, \label{Eq:Fn_res_a} \\ \!\!\!\!\!\!\!\!G^{(a)}_n &\cong& -\psi_n'(m^{\rm (res)}x)\left\{m^{\rm (res)}\chi_n'(x)x\delta m - \chi_n(x) \right\}\!. \label{Eq:Gn_res_a}\end{aligned}$$ It is seen from Eqs. , , that scattering coefficient reaches its maximum (the points of the constructive interference), $a_n =1$ at $$\label{Eq:delta_m_G_a} \delta m^{(a)}_G \cong \frac{1}{x m^{\rm (res)}}\frac{\chi_n(x)}{\chi_n'(x)},$$ and minimal value, $a_n=0$ (the points of the destructive interference) at $$\label{Eq:delta_m_F_a} \delta m^{(a)}_F \cong \frac{1}{x m^{\rm (res)}}\frac{\psi_n(x)}{\psi_n'(x)}.$$ The corresponding expressions for $b_n$ read $$\begin{aligned} F^{(b)}_n &\cong& m^{\rm (res)}\psi_n(x)\psi''_n(m^{\rm (res)}x)x\delta m \label{Eq:Fn_res_b}\\ & & - \psi'_n(x)\psi_n(m^{\rm (res)}x), \nonumber\\ G^{(b)}_n &\cong& -m^{\rm (res)}\chi_n(x)\psi''_n(m^{\rm (res)}x)x\delta m \label{Eq:Gn_res_b}\\ & & + \chi'_n(x)\psi_n(m^{\rm (res)}x) \nonumber.\end{aligned}$$ $$\label{Eq:delta_m_G_b} \delta m^{(b)}_G \cong \frac{1}{x m^{\rm (res)}}\frac{\chi'_n(x)\psi_n(m^{\rm (res)}x)}{\chi_n(x)\psi''_n(m^{\rm (res)}x)},$$ $$\label{Eq:delta_m_F_b} \delta m^{(b)}_F \cong \frac{1}{x m^{\rm (res)}}\frac{\psi'_n(x)\psi_n(m^{\rm (res)}x)}{\psi_n(x)\psi''_n(m^{\rm (res)}x)}.$$ where now $m^{{\rm (res)}}$ stands for $m^{({\rm res},H)}_{n,p}\!\!\!,$ satisfying Eq. . Note, $\psi_n(m^{\rm (res)}x)/\psi''_n(m^{\rm (res)}x) \cong -1$ at $m^{\rm (res)}x > n^2$, see Eqs. . The obtained results explain the mentioned convergence of $a_n$ to the $m$-independent form at $m \rightarrow \infty$: at an increase in $m$ the width of the resonance line $|\delta m_G - \delta m_F|$ contracts as $1/xm^{\rm (res)}$, while the amplitude of the resonances and spacing between two adjacent maxima (minima), both remain $m$-independent. Asymptotically, at $m \rightarrow \infty$ the maxima and minima merge with each other, and the resonance profile vanishes. An example of such a process for $|a_1|^2$ at $x=1$ is presented in Fig. \[fig:a1\]. ![(color online) Contraction of the Fano resonance lines for $|a_1|^2$ with an increase in $m;\;x=1;\;\zeta = m - m^{{\rm (res)}}.$ Here (solid line) and 111.5 (dashed line). A yellow horizontal line corresponds to $a_1^{\rm (PRS)}$. []{data-label="fig:a1"}](a1_contraction){width="\linewidth"} Now let us inspect coefficients $d_n$. As it follows from $$\label{Eq:a_n_trough_aPRS_and_d_n} a_n = a_n^{\rm (PRS)} -\frac{\psi_n'(mx)}{m\xi_n'(x)}d_n.$$ Then, it is convenient to introduce the rescaled coefficient $$\label{Eq:d_n_tilde} \tilde{d_n} = \frac{\psi_n'(mx)}{m\xi_n'(x)}d_n \equiv \frac{i}{m\frac{\psi_n(mx)}{\psi_n'(mx)} - \frac{\xi_n(x)}{\xi_n'(x)}}\frac{1}{\xi'^2_n(x)}.$$ Next, utilizing the identity $$\label{Eq:identity_2} \left|\xi'^2_n(x){\rm Im}\,\left(\frac{\xi_n(x)}{\xi_n'(x)}\right)\right| \equiv 1,$$ see Appendix A, its modulus may be presented as follows: $$\label{Eq:d_n_tilde_AB} |\tilde{d_n}|^2 = \frac{1}{\left(m\frac{\psi_n(mx)}{\psi'_n(mx)}-A_n^{(d)}(x)\right)^2B_n^{(d)2}(x)+1},$$ where $$\label{Eq:A_n^{(d)}_B_n^{(d)}} A_n^{(d)}(x) = {\rm Re}\,\left(\frac{\xi_n(x)}{\xi_n(x)'}\right);\;\;\; B_n^{(d)}(x) = \left|\xi_n'^2(x)\right| \equiv \left|\xi_n'(x)\right|^2.$$ The conclusion, which immediately follows from Eq. , is that the resonant values of $|\tilde{d_n}|^2$ equals unity and the resonances are achieved at the values of $m$, satisfying the equation: $$\label{Eq:resonant_cond_|d_n|} m\frac{\psi_n(mx)}{\psi'_n(mx)}=A_n^{(d)}(x).$$ We should emphasize that the resonance condition stipulated by Eq.  is an *exact result*, valid at any $m$ and $x$. To obtain the lineshape in the proximity of $m=m^{\rm (res)}$, with $m^{\rm (res)}$ defined by Eq. , as usual, we have to expand the argument of $\psi_n(mx)$ in powers of a small $\delta m = m-m^{\rm (res)}$, so that $\psi_n(mx) \cong \psi'_n(m^{\rm (res)}x)x\delta m$. Then, the general expression, Eq.  is reduced to the following simple form: $$\label{Eq:lineshape_|d_n|^2} |\tilde{d_n}|^2 = \frac{1}{\left(m^{\rm (res)}x\delta m -A_n^{(d)}(x)\right)^2B_n^{(d)2}(x)+1}.$$ It is a typical Lorentzian profile with a maximum at $$\label{Eq:delta_m_|d|_max} \delta m_{|d_n|^2}^{\rm (res)} = \frac{A_n^{(d)}(x)}{m^{\rm (res)}x}$$ and the half-maximum linewidth (FWHM) $$\label{Eq:linewidth_|d_n|^2} \gamma_{|d_n|^2} = \frac{2}{m^{({\rm res})}xB_n^{(d)}(x)}.$$ Note, that Eq.  gives the mismatch between the positions of the resonance points defined according to Eq.  and the ones corresponding to the maxima of $|d_n|^2$. It seems we have encountered a paradox. On one side, we have obtained that in contrast to the outer problem, the inner one does not have a definite limit at $m \rightarrow \infty$, see Eq. . On the other side, $\tilde{d_n} = a_n^{\rm (PRS)} - a_n$, see Eqs. , . This equality means that since $a_n \xrightarrow[m\rightarrow\infty]{}a_n^{\rm (PRS)}$, coefficient $\tilde{d_n}$ should vanish in this limit. The latter reasoning agrees well with the just obtained for $\tilde{d_n}$ lineshape. Indeed, being equal in its maximum to unity, the linewidth for $|\tilde{d_n}|^2$ vanishes at $m \rightarrow \infty$ as $1/m^{\rm (res)}$, while the value of $|\tilde{d_n}|^2$ at off-resonance regions vanishes as $1/m^2$, see Eq. . Then, at $m \rightarrow \infty$ the resonance lines become infinitesimally narrow and the entire profile $|\tilde{d_n}(m)|^2$ vanishes. Finally, because the difference between $\tilde{d_n}$ and ${d_n}$ is just in the multiplicative scaling factor the same conclusion, seemingly, may be applied to the profile $|{d_n(m)}|^2$. However, one should be careful here, because the scaling factor itself depends on $m$. Since , the maximal value of $|d_n|^2$ increases as $m^{(\rm{res})\,2}$, see Eq. . It means that the total area under a given resonance line increases with an increase in $m^{\rm (res)}$ linearly in $m^{\rm (res)}$, see Eq. , making (in contrast to $a_n$) the resonances more and more pronounced. Note, however, that it does not mean nonexistence of *any* universality in the profile $|d_n(m)|^2$ at $m \rightarrow \infty$. Let us consider *a part* of the profile from its bottom to any *fixed* value $D_n^2$. According to Eq.  $$\label{Eq:d_n_through_d_n_tilde} d_n =\frac{m\xi_n'(x)}{\psi_n'(mx)}\tilde{d}_n \equiv \frac{m(\psi_n'(x) -i\chi_n'(x))}{\psi_n'(mx)}\tilde{d}_n .$$ Then, bearing in mind that in the Fraunhofer regime for the discussed resonances $|\psi_n'(m^{({\rm res})}x)| = 1$, see Eqs. , , the width of the line at this distance from the bottom ($\gamma_D$) is given by the difference , where $\delta m_{1,2}$ are the two roots of the equation $$\label{Eq:eq_for Gamma_D} \frac{m^{(\rm{res})\,2}\left(\psi_n'^2(x)+\chi_n'^2(x)\right)}{\left(m^{\rm (res)}x\delta m -A_n^{(d)}(x)\right)^2B_n^{(d)2}(x)+1} = D^2,$$ ![(color online) Independence of $m^{{\rm (res)}}$ of the bottom parts of the resonance lines for profiles $|c_1|^2$ and $|d_1|^2$ at $x =1$ and large values of $m^{{\rm (res)}};\; \zeta = m - m^{{\rm (res)}}$. Here profiles $|c_1(\zeta)|^2$ and $|d_1(\zeta)|^2$ are presented at $m^{{\rm (res)}} = 14.1019...$ (thick blue solid line), $m^{{\rm (res)}} = 1013.1631...$ (thick yellow dashed line) and $m^{{\rm (res)}} = 15.5792$ (red solid line), $m^{{\rm (res)}} = 1014.7325...$ (black dashed line), respectively. Increase in $m^{{\rm (res)}}$ in the three orders of magnitudes, practically, does not affect the shape of the lines. Note the perfect coincidence of the corresponding lines, even in the off-resonance regions, despite Eq.  is valid only in the vicinity of the points of the resonances and, generally speaking, cannot be applied to describe the lineshape far from them.[]{data-label="fig:d1"}](c1d1__bottom){width="\linewidth"} Trivial calculations show that at the quantity $\gamma_D$ converges to the following $m$-independent expression: $$\label{Eq:Gamma_D} \gamma_D \cong \frac{2\sqrt{\psi_n'^2(x) + \chi_n'^2(x)}}{DB(x)x}.$$ This asymptotics is valid for any $D$ satisfying the aforementioned constraint. It means that at large $m$ a part of the resonance lineshape close to the bottom of the resonance line becomes universal. An example of this universality for $n=1$ and $x=1$ is shown in Fig. \[fig:d1\]. Thus, instead of vanishing [(as it should be for a PRS)]{} at $m \rightarrow \infty$ the resonance lines for coefficient $|d_n|^2$ at the bottom converge to a certain universal form, while the maximal value of $|d_n|^2$ at the peak of the resonance increases as $m^{(\rm{res})\,2}$ and the half-maximum linewidth contracts as $1/m^{\rm (res)}$. The behavior of $c_n$ is quite analogous to that of $d_n$. The expressions describing this behavior are presented below for references: $$\begin{aligned} & & b_n = b_n^{\rm (PRS)} -\tilde{c}_n,\;\;\tilde{c}_n \equiv \frac{\psi_n(mx)}{m\xi_n(x)}c_n, \label{Eq:b_n_trough_bPRS_and_c_n--c_n^tilde} \\ & & |\tilde{c_n}|^2 = \frac{1}{\left(m\frac{\psi'_n(mx)}{\psi_n(mx)}-A_n^{(c)}(x)\right)^2B_n^{(c)2}(x)+1} \label{Eq:c_n_tilde_AB} \\ & & \approx \frac{1}{\left(\frac{\psi''_n(m^{\rm (res)}x)}{\psi_n(m^{\rm (res)}x)}m^{\rm (res)}x\delta m -A_n^{(c)}(x)\right)^2B_n^{(c)2}(x)+1},\nonumber \\ & & \delta m_{|c_n|^2}^{\rm (res)} = \frac{A_n^{(c)}(x)\psi_n(m^{\rm (res)}x)}{m^{\rm (res)}x\psi''_n(m^{\rm (res)}x)},\; \label{Eq:delta_m_|c|_max}\\ & & \gamma_{|c_n|^2} = \frac{2}{m^{({\rm res})}xB_n^{(c)}(x)}\left|\frac{\psi_n(m^{\rm (res)}x)}{\psi''_n(m^{\rm (res)}x)}\right|. \label{Eq:linewidth_|c_n|^2}\end{aligned}$$ Here $m^{\rm (res)}$ is defined according to Eq. , $$\label{Eq:A_n^{(c)}_B_n^{(c)}} A_n^{(c)}(x) = {\rm Re}\,\left(\frac{\xi'_n(x)}{\xi_n(x)}\right);\;\;\; B_n^{(c)}(x) = \left|\xi_n^2(x)\right| \equiv \left|\xi_n(x)\right|^2,$$ and to obtain Eq.  the identity $$\left|\xi^2_n(x){\rm Im}\,\left(\frac{\xi'_n(x)}{\xi_n(x)}\right)\right| \equiv 1,$$ whose proof is completely analogous to that for Eq. , has been employed. Let us make numerical estimates. At and (typical values for a number of semiconductors in the visible and IR diapason) the resonant ; ; ; . It gives an estimate for the range of the growth of the electric field inside the particle with respect to the incident wave, while the corresponding values for the volume density of the electromagnetic energy will be of the orders of squires of these values. To conclude this section a certain important remark should be made. Though it is convenient for the theoretical study adopted in the present paper to inspect the resonances at varying $m$ and fixed values of the other problem parameters, such an approach seems completely irrelevant from the physical viewpoint. Really, while in an actual experiment, it is rather easy to change the value of the size parameter, $x$ just varying the wavelength of the incident light, it is very difficult to change $m$. The refractive index is a given property of the material the particle made of. To change it either materials with strong dispersion should be employed, or one has to have a set of particles with the same size but different refractive indices varying in small steps. Both the options look unrealistic. Thus, it seems our study is quite meaningless. Fortunately, this is not the case. The point is that the actual parameter of the theory is the product, $\rho = mx$. The problem in question corresponds to $\rho \gg 1$, while the spacing between two sequential resonances $\delta \rho = \pi$, see Eqs. , . Now, if instead of variations of $m$ at a fixed $x$ we consider variations of $x$ at a fixed $m$, to cover the distance between the two sequential resonances we have to consider departures of $x$ from a resonant value $x^{{\rm(res)}}$ (defined by the same conditions, Eqs. , ) of the order $\delta x \sim 1/m \ll 1$. It means that in slowly-varying functions of $x$ solely, namely in $\xi_n(x),\;\psi_n(x)$ and $\chi_n(x)$ we may neglect these small variations of $x$, replacing these functions by their values at $x = x^{{\rm(res)}}$. The only remaining step is to replace $\delta \rho = x\delta m$ by $\delta \rho = m\delta x$. Then, all the expressions obtained in this section and in what follows are readily recalculated for the case of varying $x$ and fixed $m$. Fano resonances in partial wave scattering =========================================== Up to now we have inspected the modula of the scattering coefficients. Let us focus on their phases. Following the approach, described in monograph [@Hulst::1981], it is convenient to introduce a real angle $\Delta_n^{(a)}$ according to the expression $$\begin{aligned} \label{Eq:Delta} \tan\Delta_n^{(a)} \equiv \frac{F_n^{(a)}}{G_n^{(a)}}\;.\end{aligned}$$\ Then, $a_n$ can be written in the following form $$\begin{aligned} \label{Eq:Sca2} a_n=\frac{\tan\Delta_n^{(a)}}{\tan\Delta_n^{(a)}+i}=\frac{i}{2}(1-e^{2i\Delta_n^{(a)}})\;.\end{aligned}$$ Let us recall now that $a_n$ may be presented as a sum of two terms: $a_n^{\rm (PRS)}$ and $\tilde{d}_n=d_n\psi_n'(mx)/(m\xi_n'(x))$, see Eq. . Here the first term does not depend on $m$, while the second is a sharp function of $m$. Then, it makes sense to split $\Delta_n^{(a)}$ into two corresponding parts, namely to present it as $$\label{Eq:Delta_as_a_sum} \Delta_n^{(a)} \equiv \Delta_n^{(a,{\rm PRS})} + \Delta_n^{(a,{\rm res})},$$ where $\Delta_n^{(a,{\rm PRS})}$ is defined for a PRS in the same manner as that for the just introduced $\Delta_n^{(a)}$ for a particle with arbitrary $m$, i.e., $$\label{Eq:tan_delta_PRS} \tan \Delta_n^{(a,{\rm PRS})} \equiv \frac{F_n^{(a,{\rm PRS})}}{G_n^{(a,{\rm PRS})}} = - \frac{\psi_n'(x)}{\chi_n'(x)},$$ see Eq. . Then, the quantity $\Delta_n^{(a,{\rm res)}}$ is defined by the identity $$\begin{aligned} \tan\Delta_n^{(a)} &\equiv& \tan(\Delta_n^{(a,{\rm res)}} + \Delta_n^{(a,{\rm PRS})})=\nonumber \\ & & \frac{\tan\Delta_n^{(a,{\rm res)}} +\tan\Delta_n^{(a,{\rm PRS})}}{1-\tan\Delta_n^{(a,{\rm res)}}\tan\Delta_n^{(a,{\rm PRS)}}}\;.\label{Eq:Delta_res}\end{aligned}$$ Taking into account Eqs. , and , after some algebra it is possible to show that to satisfy Eq.  identically, tangent of $\Delta_n^{(a,{\rm res)}}$ must be equal $$\label{Eq:tan_Delta_res} \tan \Delta_n^{(a,{\rm res)}} =-\frac{\psi'_n(mx)}{G^{(a,{\rm PRS})}_n G_n^{(a)}+F^{(a,{\rm PRS})}_n F_n^{(a)}}.$$ If now we introduce the notations $$\label{Eq:q_and_epsilon_a} q_n^{(a)} \equiv -\cot \Delta_n^{(a,{\rm PRS})}= \frac{\chi_n'(x)}{\psi_n'(x)},\;\; \epsilon_n^{(a)} \equiv -\cot \Delta_n^{(a,{\rm res)}},$$ the expression for $|a_n|^2$ transforms into the following: $$\begin{aligned} |a_n|^2=\frac{\left(\epsilon_n^{(a)}+q_n^{(a)}\right)^2}{\left(1+q^{(a)2}\right)\left(1+\epsilon^{(a)2}\right)}\;.\label{Eq:|a_n|_Fano}\end{aligned}$$ It is the conventional Fano profile, normalized to its maximal value, [@Fano:PR:1961]. A similar treatment of $b_n$ brings about analogous results with $$\label{Eq:qb_n} q_n^{(b)} = \frac{\chi_n(x)}{\psi_n(x)}.$$ ![(color online) Comparison of profile $|a_1(m)|^2$, given by the exact Mie solution (blue solid line), with that, produced by Eqs. , ; (black dashed line); $x=1$. A yellow horizontal line corresponds to $a_1^{\rm (PRS)}$.[]{data-label="fig:Fano_exact_vs_approx"}](Fano_exact_vs_approx "fig:"){width="\linewidth"}\ ![(color online) $q_1^{(a)}(x)$ – solid and $q_1^{(b)}(x)$ – dashed according to Eqs. , . []{data-label="fig:qab"}](qab "fig:"){width="\linewidth"}\ Once again we encounter a discrepancy, this time with our own results. It seems that by means of just identical transformations we have proven that the profiles of $|a_n|^2,\;|b_n|^2$ are of the Fano type always. On the other hand, it has been shown in our previous publication [@Tribelsky:PRL:2008] that at least in the case of small particles the Fano resonances in partial scattered coefficients cannot happen. Naturally, the discrepancy, as usual, is an illusion. The fact is that the condition $mx \ll 1$ does not allow $m$ to be large enough to reach the point of the first Fano resonance. Therefore, to prove that the profile, Eq.  does correspond to the Fano lineshape, we have to show that (i) in the specified range of variations of $m\;\;\epsilon_n^{(a)}(m)$ reaches the values corresponding to the constructive ($\epsilon_n^{(a)} = 1/q_n^{(a)}$) and destructive ($\epsilon_n^{(a)} = -q_n^{(a)}$) conditions, i.e., the problem in question may exhibit the Fano resonances, indeed, and (ii) for a given Fano profile in the vicinity of both the constructive and destructive interference $\epsilon$ is one and the same *linear* function of $m$. Regarding the former, the manifestation of the constructive and destructive resonances in the problem has been already shown in the previous sections of this paper and in other publications [@Tribelsky:EPL:2012; @Rybin:OE:2013; @Rybin:SciRep:2014]. As for the latter, employing Eqs. , , after simple calculations, it is possible to show that: ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![(color online) An example of a reduction of the Fano profile to Lorentzian at $x=0.3$. While a logarithmic plot of $|a_1(m)|^2$ (a) clearly exhibits the Fano line with the point of the constructive resonance ($|a_1(m)|^2=1$) at $m \approx 14.905$ and the destructive resonance ($|a_1(m)|^2=0$) at $m \approx 15.012$, a linear plot of the same dependence (b) is, practically, indistinguishable from the Lorentzian profile, centered about the point of the constructive resonance. The high-resolution-lineshape in the vicinity of the constructive and destructive resonances is shown in the inset. The background level $|a_1^{{\rm (PRS)}}|^2 \approx 0.0003404$. It corresponds to $q^{(a)2} \approx 2937$.[]{data-label="fig:Fano-Lorentz"}](Fano-Lorentz_a "fig:"){width="50.00000%"} ![(color online) An example of a reduction of the Fano profile to Lorentzian at $x=0.3$. While a logarithmic plot of $|a_1(m)|^2$ (a) clearly exhibits the Fano line with the point of the constructive resonance ($|a_1(m)|^2=1$) at $m \approx 14.905$ and the destructive resonance ($|a_1(m)|^2=0$) at $m \approx 15.012$, a linear plot of the same dependence (b) is, practically, indistinguishable from the Lorentzian profile, centered about the point of the constructive resonance. The high-resolution-lineshape in the vicinity of the constructive and destructive resonances is shown in the inset. The background level $|a_1^{{\rm (PRS)}}|^2 \approx 0.0003404$. It corresponds to $q^{(a)2} \approx 2937$.[]{data-label="fig:Fano-Lorentz"}](Fano-Lorentz_b "fig:"){width="50.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ $$\begin{aligned} \epsilon_n^{(a)} &=& \left(\psi_n'^2(x) + \chi_n'^2(x)\right)m^{{\rm(res})}x\delta m\nonumber\\ & & - \psi_n'(x)\psi_n(x) - \chi_n'(x)\chi_n(x).\label{Eq:epsilon_though_delta_m}\end{aligned}$$ It means that in the vicinity of the resonances $\epsilon_n^{(a)}$ is a linear function of $\delta m$, indeed. Regarding the validity of this approximation up to the points of the extrema of the profile, it follows from Eqs. , . Note that Eq  yields that the characteristic scale for the Fano profiles in $\delta m$ is $1/m$, which agrees with the previous consideration of the linewidths, see Section \[sec:3\]. It is worthwhile mentioning that the accuracy of approximation, is surprisingly good. As an example a comparison of presentation, Eqs. , , with the exact Mie solution for $|a_1|^2$ at $x=1$ in the vicinity of $m=4.5$ is presented in Fig. \[fig:Fano\_exact\_vs\_approx\]. It is important that the asymmetry parameter $q_n^{(a,b)}$ in this case is expressed by simple formulas, Eqs. , in terms of $\psi_n(x)$ and $\chi_n(x)$, i.e., it depends just on the multipolarity of the scattered partial wave $n$, radius of the scattering sphere and wavelength of the incident light (we remind that $x \equiv 2\pi R/\lambda$) and does not depend on the reflective index of the particle $m$. Note also that the expressions for $q_n^{(a,b)}$ may be obtained in a less formal way too. According to the preceding analysis in the off-resonance regions In terms of the Fano resonances these regions correspond to the limit $\epsilon \rightarrow \infty$, when the Fano profile tends to $1/(1+q^2)$, see Eq. . Equalizing $a^{({\rm PRS})}$ to $1/(1+q^{(a)2})$ and $b^{({\rm PRS})}$ to $1/(1+q^{(b)2})$ we again arrive at Eqs. , for $q_n^{(a,b)}$. Let us discuss the dependence $q_n^{(a,b)}(x)$ in detail. At $x > n^2$ we just can take the first term of the asymptotical expansions of the Riccati-Bessel functions. For $\psi_n(x)$ it is given by Eq. . The corresponding asymptotics for $\chi_n(x)$ reads [@DLMF::2015] $$\label{Eq:chi_n_at_large_x} \chi_n(x) \cong \cos\left(x-\frac{n\pi}{2}\right).$$ Then, at $x > n^2$ $$\label{Eq:qab_at_large_x} q_n^{(a)} \cong - \tan\left(x-\frac{n\pi}{2}\right),\;\; q_n^{(b)} \cong \cot\left(x-\frac{n\pi}{2}\right).$$ Note, Eq.  yields $q_{n}^{(a)}= q_{n \pm 1}^{(b)}=q_{n+2}^{(a)}$. In the opposite limit of small $x$, utilizing the known asymptotic expressions for the Bessel functions at a small value of the argument, we arrive at the following expressions for $q_n^{(a,b)}$ at $x \ll 1$: $$\begin{aligned} q_n^{(a)} & \cong & -\frac{n}{n+1} \frac{2^{1 + 2 n}\Gamma(n +\frac{1}{2}) \Gamma(n +\frac{3}{2})}{\pi x^{2n+1}}, \label{Eq:qa_at_small_x} \\ & \mbox{}&\nonumber\\ q_n^{(b)} & \cong & \frac{2^{1 + 2 n}\Gamma(n +\frac{1}{2}) \Gamma(n +\frac{3}{2})}{\pi x^{2n+1}}, \label{Eq:qb_at_small_x}\end{aligned}$$ where $\Gamma(z)$ stands for the gamma function. As an example, the dependences $q_1^{(a,b)}(x)$ in domain $0\leq x \leq 10$, are shown in Fig. \[fig:qab\]. Both expressions, Eqs.  and Eq.  diverge as $1/x^{2n+1}$ at $x \rightarrow \infty$. On the other hand, at large $q^2$ the Fano profile, Eq.  is reduced to the Lorentzian one: $$\label{Fano_to_Lorentz} \frac{\left(\epsilon+q\right)^2}{\left(1+q^{2}\right)\left(1+\epsilon^{2}\right)} \xrightarrow[q \rightarrow \infty]{}\frac{1}{1+\epsilon^{2}}.$$ ------------------------------------ ------------------------------------ {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} ------------------------------------ ------------------------------------ To understand the physical grounds for this reduction, we have to highlight the following. As it has been pointed out already, in the off-resonance regions the Fano profile tends to a non-zero constant background value, $1/(1+q^2)$, while the Lorentzian profile tends to zero, cf. the limits of the right- and left-hand-side of Eqs. . At the point of the constructive resonance ($\epsilon = 1/q$) the Fano profile exhibits its maximal value, equal to unity. So does the Lorentzian profile. However, if for the Lorentzian profile its increase from the off-resonance region also equals unity, for the Fano profile it equals the difference between unity and the background level, i.e., $1-[1/(1+q^2)] = q^2/(1+q^2)$. In the point of the destructive resonance the Fano profile drops from the background level to zero, that is to say, the amplitude of the corresponding modulation of the profile equals the background level $1/(1+q^2)$. The ratio of the amplitudes of the two modulations of the profile (constructive to destructive) is just $q^2$. The larger $q^2$, the less pronounced the destructive resonance relative to constructive and the closer the Fano profile to Lorentzian. If now we remember, that the case $x \ll 1$ corresponds to a *small* particle and a small particle in the off-resonance regions is a *weak scatterer*, we immediately understand why the background level at $x \ll 1$ is small and the Fano profile is close to Lorentzian. Thus, we have arrived at an important conclusion: *though, formally, the Mie resonances for the partial scattered waves of the outer problem for small particles still belong to the Fano type, actually, the corresponding lineshape is very close to the Lorentzian*, cf. [@Tribelsky:EPL:2012]. An example of such a profile is presented in Fig. \[fig:Fano-Lorentz\]. Scale invariance ================ The results discussed in the previous sections give rise to a simple but very important conclusion. Namely, we have obtained that at any fixed $x$ and $n$ the profiles of $|a_n(m)|^2,\;|b_n(m)|^2$ exhibit infinite sequences of the Fano resonances. All these resonances have one and the same, $m$-independent value of $q$, see Eq. , and the characteristic scale, decreasing as $1/m^{{(\rm res})}$, see Eqs. , . Since the shape of the Fano profile is completely defined by the value of $q$, the latter means that at a given $x$ all the Fano resonances are identical and *may be reduced to a single universal form* by the scale transformation $$\label{Eq:scale_m} \delta m \rightarrow m^{({\rm res})}\delta m.$$ Regarding $|c_n|^2,\;|d_n|^2$, at a fixed $m$ these profiles are Lorentzian, and, therefore, also universal. The corresponding scale transformations are Eq. , supplemented with the rescaling of the coefficients $$\label{Eq:scale_cd} c_n \rightarrow c_n/m^{({\rm res})},\;\; d_n \rightarrow d_n/m^{({\rm res})}.$$ Finally, bearing in mind that the mismatches in the positions of the points of the resonances for modes with different $n$ are also scaled as $1/m^{({\rm res})}$, we obtain that the stipulated scale transformations should reduce the entire variety of the resonances to a single universal set, *including the mutual position* of the resonances with different $n$. An example of such a collapse is shown in Fig. \[fig:Fig4\]. It is also seen from Fig. \[fig:Fig4\] that an increase in $n$ results in shifts of the points of resonances for $|c_n|^2$ and $|d_n|^2$ to the left. For the Fano resonances (Fig. \[fig:Fig4\]a,b) the increase in $n$ is also accompanied by the change of the asymmetry parameter $q$, see Eq. , and hence, by the change of the corresponding lineshape. Thus, to identify the shift one should look at the position of the local extrema. Note also, that for the lines $|c_n|^2,\;|d_n|^2$ the increase in $n$ results in an increase in the $Q$-factor of the resonances, see Fig. \[fig:Fig4\]c,d. These peculiarities are generic for the problem in question and valid for any $n$. Dissipative effects {#sec:6} =================== Remember now, that the non-dissipative limit discussed above is a nonexistent in nature abstraction. In fact, dissipative losses always remain finite, as long as a real material is a concern. Then, a natural question “How the obtained results are affected by the dissipative losses?" arises. In this section we are going to answer the question. To this end, we have to introduce a complex refractive index $$\label{Eq:mhat} \hat{m} = m + i\kappa$$ Now note that, actually, there are two cases: strong dissipation ($m$ and $\kappa$ are of the same order of magnitude) and weak dissipation ($\kappa \ll m$). The former case is trivial — the dissipation just suppresses the resonances. Thus, the most interesting is the weak-dissipation-case, especially its limit of the extremely weak dissipation ($\kappa \ll 1$), when the dissipative damping may compete with the small radiative one, and effects similar to the anomalous scattering [@Tribelsky:PRL:2006] may be observed. Examples of the extremely weak dissipation may be readily found among widely used semiconductors. For instance, at the wavelength of 532 nm (the second harmonic of a Nd:YAG laser) the complex refractive indices for gallium phosphide, silicon and gallium arsenide are respectively [@refractiveindex.info]. In what follows all numerical examples, illustrating the developed theory, will be given for a particle made of gallium phosphate embedded into a transparent matrix with a refractive index close to unity. The dispersion of will be neglected, since in the proximity of the specified wavelength it is rather weak. Let us discuss the weak dissipation case in detail. For this purpose, we have to consider small *complex* departures of $\delta\hat{m} = \delta m + i\kappa$ from a purely real $m^{{(\rm res)}}$, defined by the condition . Expansion of $\psi(\hat{m}x)=0$ in powers of small $x\delta\hat{m}$ about the point $m^{{(\rm res)}}x$ results in the following trivial generalization of expression, Eq.  for $|\tilde{d}_n|^2$: $$\label{Eq:lineshape_|tilde_d_n|^2_with_kappa} |\tilde{d_n}|^2 = \frac{1}{\left(m^{\rm (res)}x\delta m -A_n^{(d)}(x)\right)^2B_n^{(d)2}(x) +\left(1+\kappa B_n^{(d)}(x)m^{\rm (res)}x\right)^2},$$ with the connection between $|\tilde{d}_n|^2$ and $|{d}_n|^2$ $$\label{Eq:|dn|^2_through_|tilde_dn|^2} |d_n|^2 = \frac{m^2B_n^{(d)}(x)}{|\psi'_n(mx)|^2} |\tilde{d}_n|^2,$$ following from Eqs. , . Eq.  yields the linewidth $$\label{Eq:Gamma_|dn|^2_kappa} \gamma_{|d_n|^2}^{(\kappa)} = 2\frac{1+\kappa B_n^{(d)}(x)m^{\rm (res)}x}{B_n^{(d)}(x)m^{\rm (res)}x}.$$ The profile $|d_n|^2$ is maximized by the same $\delta m_{|d_n|^2}^{\rm (res)}$, given by Eq. , but the maximal value now is different: $$\label{Eq:Max_|dn|_kappa} \text{Max}\left\{ |d_n|^2\right\} \cong \frac{m^{{\rm (res)}2}B_n^{(d)}(x) }{|\psi'_n(m^{\rm(res)}x)|^2\left(1+\kappa B_n^{(d)}(x)m^{\rm (res)}x\right)^2}.$$ The obtained results give rise to important conclusions. Namely, as it is clearly seen from Eq. , there is a point of crossover, $m_{\rm cr}$ from the non-dissipative regime (at $m \ll m_{\rm cr}$) to dissipative (at $m \gg m_{\rm cr}$), where $m_{\rm cr}$ is a solution of the equation: $$\label{Eq:m_cr} \kappa m x B_n^{(d)}(x) = 1.$$ If at $m \ll m_{\rm cr}$ the linewidth is determined by Eq. , at $m \gg m_{\rm cr}$ it converges to the universal $m$-$x$-$n$-independent value $2\kappa$. However, the most dramatic changes happen with the amplitude of the resonances. While in the non-dissipative limit the amplitude of the profile $|d_n|^2$ at the resonance points at large $m$ increases as $m^2$, see Eq. , now the entire profile converges to the universal form $$\label{Eq:|dn|_kappa_large_m} |d_n|^2 \xrightarrow[{m \gg {m_{_{cr}}}}]{}\frac{1}{|\psi'_n(m^{\rm(res)}x)|^2x^2B_n^{(d)}(x)}\frac{1}{\delta m^2 + \kappa^2},$$ which becomes completely $m^{\rm(res)}$-independent in the Fraunhofer regime, $m^{\rm(res)}x > n^2$ (we remind that in this regime $|\psi'_n(m^{\rm(res)}x)| \cong 1$). Coefficients $c_n$ may be inspected exactly in the same manner. The corresponding expressions this inspection yields are quite similar to those, obtained above for $d_n$. For example, $$\begin{aligned} & & \text{Max}\left\{ |c_n|^2\right\} \label{Eq:Max_|cn|_kappa}\\ & & \cong \frac{m^{{\rm (res)}2}B_n^{(c)}(x) }{|\psi_n(m^{\rm(res)}x)|^2\left(1-\kappa\frac{\psi''(m^{\rm (res)}x)}{\psi(m^{\rm (res)}x)} B_n^{(c)}(x)m^{\rm (res)}x\right)^2}.\nonumber\end{aligned}$$ The reader should not be confused with sing minus in front of the term with the dissipative constant $\kappa$. The point it that this term includes also the factor ${\psi''(m^{\rm (res)}x)}/{\psi(m^{\rm (res)}x)}$, which is always negative and tends to minus unity in the Fraunhofer regime, see Eq. . Thus, actually, the sign in front of the dissipative constant is plus. The other expressions for $c_n$, analogous to those discussed above for $d_n$, are not presented here owing to the triviality of the corresponding calculations. They result in the behavior of $c_n$ quite similar to that for $d_n$. These peculiarities of $c_n$ and $d_n$ brings about a completely different scenario for vanishing of the Fano resonances for coefficients $a_n$ and $b_n$. For definiteness, let us focus on $a_n$. This coefficient is expressed in terms of $m$-independent $a^{{\rm (PRS)}}_n$ and $m$-dependent $d_n$ according to Eq. . Thus, the entire $m$-dependence of $a_n$ is given by the second term in the right-hand-side of Eq. . We remind the reader that in the non-dissipative limit $|a_n|^2$ always vanishes at the points of the destructive Fano resonances and reaches unity at the points of the constructive resonances, see Fig. \[fig:a1\] and Eq. . Then, the vanishing of the Fano resonances with an increase in $m$ occurs owing to the contraction of the resonance lines. Asymptotically the points of the constructive and destructive resonances merge, while *the amplitude* of modulations of $|a_n|^2$ caused by the Fano resonances all the time *remains the same*: each resonance forces $|a_n|^2$ to vary from zero to unity. In contrast, now at large enough $m$ the width of the resonances for $|a_n|^2$ becomes $m$-independent and equal to $2\kappa$ (according to Eq.  the characteristic widths of the resonances for $a_n$ and $d_n$ have the same order of magnitude), while the amplitude of the modulations of $|a_n - a^{{\rm (PRS)}}_n|^2$ decreases as $1/m^2$, see Eqs. , . Thus, the vanishing of the Fano resonances (convergence of $a_n$ to $a^{{\rm (PRS)}}_n$) occurs owing to the vanishing of the *amplitude* of the modulations of $a_n$. A crossover from the non-dissipative scenario to dissipative is again determined by $m_{{\rm cr}}$, see Eq. . The behavior of $b_n$ is analogous. Resonances at varying size parameter and fixed refractive index =============================================================== Amplitudes of resonances ------------------------ Up to now the resonances have been studied at a fixed $x$ (and $\kappa$) at varying $m$. On the other hand, as it has been mentioned above, the most interesting from the experimental viewpoint is the dependence of the resonances on the size of the particle at a fixed value of the refractive index. In this case at any fixed $n$ an increase in $x$ again results in a cascade of resonances, whose position is determined by the same conditions Eqs. , , regarded now as equations At $mx \gg 1$ these two problem formulations (fixed $x$ at varying $m$ and fixed $m$ at varying $x$) are easily reduced to each other by the set of transformations as long as the shape of a single resonance line is a concern, see the corresponding remark in the end of Sec. \[sec:3\]. However, this is not the case anymore, if we are interested in rather a large range of variations of the size parameter. The corresponding behavior of the scattering coefficients is discussed in the present section. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(color online) The behavior of $B_n^{(d)}(x) \equiv |\xi_n'(x)|^2$ and $B_n^{(c)}(x) \equiv |\xi_n(x)|^2$ for the first five multipoles. The curves from left to right correspond to $n=1,\;2,\;3,\;4$ and 5, respectively.[]{data-label="fig:Bn(x)"}](Bdn "fig:"){width="50.00000%"} ![(color online) The behavior of $B_n^{(d)}(x) \equiv |\xi_n'(x)|^2$ and $B_n^{(c)}(x) \equiv |\xi_n(x)|^2$ for the first five multipoles. The curves from left to right correspond to $n=1,\;2,\;3,\;4$ and 5, respectively.[]{data-label="fig:Bn(x)"}](Bcn "fig:"){width="50.00000%"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The results obtained in the previous Sec. \[sec:6\] show that the key function determining the behavior of the resonances at large variations of $x$, is $B_n^{(d)}(x) = |\xi_n'(x)|^2$. The known expansions of the Bessel functions in powers of their small argument yield the following behavior of $B_n^{(d)}(x)$ at $x \rightarrow 0$: $$\label{Eq:Bn_at_x->0} B_n^{(d)}(x) \cong \frac{n^2 4^n \Gamma^2(n+\frac{1}{2})}{\pi x^{2(n+1)}}.$$ An increase in $x$ is supplemented with a monotonic decrease in $B_n^{(d)}$ until this function reaches its minimal value at $x \approx n$. A further increase in $x$ results in a slow monotonic growth of $B_n^{(d)}$, asymptotically approaching unity at $x \rightarrow \infty$, see Eqs. , . Accordingly, the entire domain $0 \leq x < \infty$ of variations of $x$ is partitioned into two subdomains: the subdomain $0 < x \leq n$ of a sharp fall of $B_n^{(d)}(x)$ from infinity to its minimal value below unity and the one of a slow asymptotical growth $B_n^{(d)}(x)$ to unity $(n < x <\infty)$. As an example, functions $B_n^{(d)}(x)$ for the first five multipoles are presented in Fig. \[fig:Bn(x)\]a. Thus, at any $n$ both $B_n^{(d)}(x)$ and $xB_n^{(d)}(x)$ are singular functions at $x\rightarrow 0$. It brings about a dramatic enhancement of the dissipative effects at small $x$, see Eq. . On the other hand, an increase in $B_n^{(d)}(x)$ increases the numerator in the right-hand-side of Eq. , defining the amplitude of the resonance. That is to say, at small $x$ there is a competition between the growth of the amplitude of the resonance owing to the increase in $B_n^{(d)}(x)$ in the numerator of Eq.  and its suppression because of the growth of the same quantity in the denominator of the same equation. In this case we have to distinguish to limits: \(i) Despite the large value of $B_n^{(d)}$ the first (smallest) resonant $x^{({\rm res})}_{n,1}<n$ still corresponds to weak dissipation . Then, the amplitude of this resonance is the largest for the given $n$. An increase in $x$ gives rise to the fall of the amplitudes of the sequential resonances, first due to the decrease of $B_n^{(d)}(x)$ in the numerator of Eq. \[Eq:Max\_|dn|\_kappa\], and then due to the transition to the dissipation-controlled region due to increase of $\kappa B_n^{(d)}(x)m x$ in the denominator of this expression. \(ii) $x^{({\rm res})}_{n,1}<n$ is so small (i.e., $B_n^{(d)}(x^{\rm (res)}_{n,1})$ is so large), that $\kappa B_n^{(d)}(x^{\rm (res)}_{n,1})m x^{\rm (res)}_{n,1} \gg 1$, despite smallness of $\kappa$. Then, the amplitude of the corresponding resonance approximately equals $$\text{Max}\{|d_n|^2\} \cong \frac{1}{(\psi'_n(mx^{\rm (res)})\kappa x^{\rm (res)})^2B_n^{(d)}(x^{\rm (res)})},$$ see, Eq. . Since, by definition, in a given cascade $x^{\rm (res)}_{n,1}<x^{\rm (res)}_{n,2}<x^{\rm (res)}_{n,3}<\ldots,$ the next resonances in the same cascade result in a decrease of $B_n^{(d)}(x^{\rm (res)}_{n,p})$ and, hence, in an *increase* of their amplitude. It goes on in this manner until the decease of $B_n^{(d)}$, eventually, drives the particle out of the dissipation-controlled regime. Then, a further increase in $x^{\rm (res)}$ results in the effects, described above in Thus, now we have two dissipation-controlled domains: the first at small $x$ and the second at large, separated by a non-dissipative domain. The maximal amplitude of the resonance mode is achieved at $x^{\rm (res)}$ situated at the boundary between the first dissipation-control domain the non-dissipative domain. It is important to stress also the quite different asymptotic behavior of profiles $|d_n(m)|^2$ at a fixed $x$, and $|d_n(x)|^2$ at a fixed $m$, respectively. If $|d_n(m)|^2$ at $m \rightarrow \infty$ converges to a certain universal periodic function, see Eq. , $|d_n(m)|^2$ at $x \rightarrow \infty$ vanishes as $1/(\kappa x)^2$, owing to Eq.  and limits $$|\psi'_n(mx^{\rm (res)})| \xrightarrow[{x^{\rm (res)} \to \infty }]{}1,\;\; B_n^{(d)}(x) \xrightarrow[{x \to \infty }]{}1.$$ The behavior of $|c_n|^2$ in general is analogous to the discussed above for $|d_n|$. However, in contrast to $B_n^{(d)}(x)$, function $B_n^{(c)}(x)$ is monotonically decreasing, see Fig. \[fig:Bn(x)\]b. This difference in $B_n$ results in a certain difference in the shape of the envelopes of the resonances for $|d_n(x)|$ and $|c_n(x)|$, see Fig. \[fig:GaP\_d20\]. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(color online) A typical behavior of $|d_n(x)|$ and $|c_n|$ on the example of $n=20$ for a particle made of GaP. Blue solid lines corresponds to the exact solution, Eqs.  ; the envelopes, calculated according to the approximate Eq.   are shown as red dashed lines. Note strongly non-monotonic dependence of the amplitude of the resonant oscillations on $x$ due to the enhancement of the dissipative effects at certain domains of variations of $x$ and a “bottle neck" in the dependence $|d_n(x)|$ at the proximity of $x=n=20$. The bottle neck is originated in the corresponding minimum in $B_n^{(d)}(x)$, which does not have $B_n^{(c)}(x)$. For more details, see the text. []{data-label="fig:GaP_d20"}](GaP_d20 "fig:"){width="50.00000%"} ![(color online) A typical behavior of $|d_n(x)|$ and $|c_n|$ on the example of $n=20$ for a particle made of GaP. Blue solid lines corresponds to the exact solution, Eqs.  ; the envelopes, calculated according to the approximate Eq.   are shown as red dashed lines. Note strongly non-monotonic dependence of the amplitude of the resonant oscillations on $x$ due to the enhancement of the dissipative effects at certain domains of variations of $x$ and a “bottle neck" in the dependence $|d_n(x)|$ at the proximity of $x=n=20$. The bottle neck is originated in the corresponding minimum in $B_n^{(d)}(x)$, which does not have $B_n^{(c)}(x)$. For more details, see the text. []{data-label="fig:GaP_d20"}](GaP_c20 "fig:"){width="50.00000%"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Overlap of resonances --------------------- One of the key features of the problem in question is an overlap of a large number of resonances, which may result in a creation within the particle “hot spots" with a giant concentration of the electromagnetic field. The corresponding field structure is determined by the coordinate dependence of the resonant modes. Its detailed inspection requires a separate consideration and will be reported elsewhere. In the present paper we discuss just the necessary conditions for such hot spots to come into being, namely the overlap of profiles of the modula of the scattering coefficients, describing the field within the particle. As usual, for the sake of briefness we restrict the discussion with the properties of $|d_n|^2$ solely. The behavior of $|c_n|^2$ may be studied in the same manner and exhibits similar peculiarities. Overlap of two resonances means that the mismatch between the positions of the maxima of their lines is smaller than the largest linewidth. Therefore, to begin with, we have to discuss the linewidths of the resonances at a fixed $m$ and varying $x$. To this end, according to the stipulated above general rules, we have to replace in Eq.  $x\delta m \rightarrow m\delta x,\; m^{\rm (res)} \rightarrow m$ and $x \rightarrow x^{\rm (res)}$. This brings about the following expression for the linewidth: $$\label{Eq:gamma_x} \gamma_{|d_n|^2}^{(\kappa,x)} = 2\frac{1+\kappa B_n^{(d)}(x^{\rm (res)})mx^{\rm (res)}}{B_n^{(d)}(x^{\rm (res)})m^2}.$$ Thus, in the non-dissipative region ($\kappa B_n^{(d)}(x)mx \ll 1$) the linewidth is $2/[B_n^{(d)}(x)m^2]$, while in the dissipation-controlled region ($\kappa B_n^{(d)}(x)mx \gg 1$) it is $2\kappa x/m$. On the other hand, the resonant values of $x$ are defined by the condition $\psi_n(mx^{{\rm (res,}E)}) = 0$ for the electric modes and for magnetic, see Eqs. , . Below the Fraunhofer regime, at $mx < n^2$ different solutions of these equations are situated at distances of the order of $1/m$. It is always much larger than the linewidth in the non-dissipative region and still larger than that in the dissipation-controlled case, provided $x$ is of the order of unity, or smaller. Therefore, the overlap of the resonances in these regions may occur just accidentally. However, the case is changed drastically in the Fraunhofer regime, when $mx$ becomes larger than $n^2$. In this regime points of different resonances are going to merge. In the leading (in $1/mx$) approximation they are just coincide, see Sec. \[sec:2\]. To resolve the mismatch between the points of different resonances we have to go beyond the leading approximation. Employing the asymptotic expansion of the Riccati-Bessel functions at large values of their argument [@DLMF::2015] and taking into account the first subleading term, instead of Eq.  we arrive at the following equation, determining the points of the electric mode resonances: $$\label{Eq:sin=0_subleading} \sin\left(\rho_{_E}-\frac{n\pi}{2}\right) + \frac{n(n+1)}{2\rho_{_E}}\cos\left(\rho_{_E}-\frac{n\pi}{2}\right) \cong 0,$$ where $\rho_{_E} > n^2$ stands for $mx^{{\rm (res,}E)}$. Looking for a solution of this equation in the form $\rho_{_E} = \rho_{_E}^{(0)} + \delta \rho_{_E};$ $|\delta\rho_{_E}| \ll 1$, where $\rho_{_E}^{(0)}=[(2p+n)\pi/2]$ it is easy to obtain that $$\label{Eq:delta_rho_E} \delta\rho_{_E} \cong - \frac{n(n+1)}{\pi(n+2p)}.$$ The solution is valid at $n(n+1) \ll \pi(n+2p)$. The corresponding mismatch between different resonances is $\delta \rho_{_E}/m$. Strictly speaking, we have to add to this mismatch another one, caused by the departure of the position of the maxima of $|d_n|^2$ from the resonance points determined according to the employed condition $\psi_n(mx) =0$. However, according to Eq.  this mismatch equals $A_n^{(d)}(x)/m^2$ (we recall, that to apply Eq.  to our case we have to replace $x\delta m \to m\delta x$). At large $m$ this quantity is small relative to $\delta\rho_{_E}/m$ and may be neglected. ![(color online) Cascades of resonances for $|c_n|$ and $|d_n|$ at a fixed refractive index and varying size parameter, $x$ for a spherical particle made of GaP, embedded in a transparent medium with the refractive index close to unity. Calculations according to Eqs. , . Gradual overlap of resonances for and at an increase in $x$ is seen clearly. Note broadening of the resonance lines with an increase in $x$ and an increase of the amplitudes of the first resonances in each cascade with an increase in $n$. The first resonances in each cascades have the largest amplitudes. The large mismatches between the first resonances in the cascades for $|c_1|$ – $|d_2|$ and $|c_2|$ – $|d_3|$ are generic, while the overlaps of the first resonances in the cascades for $|d_1|$ – $|c_2|$ and $|d_2|$ – $|c_3|$ are accidental. For more details, see the text.[]{data-label="fig:GaP_ovlp"}](GaP_ovlp){width="50.00000%"} Similar inspection of $|c_n|^2$ gives rise to $\rho_{_H} = \rho_{_H}^{(0)} + \delta \rho_{_H}$ with $$\label{Eq:delta_rho_H} \rho_{_H}^{(0)} = \frac{(n+2p+1)\pi}{2},\;\; \delta\rho_{_H} \cong \frac{n(n+1)}{\pi(n+2p+1)}.$$ Let us try to collect all together. According to the results discussed in Sec. \[sec:2\], there are two types of possible overlaps. First, corresponds to the overlap of the modes of the same type (e.g., electric – electric, or magnetic – magnetic) with different $n$ and $p$, changed in such a manner that sum $n+2p$ remains fixed. The adjacent candidates for this overlap have the difference in $n$ equals 2 and the difference in $p$ equals 1. According to Eqs. , the mismatches between the resonance points for these two modes are $$\label{Delta_x_n_n+2} (\Delta x_{_E})_{n,n+2} \cong (\Delta x_{_H})_{n,n+2} \cong \frac{2(3+2n)}{m\pi N},$$ where integer $N$ satisfies the condition $\pi N/2 = \rho_{_H}^{(0)}=\rho_{_H}^{(0)} = mx^{\rm (res)}$. The candidates for the second type of the overlap are modes with the different nature: electric – magnetic. In this case the adjacent modes have both $n$ and $p$ differed in unity, however, the signs of these mismatches are opposite, see Eqs. , . Then, the corresponding mismatch is $$\label{Delta_x_EH_n_n+1} (\Delta x_{_{E,H}})_{n,n+1} \cong \frac{2(n+1)^2}{m\pi N}.$$ In all the cases the mismatches monotonically increase with an increase in $n$. A rough estimate of the number of modes, which may overlap yields from the applicability condition of the Fraunhofer regime: $\rho > n^2$. For a given $x^{\rm (res)}$ it results in the inequality $n < \sqrt{N} \sim \sqrt{mx^{\rm (res)}}$. Thus, at $n \gg 1$ the mismatches between the modes of the same type are scaled as while the ones between the modes of different types are scaled as $n^2/(mN)$. On the other hand, the linewidth in the dissipation-controlled region is scaled as $\kappa x/m$, see Eq. . An increase in $x$ decreases $\Delta x$ and increases the linewidth. The overlap begins, when the later becomes larger than the former. First, it happens with the modes of the same type at $x = x_{\rm ovlp}$, where $$\label{Eq:x_ovlp} x_{\rm ovlp} \sim \sqrt{\frac{n}{m\kappa}}.$$ A further growth of $x$ increases the number of the overlapping modes, until all modes lying in the Fraunhofer range overlap. It happens at $x>X_{\rm ovlp}$, where $X_{\rm ovlp}$ is such a value of $x$, when $(\Delta x)_{1,\sqrt{N}}$ becomes equal, or smaller than the linewidth. Here $(\Delta x)_{1,\sqrt{N}}$ stands for the mismatch between the mode with $n=1$ and the mode with the maximal $n$ still belonging to the Fraunhofer range, i.e., with $n \sim \sqrt{N}$. According to Eqs. , , $(\Delta x)_{1,\sqrt{N}} \sim (n^2/mN)_{_{n=\sqrt{N}}} = 1/m$. It yields a simple estimate $X_{\rm ovlp} \sim 1/\kappa$. As it follows from Eq.  Max$\{|d_n(X_{\rm ovlp})|\} \sim 1$. Then, in the point, where all the overlapping modes are summarized coherently with the weight factor of the order of unity, the resulting amplitude of the electric (magnetic) field is $\sqrt{N}$ times greater (the density of energy N times greater) the one in the incident wave. To give a certain impression about a manifestation of the discussed resonances in a possible real experiment we produce some results for a particle made of gallium phosphide. The corresponding plots are presented in Figs. \[fig:GaP\_d20\], \[fig:GaP\_ovlp\]. Our calculations for GaP show also that the absolute maximum for a single resonant mode is achieved for the first resonance of mode $c_5$ at $x \approx 2.318$. The corresponding value of $|c_5|$ equals 34.641. The first resonant of $|d_5|$ gives rise to the close value: 33.161 at $x \approx 2.626$. The density of the electromagnetic energy in the hot spots corresponding to these solitary resonances is about $10^3$ greater than that for the incident wave. Meanwhile, multiple overlap of not so sharp resonances for a particle with $x \approx 25$ may produce hot spots with the density of energy $10^4$ relative to that in the incident wave. Conclusions =========== Summarizing the results obtained, we may say that the detailed study of light scattering by a high refractive index particle with low dissipation discussed in the present paper, has revealed a number of new important features of the problem. In particular, we have shown that while at increasing $m$ the partial scattered waves outside the particle at any fixed $x$ and $n$ tend to the limits, corresponding to the light scattering by a perfectly reflective sphere, the field within the particle does not have any limit at all. The reason for this difference in the behavior between the solutions of the outer and inner problems is related to the different manifestation of the infinite sequences of cascades of the Mie resonances in the two problems. For the scattered field outside the particle each Mie resonance in a cascade has the asymmetric Fano lineshape. As $m$ increases the resonances are suppressed, and the expressions for the scattering coefficients converge to the corresponding $m$-independent quantities for the perfectly reflecting sphere. For the field within the particle, each Mie resonance in a given cascade has the Lorentzian lineshape. In this case, while at large $m$ the profile of the cascade at a fixed distance from its bottom converges to a certain universal form, the peak value of the modulus of the electric (magnetic) field amplitude increases with an increase in $m$. Thus, the increase in $m$ makes the resonances more pronounced. At finite dissipation the growth of the amplitude of the resonances eventually saturates, and the lineshape becomes a periodic function It is important that at large enough $m$ the positions of the electric resonances for a partial mode with multipolarity $n$ correspond to those for magnetic resonances with the multipolarity $n+1$, the electric mode with the multipolarity $n+2$, etc. The same is true for the magnetic modes. For a given multipolarity the points of both the types of the resonances together (i.e., electric and magnetic) are situated in the $m$-axis periodically with the period equal to the half of the period for each type of the resonances separately. It means the points of a magnetic resonance are situated just in the middle of the two adjacent points of the electric resonances and vice versa. We have shown that the cascade of the Fano resonances, exhibited with an increase in the refractive index of a scatterer, is a general, intrinsic feature of the light scattering problem. In this problem the field scattering by the same scatterer, but with the perfectly reflecting properties plays the role of a background partition, while the resonantly excited in the scatterer Mie modes correspond to the resonant ones. The characteristics of the Fano profiles, including a simple expression for the asymmetry parameter $q$, see Eqs. , , have been obtained from “the first principles" based upon the identical transformations of the exact Mie solution. We have demonstrated that in the non-dissipative limit the discussed resonances (for both the inner and outer problems) possess a scale invariance, so that at any fixed value of the size parameter $x$ any resonance line in the infinite cascades of the resonances may be reduced to the universal, $m^{({\rm res})}$-independent form by the scale transformations, Eqs. , . It should be stressed that the universality extends to both the shape of the lines and the mutual positions of lines with different $n$ with respect to each other. The quantitative applicability conditions for the non-dissipative and dissipation-controlled regimes as well as the corresponding crossover points have been obtained in the explicit form. The peculiarities of the resonances at a fixed refractive index and a varying size parameter, $x$ have been revealed. It has been shown that, generally speaking, the entire domain $0 \leq x < \infty$ is partitioned into three subdomains: two dissipation-controlled subdomains (at small and very large $x$) are separated by a non-dissipative one. The explicit expressions, determining the boundaries between the subdomains as well as the formulas for the lineshape and linewidth have been derived. The linewidth is minimal in the non-dissipative domain and increases in the dissipation-controlled subdomains with an increase in the departure of $x$ from the corresponding boundary. For the size parameter lying in the non-dissipative subdomain a high concentration of the electromagnetic field within the particle may be achieved owing to individual narrow-line partial resonances of modes with a high $Q$-factor. For the realistic values of the complex refractive index, typical for a number of common semiconductors in the visible and near IR range of the spectrum, the peak value of the density of electromagnetic energy in this case may exceed the one in the incident plane wave in the three orders of magnitude. In contrast, for the size parameter, lying in the dissipation-controlled subdomains even larger concentration may be realized due to multiple overlap of rather broad resonance lines of modes with of different types (magnetic and electric) and different multipolarity. The discussed effect of the giant concentration of the electromagnetic field within a particle with a high refractive index may have great importance in medical applications (such as, e.g., cancer therapy [@El-Sayed:LMS:2008]), in the design and fabrication of high-nonlinear nanostructures, etc. A detailed study of this important issue is a separate problem and will be reported elsewhere. The authors are grateful to B.S. Luk’yanchuk for the critical reading of the manuscript and valuable comments. The work of AEM was supported by the Australian Research Council via Future Fellowship\ Proof of identity, Eq.  ======================= The proof is based upon Eq. . According to it and presentation of $\xi_n$ in the form with real functions $\psi_n,\;\chi_n$, we may write the following chain of identities: $$\begin{aligned} \frac{\xi_n(x)}{\xi_n'(x)} \equiv \frac{\psi_n'(x)\xi_n(x)}{\psi_n'(x)\xi_n'(x)} \equiv \frac{\psi_n(x)\xi_n'(x) -i}{\psi_n'(x)\xi_n'(x)} \tag{A.1}\label{Eq:xsi_n/xsi_n_prime}\\ \equiv \frac{\psi_n(x)}{\psi_n'(x)} - \frac{i\xi_n'^\ast(x)}{\psi_n'(x)|\xi_n'(x)|^2}, \nonumber\end{aligned}$$ where asterisk means complex conjugation. Then, bearing in mind the same presentation $\xi_n = \psi_n - i\chi_n$, and taking the imaginary part of Eq.  we obtain $${\rm Im}\frac{\xi_n(x)}{\xi_n'(x)} = -\frac{1}{|\xi_n'(x)|^2}.$$\ Finally, multiplying it by $\xi_n'^2(x)$ and taking modulus, we arrive at identity, Eq. . [^1]: The problem of light scattering by a PRS is a bit tricky. The point is that the field inside the PRS is zero identically. Then, instead of the four independent scattering coefficients: two for the scattered field outside the particle and two for the field within the particle, only the former two remain. The reduction of the independent constants of integration of the Maxwell equations, requires the corresponding reduction of the boundary conditions. It could be shown, that for the electric modes discussed here only boundary condition, Eq.  remains, while the boundary condition, Eq.  should be removed at this limit.

--- abstract: | We generalize the unique decoding algorithm for one-point AG codes over the Miura-Kamiya $C_{ab}$ curves proposed by @lee11 to general one-point AG codes, without any assumption. We also extend their unique decoding algorithm to list decoding, modify it so that it can be used with the Feng-Rao improved code construction, prove equality between its error correcting capability and half the minimum distance lower bound by @andersen08 that has not been done in the original proposal except for one-point Hermitian codes, remove the unnecessary computational steps so that it can run faster, and analyze its computational complexity in terms of multiplications and divisions in the finite field. As a unique decoding algorithm, the proposed one is as fast as the BMS algorithm for one-point Hermitian codes, and as a list decoding algorithm it is much faster than the algorithm by @beelen10.\ **Keywords:** algebraic geometry code, Gröbner basis, list decoding\ **MSC 2010:** Primary: 94B35; Secondary: 13P10, 94B27, 14G50 author: - 'Ryutaroh Matsumoto[^1], Diego Ruano[^2],  and Olav Geil' date: 'April 22, 2013' title: 'List Decoding Algorithm based on Voting in Gröbner Bases for General One-Point AG Codes' --- Introduction ============ We consider the list decoding of one-point algebraic geometry (AG) codes. @guruswami99 proposed the well-known list decoding algorithm for one-point AG codes, which consists of the interpolation step and the factorization step. The interpolation step has large computational complexity and many researchers have proposed faster interpolation steps, see [@beelen10 Figure 1]. By modifying the unique decoding algorithm [@lee11] for primal one-point AG codes, we propose another list decoding algorithm based on voting in Gröbner bases whose error correcting capability is higher than [@guruswami99] and whose computational complexity is smaller than [@beelen10; @guruswami99] in many cases. A decoding algorithm for primal one-point AG codes was proposed in [@ldecodepaper], which was a straightforward adaptation of the original Feng-Rao majority voting for the dual AG codes [@fengrao93] to the primal ones. The Feng-Rao majority voting in [@ldecodepaper] for one-point primal codes was generalized to multi-point primal codes in [@beelen08 Section 2.5]. The one-point primal codes can also be decoded as multi-point dual codes with majority voting [@beelen07; @duursma11; @duursma10], whose faster version was proposed in [@sakata11] for the multi-point Hermitian codes. @lee11 proposed another unique decoding (not list decoding) algorithm for primal codes based on the majority voting inside Gröbner bases. The module used by them [@lee11] is a curve theoretic generalization of one used for Reed-Solomon codes in [@kuijper11] that is a special case of the module used in [@lee08]. An interesting feature in [@lee11] is that it did not use differentials and residues on curves for its majority voting, while they were used in [@beelen08; @ldecodepaper]. The above studies [@beelen08; @lee11; @ldecodepaper] dealt with the primal codes. We recently proved in [@gmr13] that the error-correcting capabilities of [@lee11; @ldecodepaper] are the same. The earlier papers [@duursma94; @pellikaan93] suggest that central observations in [@andersen08; @gmr13; @ldecodepaper] were known to the Dutch group, which is actually the case [@duursma12pcomm]. @chen99, @elbrondjensen99 and @amoros06 studied the error-correcting capability of the Feng-Rao [@fengrao93] or the BMS algorithm [@sakata95b; @sakata95a] with majority voting beyond half the designed distance that are applicable to the dual one-point codes. There was room for improvements in the original result [@lee11], namely, (a) they have not clarified the relation between its error-correcting capability and existing minimum distance lower bounds except for the one-point Hermitian codes, (b) they have not analyzed the computational complexity, (c) they assumed that the maximum pole order used for code construction is less than the code length, and (d) they have not shown how to use the method with the Feng-Rao improved code construction [@feng95]. We shall (1) prove that the error-correcting capability of the original proposal is always equal to half of the bound in [@andersen08] for the minimum distance of one-point primal codes (Proposition \[prop:AG\]), (2) generalize their algorithm to work with any one-point AG codes, (3) modify their algorithm to a list decoding algorithm, (4) remove the assumptions (c) and (d) above, (5) remove unnecessary computational steps from the original proposal, (6) analyze the computational complexity in terms of the number of multiplications and divisions in the finite field. The proposed algorithm is implemented on the Singular computer algebra system [@singular313], and we verified that the proposed algorithm can correct more errors than [@beelen10; @guruswami99] with manageable computational complexity. This paper is organized as follows: Section \[sec2\] introduces notations and relevant facts. Section \[sec3\] improves [@lee11] in various ways, and the differences to the original [@lee11] are summarized in Section \[sec:diff\]. Section \[sec4\] shows that the proposed modification to [@lee11] works as claimed. Section \[sec:experiment\] compares its computational complexity with the conventional methods. Section \[sec6\] concludes the paper. Part of this paper was presented at 2012 IEEE International Symposium on Information Theory, Cambridge, MA, USA, July 2012 [@gmr12isit]. Notation and Preliminary {#sec2} ======================== Our study heavily relies on the standard form of algebraic curves introduced independently by @geilpellikaan00 and @miura98, which is an enhancement of earlier results [@miura92; @saints95]. Let $F/\mathbf{F}_q$ be an algebraic function field of one variable over a finite field $\mathbf{F}_q$ with $q$ elements. Let $g$ be the genus of $F$. Fix $n+1$ distinct places $Q$, $P_1$, …, $P_n$ of degree one in $F$ and a nonnegative integer $u$. We consider the following one-point algebraic geometry (AG) code $$C_u = \{ {\mathrm{ev}}(f) \mid f \in \mathcal{L}(uQ)\} \label{eq:cu}$$ where ${\mathrm{ev}}(f) = (f(P_1)$, …, $f(P_n))$. Suppose that the Weierstrass semigroup $H(Q)$ at $Q$ is generated by $a_1$, …, $a_t$, and choose $t$ elements $x_1$, …, $x_t$ in $F$ whose pole divisors are $(x_i)_\infty = a_iQ$ for $i=1$, …, $t$. We do *not* assume that $a_1$ is the smallest among $a_1$, …, $a_t$. Without loss of generality we may assume the availability of such $x_1$, …, $x_t$, because otherwise we cannot find a basis of $C_u$ for every $u$. Then we have that $\mathcal{L}(\infty Q) = \cup_{i=1}^\infty\mathcal{L}(iQ)$ is equal to $\mathbf{F}_q[x_1$, …, $x_t]$ [@saints95]. We express $\mathcal{L}(\infty Q)$ as a residue class ring $\mathbf{F}_q[X_1$, …, $X_t]/I$ of the polynomial ring $\mathbf{F}_q[X_1$, …, $X_t]$, where $X_1$, …, $X_t$ are transcendental over $\mathbf{F}_q$, and $I$ is the kernel of the canonical homomorphism sending $X_i$ to $x_i$. @geilpellikaan00 [@miura98] identified the following convenient representation of $\mathcal{L}(\infty Q)$ by using Gröbner basis theory [@adams94]. The following review is borrowed from [@miuraform]. Hereafter, we assume that the reader is familiar with the Gröbner basis theory in [@adams94]. Let $\mathbf{N}_0$ be the set of nonnegative integers. For $(m_1$, …, $m_t)$, $(n_1$, …, $n_t) \in \mathbf{N}_0^t$, we define the weighted reverse lexicographic monomial order $\succ$ such that $(m_1$, …, $m_t)$ $\succ$ $(n_1$, …, $n_t)$ if $a_1 m_1 + \cdots + a_t m_t > a_1 n_1 + \cdots + a_t n_t$, or $a_1 m_1 + \cdots + a_t m_t = a_1 n_1 + \cdots + a_t n_t$, and $m_1 = n_1$, $m_2 = n_2$, …, $m_{i-1} = n_{i-1}$, $m_i<n_i$, for some $1 \leq i \leq t$. Note that a Gröbner basis of $I$ with respect to $\succ$ can be computed by [@saints95 Theorem 15], [@schicho98], [@tang98 Theorem 4.1] or [@bn:vasconcelos Proposition 2.17], starting from any affine defining equations of $F/\mathbf{F}_q$. \[ex1\] According to @hoholdt95 [Example 3.7], $$u^3 v + v^3 + u = 0$$ is an affine defining equation for the Klein quartic over $\mathbf{F}_8$. There exists a unique $\mathbf{F}_8$-rational place $Q$ such that $(v)_\infty= 3Q$, $(uv)_\infty = 5Q$, and $(u^2v)_\infty = 7Q$. The numbers $3$, $5$ and $7$ is the minimal generating set of the Weierstrass semigroup at $Q$. Choosing $v$ as $x_1$, $uv$ as $x_2$ and $u^2v$ as $x_3$, by [@tang98 Theorem 4.1] we can see that the standard form of the Klein quartic is given by $$X_2^2+X_3X_1, X_3X_2+X_1^4+X_2, X_3^2+X_2X_1^3+X_3,$$ which is the reduced Gröbner basis with respect to the monomial order $\succ$. We can see that $a_1=3$, $a_2=5$, and $a_3=7$. \[ex:gs1\] Consider the function field $\mathbf{F}_9(u_1,v_2,v_3)$ with relations $$v_2^3+v_2 = u_1^4, \quad v_3^3 + v_3 = (v_2/u_1)^4. \label{eq:gs1}$$ This is the third function field in the asymptotically good tower introduced by @garcia95. Substituting $v_2$ with $u_1u_2$ and $v_3$ with $u_2 u_3$ in Eq. (\[eq:gs1\]) we have affine defining equations $$u_1^2u_2^3+u_2-u_1^3=0, \quad u_2^2u_3^3+u_3-u_2^3=0.$$ The function $u_1$ has a unique pole $Q$ in $\mathbf{F}_9(u_1,u_2,u_3) =\mathbf{F}_9(u_1,v_2,v_3)$. The minimal generating set of the Weiestrass semigroup $H(Q)$ at $Q$ is $9$, $12$, $22$, $28$, $32$ and $35$ [@voss97 Example 4.11]. It has genus $22$ and $77$ $\mathbf{F}_9$-rational points different from $Q$ [@garcia95]. Define six functions $x_1 = u_1$, $x_2 = u_1 u_2$, $x_3 = u_1^2u_2u_3$, $x_4 = u_1^3u_2^2u_3^2$, $x_5 = ((u_1u_2)^2+1)u_2u_3$ and $x_6 = ((u_1u_2)^2+1)u_2^2u_3^2$. We have $(x_1)_\infty = 9Q$, $(x_2)_\infty = 12Q$, $(x_3)_\infty = 22Q$, $(x_4)_\infty = 35Q$, $(x_5)_\infty = 28Q$ and $(x_6)_\infty = 32Q$ [@umehara98]. From this information and [@tang98 Theorem 4.1] we can compute the $15$ polynomials in the reduced Gröbner basis of the ideal $I \subset \mathbf{F}_9[X_1$, …, $X_6]$ defining $\mathcal{L}(\infty Q)$ as $\{X_2^3-X_1^4+X_2$, $X_5X_2-X_3X_1^2$, $X_6X_2-X_4X_1$, $X_3^2-X_4X_1$, $X_3X_2^2-X_5X_1^2+X_3$, $X_5X_3-X_6X_1^2$, $X_6X_3-X_1^6+X_5X_1^2+X_2X_1^2$, $X_5^2-X_4X_2X_1-X_6$, $X_4X_3-X_2X_1^5+X_3X_1^3+X_2^2X_1$, $X_4X_2^2-X_6X_1^3+X_4$, $X_6X_5-X_2^2X_1^4+X_3X_2X_1^2+X_5$, $X_5X_4-X_1^7+X_5X_1^3+X_2X_1^3$, $X_6^2-X_5X_1^4+X_4X_2X_1+X_3X_1^2+X_6$, $X_6X_4-X_3X_1^5+X_6X_1^3+X_3X_2X_1$, $X_4^2-X_3X_2X_1^4+X_4X_1^3+X_5X_1^2-X_3\}$. Note that polynomials in the above Gröbner basis are in the ascending order with respect to the monomial order $\prec$ while terms in each polynomial are in the descending order with respect to $\prec$. For $i=0$, …, $a_1-1$, we define $b_i = \min\{ m \in H(Q) \mid m \equiv i \pmod{a_1}\}$, and $L_i$ to be the minimum element $(m_1$, …, $m_t) \in \mathbf{N}_0^t$ with respect to $\prec$ such that $a_1 m_1 + \cdots + a_t m_t = b_i$. Note that $b_i$’s are the well-known Apéry set [@MR2549780 Lemmas 2.4 and 2.6] of the numerical semigroup $H(Q)$. Then we have $\ell_1 = 0$ if we write $L_i$ as $(\ell_1$, …, $\ell_t)$. For each $L_i = (0$, $\ell_{i2}$, …, $\ell_{it})$, define $y_i = x_2^{\ell_{i2}} \cdots x_t^{\ell_{it}} \in \mathcal{L}(\infty Q)$. The footprint of $I$, denoted by $\Delta(I)$, is $\{ (m_1$, …, $m_t) \in \mathbf{N}_0^t \mid X_1^{m_1} \cdots X_t^{m_t}$ is not the leading monomial of any nonzero polynomial in $I$ with respect to $\prec\}$, and define $\Omega_0 = \{x_1^{m_1} \cdots x_t^{m_t} \mid (m_1$, …, $m_t) \in \Delta(I)\}$. Then $\Omega_0$ is a basis of $\mathcal{L}(\infty Q)$ as an $\mathbf{F}_q$-linear space [@adams94], two distinct elements in $\Omega_0$ have different pole orders at $Q$, and $$\begin{aligned} \Omega_0 &=& \{ x_1^m x_2^{\ell_2} \cdots, x_t^{\ell_t} \mid m \in \mathbf{N}_0, (0, \ell_2, \ldots, \ell_t) \in \{L_0, \ldots, L_{a_1-1}\}\}\nonumber\\ &=& \{ x_1^m y_i \mid m \in \mathbf{N}_0, i=0, \ldots, a_1-1\}. \label{eq:footprintform}\end{aligned}$$ Equation (\[eq:footprintform\]) shows that $\mathcal{L}(\infty Q)$ is a free $\mathbf{F}_q[x_1]$-module with a basis $\{y_0$, …, $y_{a_1-1}\}$. Note that the above structured shape of $\Omega_0$ reflects the well-known property of every weighted reverse lexicographic monomial order, see the paragraph preceding to [@bn:eisenbud Proposition 15.12]. \[ex2\] For the curve in Example \[ex1\], we have $y_0 = 1$, $y_1=x_3$, $y_2=x_2$. Let $v_Q$ be the unique valuation in $F$ associated with the place $Q$. The semigroup $H(Q)$ is equal to $\{i a_1 - v_Q (y_j) \mid 0\le i,0\le j<a_1\}$ [@MR2549780 Lemma 2.6]. By [@miuraform Proposition 3.18], for each nongap $s\in H(Q)$ there is a unique monomial $x_1^i y_j \in \Omega_0$ with $0\le j<a_1$ such that $-v_Q (x_1^i y_j )=s$, and let us denote this monomial by $\varphi_s$. Let $\Gamma \subset H(Q)$, and we may consider the one-point codes $$C_\Gamma = \langle \{{\mathrm{ev}}(\varphi_s) \mid s \in \Gamma \}\rangle,\label{eq:cgamma}$$ where $\langle \cdot \rangle$ denotes the $\mathbf{F}_q$-linear space spanned by $\cdot$. Since considering linearly dependent rows in a generator matrix has no merit, we assume $$\Gamma \subseteq \widehat{H}(Q), \label{eq:gammaassume}$$ where $\widehat{H}(Q) = \{ u \in H(Q) \mid C_u \neq C_{u-1} \}$. One motivation for considering these codes is that it was shown in [@andersen08] how to increase the dimension of the one-point codes without decreasing the lower bound $d_{\mathrm{AG}}$ for the minimum distance. The bound $d_{\mathrm{AG}}(C_\Gamma)$ is defined for $C_\Gamma$ as follows [@andersen08]: For $s \in \Gamma$, let $$\lambda(s) = \sharp \{ j \in H(Q) \mid j+s \in \widehat{H}(Q) \}. \label{eq:lambda}$$ Then $d_{\mathrm{AG}}(C_\Gamma) = \min\{ \lambda(s) \mid s\in \Gamma \}$. It is proved in [@geil11] that $d_{\mathrm{AG}}$ gives the same estimate for the minimum distance as the Feng-Rao bound [@fengrao93] for one-point dual AG codes when both $d_{\mathrm{AG}}$ and the Feng-Rao bound can be applied, that is, when the dual of a one-point code is isometric to a one-point code. Furthermore, it is also proved in [@geil11] that $d_{\mathrm{AG}}(C_\Gamma)$ can be obtained from the bounds in [@beelen07; @duursma11; @duursma10], hence $d_{\mathrm{AG}}$ can be understood as a particular case of these bounds [@beelen07; @duursma11; @duursma10]. Procedure of New List Decoding based on Voting in Gröbner Bases {#sec3} =============================================================== Overall Structure ----------------- Suppose that we have a received word $\vec{r} \in \mathbf{F}_q^n$. We shall modify the unique decoding algorithm proposed by @lee11 so that we can find all the codewords in $C_\Gamma$ in Eq. (\[eq:cgamma\]) within the Hamming distance $\tau$ from $\vec{r}$. The overall structure of the modified algorithm is as follows: 1. \[l1\] Precomputation before getting a received word $\vec{r}$, 2. \[l2\] Initialization after getting a received word $\vec{r}$, 3. \[l3\] Termination criteria of the iteration, and 4. \[l4\] Main part of the iteration. Steps \[l2\] and \[l4\] are based on [@lee11]. Steps \[l1\] and \[l3\] are not given in [@lee11]. Each step is described in the following subsections in Section \[sec3\]. We shall analyze time complexity except the precomputation part of the algorithm. Modified Definitions for the Proposed Modification -------------------------------------------------- We retain notations from Section \[sec2\]. In this subsection, we modify notations and definitions in [@lee11] to describe the proposed modification to their algorithm. We also introduce several new notations. Define a set $\Omega_1 = \{ x_1^i y_jz^k \mid 0 \leq i$, $0 \leq j < a_1$, $k=0,1\}$. Our $\Omega_1$ is $\Omega$ in [@lee11]. Recall also that $\Omega_0 = \{ \varphi_s \mid s \in H(Q)\}$. Since the $\mathbf{F}_q[x_1]$-module ${\mathcal{L}(\infty Q)}z \oplus {\mathcal{L}(\infty Q)}$ has a free basis $\{ y_j z, y_j \mid 0 \leq j < a_1\}$, we can regard $\Omega_1$ as the set of monomials in the Gröbner basis theory for modules. We introduce a monomial order on $\Omega_1$ as follows. For given two monomials $x_1^{i_1} y_j z^{i_{t+1}}$ and $x_1^{i'_1} y_{j'} z^{i'_{t+1}}$, first rewrite $y_j$ and $y_{j'}$ by $x_2$, …, $x_t$ defined in Section \[sec2\] and get $x_1^{i_1} y_j z^{i_{t+1}} = x_1^{i_1} x_2^{i_2} \cdots x_t^{i_t} z^{i_{t+1}}$ and $x_1^{i'_1} y_{j'} z^{i'_{t+1}} = x_1^{i'_1} x_2^{i'_2} \cdots x_t^{i'_t} z^{i'_{t+1}}$. For a nongap $s \in H(Q)$, we define the monomial order $x_1^{i_1} x_2^{i_2} \cdots x_t^{i_t} z^{i_{t+1}}<_s x_1^{i'_1} x_2^{i'_2} \cdots x_t^{i'_t} z^{i'_{t+1}}$ parametrized by $s$ if $i_{t+1}s - v_Q(x_1^{i_1} x_2^{i_2} \cdots x_t^{i_t}) < i'_{t+1}s - v_Q(x_1^{i'_1} x_2^{i'_2} \cdots x_t^{i'_t})$ or $i_{t+1}s - v_Q(x_1^{i_1} x_2^{i_2} \cdots x_t^{i_t}) = i'_{t+1}s - v_Q(x_1^{i'_1} x_2^{i'_2} \cdots x_t^{i'_t})$ and $i_1 = i'_1$, $i_2 = i'_2$, …, $i_{\ell-1}=i'_{\ell-1}$ and $i_\ell > i'_\ell$ for some $1 \leq \ell \leq t+1$. Observe that the restriction of $<_s$ to $\Omega_0$ is equal to $\prec$ defined in Section \[sec2\]. In what follows, every Gröbner basis, leading term, and leading coefficient is obtained by considering the Gröbner basis theory for modules, not for ideals. For $f \in {\mathcal{L}(\infty Q)}z \oplus {\mathcal{L}(\infty Q)}$, $\gamma(f)$ denotes the number of nonzero terms in $f$ when $f$ is expressed as an $\mathbf{F}_q$-linear combination of monomials in $\Omega_1$. $\gamma_{\neq 1}(f)$ denotes the number of nonzero terms whose coefficients are not $1 \in \mathbf{F}_q$. For the code $C_\Gamma$ in Eq. (\[eq:cgamma\]), define the divisor $D=P_1+ \cdots + P_n$. Define $\mathcal{L}(-G+\infty Q) = \bigcup_{i=1}^\infty \mathcal{L}(-G+iQ)$ for a positive divisor $G$ of $F/\mathbf{F}_q$. Then $\mathcal{L}(-D+\infty Q)$ is an ideal of ${\mathcal{L}(\infty Q)}$ [@matsumoto99ldpaper]. Let $\eta_i$ be any element in $\mathcal{L}(-D+\infty Q)$ such that $\textsc{lm}(\eta_i) = x_1^j y_i$ with $j$ being the minimal given $i$. Then by [@lee11 Proposition 1], $\{\eta_0$, …, $\eta_{a_1-1}\}$ is a Gröbner basis for $\mathcal{L}(-D+\infty Q)$ with respect to $<_s$ as an $\mathbf{F}_q[x_1]$-module. For a nonnegative integer $s$, define $\Gamma^{(\leq s)} = \{ s' \in \Gamma \mid s' \leq s \}$, $\Gamma^{(> s)} = \{ s' \in \Gamma \mid s' > s \}$, and $\mathrm{prec}(s)= \max\{s' \in H(Q) \mid s'<s \}$. We define $\mathrm{prec}(0) = -1$. Precompuation before Getting a Received Word {#sec:precomput} -------------------------------------------- Before getting $\vec{r}$, we need to compute the Pellikaan-Miura standard form of the algebraic curve, $y_0(=1)$, $y_1$, …, $y_{a_1-1}$, and $\varphi_s$ for $s \in H(Q)$ as defined in Section \[sec2\]. Also compute $\eta_0$, …, $\eta_{a_1-1}$, which can be done by [@matsumoto99ldpaper]. For each $(i,j)$, express $y_i y_j$ as an $\mathbf{F}_q$-linear combination of monomials in $\Omega_0$. Such expressions will be used for computing products and quotients in ${\mathcal{L}(\infty Q)}$ as explained in Section \[sec:product\]. [From]{} the above data, we can easily know $\textsc{lc}(y_i y_j)$, which will be used in Eqs. (\[eq:quotientpre\]) and (\[eq:mui\]). Find elements $\varphi_s \in \Omega_0$ with $s \in \widehat{H}(Q)$. There are $n$ such elements, which we denote by $\psi_1$, …$\psi_n$ such that $-v_Q(\psi_i) < -v_Q(\psi_{i+1})$. Compute the $n \times n$ matrix $$M=\left( \begin{array}{ccc} \psi_1(P_1)&\cdots&\psi_1(P_n)\\ \vdots&\vdots& \vdots\\ \psi_n(P_1)&\cdots&\psi_n(P_n) \end{array}\right)^{-1}.\label{eq:invgenmatrix}$$ Multiplication and Division in an Affine Coordinate Ring -------------------------------------------------------- In both original unique decoding algorithm [@lee11] and our modified version, we need to quickly compute the product $gh$ of two elements $g,h$ in the affine coordinate ring ${\mathcal{L}(\infty Q)}$. In our modified version, we also need to compute the quotient $g/h$ depending on the choice of iteration termination criterion described in Section \[sec:termination\]. Since the authors could not find quick computational procedures for those tasks in ${\mathcal{L}(\infty Q)}$, we shall present such ones here. ### Multiplication in an Affine Coordinate Ring {#sec:product} The normal form of $g$, for $g \in {\mathcal{L}(\infty Q)}$, is the expression of $g$ written as an $\mathbf{F}_q$-linear combination of monomials $\varphi_s \in \Omega_0$. $g,h$ are assumed to be in the normal form. We propose the following procedure to compute the normal form of $gh$. Let the normal form of $y_i y_j$ be $$\sum_{k=0}^{a_1-1} y_k f_{i,j,k}(x_1).$$ with $f_{i,j,k}(x_1) \in \mathbf{F}_q[x_1]$, which is computed in Section \[sec:precomput\]. We denote by $X_1$, $Y_1$, …, $Y_{a_1-1}$ algebraically independent variables over $\mathbf{F}_q$. 1. Assume that $g$ and $h$ are in their normal forms. Change $y_i$ to $Y_i$ and $x_1$ to $X_1$ in $g,h$ for $i=1$, …, $a_1-1$. Recall that $y_0=1$. Denote the results by $G,H$. 2. Compute $GH$. This step needs $$\gamma(g)\times \gamma(h)\label{eq:NFmulti1}$$ multiplications in $\mathbf{F}_q$. 3. Let $GH = \sum_{0\leq i,j <a_1} Y_i Y_j F_{G,H,i,j}(X_1)$. Then we have $$gh = \sum_{0\leq i,j<a_1} F_{G,H,i,j}(x_1) \sum_{k=0}^{a_1-1} y_k f_{i,j,k}(x_1). \label{eq:gh}$$ Computation of $F_{G,H,i,j}(X_1)\sum_{k=0}^{a_1-1} y_k f_{i,j,k}(x_1)$ needs at most $\gamma_{\neq 1}(\sum_{k=0}^{a_1-1} y_k f_{i,j,k}(x_1)) \allowbreak \gamma( F_{G,H,i,j}(x_1))$ multiplications in $\mathbf{F}_q$. Therefore, the total number of multiplications in $\mathbf{F}_q$ in this step is at most $$\sum_{0 \leq i,j < a_1} \gamma( F_{G,H,i,j}(x_1)) \gamma_{\neq 1}(\sum_{k=0}^{a_1-1} y_k f_{i,j,k}(x_1)).\label{eq:NFmulti2}$$ Therefore, the total number of multiplications in $\mathbf{F}_q$ is at most $$\gamma(g)\times \gamma(h) + \sum_{0 \leq i,j < a_1} \gamma( F_{G,H,i,j}(x_1)) \gamma_{\neq 1}(\sum_{k=0}^{a_1-1} y_k f_{i,j,k}(x_1)) . \label{eq:NFmulti3}$$ Define Eq. (\[eq:NFmulti3\]) as $\mathrm{multi}(g,h)$. We emphasize that when the characteristic of $\mathbf{F}_q$ is 2 and all the coefficients of defining equations belong to $\mathbf{F}_2$, which is almost always the case for those cases of interest for applications in coding theory, then $\gamma_{\neq 1}(\sum_{k=0}^{a_1-1} y_k f_{i,j,k}(x_1))$ in Eq. (\[eq:NFmulti3\]) is zero. This means that $\mathcal{L}(\infty Q)$ has little additional overhead over $\mathbf{F}_q[X]$ for computing products of their elements in terms of the number of $\mathbf{F}_q$-multiplications and divisions. Define $(i,j)$ to be equivalent to $(i',j')$ if $y_i y_j = y_{i'}y_{j'} \in \mathcal{L}(\infty Q)$. Denote by $[i,j]$ the equivalence class represented by $(i,j)$. For $(i,j), (i',j') \in [i,j]$ we have $f_{i,j,k}(x_1) = f_{i',j',k}(x_1)$, which is denoted by $f_{[i,j],k}(x_1)$. The right hand side of Eq. (\[eq:gh\]) can be written as $$\sum_{[i,j]}\left(\sum_{(i',j')\in[i,j]} F_{G,H,i',j'}(x_1)\right) \sum_{k=0}^{a_1-1} y_k f_{[i,j],k}(x_1). \label{eq:gh2}$$ By using Eq. (\[eq:gh2\]) instead of Eq. (\[eq:gh\]), we have another upper bound on the number of multiplications as $$\gamma(g)\times \gamma(h) + \sum_{[i,j]} \gamma\left( \sum_{(i',j')\in[i,j]} F_{G,H,i',j'}(x_1)\right) \gamma_{\neq 1}(\sum_{k=0}^{a_1-1} y_k f_{[i,j],k}(x_1)) . \label{eq:NFmulti4}$$ Since $$\gamma\left( \sum_{(i',j')\in[i,j]} F_{G,H,i',j'}(x_1)\right) \leq \sum_{(i',j')\in[i,j]} \gamma( F_{G,H,i',j'}(x_1)),$$ we have Eq. (\[eq:NFmulti4\]) $\leq$ Eq. (\[eq:NFmulti3\]). However, Eq. (\[eq:NFmulti4\]) is almost always the same as Eq. (\[eq:NFmulti3\]) over the curve in Example \[ex:gs1\], and Eq. (\[eq:NFmulti4\]) will not be used in our computer experiments in Section \[sec:experiment\]. ### Computation of the Quotient Assume $h\neq 0$. The following procedure computes the quotient $g/h \in \mathcal{L}(\infty Q)$ or declares that $g$ does not belong to the principal ideal of ${\mathcal{L}(\infty Q)}$ generated by $h$. 1. Initialize $\sigma=0$. Also initialize $\zeta=0$. 2. \[lab\] Check if $-v_Q(g) \in -v_Q(h) + H(Q)$. If not, declare that $g$ does not belong to the principal ideal of ${\mathcal{L}(\infty Q)}$ generated by $h$, and finish the procedure. 3. Let $\varphi_s \in \Omega_0$ such that $-v_Q(g) = -v_Q(\varphi_sh)$. Observe that $\textsc{lc}(\varphi_s\textsc{lm}(h)) = \textsc{lc}(y_{s \bmod a_1} y_{-v_Q(h) \bmod a_1})$ and that $\textsc{lc}(y_{s \bmod a_1} y_{-v_Q(h) \bmod a_1})$ is precomputed as Section \[sec:precomput\]. Let $$\mathbf{F}_q \ni t = \textsc{lc}(g)/(\textsc{lc}(h) \times \underbrace{\textsc{lc}(\varphi_s\textsc{lm}(h))}_{\textrm{Precomputed in Section \ref{sec:precomput}}}). \label{eq:quotientpre}$$ Computation of $t\varphi_s$ needs one multiplication and one division in $\mathbf{F}_q$. Observe that $-v_Q(g - t\varphi_sh) < -v_Q(g)$. 4. Compute the normal form of $t \varphi_s h$, which requires at most $\mathrm{multi}(t\varphi_s, h)$ multiplications in $\mathbf{F}_q$. Increment $\zeta$ by $2+\mathrm{multi}(t\varphi_s, h)$. 5. Update $\sigma \leftarrow \sigma + t \varphi_s$ and $g \leftarrow g - t\varphi_sh$. If the updated $g$ is zero, then output the updated $\sigma$ as the quotient and finish the procedure. Otherwise go to Step \[lab\]. This step has no multiplication nor division. Define $\mathrm{quot}(g,h)$ as $\zeta$ after finishing the above procedure. $\mathrm{quot}(g,h)$ is an upper bound on the number of multiplications and divisions in $\mathbf{F}_q$ in the above procedure. The program variable $\zeta$ is just to define $\mathrm{quot}(g,h)$, and the decoding algorithm does not need to update $\zeta$. Observe also that the above procedure is a straightforward generalization of the standard long division of two univariate polynomials [@mckeague11]. Initialization after Getting a Received Word $\vec{r}$ {#sec:initialization} ------------------------------------------------------ Let $(i_1$, …, $i_n)^T = M\vec{r}$, where $M$ is defined in Eq. (\[eq:invgenmatrix\]). Define $h_{\vec{r}} = \sum_{j=1}^n i_j \psi_j$. Then we have ${\mathrm{ev}}(h_{\vec{r}}) = \vec{r}$. The computation of $h_{\vec{r}}$ from $\vec{r}$ needs at most $n^2$ multiplications in $\mathbf{F}_q$. Let $N=-v_Q(h_{\vec{r}})$. For $i=0$, …, $a_1-1$, compute $g_i^{(N)} = \eta_i \in {\mathcal{L}(\infty Q)}$ and $f_i^{(N)} = y_i(z-h_{\vec{r}}) \in {\mathcal{L}(\infty Q)}z \oplus {\mathcal{L}(\infty Q)}$. The computation of $f_i^{(N)}$ needs at most $\mathrm{multi}(y_i, h_{\vec{r}})$ multiplications in $\mathbf{F}_q$. Therefore, the total number of multiplications in the initialization is at most $$n^2 + \sum_{i=0}^{a_1-1} \mathrm{multi}(y_i, h_{\vec{r}}). \label{eq:compl:ini}$$ Let $s=N$ and execute the following steps. Three Termination Criteria of the Iteration {#sec:termination} ------------------------------------------- After finishing the initialization step in Section \[sec:initialization\], we iteratively compute $f_i^{(s)}$ and $g_i^{(s)}$ with $N \geq s \in H(Q) \cup \{-1\}$ and $w_s$ with $N \geq s \in H(Q)$ from larger $s$ to smaller $s$. The single iteration consists of two parts: The first part is to check if an iteration termination criterion is satisfied. The second part is computation of $f_i^{(s)}$ and $g_i^{(s)}$ for $N \geq s \in H(Q) \cup \{-1\}$. We describe the first part in Section \[sec:termination\]. Let $f_{\mathrm{min}} = \alpha_0 + z \alpha_1$ having the smallest $-v_Q(\alpha_1)$ among $f^{(s)}_0$, …, $f^{(s)}_{a_1-1}$. In the following subsections, we shall propose three different procedures to judge whether or not iterations in the proposed algorithm can be terminated. In an actual implementation of the proposed algorithm, one criterion is chosen and the chosen one is consistently used throughout the iterations. The first one and the second one are different generalizations of [@kuijper11 Theorem 12] for the case $g=0$ to $g > 0$. @kuijper11 [Theorem 12] proved that if the number $\delta$ of errors satisfies $2 \delta < d_{\mathrm{RS}}(C_s)$, where $d_{\mathrm{RS}}(C_s)$ is the minimum distance $n-s$ of the $[n,s+1]$ Reed-Solomon code $C_s$, then the transmitted information word is obtained by Ali-Kuijper’s algorithm as $-\alpha_0/\alpha_1$. To one-point primal AG codes, $d_{\mathrm{RS}}(C_s)$ can be generalized as either $d_{\mathrm{AG}}(C_s)$ or $n-s-g$. The former generalization $d_{\mathrm{AG}}(C_s)$ corresponds to the first criterion in Section \[sec:first\] and the latter $n-s-g$ corresponds to the second in Section \[sec:second\]. The third one is almost the same as the original procedure in [@lee11]. The first one was proposed in [@gmr12isit] while the second and the third ones are new in this paper. We shall compare the three criteria in Section \[sec:comparethree\]. Throughout this paper, ${\mathrm{wt}}(\vec{x})$ denotes the Hamming weight of a vector $\vec{x}\in \mathbf{F}_q^n$. ### First Criterion for Judging Termination {#sec:first} If - $s \in \Gamma$, - $d_{\mathrm{AG}}(C_{\Gamma^{(\leq s)}}) > 2 \tau$, and - $-v_Q(\alpha_1) \leq \tau + g$ then do the following: 1. \[first1\] Compute $\alpha_0/\alpha_1 \in F$. This needs at most $$\mathrm{quot}(\alpha_0, \alpha_1)\label{eq:compl3}$$ multiplications and divisions in $\mathbf{F}_q$. 2. If $\alpha_0/\alpha_1 \in {\mathcal{L}(\infty Q)}$ and $\alpha_0/\alpha_1$ can be written as a linear combination of monomials in $\{\varphi_{s'}\in s'\in \Gamma^{(\leq s)}\}$, then do the following: 1. \[first11\] If $d_{\mathrm{AG}}(C_{\Gamma}) > 2 \tau$ or $-v_Q(\alpha_1) \leq \tau$ then include the coefficients of $-\alpha_0/\alpha_1+\sum_{s' \in \Gamma^{(> s)}} w_{s'} \varphi_{s'}$ into the list of transmitted information vectors, and avoid proceeding with the rest of the decoding procedure. 2. \[first12\] Otherwise compute ${\mathrm{ev}}(-\alpha_0/\alpha_1 +\sum_{s' \in \Gamma^{(> s)}} w_{s'} \varphi_{s'})$. This needs at most $$n \gamma(-\alpha_0/\alpha_1+\sum_{s' \in \Gamma^{(> s)}} w_{s'} \varphi_{s'}) \label{eq:compl4}$$ multiplications and divisions in $\mathbf{F}_q$. 3. \[first13\] If $${\mathrm{wt}}\left({\mathrm{ev}}(-\alpha_0/\alpha_1+\sum_{s' \in \Gamma^{(> s)}} w_{s'} \varphi_{s'})- \vec{r}\right) \leq \tau,$$ then include the coefficients of $-\alpha_0/\alpha_1$ $+$ $\sum_{s' \in \Gamma^{(> s)}} w_{s'} \varphi_{s'}$ into the list of transmitted information vectors, and avoid proceeding with $s$. Otherwise, continue the iterations unless $s < n - g - 2\tau$. ### Second Criterion for Judging Termination {#sec:second} If $s = \max\{ s' \in \Gamma \mid s' < n-2 \tau - g\}$, then do the following: 1. \[second1\] If $-v_Q(\alpha_1) > \tau + g$ then stop proceeding with iteration. 2. Otherwise compute $\alpha_0/\alpha_1 \in F$. This needs at most $$\mathrm{quot}(\alpha_0, \alpha_1) \label{eq:compl5}$$ multiplications and divisions in $\mathbf{F}_q$. 3. If $2 \tau < d_{\mathrm{AG}} (C_\Gamma )$, $\alpha_0/\alpha_1 \in {\mathcal{L}(\infty Q)}$, and $\alpha_0/\alpha_1$ can be written as a linear combination of monomials in $\{\varphi_{s'}\in s'\in \Gamma^{(\leq s)}\}$ then declare the coefficients of $-\alpha_0/\alpha_1+\sum_{s' \in \Gamma^{(> s)}} w_{s'} \varphi_{s'}$ as the only transmitted information and finish. Otherwise declare “decoding failure” and finish. 4. If $2 \tau \geq d_{\mathrm{AG}} (C_\Gamma )$, $\alpha_0/\alpha_1 \in {\mathcal{L}(\infty Q)}$ and $\alpha_0/\alpha_1$ can be written as a linear combination of monomials in $\{\varphi_{s'}\in s'\in \Gamma^{(\leq s)}\}$, then do the following: 1. \[second11\] If $-v_Q(\alpha_1) \leq \tau$ then include the coefficients of $-\alpha_0/\alpha_1+\sum_{s' \in \Gamma^{(> s)}} w_{s'} \varphi_{s'}$ into the list of transmitted information vectors, and avoid proceeding with $s$. 2. \[second12\] Otherwise compute ${\mathrm{ev}}(-\alpha_0/\alpha_1 +\sum_{s' \in \Gamma^{(> s)}} w_{s'} \varphi_{s'})$. This needs at most $$n \gamma(-\alpha_0/\alpha_1+\sum_{s' \in \Gamma^{(> s)}} w_{s'} \varphi_{s'})\label{eq:compl6}$$ multiplications and divisions in $\mathbf{F}_q$. 3. If $${\mathrm{wt}}\left({\mathrm{ev}}(-\alpha_0/\alpha_1+\sum_{s' \in \Gamma^{(> s)}} w_{s'} \varphi_{s'})- \vec{r}\right) \leq \tau,$$ then include the coefficients of $-\alpha_0/\alpha_1$ $+$ $\sum_{s' \in \Gamma^{(> s)}} w_{s'} \varphi_{s'}$ into the list of transmitted information vectors. 5. Finish the iteration no matter what happened in the above steps. ### Third Criterion for Judging Termination {#sec:3rd} Just repeat the iteration until finding $f_i^{(s)}$ at $s=-1$ for $i=0$, …, $a_1-1$. If $2 \tau < d_{\mathrm{AG}} (C_\Gamma )$ then declare the vector $(w_s : s \in \Gamma)$ as the only transmitted information and finish. If $2 \tau \geq d_{\mathrm{AG}} (C_\Gamma )$ then do the following: 1. \[third1\] If $\alpha_0=0$ and $-v_Q(\alpha_1) \leq \tau$ then include the vector $(w_s : s \in \Gamma)$ into the list of transmitted information vectors. Finish the iteration. 2. \[third2\] If $-v_Q(\alpha_1) > \tau+g$ then finish the iteration. 3. \[third3\] Otherwise compute ${\mathrm{ev}}(\sum_{s \in \Gamma} w_{s} \varphi_{s})$. This needs at most $$n \gamma(\sum_{s \in \Gamma} w_{s} \varphi_{s})\label{eq:compl7}$$ multiplications and divisions in $\mathbf{F}_q$. 4. \[third4\] If $${\mathrm{wt}}\left({\mathrm{ev}}(\sum_{s \in \Gamma} w_{s} \varphi_{s})- \vec{r}\right) \leq \tau,$$ then include the vector $(w_s : s \in \Gamma)$ into the list of transmitted information vectors. Finish the iteration. Iteration of Pairing, Voting, and Rebasing {#sec:mainiteration} ------------------------------------------ The iteration of the original algorithm [@lee11] consists of three steps, called pairing, voting, and rebasing. We will make a small change to the original. Our modified version is described below. ### Pairing Let $$g_i^{(s)} = \sum_{0\le j<a_1}c_{i,j} y_j z+ \sum_{0\le j<a_1} d_{i,j} y_j, \mathrm{~with~} c_{i,j}, d_{i,j} \in\mathbf{F}_q[x_1],$$ $$f_i^{(s)} = \sum_{0\le j<a_1}a_{i,j} y_j z+ \sum_{0\le j<a_1}b_{i,j} y_j, \mathrm{~with~} a_{i,j}, b_{i,j} \in\mathbf{F}_q[x_1],$$ and let $\nu_i^{(s)}=\textsc{lc} (d_{i,i})$. We assume that $\textsc{lt}(f_i^{(s)}) = a_{i,i} y_i z$ and $\textsc{lt}(g_i^{(s)}) = d_{i,i} y_i$. For $0 \le i <a_1$, as in [@lee11] there are unique integers $0\le i'< a_1$ and $k_i$ satisfying $$-v_Q(a_{i,i} y_i) + s = a_1 k_i -v_Q(y_{i'}).$$ Note that by the definition above $$\label{eq:iprime}i' = i + s \bmod a_1,$$ and the integer $-v_Q (a_{i,i} y_i )+s$ is a nongap if and only if $k_i\ge 0$. Now let $c_i=\deg_{x_1}(d_{i',i'})-k_i$. Note that the map $i\mapsto i'$ is a permutation of $\{0,1,\dots,a-1\}$ and that the integer $c_i$ is defined such that $a_1 c_i= -v_Q (d_{i',i'} y_{i'} )+v_Q (a_{i,i} y_i )-s$. ### Voting For each $i \in \{0, \ldots a_1 -1\}$, we set $$\mu_i= \textsc{lc}( a_{i,i} y_i \varphi_s), ~ w_{s,i}=-\frac{b_{i,i'}[x_1^{k_i}]}{\mu_i}, ~ \bar{c}_i=\max\{c_i,0\}, \label{eq:mui}$$ where $b_{i,i'}[x_1^{k_i}]$ denotes the coefficient of $x_{k_i}$ of the univariate polynomial $b_{i,i'} \in \mathbf{F}_q[x_1]$. We remark that the leading coefficient $\mu_i$ must be considered after expressing $a_{i,i} y_i \varphi_s$ by monomials in $\Omega_0$. Observe that $\textsc{lc}(y_i \varphi_s) = \textsc{lc}(y_i y_{s \bmod a_1})$ and that $\textsc{lc}(y_i y_{s \bmod a_1})$ is already precomputed as Section \[sec:precomput\]. By using that precomputed table, computation of $\mu_i$ needs one multiplication. The total number of multiplications and divisions in Eq. (\[eq:mui\]) is $$2 a_1 \label{eq:compl1}$$ excluding negation from the number of multiplication. Let $$\nu(s)=\frac{1}{a_1}\sum_{0\le i<a_1} \max\{-v_Q(\eta_{i'})+v_Q( y_i )-s,0\}.\label{eq:nu}$$ We consider two different candidates depending on whether $s \in \Gamma$ or not: - If $s \in H(Q) \setminus \Gamma$, set $$w=0. \label{eq:w0}$$ - If $s \in \Gamma$, let $w$ be one of the element(s) in $\mathbf{F}_q$ with $$\label{eq:acceptedvote} \sum_{w=w_{s,i}}\bar{c}_i \geq \sum_{w\neq w_{s,i}}\bar{c}_i - 2\tau+\nu(s).$$ Let $w_s=w$. If several $w$’s satisfy the condition above, repeat the rest of the algorithm for each of them. ### Rebasing {#sec:rebasing} In all of the following cases, we need to compute the normal form of the product $w\varphi_s \times \sum_{j=0}^{a_1-1} a_{i,j} y_j$, and the product $w\varphi_s \times \sum_{j=0}^{a_1-1} c_{i,j} y_j$. For each $i$, the number of multiplications is $$\leq \mathrm{multi}(w\varphi_s, \sum_{j=0}^{a_1-1} a_{i,j} y_j) + \mathrm{multi}(w\varphi_s, \sum_{j=0}^{a_1-1} c_{i,j} y_j), \label{eq:multirebasing}$$ where $\mathrm{multi}(\cdot, \cdot)$ is defined in Section \[sec:product\]. - If $w_{s,i}=w$, then let $$\begin{aligned} \label{fkfmvd} &&g_{i'}^{( \mathrm{prec}(s))}=g_{i'}^{(s)}(z+w\varphi_s),\\ &&f_i^{( \mathrm{prec}(s))}=f_i^{(s)}(z+w\varphi_s)\end{aligned}$$where the parentheses denote substitution of the variable $z$ and let $\nu_{i'}^{( \mathrm{prec}(s))}=\nu_{i'}^{(s)}$. The number of multiplications in this case is bounded by Eq. (\[eq:multirebasing\]). - If $w_{s,i}\neq w$ and $c_i>0$, then let $$\begin{aligned} && g_{i'}^{( \mathrm{prec}(s))}=f_i^{(s)}(z+w\varphi_s), \\ && f_i^{( \mathrm{prec}(s))}=x_1^{c_i}f_i^{(s)}(z+w\varphi_s)-\frac{\mu_i(w-w_{s,i}) }{\nu_{i'}^{(s)}}g_{i'}^{(s)}(z+w\varphi_s)\end{aligned}$$ and let $\nu_{i'}^{( \mathrm{prec}(s))}=\mu_i(w-w_{s,i})$. Computation of $\frac{\mu_i(w-w_{s,i}) }{\nu_{i'}^{(s)}}$ needs one multiplication and one division. The product of $\frac{\mu_i(w-w_{s,i}) }{\nu_{i'}^{(s)}}$ and $g_{i'}^{(s)}(z+w\varphi_s)$ needs $\gamma(g_{i'}^{(s)}(z+w\varphi_s))$ multiplications, where $\gamma$ is defined in Section \[sec:product\]. Thus, the number of multiplications and divisions is $$\leq 2 + \gamma(g_{i'}^{(s)}(z+w\varphi_s)) + \mbox{Eq.\ (\ref{eq:multirebasing})}. \label{eq:compl2}$$ - If $w_{s,i}\neq w$ and $c_i\le 0$, then let $$\begin{aligned} && g_{i'}^{({ \mathrm{prec}(s)})}=g_{i'}^{(s)}(z+w\varphi_s),\\ && f_i^{({ \mathrm{prec}(s)})}=f_i^{(s)}(z+w\varphi_s)-\frac{\mu_i(w-w_{s,i})}{\nu_{i'}^{(s)}}x_1^{-c_i}g_{i'}^{(s)}(z+w\varphi_s)\end{aligned}$$ and let $\nu_{i'}^{( \mathrm{prec}(s))}=\nu_{i'}^{(s)}$. Computation of $\frac{\mu_i(w-w_{s,i}) }{\nu_{i'}^{(s)}}$ needs one multiplication and one division. The product of $\frac{\mu_i(w-w_{s,i}) }{\nu_{i'}^{(s)}}$ and $g_{i'}^{(s)}(z+w\varphi_s)$ needs $\gamma(g_{i'}^{(s)}(z+w\varphi_s))$ multiplications, where $\gamma$ is defined in Section \[sec:product\]. Thus, the number of multiplications and divisions is $\leq $ Eq. (\[eq:compl2\]). After computing $f_i^{(\mathrm{prec}(s))}$ and $g_i^{(\mathrm{prec}(s))}$ as above, update the program variable $s$ to $\mathrm{prec}(s)$ and go to the beginning of Section \[sec:termination\]. Difference to the Original Method {#sec:diff} --------------------------------- In this subsection, we review advantages of our modified algorithm over the original [@lee11]. - Our version can handle any one-point primal AG codes, while the original can handle codes only coming from the $C_{ab}$ curves [@miura92]. This generalization is enabled only by replacing $y^j$ in [@lee11] by $y_j$ defined in Section \[sec2\]. - Our version can find all the codeword within Hamming distance $\tau$ from the received word $\vec{r}$, while the original is a unique decoding algorithm. - Our version does not compute $f^{(s)}_i$, $g^{(s)}_i$ for a Weiestrass gap $s \notin H(Q)$, while the original computes them for $N\geq s\notin H(Q)$. - The original algorithm assumed $u<n$, where $u$ is as defined in Eq. (\[eq:cu\]). This assumption is replaced by another less restrictive assumption (\[eq:gammaassume\]) in our version. - Our version supports the Feng-Rao improved code construction [@feng95], while the original does not. This extension is made possible by the change at Eq. (\[eq:w0\]). - The first and the second termination criteria come from [@kuijper11 Theorem 12] and do not exist in the original [@lee11]. - The third termination criterion is essentially the same as the original [@lee11], but examination of the Hamming distance between the decoded codeword and $\vec{r}$ is added when $2 \tau \geq d_{\mathrm{AG}}(C_\Gamma)$. - The original [@lee11] is suitable for parallel implementation on electric circuit similar to the Kötter architecture [@koetter98]. Our modified version retains this advantage. Theoretical Analysis of the Proposed Modification {#sec4} ================================================= In this section we prove that our modified algorithm can find all the codewords within Hamming distance $\tau$ from the received word $\vec{r}$. We also give upper bounds on the number of iterations in Section \[sec:nriteration\]. Supporting Lemmas {#sec:sl} ----------------- In Section \[sec:sl\] we shall introduce several lemmas necessary in Sections \[sec:nrvotes\]–\[sec:firstwhy\]. Recall that the execution of our modified algorithm can branch when there are multiple candidates satisfying the condition (\[eq:acceptedvote\]). For a fixed sequence of determined $w_s$, define $\vec{r}^{(N)} = \vec{r}$ and recursively define $\vec{r}^{(\mathrm{prec}(s))} = \vec{r}^{(s)} - {\mathrm{ev}}(w_s \varphi_s)$. By definition $\vec{r}^{(-1)} = \vec{r}-{\mathrm{ev}}(\sum_{s\in \Gamma} w_s \varphi_s)$. The following lemma explains why the authors include “Gröbner bases” in the paper title. The module $I_{\vec{r}^{(N)}}$ was used in [@beelen10; @fujisawa06; @lax12; @lee09; @mrg13jsc; @sakata01; @sakata03] but the use of $I_{\vec{r}^{(s)}}$ with $s < N$ was new in [@lee11]. \[lem:ir\] Fix $s \in H(Q) \cup \{-1\}$. Let $\vec{r}^{(s)}$ correspond to $w_s$ ($s \in \Gamma$) chosen by the decoding algorithm. Define the $\mathbf{F}_q[x_1]$-submodule $I_{\vec{r}^{(s)}}$ of ${\mathcal{L}(\infty Q)}z \oplus {\mathcal{L}(\infty Q)}$ by $$I_{\vec{r}^{(s)}}=\{ \alpha_0 + \alpha_1 z \mid \alpha_0, \alpha_1 \in {\mathcal{L}(\infty Q)}, v_{P_i}(\alpha_0+r_i^{(s)}\alpha_1) \geq 1, 1 \leq i \leq n \},\label{eq:ir}$$ where $\vec{r}^{(s)} = (r_1^{(s)}$, …, $r_n^{(s)})$. Then $\{ f_i^{(s)}$, $g_i^{(s)} \mid i=0$, …, $a_1-1\}$ is a Gröbner basis of $I_{\vec{r}^{(s)}}$ with respect to $<_s$ as an $\mathbf{F}_q[x_1]$-module. [**Proof. **]{}This lemma is a generalization of [@lee11 Proposition 11]. We can prove this lemma in exactly the same way as the proof of [@lee11 Proposition 11] with replacing $y^j$ in [@lee11] with $y_j$ and $s-1$ in [@lee11] by $\mathrm{prec}(s)$. [ ]{} The following proposition shows that the original decoding algorithm [@lee11] can correct errors up to half the bound $d_{\mathrm{AG}}(C_\Gamma)$, which was not claimed in [@lee11]. \[prop:AG\] Fix $s \in \Gamma$. Let $\lambda(s)$ as defined in Eq. (\[eq:lambda\]) and $\nu(s)$ as defined in Eq. (\[eq:nu\]). Then $\nu(s) = \lambda(s)$. [**Proof. **]{}Let $T_i = \{ j \in H(Q) \mid j \equiv i \pmod{a_1}, j+s \in \widehat{H}(Q)\}$, then we have $\lambda(s) = \sharp T_0 + \cdots + \sharp T_{a_1-1}$. Moreover, observe that $$H(Q) \setminus \widehat{H}(Q) = \{ -v_Q (\eta_i x_1^k) \mid i=0, \ldots, a_1-1, k=0,1,\ldots \}.$$ Therefore, for $s \in \Gamma$ we have $$\begin{aligned} T_i &=& \{ j \in H(Q) \mid j \equiv i \pmod{a_1}, j+s \in \widehat{H}(Q) \} \\ &=& \{ j \in H(Q) \mid j \equiv i \pmod{a_1}, j+s \notin H(Q) \setminus \widehat{H}(Q) \}\\ &=& \{ j \in H(Q) \mid j \equiv i \pmod{a_1}, j+s \notin \{ -v_Q(\eta_{i'} x_1^k) \mid k \ge 0 \} \}\\ &=& \{ -v_Q(y_i x_1^m) \mid s -v_Q(y_i x_1^m) \notin \{ -v_Q(\eta_{i'} x_1^k) \mid k \ge 0 \} \},\end{aligned}$$ where the third equality holds by Eq. (\[eq:iprime\]). By the equalities above, we see $$\sharp T_i = \max\left\{0, \frac{-v_Q(\eta_{i'}) + v_Q (y_i)-s}{-v_Q(x_1)}\right\},$$ which proves the equality $\nu(s)=\lambda(s)$. [ ]{} @lee11 showed that their original decoding algorithm can correct up to $\lfloor (d_{\mathrm{LBAO}}(C_u)-1)/2 \rfloor$ errors, where $d_{\mathrm{LBAO}}(C_u) =\min\{\nu(s)\mid s\in H(Q)$, $s\le u\}$. Proposition \[prop:AG\] implies that $d_{\mathrm{LBAO}}(C_u)$ is equivalent to $d_{\mathrm{AG}} (C_u)$ for every one-point primal code $C_u$, and therefore [@andersen08 Theorem 8] implies [@lee11 Proposition 12]. In another recent paper [@gmr13] we proved that $d_{\mathrm{AG}}$ and $d_{\mathrm{LBAO}}$ are equal to the Feng-Rao bound as defined in [@beelen08; @ldecodepaper] for $C_u$. Lower Bound for the Number of Votes {#sec:nrvotes} ----------------------------------- In Section \[sec:nrvotes\] we discuss the number of votes (\[eq:acceptedvote\]) which a candidate $w_{s,i}$ receives. Since we study list decoding, we cannot assume the original transmitted codeword nor the error vector as in [@lee11]. Nevertheless, the original theorems in [@lee11] allow natural generalizations to the list decoding context. \[lem:nrvotes\] Fix $s \in \Gamma$. For $s' \in \Gamma^{(> s)}$, fix a sequence of $w_{s'}$ chosen by the decoding algorithm, and define $\vec{r}^{(s)}$ corresponding to the chosen sequence of $w_{s'}$. Fix $\omega_s \in \mathbf{F}_q$. Let $\vec{e} = (e_1$, …, $e_n)^T$ be a nonzero vector with the minimum Hamming weight in the coset $\vec{r}^{(s)} -{\mathrm{ev}}(\omega_s\varphi_s)+ C_{s-1}$, where $C_{s-1}$ is as defined in Eq. (\[eq:cu\]). Define $$\begin{aligned} J_{\vec{e}} &=& \bigcap_{e_i\neq 0} \mathcal{L}(-P_i+\infty Q)\\ &=& \mathcal{L}\left(\infty Q - \sum_{e_i\neq 0} P_i\right) \mbox{ (by \citep{matsumoto99ldpaper})}.\end{aligned}$$ Let $\{\epsilon_0$, …, $\epsilon_{a_1-1}\}$ be a Gröbner basis for $J_{\vec{e}}$ as an $\mathbf{F}_q[x_1]$-module with respect to $<_s$ (for any integer $s$), such that $\textsc{lm}(\epsilon_j) = x_1^{k_j} y_j$. Under the above notations, we have $$\begin{aligned} -v_Q(\epsilon_i)+v_Q(a_{i,i} y_i )&\geq&a_1\bar{c}_i,\\ \min{\{ -v_Q(\epsilon_i)+s,-v_Q(\eta_{i'}) \}}&\geq&-v_Q(d_{i',i'} y_{i'}),\end{aligned}$$ for $i$ with $w_{s,i}\neq \omega_s$, and $$\min\{-v_Q(\epsilon_i)+s, -v_Q(\eta_{i'})\} \geq -v_Q(d_{i',i'}y_{i'})-a_1 \bar{c}_i,$$ for $i$ with $w_{s,i}= \omega_s$. [**Proof. **]{}The proof is the same as those of [@lee11 Propositions 7 and 8], with replacing $y^j$ in [@lee11] by $y_j$, $\delta(\cdot)$ in [@lee11] by $-v_Q(\cdot)$. [ ]{} The following lemma is a modification to [@lee11 Proposition 9] for the list decoding. We retain notations from Lemma \[lem:nrvotes\]. We have $$\begin{aligned} \lefteqn{a_1 \sum_{w_{s,i}=\omega_s}\bar{c}_i\geq a_1 \sum_{w_{s,i}\neq\omega_s}\bar{c}_i - 2a_1{\mathrm{wt}}(\vec{e})} \\ && \mbox{ } + \sum_{0\le i<a_1}\max{\{ -v_Q(\eta_{i'})+v_Q( y_i)-s,-v_Q(\epsilon_i)+v_Q( y_i ) \}}.\end{aligned}$$ [**Proof. **]{}Lemma \[lem:nrvotes\] implies $$\begin{aligned} \sum_{w_{s,i}=\omega_s}a_1\bar{c}_i &\geq&\sum_{w_{s,i}=\omega_s}-v_Q(d_{i',i'} y_{i'} )-\min{\{ -v_Q(\epsilon_i)+s,-v_Q(\eta_{i'}) \}}\\ &\geq&\sum_{0\le i<a_1}-v_Q(d_{i',i'} y_{i'} )-\min{\{ -v_Q(\epsilon_i)+s,-v_Q(\eta_{i'}) \}}\end{aligned}$$ and $$\begin{aligned} \sum_{w_{s,i}\neq\omega_s}a_1\bar{c}_i&\le&\sum_{w_{s,i}\neq\omega_s}-v_Q(\epsilon_i)+v_Q(a_{i,i} y_i )\\ &\le&\sum_{0\le i<a_1}-v_Q(\epsilon_i)+v_Q(a_{i,i} y_i ).\end{aligned}$$ Now we have a chain of inequalities $$\begin{aligned} && \sum_{w_{s,i}=\omega_s}a_1\bar{c}_i - \sum_{w_{s,i}\neq\omega_s}a_1\bar{c}_i\nonumber\\ & \geq & \sum_{0\le i<a_1}-v_Q(d_{i',i'} y_{i'})-\min{\{ -v_Q(\epsilon_i)+s,-v_Q(\eta_{i'}) \}}\nonumber\\ &&- \sum_{0\le i<a_1}-v_Q(\epsilon_i)+v_Q(a_{i,i} y_i )\nonumber\\ &=& \sum_{0\le i<a_1}-v_Q(d_{i',i'} y_{i'} ) -v_Q(a_{i,i} y_i)\nonumber \\&&\mbox{ }-\min{\{ -v_Q(\epsilon_i)+s,-v_Q(\eta_{i'}) \}}+v_Q(\epsilon_i)\nonumber\\ &= & \sum_{0\le i<a_1}-v_Q(\eta_{i'})-v_Q( y_i )\label{eq:this}\\ &&\mbox{ }+\max{\{ +v_Q(\epsilon_i)-s,+v_Q(\eta_{i'}) \}}+v_Q(\epsilon_i)\nonumber\\ &= & \sum_{0\le i<a_1}\max{\{ -v_Q(\eta_{i'})+v_Q( y_i )-s,-v_Q(\epsilon_i)+v_Q( y_i ) \}}\nonumber\\ && -\sum_{0\le i<a_1}2(-v_Q(\epsilon_i)+v_Q( y_i ))\nonumber\end{aligned}$$ where at Eq. (\[eq:this\]) we used the equality $$\begin{aligned} && \sum_{0\le i<a_1}-v_Q(d_{i',i'} y_{i'} )+\sum_{0\le i<a_1}-v_Q(a_{i,i} y_i)\\ &=&\sum_{0\le i<a_1}-v_Q(d_{i,i} y_i )+\sum_{0\le i<a_1}-v_Q(a_{i,i} y_i )\\ &=&\sum_{0\le i<a_1}(-v_Q(d_{i,i})-v_Q(a_{i,i}))+\sum_{0\le i<a_1}-2v_Q( y_i)\\ &=&a_1n+\sum_{0\le i<a_1}-2v_Q( y_i )\\ &=&\sum_{0\le i<a_1}(-v_Q(\eta_{i})+v_Q(y_i))+\sum_{0\le i<a_1}-2v_Q(y_i)\\ &=&\sum_{0\le i<a_1}-v_Q(\eta_{i'})+\sum_{0\le i<a_1}-v_Q(y_i)\end{aligned}$$ shown in [@lee11 Lemma 2 and Eq. (1)]. Finally note that $$\begin{aligned} &&\sum_{0\le i<a_1}2(-v_Q(\epsilon_i)+v_Q(y_i))\\ &=&\sum_{0\le i<a_1}2a_1\deg_{x_1}(\epsilon_i)=2a_1{\mathrm{wt}}(\vec{e})\end{aligned}$$ by [@lee11 Eq. (3)]. [ ]{} The following lemma is a modification to [@lee11 Proposition 10] for list decoding, and provides a lower bound for the number of votes (\[eq:acceptedvote\]) received by any candidate $\omega_s \in \mathbf{F}_q$, as indicated in the section title. \[prop:nrvotes\] We retain notations from Lemma \[lem:nrvotes\]. Let $\nu(s)$ be as defined in Eq. (\[eq:nu\]). We have $$\sum_{w_{s,i}=\omega_s}\bar{c}_i\geq \sum_{w_{s,i}\neq\omega_s}\bar{c}_i - 2{\mathrm{wt}}(\vec{e})+\nu(s).$$ [**Proof. **]{}We have $$\begin{aligned} &&\sum_{0\le i<a_1}\max{\{ -v_Q(\eta_{i'})+v_Q(y_i)-s,-v_Q(\epsilon_i)+v_Q(y_i) \}}\\ &\ge&\sum_{0\le i<a_1}\max{\{ -v_Q(\eta_{i'})+v_Q(y_i)-s,0 \}}\end{aligned}$$ as $-v_Q(\epsilon_i)+v_Q(y_i)\ge 0$ for $0\le i<a_1$. [ ]{} Correctness of the Modified List Decoding Algorithm with the Third Iteration Termination Criterion {#sec:thirdwhy} -------------------------------------------------------------------------------------------------- In this subsection and the following sections, we shall prove that the proposed list decoding algorithm will find all the codewords within the Hamming distance $\tau$ from the received word $\vec{r}$. Since the third iteration termination criterion is the easiest to analyze, we start with the third one. Fix a sequence $w_s$ for $s\in \Gamma$. If ${\mathrm{wt}}(\vec{r} - {\mathrm{ev}}(\sum_{s\in \Gamma} w_s \varphi_s))\leq \tau$ then the sequence $w_s$ is found by the algorithm because of Proposition \[prop:nrvotes\]. When $2 \tau < d_{\mathrm{AG}} (C_\Gamma )$, by Proposition \[prop:AG\] the decoding is not list decoding, and the algorithm just declares the sequence $w_s$ as the transmitted information. On the other hand, if $2 \tau \geq d_{\mathrm{AG}} (C_\Gamma )$, then the found sequence could correspond to a codeword more distant than Hamming distance $\tau$, and the algorithm examines the Hamming distance between the found codeword and the received word $\vec{r}$. Since computing ${\mathrm{ev}}(f)$ for $f\in {\mathcal{L}(\infty Q)}$ needs many multiplications in $\mathbf{F}_q$, the algorithm checks some sufficient conditions to decide the Hamming distance between the found codeword and the received word $\vec{r}$. Let $\vec{r}^{(-1)} = (r^{(-1)}_1$, …, $r^{(-1)}_n)$. When $\alpha_0=0$ in Section \[sec:3rd\], by Lemma \[lem:ir\], we have $$\begin{aligned} {\mathrm{wt}}(\vec{r} - {\mathrm{ev}}(\sum_{s\in \Gamma} w_s \varphi_s))&=& {\mathrm{wt}}(\vec{r}^{(-1)})\\ &\leq& \sum_{r^{(-1)}_i \neq 0} v_{P_i}(\alpha_1)\\ &\leq&-v_Q(\alpha_1),\end{aligned}$$ because Eq. (\[eq:ir\]) and $\alpha_0=0$ implies that $v_{P_i}(\alpha_1) \geq 1$ for $r^{(-1)}_i \neq 0$. By the above equation, $-v_Q(\alpha_1) \leq \tau$ implies that the found codeword is within Hamming distance $\tau$ from $\vec{r}$. This explains why the algorithm can avoid computation of the evaluation map ${\mathrm{ev}}$ in Step \[third1\] in Section \[sec:3rd\]. In order to explain Step \[third2\] in Section \[sec:3rd\], we shall show that the condition of Step \[third2\] in Section \[sec:3rd\] implies that ${\mathrm{wt}}(\vec{r} - {\mathrm{ev}}(\sum_{s\in \Gamma} w_s \varphi_s))> \tau$. Suppose that ${\mathrm{wt}}(\vec{r} - {\mathrm{ev}}(\sum_{s\in \Gamma} w_s \varphi_s)) \leq \tau$. Then there exists $\beta_1 \in {\mathcal{L}(\infty Q)}$ such that $v_{P_i}(\beta) \geq 1$ for $r^{(-1)}_i \neq 0$, $-v_Q(\beta) \leq \tau + g$, and $\beta_1 z \in I_{\vec{r}^{(-1)}}$. Because the leading term of $\beta_1 z$ must be divisible by $\textsc{lt}(f_i^{(-1)})$ for some $i$ by the property of Gröbner bases, we must have $-v_Q(\alpha_1) \leq -v_Q(\beta_1)$. This explains why the algorithm can avoid computation of the evaluation map ${\mathrm{ev}}$ in Step \[third2\] in Section \[sec:3rd\]. Otherwise, the algorithm computes the Hamming distance between the found codeword and $\vec{r}$ in Steps \[third3\] and \[third4\] in Section \[sec:3rd\]. Correctness of the Modified List Decoding Algorithm with the Second Iteration Termination Criterion {#sec:secondwhy} --------------------------------------------------------------------------------------------------- We shall explain why the second criterion in Section \[sec:second\] correctly finds the required codewords. For explanation, we present slightly rephrased version of facts in [@beelen08]. \[lem:23\][@beelen08 Lemma 2.3] Let $\beta_1 z + \beta_0 \in I_{\vec{r}^{(s)}}$ with $\textsc{lt}(\beta_1 z + \beta_0) = \textsc{lt}(\beta_1 z)$ with respect to $<_s$ and $-v_Q(\beta_1) < n-\tau-s$. If there exists $f \in \mathcal{L}(sQ)$ such that ${\mathrm{wt}}({\mathrm{ev}}(f)-\vec{r}^{(s)}) \leq \tau$, then we have $f = - \beta_0/\beta_1$. [**Proof. **]{}Observe that $\textsc{lt}(\beta_1 z + \beta_0) = \textsc{lt}(\beta_1 z)$ implies that $-v_Q(\beta_0) \leq -v_Q(\beta_1)+s < n-\tau$. The claim of Lemma \[lem:23\] is equivalent to [@beelen08 Lemma 2.3] with $A = (n-\tau-1)Q$ and $G = sQ$. Note that the assumption $\deg A > (n+\deg G)/2+g-1$ was not used in [@beelen08 Lemma 2.3] but only in [@beelen08 Lemma 2.4]. [ ]{}Note that the following proposition was essentially proved in [@beelen08 Proposition 2.10], [@justesen04 Section 14.2], and [@shokrollahi99 Theorem 2.1] with $b=1$. \[prop:candivide\] Let $\alpha_0$ and $\alpha_1$ be as in Section \[sec:second\]. If $s < n -g -2\tau$ and there exists $f \in \mathcal{L}(sQ)$ such that ${\mathrm{wt}}({\mathrm{ev}}(f)-\vec{r}^{(s)}) \leq \tau$, then we have $f = -\alpha_0/\alpha_1$. [**Proof. **]{}Let $g \in {\mathcal{L}(\infty Q)}$ such that $g(P_i) = 0$ if $f(P_i) \neq r_i^{(s)}$, and assume that $g$ has the minimum pole order at $Q$ among such elements in ${\mathcal{L}(\infty Q)}$. Then $-v_Q(g) \leq \tau + g$. One has that $gz - fg \in I_{\vec{r}^{(s)}}$ and $\textsc{lt}(gz - fg) = \textsc{lt}(gz)$ with respect to $<_s$. By the property of Gröbner bases, $\textsc{lt}(gz)$ is divisible by $\textsc{lt}(f^{(s)}_i)$ for some $i$, which implies $-v_Q(\alpha_1) \leq -v_Q(g) \leq \tau + g$. By Lemma \[lem:23\] we have $f = -\alpha_0/\alpha_1$. [ ]{} We explain how the procedure in Section \[sec:second\] works as desired. When the condition in Step \[second1\] in Section \[sec:second\] is true, then there cannot be a codeword within Hamming distance $\tau$ from $\vec{r}^{(s)}$ by the same reason as Section \[sec:thirdwhy\]. So the algorithm stops processing with $\vec{r}^{(s)}$. When $2 \tau < d_{\mathrm{AG}} (C_\Gamma )$, then the algorithm declares $-\alpha_0/\alpha_1+\sum_{s' \in \Gamma^{(> s)}} w_{s'} \varphi_{s'}$ as the unique codeword. When $2 \tau \geq d_{\mathrm{AG}} (C_\Gamma )$, then the algorithm examines the found codeword close enough to $\vec{r}$ in Steps \[second11\] and \[second12\] in Section \[sec:second\]. When $-v_Q(\alpha_1) \leq \tau$ we can avoid computation of the evaluation map ${\mathrm{ev}}$ by the same reason as Section \[sec:thirdwhy\], which is checked at Step \[second11\]. Otherwise we compute the codeword vector at Step \[second12\] and examine its Hamming distance to $\vec{r}^{(s)}$. By Proposition \[prop:candivide\], the codeword must be found at $s= \max\{ s' \in \Gamma \mid s' < n-2 \tau + g\}$. Therefore, we do not execute the iteration at $s< \max\{ s' \in \Gamma \mid s'< n-2 \tau + g \}$. Correctness of the Modified List Decoding Algorithm with the First Iteration Termination Criterion {#sec:firstwhy} -------------------------------------------------------------------------------------------------- We shall explain why the first criterion in Section \[sec:first\] correctly finds the required codewords. The idea behind the first criterion is that there cannot be another codeword within Hamming distance $\tau$ from $\vec{r}^{(s)}$ when the algorithm already found one. So the algorithm can stop iteration with smaller $s$ once a codeword is found as ${\mathrm{ev}}(-\alpha_0/\alpha_1 +\sum_{s' \in \Gamma^{(> s)}} w_{s'} \varphi_{s'})$. The algorithm does not examine conditions when $-v_Q(\alpha_1) > \tau + g$ by the same reason as Sections \[sec:thirdwhy\] and \[sec:secondwhy\]. When $2 \tau < d_{\mathrm{AG}} (C_\Gamma )$, then the algorithm declares $-\alpha_0/\alpha_1+\sum_{s' \in \Gamma^{(> s)}} w_{s'} \varphi_{s'}$ as the unique codeword. When $2 \tau \geq d_{\mathrm{AG}} (C_\Gamma )$, then the algorithm examines the found codeword close enough to $\vec{r}$ in Steps \[first11\]–\[first13\] in Section \[sec:first\]. When $-v_Q(\alpha_1) \leq \tau$ we can avoid computation of the evaluation map ${\mathrm{ev}}$ by the same reason as Section \[sec:thirdwhy\], which is checked at Step \[first11\]. By Proposition \[prop:candivide\], the codeword must be found at some $s\geq \max\{ s' \in \Gamma \mid s' < n-2 \tau + g\}$. Therefore, we do not execute the iteration at $s< \max\{ s' \in \Gamma \mid s' < n-2 \tau + g\}$. Upper Bound on the Number of Iterations {#sec:nriteration} --------------------------------------- Observe that for $s > \max \Gamma$ we set always $w_s$ to $0$. For each $s \in \Gamma$ satisfying $\nu(s) \leq 2 \tau$, the number of accepted candidates satisfying Eq. (\[eq:acceptedvote\]) can be at most $q$. On the other hand, for $s$ with $\nu(s) > 2\tau$, the number of candidates is either zero or one, because at most one $w \in \mathbf{F}_q$ can satisfy Eq.  (\[eq:acceptedvote\]). Therefore, we have upper bounds for the number of iterations, counting executions of Rebasing in Section \[sec:rebasing\], as $$\begin{aligned} &&\sharp \{s\in H(Q) \mid \max \Gamma \leq s < N\} + \exp_q (\sharp \{s \in \Gamma \mid \nu(s) \leq 2 \tau\})\nonumber\\ &&\times \sharp \{s \in H(Q)\cup\{-1\} \mid s < \max \Gamma \}\label{iteration-ub1}\end{aligned}$$ for the third criterion for judging termination, where $\exp_q(x) = q^x$, and $$\begin{aligned} &\sharp \{s\in H(Q) \mid \max \Gamma \leq s < N)\} + \exp_q (\sharp \{s \in \Gamma \mid \nu(s) \leq 2 \tau\})\nonumber\\ &\times \sharp \{s \in H(Q) \mid \max\{ s' \in \Gamma \mid s' < n-2 \tau - g\} \leq s < \max \Gamma \}\label{iteration-ub2}\end{aligned}$$ for the first and the second criteria for judging termination. We will use $\max \widehat{H}(Q)$ in place of $N$ in Eqs. (\[iteration-ub1\]) and (\[iteration-ub2\]) for computation in Tables \[tab1\]–\[tab3\], because $N$ depents on $\vec{r}$ and $N \leq \max \widehat{H}(Q)$. Observe that the list decoding can be implemented as $\exp_q (\sharp \{s \in \Gamma \mid \nu(s) \leq 2 \tau\})$ parallel execution of the unique decoding. Therefore, when one can afford $\exp_q (\sharp \{s \in \Gamma \mid \nu(s) \leq 2 \tau\})$ parallel implementation, which increases the circuit size, the decoding time of list decoding is the same as that of the unique decoding. Comparison to Conventional Methods {#sec:experiment} ================================== Simulation Condition and Results {#sec51} -------------------------------- We have provided an upper bound on the number of multiplications and divisions at each step of the proposed algorithm. We simulated $1,000$ transmissions of codewords with the one-point primal codes on Klein quartic over $\mathbf{F}_8$ with $n=23$ by using Examples \[ex1\] and \[ex2\], the one-point Hermitian codes over $\mathbf{F}_{16}$ with $n=64$, and the one-point primal codes on the curve in Example \[ex:gs1\] over $\mathbf{F}_9$ with $n=77$. The program is implemented on the Singular computer algebra system [@singular313]. The program used for this simulation is available from <http://arxiv.org/src/1203.6127/anc>. In the execution, we counted the number of iterations, (executions of Rebasing in Section \[sec:rebasing\]), the sum of upper bounds on the number of multiplications and divisions given in Eqs. (\[eq:compl:ini\]), (\[eq:compl3\]), (\[eq:compl4\]), (\[eq:compl5\]), (\[eq:compl6\]), (\[eq:compl7\]), (\[eq:compl1\]), (\[eq:multirebasing\]) and (\[eq:compl2\]), and the number of codewords found. Note also that Eq. (\[eq:NFmulti3\]) instead of Eq. (\[eq:NFmulti4\]) is used. The parameter $\tau$ is set to the same as the number of generated errors in each simulation condition. $N$ or $R$ in the number of errors in Tables \[tab2\] and \[tab22\] indicates that the error vector is generated toward another codeword nearest from the transmitted codeword or completely randomly, respectively. The distribution of codewords is uniform on $C_\Gamma$. That of error vectors is uniform on the vectors of Hamming weight $\tau$. In the code construction, we always try to use the Feng-Rao improved construction. Specifically, for a given designed distance $\delta$, we choose $\Gamma = \{ s \in \widehat{H}(Q) \mid \lambda(s)=\nu(s) \geq \delta \}$, and construct $C_\Gamma$ of Eq. (\[eq:cgamma\]). In the following, the designed distance is denoted by $d_{\mathrm{AG}}(C_\Gamma)$. It can be seen from Tables \[tab1\]–\[tab3\] and the following subsections that the computational complexity of the proposed algorithm tends to explode when the number of errors exceeds the error-correcting capability of the Guruswami-Sudan algorithm [@guruswami99]. Comparison among the Three Proposed Termination Criteria {#sec:comparethree} -------------------------------------------------------- In Section \[sec:termination\] we proposed three criteria for terminating iteration of the proposed algorithm. From Tables \[tab1\]–\[tab3\], one can see the following. The first criterion has the smallest number of iterations, and the second is the second smallest. On the other hand, the first criterion has the largest number of multiplications and divisions. The second and the third have the similar numbers. Only the first criterion was proposed in [@gmr12isit] and we see that the new criteria are better than the old one. The reason is as follows: The computation of quotient $\alpha_0/\alpha_1$ at Step \[first1\] in Section \[sec:first\] is costlier than updating $f_i^{(s)}$ and $g_i^{(s)}$ in Section \[sec:rebasing\] and the first criterion computes $\alpha_0/\alpha_1$ many times, which cancels the effect of decrease in the number of iterations. On the other hand, the second criterion computes $\alpha_0/\alpha_1$ only once, so it has the smaller number of multiplications and divisions than the first. The second criterion is faster when $2 \tau < d_{\mathrm{AG}}(C_\Gamma)$, while the third tends to be faster when $2 \tau \geq d_{\mathrm{AG}}(C_\Gamma)$. In addition to this, the ratio of the number of iterations in the second criterion to that of the third is smaller with $2 \tau < d_{\mathrm{AG}}(C_\Gamma)$ than with $2 \tau \geq d_{\mathrm{AG}}(C_\Gamma)$. We speculate the reason behind them as follows: When $2 \tau \geq d_{\mathrm{AG}}(C_\Gamma)$ and a wrong candidate is chosen at Eq. (\[eq:acceptedvote\]), after several iterations of Sections \[sec:termination\] and \[sec:mainiteration\], we often observe in our simulation that no candidate satisfies Eq. (\[eq:acceptedvote\]) and the iteration stops automatically. Under such situation, the second criterion does not help much to decrease the number of iterations nor the computational complexity when a wrong candidate is chosen at Eq. (\[eq:acceptedvote\]), and there are many occasions at which a wrong candidate is chosen at Eq. (\[eq:acceptedvote\]) when $2 \tau \geq d_{\mathrm{AG}}(C_\Gamma)$. On the other hand, when $2 \tau < d_{\mathrm{AG}}(C_\Gamma)$, the second criterion helps to determine the transmitted information earlier than the third. Tightness of Upper Bounds (\[iteration-ub1\]) and (\[iteration-ub2\]) --------------------------------------------------------------------- In Table \[tab3\], we observe that the upper bounds (\[iteration-ub1\]) and (\[iteration-ub2\]) are much larger than the actual number of iterations for $\tau=5$. The disappearance of candidates satisfying Eq. (\[eq:acceptedvote\]) in the last paragraph may also explain the reason behind the large differences for $\tau=5$. On the other hand, we observe that the upper bound (\[iteration-ub2\]) is quite tight for $\tau=5$ in Table \[tab1\] and $\tau=10$N in Table \[tab22\]. This suggests that improvement of Eq. (\[iteration-ub2\]) may need some additional assumption. Klein Quartic, $(d_{\mathrm{AG}}(C_\Gamma),\tau)=(4,1)$ or $(10,4)$ ------------------------------------------------------------------- We can use [@beelen07; @beelen08; @duursma11; @duursma10; @ldecodepaper] to decode this set of parameters. It is essentially the forward elimination in the Gaussian elimination, and it takes roughly $n^3/3$ multiplications. In this case $n^3/3= 4,055$. The proposed algorithm has the lower complexity than [@beelen07; @beelen08; @duursma11; @duursma10; @ldecodepaper]. Klein Quartic, $(d_{\mathrm{AG}}(C_\Gamma),\tau)=(4,2)$ or $(4,3)$ ------------------------------------------------------------------ The code is $C_u$ with $u=20$, $\dim C_u = 18$. There is no previously known algorithm that can handle this case. Klein Quartic, $(d_{\mathrm{AG}}(C_\Gamma),\tau)=(10,5)$ -------------------------------------------------------- The code is $C_u$ with $u=13$, $\dim C_u = 11$. According to @beelen10 [Figure 1], we can use the original Guruswami-Sudan [@guruswami99] but it seems that its faster variants cannot be used. We need the multiplicity $7$ to correct $5$ errors. We have to solve a system of $23(7+1)7/2 = 644$ linear equations. It takes $644^3/3 =89,029,994$ multiplications in $\mathbf{F}_8$. The proposed algorithm is much faster. Klein Quartic, $(d_{\mathrm{AG}}(C_\Gamma),\tau)=(10,6)$ -------------------------------------------------------- The code is $C_u$ with $u=13$, $\dim C_u = 11$. There is no previously known algorithm that can handle this case. Hermitian, $(d_{\mathrm{AG}}(C_\Gamma),\tau)=(6,2)$ or $(20,9)$ --------------------------------------------------------------- We can use the BMS algorithm [@sakata95b; @sakata95a] for this case. The complexity of [@sakata95b; @sakata95a] is estimated as $O(a_1 n^2)$ and $a_1 n^2 = 24,576$. The complexity of the proposed algorithm seems comparable to [@sakata95b; @sakata95a]. However, we are not sure which one is faster. Hermitian, $(d_{\mathrm{AG}}(C_\Gamma),\tau)=(6,3)$ or $(6,4)$ -------------------------------------------------------------- The code becomes the Feng-Rao improved code with designed distance $6$. Its dimension is $55$. In order to have the same dimension by $C_u$ we have to set $u=60$, whose AG bound [@andersen08] is $4$ and the Guruwsami-Sudan can correct up to 2 errors. The proposed algorithm finds all codewords in the improved code with $3$ and $4$ errors. Hermitian, $(d_{\mathrm{AG}}(C_\Gamma),\tau)=(20,10)$ ----------------------------------------------------- The code is $C_u$ with $u=44$. The required multiplicity is $11$, and the required designed list size is $14$. The fastest algorithm for the interpolation step seems [@beelen10]. @beelen10 [Example 4] estimates the complexity of their algorithm as $O(\lambda^5 n^2 (\log \lambda n)^2 \log(\log \lambda n))$, where $\lambda$ is the designed list size. Ignoring the log factor and assuming the scaling factor one in the big-$O$ notation, the number of multiplications and divisions is $\lambda^5 n^2 = 2,202,927,104$. The proposed algorithm needs much fewer number of multiplications and divisions in $\mathbf{F}_{16}$. Hermitian, $(d_{\mathrm{AG}}(C_\Gamma),\tau)=(20,11)$ ----------------------------------------------------- The Guruwsami-Sudan algorithm [@guruswami99] can correct up to $10$ errors and there seems no previously known algorithm that can handle this case. Garcia-Stichtenoth (Example \[ex:gs1\]), $(d_{\mathrm{AG}}(C_\Gamma),\tau) = (6,2)$, $(10,4)$, or $(20,9)$ ---------------------------------------------------------------------------------------------------------- We can use [@beelen07; @beelen08; @duursma11; @duursma10; @ldecodepaper] to decode this set of parameters. It is essentially the forward elimination in the Gaussian elimination, and it takes roughly $n^3/3$ multiplications. In this case $n^3/3= 152,177$. The proposed algorithm has the lower complexity than [@beelen07; @beelen08; @duursma11; @duursma10; @ldecodepaper]. Garcia-Stichtenoth (Example \[ex:gs1\]), $(d_{\mathrm{AG}}(C_\Gamma),\tau) = (6,3)$ ----------------------------------------------------------------------------------- This is a Feng-Rao improved code with dimension $58$. In order to realize a code with the same dimension, we have to set $u=79$ in $C_u$. The Guruswami-Sudan algorithm [@guruswami99] can correct no error in this set of parameters. There seems no previously known algorithm that can handle this case. Garcia-Stichtenoth (Example \[ex:gs1\]), $(d_{\mathrm{AG}}(C_\Gamma),\tau) = (10,5)$ ------------------------------------------------------------------------------------ This is a Feng-Rao improved code with dimension $52$. In order to realize a code with the same dimension, we have to set $u=73$ in $C_u$. The Guruswami-Sudan algorithm [@guruswami99] can correct $2$ errors in this set of parameters. There seems no previously known algorithm that can handle this case. Garcia-Stichtenoth (Example \[ex:gs1\]), $(d_{\mathrm{AG}}(C_\Gamma),\tau) = (20,10)$ ------------------------------------------------------------------------------------- This is an ordinary one-point AG code $C_u$ with $u=58$ and dimension $37$. The Guruswami-Sudan algorithm [@guruswami99] can correct $10$ errors with the multiplicity $154$ and the designed list size $178$. We have to solve a system of $77 \times (154+1)154/2 = 918,995$ linear equations. It takes $918,995^3/3 =258,712,963,551,308,291$ multiplications in $\mathbf{F}_9$. The proposed algorithm is much faster. Garcia-Stichtenoth (Example \[ex:gs1\]), $(d_{\mathrm{AG}}(C_\Gamma),\tau) = (20,11)$ ------------------------------------------------------------------------------------- The Guruswami-Sudan algorithm [@guruswami99] can correct up to $10$ errors and there seems no previously known algorithm that can handle this case. Conclusion {#sec6} ========== In this paper, we modified the unique decoding algorithm for plane AG codes in [@lee11] so that it can support one-point AG codes on *any* curve, and so that it can do the list decoding. The error correction capability of the original [@lee11] and our modified algorithms are also expressed in terms of the minimum distance lower bound in [@andersen08]. We also proposed procedures to compute products and quotients in coordinate ring of affine algebraic curves, and by using those procedures we demonstrated that the modified decoding algorithm can be executed quickly. Specifically, its computational complexity is comparable to the BMS algorithm [@sakata95b; @sakata95a] for one-point Hermitian codes, and much faster than the standard list decoding algorithms [@beelen10; @guruswami99] for several cases. The original decoding algorithm [@lee11] allows parallel implementation on circuits like the Kötter architecture [@koetter98]. Our modified algorithm retains this advantage. Moreover, if one can afford large circuit size, the proposed list decoding algorithm can be executed as quickly as the unique decoding algorithm by parallel implementation on a circuit. Acknowledgment {#acknowledgment .unnumbered} ============== The authors thank an anonymous reviewer for his/her careful reading that improved the presentation. This research was partially supported by the MEXT Grant-in-Aid for Scientific Research (A) No. 23246071, the Villum Foundation through their VELUX Visiting Professor Programme 2011–2012, the Danish National Research Foundation and the National Science Foundation of China (Grant No. 11061130539) for the Danish-Chinese Center for Applications of Algebraic Geometry in Coding Theory and Cryptography, the Spanish grant MTM2007-64704, and the Spanish MINECO grant No. MTM2012-36917-C03-03. The computer experiments in this research was conducted on Singular 3.1.3 [@singular313]. 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--- abstract: 'In this paper we study the Cauchy problem for doubly dissipative elastic waves in two space dimensions, where the damping terms consist of two different friction or structural damping. We derive energy estimates and diffusion phenomena with different assumptions on initial data. Particularly, we find the dominant influence on diffusion phenomena by introducing a new threshold of diffusion structure.' address: | Institute of Applied Analysis, Faculty for Mathematics and Computer Science\ Technical University Bergakademie Freiberg\ Prüferstra[ß]{}e 9\ 09596 Freiberg\ Germany author: - Wenhui Chen date: 'January 1, 2004' title: Dissipative structure and diffusion phenomena for doubly dissipative elastic waves in two space dimensions --- Introduction {#Introduction} ============ In this paper we consider the following Cauchy problem for doubly dissipative elastic waves in two space dimensions: $$\label{Eq.DoublyDissElasticWaves} \left\{ \begin{aligned} &u_{tt}-a^2\Delta u-\left(b^2-a^2\right)\nabla\operatorname{div}u+(-\Delta)^{\rho}u_t+(-\Delta)^{\theta}u_t=0,&&x\in{\mathbb}{R}^2,\,\,t>0,\\ &(u,u_t)(0,x)=(u_0,u_1)(x),&&x\in{\mathbb}{R}^2, \end{aligned} \right.$$ where the unknown $u=u(t,x)\in{\mathbb}{R}^2$ denotes the elastic displacement. The positive constants $a$ and $b$ in are related to the Lamé constants and fulfill $b>a>0$. Moreover, the parameters $\rho$ and $\theta$ in satisfy $0\leq\rho<1/2<\theta\leq1$. Let us recall some related works to our problem . Taking $a=b=1$, $\rho=0$ and $\theta=1$ in , then we immediately turn to doubly dissipative wave equation, where the damping terms consist of *friction* $u_t$ as well as *viscoelastic damping* $-\Delta u_t$ $$\label{Eq.DoublyDissWave} \left\{ \begin{aligned} &u_{tt}-\Delta u+u_t-\Delta u_t=0,&&x\in{\mathbb}{R}^n,\,\,t>0,\\ &(u,u_t)(0,x)=(u_0,u_1)(x),&&x\in{\mathbb}{R}^n, \end{aligned} \right.$$ with $n\geq1$. The recent paper [@IkehataSawada2016] derived asymptotic profiles of solutions to in a framework of weighted $L^1$ data. Precisely, the authors found that from asymptotic profiles of solutions point of view, friction $u_t$ is more dominant than viscoelastic damping $-\Delta u_t$ as $t\rightarrow\infty$. Later in [@IkehataMichihisa2018], the authors obtained higher-order asymptotic expansions of solutions to and gave some lower bounds estimates to show the optimality of these expansions. For the other related works on , we refer the reader to the recent papers [@IkehataTakeda2017; @DAbbicco2017; @IkehataTakeda2019]. However, asymptotic profiles of solutions to general doubly dissipative wave equation, where the damping terms consist of friction or structural damping (i.e., taking $a=b=1$ in ), are still open. This open problem is proposed in [@IkehataSawada2016]. The main difficulty is to answer what is the dominant profile of solutions, due to the fact that the asymptotic profiles for wave equation with damping term $(-\Delta)^{\rho}u_t$ for $0\leq \rho<1/2$, or with damping term $(-\Delta)^{\theta}u_t$ for $1/2<\theta\leq 1$, are quite different. One may see, for example, [@Matsumura1976; @Karch2000; @MarcatiNishihara2003; @HosonoOgawa2004; @Narazaki2004; @Nishihara2003; @Takeda2015; @Ikehata2014; @DabbiccoEbert2014; @IkehataOnodera2017; @Michihisa2017; @Michihisa2018; @IkehataTakeda2019NEW; @Shibata2000; @Ponce1985; @DabbiccoReissig2014]. Let us come back to dissipative elastic waves. In recent years the Cauchy problem for dissipative elastic waves have aroused wide concern, which can be modeled by $$\label{Eq.DissElasticWaves} \left\{ \begin{aligned} &u_{tt}-a^2\Delta u-\left(b^2-a^2\right)\nabla\operatorname{div}u+{\mathcal}{A}u_t=0,&&x\in{\mathbb}{R}^n,\,\,t>0,\\ &(u,u_t)(0,x)=(u_0,u_1)(x),&&x\in{\mathbb}{R}^n, \end{aligned} \right.$$ where $b>a>0$ and the term ${\mathcal}{A}u_t$ describes several kinds of damping mechanisms.\ In the case when $$\begin{aligned} {\mathcal}{A}u_t=u_t,\,\,\,\,\text{i.e., \emph{friction} or \emph{external damping}},\end{aligned}$$ the authors of [@IkehataCharaodaLuz2014] proved almost sharp energy estimates for $n\geq2$ by using energy methods in the Fourier space and the Haraux-Komornik inequality, and then the recent paper [@ChenReissig2019SD] investigated propagation of singularities, sharp energy estimates and diffusion phenomenon for $n=3$.\ Furthermore, in the case when $$\begin{aligned} {\mathcal}{A}u_t=(-\Delta)^{\theta}u_t\,\,\,\,\text{with}\,\,\,\,\theta\in(0,1],\,\,\,\,\text{i.e., \emph{structural damping}}, \end{aligned}$$ energy estimates are derived with different data spaces in [@IkehataCharaodaLuz2014] for $n\geq2$, and in [@Reissig2016] for $n=2$. Moreover, some qualitative properties of solutions, including smoothing effect, sharp energy estimate and diffusion phenomena (especially, *double diffusion phenomena* when $\theta\in(0,1/2)$) are obtained for $n=3$.\ Finally, in the case when $$\begin{aligned} {\mathcal}{A}u_t=(-a^2\Delta-(b^2-a^2)\nabla\operatorname{div})u_t,\,\,\,\,\text{i.e., \emph{Kelvin-Voigt damping}}, \end{aligned}$$ by applying energy methods in the Fourier space, almost sharp energy estimates for $n\geq2$ have been obtained in [@WuChaiLi2017]. Then, sharp energy estimates, $L^p-L^q$ estimates as well as asymptotic profiles of solutions are derived for $n=2$ in [@Chen2019KV]. Other studies on dissipative elastic waves can be found in literatures [@CharaoIkehata2007; @CharaoIkehata2011]. Nevertheless, concerning about decay properties and diffusion phenomena for the Cauchy problem for doubly dissipative elastic waves it seems that we still do not have any previous research manuscripts. Moreover, this problem is strongly related to the open problem proposed in [@IkehataSawada2016]. In this paper we give the answer to the two-dimensional case. Let us point out that the study of the Cauchy problem is not simply a generalization of elastic waves with friction or structural damping in [@Reissig2016; @ChenReissig2019SD]. On one hand, because there exists two different damping terms $(-\Delta)^{\rho}u_t$ and $(-\Delta)^{\theta}u_t$ with $0\leq\rho<1/2<\theta\leq1$ in our problem , it is not clear which damping term has a dominant influence on dissipative structure. On the other hand, from the paper [@ChenReissig2019SD], the authors derived diffusion phenomena to elastic waves with the damping term $(-\Delta)^{\theta}u_t$ where $\theta\in[0,1/2)\cup(1/2,1]$, which are described by the following so-called *reference system*. - In the case when $\theta=0$, the reference system consist of heat-type system with mass term as follows: $$\begin{aligned} \widetilde{V}_t-{\mathcal}{D}_1\Delta\widetilde{V}+{\mathcal}{D}_2\widetilde{V}=0, \end{aligned}$$ with real diagonal matrices ${\mathcal}{D}_1$ and ${\mathcal}{D}_2$. - In the case when $\theta\in(0,1/2)$, the reference system consist of two different parabolic systems as follows: $$\begin{aligned} \widetilde{V}_t+{\mathcal}{D}_3(-\Delta)^{1-\theta}\widetilde{V}+{\mathcal}{D}_4(-\Delta)^{\theta}\widetilde{V}=0, \end{aligned}$$ with real diagonal matrices ${\mathcal}{D}_3$ and ${\mathcal}{D}_4$. - In the case when $\theta\in(1/2,1]$, the reference system consist of parabolic system and half-wave system as follows: $$\begin{aligned} \widetilde{V}_t+{\mathcal}{D}_5(-\Delta)^{\theta}\widetilde{V}+i{\mathcal}{D}_6(-\Delta)^{\frac{1}{2}}\widetilde{V}=0, \end{aligned}$$ with real diagonal matrices ${\mathcal}{D}_5$ and ${\mathcal}{D}_6$. Hence, for different choices of damping terms, which mainly depend on the value of the parameter $\theta$ in the damping term, the diffusion phenomena are quite different. In the Cauchy problem , the damping terms consist of $(-\Delta)^{\rho}u_t$ with $\rho\in[0,1/2)$, and $(-\Delta)^{\theta}u_t$ with $\theta\in(1/2,1]$. Thus, it is not clear that the reference system is make up of what kind of evolution systems, and how do two different damping terms influence on diffusion structure. Furthermore, from [@ChenReissig2019SD] we know *the threshold of diffusion structure* is $\theta=1/2$ for elastic waves with structural damping. In other words, the structure of reference system will be changed from $\theta\in(0,1/2)$ to $\theta\in(1/2,1]$. Then, the natural question is what is the threshold of diffusion structure for doubly dissipative elastic waves. Again, we give the answers for these questions in two dimensions. Our main purpose of the present paper is to investigate dissipative structure and diffusion phenomena for doubly dissipative elastic waves with different assumptions on initial data. We find that the damping term $(-\Delta)^{\rho}u_t$ with $0\leq\rho<1/2$ has the dominant influence on energy estimates (see Theorems \[Thm.EnergyEst\] and \[Thm.EnergyWeighted.L1\]). Furthermore, in the case when $\rho+\theta<1$, the damping terms $(-\Delta)^{\rho}u_t$ and $(-\Delta)^{\theta}u_t$ with $0\leq\rho<1/2<\theta\leq1$ have the influence on diffusion structure at the same time. However, in the case when $\rho+\theta\geq1$, the diffusion structure is determined by the damping term $(-\Delta)^{\rho}u_t$ with $0\leq\rho<1/2$ only. Hence, one of our novelties is to derive a threshold $\rho+\theta=1$ of diffusion structure for doubly dissipative elastic waves. This paper is organized as follows. In Section \[Sec.AsymptoticBehavior\] we derive representation of solutions by applying WKB analysis and multistep diagonalization procedure. In Section \[Sec.EnergyEstimate\] we obtain pointwise estimate in the Fourier space and energy estimates by using this representation. In Section \[Sec.AsymptoticProfiles\] we derive diffusion phenomena with different assumptions on initial data. Finally, in Section \[Sec.ConcludingRemarks\] some concluding remarks complete the paper. **Notations:** In this paper $f\lesssim g$ means that there exists a positive constant $C$ such that $f\leq Cg$. We write $f\asymp g$ when $g\lesssim f\lesssim g$ Moreover, $H^s$ and $\dot{H}^s$ with $s\geq0$, denote Bessel and Riesz potential spaces based on $L^2$, respectively. Furthermore, $\langle D\rangle^s$ and $|D|^s$ stand for the pseudo-differential operators with symbols $\langle\xi\rangle^s$ and $|\xi|^s$, respectively, where $\langle\xi\rangle^2:=1+|\xi|^2$. We denote the identity matrix of dimensions $k\times k$ by $I_{k\times k}$. We denote the diagonal matrix by $$\begin{aligned} \operatorname{diag}\left(e^{-\lambda_jt}\right)_{j=1}^4:=\operatorname{diag}\left(e^{-\lambda_1t},e^{-\lambda_2t},e^{-\lambda_3t},e^{-\lambda_4t}\right).\end{aligned}$$ The weighted spaces $L^{1,\gamma}$ for $\gamma\in[0,\infty)$ are defined by $$\begin{aligned} L^{1,\gamma}:=\left\{f\in L^1:\|f\|_{L^{1,\gamma}}:=\int_{{\mathbb}{R}^n}(1+|x|)^{\gamma}|f(x)|dx<\infty\right\}.\end{aligned}$$ Finally, let us define the cut-off functions $\chi_{\operatorname{int}}(\xi),\chi_{\text{bdd}}(\xi),\chi_{\operatorname{ext}}(\xi)\in \mathcal{C}^{\infty}$ having their supports in the following zones: $$\begin{aligned} {\mathcal}{Z}_{\operatorname{int}}(\varepsilon)&:=\left\{\xi\in{\mathbb}{R}^2:|\xi|<\varepsilon\ll1\right\},\\ {\mathcal}{Z}_{\text{bdd}}(\varepsilon,N)&:=\left\{\xi\in{\mathbb}{R}^2:\varepsilon\leq |\xi|\leq N\right\},\\ {\mathcal}{Z}_{\operatorname{ext}}(N)&:=\left\{\xi\in{\mathbb}{R}^2:|\xi|> N\gg1\right\},\end{aligned}$$ respectively, so that $\chi_{\operatorname{int}}(\xi)+\chi_{\text{bdd}}(\xi)+\chi_{\operatorname{ext}}(\xi)=1$. Asymptotic behavior of solutions in the Fourier space {#Sec.AsymptoticBehavior} ===================================================== In this section we will derive asymptotic behavior of solutions and representation of solutions in the Fourier space. Let us apply the partial Fourier transform with respect to spatial variable such that $\hat{u}(t,\xi)={\mathcal}{F}_{x\rightarrow \xi}(u(t,x))$ to obtain $$\label{Eq.FourierDoublyDiss} \left\{ \begin{aligned} &\hat{u}_{tt}+|\xi|^{2}A(\eta)\hat{u}+\left(|\xi|^{2\rho}+|\xi|^{2\theta}\right)\hat{u}_t=0,&&\xi\in{\mathbb}{R}^2, \,\,t>0,\\ &(\hat{u},\hat{u}_t)(0,\xi)=(\hat{u}_0,\hat{u}_1)(\xi),&&\xi\in{\mathbb}{R}^2, \end{aligned}\right.$$ where $$\begin{aligned} A(\eta)=\left( {\begin{array}{*{20}c} a^2+\left(b^2-a^2\right)\eta_1^2 & \left(b^2-a^2\right)\eta_1\eta_2\\ \left(b^2-a^2\right)\eta_1\eta_2 & a^2+\left(b^2-a^2\right)\eta_2^2\\ \end{array}} \right)\end{aligned}$$ with $\eta=\xi/|\xi|\in{\mathbb}{S}^1$. Similar as [@Reissig2016; @Chen2019KV], we introduce the matrix $$\begin{split} M(\eta):=\left( {\begin{array}{*{20}c} \eta_1 & \eta_2\\ \eta_2 & -\eta_1 \\ \end{array}} \right), \end{split}$$ and define a new variable $W=W(t,\xi)$ such that $$W:=\left( \begin{aligned} &v_t+i|\xi|\text{diag}(b,a)v\\ &v_t-i|\xi|\text{diag}(b,a)v \end{aligned}\right),$$ where $v:={M}^{-1}(\eta)\hat{u}$. Moreover, we have $U(t,x)={\mathcal}{F}^{-1}_{\xi\rightarrow x}(W(t,\xi))$. Next, the following first-order system can be derived: $$\label{Eq.FirstOrderSystem} \left\{\begin{aligned} &W_t+\left(\frac{1}{2}{B}_0|\xi|^{2\rho}+i{B}_1|\xi|+\frac{1}{2}{B}_0|\xi|^{2\theta}\right)W=0,&&\xi\in{\mathbb}{R}^2,\,\,t>0,\\ &W(0,\xi)=W_0(\xi),&&\xi\in{\mathbb}{R}^2, \end{aligned}\right.$$ where the coefficient matrices ${B}_0$ and ${B}_1$ are respectively given by $$\label{Matrix.B0.B1} \begin{split} {B}_0=\left( {\begin{array}{*{20}c} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1\\ \end{array}} \right) \,\,\,\, \text{and} \,\,\,\, {B}_1=\left( {\begin{array}{*{20}c} -b & 0 & 0 & 0 \\ 0 & -a & 0 & 0\\ 0 & 0 & b & 0 \\ 0 & 0 & 0 & a\\ \end{array}} \right). \end{split}$$ Let us point out that throughout this section, we will study representation of solutions $U=U(t,x)$ to the following Cauchy problem by deriving representation of its partial Fourier transform $W(t,\xi)={\mathcal}{F}_{x\rightarrow \xi}(U(t,x))$: $$\label{Eq.InvFirstOrderSystem} \left\{\begin{aligned} &U_t+\frac{1}{2}{B}_0(-\Delta)^{\rho}U+i{B}_1(-\Delta)^{\frac{1}{2}}U+\frac{1}{2}{B}_0(-\Delta)^{\theta}U=0,&&x\in{\mathbb}{R}^2,\,\,t>0,\\ &U(0,x)=U_0(x),&&x\in{\mathbb}{R}^2, \end{aligned}\right.$$ where the coefficient matrices $B_0$ and $B_1$ are given in . Moreover, to derive qualitative properties of solutions to , we only need to study the solutions to . With the aim of deriving representation of solutions, we may apply WKB analysis and multistep diagonalization procedure (see for example [@ReissigWang2005; @Yagdjian1997; @Jachmann2009; @JachmannReissig2008; @JachmannReissig2009; @Reissig2016; @Chen2019TEP]). Before doing these, we should understand the influence of the parameter $|\xi|$ on the asymptotic behavior of solutions to . Due to our assumption $0\leq 2\rho<1<2\theta\leq2$, we now discuss the influence of $|\xi|$ by three parts. Specifically, we will apply diagonalization procedure for small frequencies $\xi\in{\mathcal}{Z}_{\operatorname{int}}(\varepsilon)$ and large frequencies $\xi\in{\mathcal}{Z}_{\operatorname{ext}}(N)$ in Subsections \[SubSec.SmallFreq\] and \[SubSec.LargeFreq\], respectively. Then, the contradiction argument will be applied to prove an exponential stability of solutions for bounded frequencies $\xi\in{\mathcal}{Z}_{\text{bdd}}(\varepsilon,N)$ in Subsection \[SubSec.BoundedFreq.\]. Treatment for small frequencies {#SubSec.SmallFreq} ------------------------------- In the case when $\xi\in{\mathcal}{Z}_{\operatorname{int}}(\varepsilon)$, it is clear that the matrix $\frac{1}{2}|\xi|^{2\rho}B_0$ has a dominant influence comparing with the matrices $i|\xi|B_1$ and $\frac{1}{2}|\xi|^{2\theta}B_0$. For this reason, by defining $$\begin{aligned} \label{Matrix.T1} T_1:=\left( {\begin{array}{*{20}c} -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1\\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1\\ \end{array}} \right),\end{aligned}$$ we introduce $W^{(1)}:=T_1^{-1}W$. Then, we may derive $$\begin{aligned} W^{(1)}_{t}+\Lambda_1(|\xi|)W^{(1)}+\left(B_0^{(1)}(|\xi|)+B_1^{(1)}(|\xi|)\right)W^{(1)}=0,\end{aligned}$$ where $$\begin{aligned} \Lambda_1(|\xi|)&=|\xi|^{2\rho}\operatorname{diag}(0,0,1,1)={\mathcal}{O}\left(|\xi|^{2\rho}\right),\\ B_0^{(1)}(|\xi|)&=|\xi|^{2\theta}\operatorname{diag}(0,0,1,1)={\mathcal}{O}\left(|\xi|^{2\theta}\right),\\ B_1^{(1)}(|\xi|)&=i|\xi|T_1^{-1}B_1T_1={\mathcal}{O}(|\xi|).\end{aligned}$$ Here $$\begin{aligned} T_1^{-1}B_1T_1=\left( {\begin{array}{*{20}c} 0 & 0 & b & 0 \\ 0 & 0 & 0 & a\\ b & 0 & 0 & 0 \\ 0 & a & 0 & 0\\ \end{array}} \right).\end{aligned}$$ In the second step we introduce $W^{(2)}:=T_2^{-1}W^{(1)}$, where $$\begin{aligned} \label{Matrix.N2} T_2:=I_{4\times4}+N_2(|\xi|) \quad\text{with}\quad N_2(|\xi|):=i|\xi|^{1-2\rho}\left( {\begin{array}{*{20}c} 0 & 0 & b & 0 \\ 0 & 0 & 0 & a\\ -b & 0 & 0 & 0 \\ 0 & -a & 0 & 0\\ \end{array}} \right).\end{aligned}$$ The following first-order system comes: $$\begin{aligned} W_t^{(2)}+\Lambda_1(|\xi|)W^{(2)}+R_2(|\xi|)W^{(2)}=0,\end{aligned}$$ where $$\begin{aligned} R_2=\underbrace{T_2^{-1}B_1^{(1)}(|\xi|)N_2(|\xi|)}_{={\mathcal}{O}(|\xi|^{2-2\rho})}+\underbrace{T_2^{-1}B_0^{(1)}(|\xi|)T_2}_{={\mathcal}{O}(|\xi|^{2\theta})}={\mathcal}{O}\left(|\xi|^{\min\{2-2\rho;2\theta\}}\right).\end{aligned}$$ To understand the dominant term in the remainder $R_2(|\xi|)$, we distinguish between three cases. *Case 2.1.1:* $\rho+\theta<1$.\ In this case the matrix $T_2^{-1}B_0^{(1)}(|\xi|)T_2$ has a dominant influence. We find that this matrix can be rewritten by the following way: $$\begin{aligned} T_2^{-1}B_0^{(1)}(|\xi|)T_2=B_0^{(1)}(|\xi|)+T_2^{-1}\left[B_0^{(1)}(|\xi|),N_2(|\xi|)\right].\end{aligned}$$ Thus, setting $W^{(3)}:=W^{(2)}$ implies $$\begin{aligned} W^{(3)}_{t}+(\Lambda_1(|\xi|)+\Lambda_2(|\xi|))W^{(3)}+R_3(|\xi|)W^{(3)}=0,\end{aligned}$$ where $\Lambda_2(|\xi|)=B_0^{(1)}(|\xi|)={\mathcal}{O}\left(|\xi|^{2\theta}\right)$ and $$\begin{aligned} R_3(|\xi|)=\underbrace{T_2^{-1}B_1^{(1)}(|\xi|)N_2(|\xi|)}_{={\mathcal}{O}(|\xi|^{2-2\rho})}+\underbrace{T_2^{-1}[\Lambda_2(|\xi|),N_2(|\xi|)]}_{={\mathcal}{O}(|\xi|^{1+2\theta-2\rho})}={\mathcal}{O}\left(|\xi|^{2-2\rho}\right).\end{aligned}$$ Because $2\theta>1$, the term $T_2^{-1}B_1^{(1)}(|\xi|)N_2(|\xi|)$ has a dominant influence in comparison with the term $T_2^{-1}[\Lambda_2(|\xi|),N_2(|\xi|)]$ in the remainder $R_3(|\xi|)$. We observe that $$\begin{aligned} T_2^{-1}B_1^{(1)}(|\xi|)N_2(|\xi|)&=B_1^{(1)}(|\xi|)N_2(|\xi|)-N_2(|\xi|)T_2^{-1}B_1^{(1)}(|\xi|)N_2(|\xi|),\\ B_1^{(1)}(|\xi|)N_2(|\xi|)&=|\xi|^{2-2\rho}\operatorname{diag}\left(b^2,a^2,-b^2,-a^2\right).\end{aligned}$$ So, by taking $W^{(4)}:=W^{(3)}$ we have $$\begin{aligned} W^{(4)}_{t}+(\Lambda_1(|\xi|)+\Lambda_2(|\xi|)+\Lambda_3(|\xi|))W^{(4)}+R_4(|\xi|)W^{(4)}=0,\end{aligned}$$ where $\Lambda_3(|\xi|)=B_1^{(1)}(|\xi|)N_2(|\xi|)={\mathcal}{O}\left(|\xi|^{2-2\rho}\right)$ and $$\begin{aligned} R_4(|\xi|)=-N_2(|\xi|)T_2^{-1}B_1^{(1)}(|\xi|)N_2(|\xi|)+T_2^{-1}[\Lambda_2(|\xi|),N_2(|\xi|)]={\mathcal}{O}\left(|\xi|^{1+2\theta-2\rho}\right).\end{aligned}$$ Up to now, we have derived pairwise distinct eigenvalues and $R_4(|\xi|)={\mathcal}{O}\left(|\xi|^{1+2\theta-2\rho}\right)$. *Case 2.1.2:* $\rho+\theta=1$.\ In this case the matrices $T_2^{-1}B_0^{(1)}(|\xi|)T_2$ and $T_2^{-1}B_1^{(1)}(|\xi|)N_2(|\xi|)$ have the same influence. For this reason, we set $$\begin{aligned} \Lambda_2(|\xi|)=B_1^{(1)}(|\xi|)N_2(|\xi|)+B_0^{(1)}(|\xi|)=|\xi|^{2-2\rho}\operatorname{diag}\left(b^2,a^2,1-b^2,1-a^2\right)={\mathcal}{O}\left(|\xi|^{2-2\rho}\right).\end{aligned}$$ Then, taking $W^{(3)}:=W^{(2)}$ again we derive $$\begin{aligned} W_t^{(3)}+(\Lambda_1(|\xi|)+\Lambda_2(|\xi|))W^{(3)}+R_3(|\xi|)W^{(3)}=0,\end{aligned}$$ where $$\begin{aligned} R_3(|\xi|)=-N_2(|\xi|)T_2^{-1}B_1^{(1)}(|\xi|)N_2(|\xi|)+T_2^{-1}\left[B_0^{(1)}(|\xi|),N_2(|\xi|)\right]={\mathcal}{O}\left(|\xi|^{3-4\rho}\right).\end{aligned}$$ Up to now, we have derived pairwise distinct eigenvalues and $R_3(|\xi|)={\mathcal}{O}\left(|\xi|^{3-4\rho}\right)$. *Case 2.1.3:* $\rho+\theta>1$.\ In this case the matrix $T_2^{-1}B_1^{(1)}(|\xi|)N_2(|\xi|)$ has a dominant influence. Following the idea from *Case 2.1.1* and setting $W^{(3)}:=W^{(2)}$ again, we may derive $$\begin{aligned} W_t^{(3)}+(\Lambda_1(|\xi|)+\Lambda_2(|\xi|))W^{(3)}+R_3(|\xi|)W^{(3)}=0,\end{aligned}$$ where $\Lambda_2(|\xi|)=B_1^{(1)}(|\xi|)N_2(|\xi|)={\mathcal}{O}(|\xi|^{2-2\rho})$ and $$\begin{aligned} R_3(|\xi|)=-N_2(|\xi|)T_2^{-1}B_1^{(1)}(|\xi|)N_2(|\xi|)+T_2^{-1}B_0^{(1)}(|\xi|)T_2={\mathcal}{O}\left(|\xi|^{\min\{3-4\rho;2\theta\}}\right).\end{aligned}$$ Up to now, we have derived pairwise distinct eigenvalues and $R_3(|\xi|)={\mathcal}{O}\left(|\xi|^{\min\{3-4\rho;2\theta\}}\right)$. Summarizing above diagonalization procedure, according to [@Jachmann2009] we obtain the next proposition, which tells us the asymptotic behavior of eigenvalues and representation of solutions. \[Prop.INT\] The eigenvalues $\lambda_{j}=\lambda_{j}(|\xi|)$ of the coefficient matrix $$\begin{aligned} B(|\xi|;\rho,\theta)=\frac{1}{2}\left(|\xi|^{2\rho}+|\xi|^{2\theta}\right)B_0+i|\xi|B_1 \end{aligned}$$ from behave for $|\xi|<\varepsilon\ll1$ as - if $\rho+\theta<1$, then $$\begin{aligned} &\lambda_{1}(|\xi|)=b^2|\xi|^{2-2\rho}+{\mathcal}{O}\left(|\xi|^{1+2\theta-2\rho}\right),\\ &\lambda_{2}(|\xi|)=a^2|\xi|^{2-2\rho}+{\mathcal}{O}\left(|\xi|^{1+2\theta-2\rho}\right),\\ &\lambda_{3}(|\xi|)=|\xi|^{2\rho}+|\xi|^{2\theta}-b^2|\xi|^{2-2\rho}+{\mathcal}{O}\left(|\xi|^{1+2\theta-2\rho}\right),\\ &\lambda_{4}(|\xi|)=|\xi|^{2\rho}+|\xi|^{2\theta}-a^2|\xi|^{2-2\rho}+{\mathcal}{O}\left(|\xi|^{1+2\theta-2\rho}\right); \end{aligned}$$ - if $\rho+\theta=1$, then $$\begin{aligned} &\lambda_{1}(|\xi|)=b^2|\xi|^{2-2\rho}+{\mathcal}{O}\left(|\xi|^{3-4\rho}\right),\\ &\lambda_{2}(|\xi|)=a^2|\xi|^{2-2\rho}+{\mathcal}{O}\left(|\xi|^{3-4\rho}\right),\\ &\lambda_{3}(|\xi|)=|\xi|^{2\rho}+\left(1-b^2\right)|\xi|^{2-2\rho}+{\mathcal}{O}\left(|\xi|^{3-4\rho}\right),\\ & \lambda_{4}(|\xi|)=|\xi|^{2\rho}+\left(1-a^2\right)|\xi|^{2-2\rho}+{\mathcal}{O}\left(|\xi|^{3-4\rho}\right); \end{aligned}$$ - if $\rho+\theta>1$, then $$\begin{aligned} &\lambda_{1}(|\xi|)=b^2|\xi|^{2-2\rho}+{\mathcal}{O}\left(|\xi|^{\min\{3-4\rho;2\theta\}}\right),\\ &\lambda_{2}(|\xi|)=a^2|\xi|^{2-2\rho}+{\mathcal}{O}\left(|\xi|^{\min\{3-4\rho;2\theta\}}\right),\\ & \lambda_{3}(|\xi|)=|\xi|^{2\rho}-b^2|\xi|^{2-2\rho}+{\mathcal}{O}\left(|\xi|^{\min\{3-4\rho;2\theta\}}\right),\\ & \lambda_{4}(|\xi|)=|\xi|^{2\rho}-a^2|\xi|^{2-2\rho}+{\mathcal}{O}\left(|\xi|^{\min\{3-4\rho;2\theta\}}\right). \end{aligned}$$ Furthermore, the solution to the Cauchy problem has in ${\mathcal}{Z}_{\operatorname{int}}(\varepsilon)$ the representation $$\begin{aligned} W(t,\xi)=T_{\operatorname{int}}^{-1}(|\xi|)\operatorname{diag}\left(e^{-\lambda_j(|\xi|)t}\right)_{j=1}^4T_{\operatorname{int}}(|\xi|)W_0(\xi), \end{aligned}$$ where $T_{\operatorname{int}}(|\xi|)=(I_{4\times4}+N_2(|\xi|))^{-1}T_1^{-1}$ with a matrix $N_2(|\xi|)={\mathcal}{O}\left(|\xi|^{1-2\rho}\right)$ for $|\xi|\rightarrow0$. Here the matrix $T_1$ is defined in . Treatment for large frequencies {#SubSec.LargeFreq} ------------------------------- We observe that the symmetric of the system with respective to the parameters $\rho$ and $\theta$. Thus, by similar procedure we can obtain pairwise distinct eigenvalues. Before stating our result for large frequencies, we define $$\begin{aligned} T_3:=I_{4\times4}+N_3(|\xi|) \quad\text{with}\quad N_3(|\xi|)=i|\xi|^{1-2\theta}\left( {\begin{array}{*{20}c} 0 & 0 & b & 0 \\ 0 & 0 & 0 & a\\ -b & 0 & 0 & 0 \\ 0 & -a & 0 & 0\\ \end{array}} \right).\end{aligned}$$ Then, following the similar procedure as the case for small frequencies and according to the thesis [@Jachmann2009] we obtain the next proposition. \[Prop.EXT\] The eigenvalues $\mu_{j}=\mu_{j}(|\xi|)$ of the coefficient matrix $$\begin{aligned} B(|\xi|;\rho,\theta)=\frac{1}{2}\left(|\xi|^{2\rho}+|\xi|^{2\theta}\right)B_0+i|\xi|B_1 \end{aligned}$$ from behave for $|\xi|> N\gg1$ as - if $\rho+\theta<1$, then $$\begin{aligned} &\mu_{1}(|\xi|)=b^2|\xi|^{2-2\theta}+{\mathcal}{O}\left(|\xi|^{\min\{3-4\theta;2\rho\}}\right),\\ &\mu_{2}(|\xi|)=a^2|\xi|^{2-2\theta}+{\mathcal}{O}\left(|\xi|^{\min\{3-4\theta;2\rho\}}\right),\\ & \mu_{3}(|\xi|)=|\xi|^{2\theta}-b^2|\xi|^{2-2\theta}+{\mathcal}{O}\left(|\xi|^{\min\{3-4\theta;2\rho\}}\right),\\ & \mu_{4}(|\xi|)=|\xi|^{2\theta}-a^2|\xi|^{2-2\theta}+{\mathcal}{O}\left(|\xi|^{\min\{3-4\theta;2\rho\}}\right). \end{aligned}$$ - if $\rho+\theta=1$, then $$\begin{aligned} &\mu_{1}(|\xi|)=b^2|\xi|^{2-2\theta}+{\mathcal}{O}\left(|\xi|^{3-4\theta}\right),\\ &\mu_{2}(|\xi|)=a^2|\xi|^{2-2\theta}+{\mathcal}{O}\left(|\xi|^{3-4\theta}\right),\\ &\mu_{3}(|\xi|)=|\xi|^{2\theta}+\left(1-b^2\right)|\xi|^{2-2\theta}+{\mathcal}{O}\left(|\xi|^{3-4\theta}\right),\\ & \mu_{4}(|\xi|)=|\xi|^{2\theta}+\left(1-a^2\right)|\xi|^{2-2\theta}+{\mathcal}{O}\left(|\xi|^{3-4\theta}\right); \end{aligned}$$ - if $\rho+\theta>1$, then $$\begin{aligned} &\mu_{1}(|\xi|)=b^2|\xi|^{2-2\theta}+{\mathcal}{O}\left(|\xi|^{1+2\rho-2\theta}\right),\\ &\mu_{2}(|\xi|)=a^2|\xi|^{2-2\theta}+{\mathcal}{O}\left(|\xi|^{1+2\rho-2\theta}\right),\\ &\mu_{3}(|\xi|)=|\xi|^{2\theta}+|\xi|^{2\rho}-b^2|\xi|^{2-2\theta}+{\mathcal}{O}\left(|\xi|^{1+2\rho-2\theta}\right),\\ & \mu_{4}(|\xi|)=|\xi|^{2\theta}+|\xi|^{2\rho}-a^2|\xi|^{2-2\theta}+{\mathcal}{O}\left(|\xi|^{1+2\rho-2\theta}\right); \end{aligned}$$ Furthermore, the solution to the Cauchy problem has in ${\mathcal}{Z}_{\operatorname{ext}}(N)$ the representation $$\begin{aligned} W(t,\xi)=T_{\operatorname{ext}}^{-1}(|\xi|)\operatorname{diag}\left(e^{-\mu_j(|\xi|)t}\right)_{j=1}^4T_{\operatorname{ext}}(|\xi|)W_0(\xi), \end{aligned}$$ where $T_{\operatorname{ext}}(|\xi|)=(I_{4\times4}+N_3(|\xi|))^{-1}T_1^{-1}$ with a matrix $N_3(|\xi|)={\mathcal}{O}\left(|\xi|^{1-2\theta}\right)$ for $|\xi|\rightarrow\infty$. Here the matrix $T_1$ is defined in . Treatment for bounded frequencies {#SubSec.BoundedFreq.} --------------------------------- Finally, we only need to derive an exponential decay of solutions to for bounded frequencies to guarantee the exponential stability of solutions. \[Prop.BDD\]The solution $W=W(t,\xi)$ to the Cauchy problem with $0\leq \rho<1/2<\theta\leq1$ fulfills the following exponential decay estimate: $$\begin{aligned} |W(t,\xi)|\lesssim e^{-ct}|W_0(\xi)|, \end{aligned}$$ for $(t,\xi)\in(0,\infty)\times {\mathcal}{Z}_{\text{bdd}}(\varepsilon,N)$, where $c$ is a positive constant. Let us recall that $$\begin{aligned} B(|\xi|;\rho,\theta)=\frac{1}{2}\left(|\xi|^{2\rho}+|\xi|^{2\theta}\right)B_0+i|\xi|B_1.\end{aligned}$$ It is clear that the eigenvalues of $B(|\xi|;\rho,\theta)$ satisfy $$\begin{aligned} 0&=\det(B(|\xi|;\rho,\theta)-\lambda I_{4\times 4})\\ &=\lambda^4-2\left(|\xi|^{2\rho}+|\xi|^{2\theta}\right)\lambda^3+\left(\left(|\xi|^{2\rho}+|\xi|^{2\theta}\right)^2+\left(a^2+b^2\right)|\xi|^2\right)\lambda^2\\ &\quad-\left(a^2+b^2\right)|\xi|^2\left(|\xi|^{2\rho}+|\xi|^{2\theta}\right)\lambda+a^2b^2|\xi|^4.\end{aligned}$$ Now, we assume there exists an eigenvalue $\lambda=id$ with $d\in{\mathbb}{R}\backslash\{0\}$. Therefore, the real number $d$ should satisfy the equations $$\label{MIDDLE.EQ} \left\{ \begin{aligned} &d^4-\left(\left(|\xi|^{2\rho}+|\xi|^{2\theta}\right)^2+\left(a^2+b^2\right)|\xi|^2\right)d^2+a^2b^2|\xi|^4=0,\\ &id\left(|\xi|^{2\rho}+|\xi|^{2\theta}\right)\left(2d^2-\left(a^2+b^2\right)|\xi|^2\right)=0. \end{aligned} \right.$$ Due to the facts that $d\neq0$ and $\xi\in {\mathcal}{Z}_{\text{bdd}}(\varepsilon,N)$, the equations leads to $$\begin{aligned} -\left(b^2-a^2\right)^2|\xi|^2=2\left(a^2+b^2\right)\left(|\xi|^{2\rho}+|\xi|^{2\theta}\right)^2.\end{aligned}$$ From our assumption $b>a>0$, we immediately find a contradiction. Thus, there not exists pure imaginary eigenvalue of $B(|\xi|;\rho,\theta)$ for any $0\leq\rho<1/2<\theta\leq 1$ and $\xi\in {\mathcal}{Z}_{\text{bdd}}(\varepsilon,N)$. Lastly, by using the compactness of the bounded zone ${\mathcal}{Z}_{\text{bdd}}(\varepsilon,N)$ and the continuity of the eigenvalues, the proof is complete. Energy estimates {#Sec.EnergyEstimate} ================ The aim of the section is to study the dissipative structure and sharp energy estimates to doubly dissipative elastic waves, where initial data belongs to Bessel potential space with additional $L^m$ regularity ($m\in[1,2]$) or with additional weighted $L^1$ regularity. The crucial point of sharp energy estimates is to derive the sharp pointwise estimate. By summarizing the results in Propositions \[Prop.INT\], \[Prop.EXT\] and \[Prop.BDD\], we obtain the result on the sharp pointwise estimate of solutions to . \[Prop.PointwiseEst\] The solution $W=W(t,\xi)$ to the Cauchy problem with $0\leq \rho<1/2<\theta\leq1$ satisfies the following pointwise estimates for any $\xi\in{\mathbb}{R}^2$ and $t\geq0$: $$\begin{aligned} |W(t,\xi)|\lesssim e^{-c\eta(|\xi|)t}|W_0(\xi)|,\end{aligned}$$ where $\eta(|\xi|):=\frac{|\xi|^{2-2\rho}}{1+|\xi|^{2\theta-2\rho}}$ and $c$ is positive constant. The pointwise estimate in Proposition \[Prop.PointwiseEst\] gives the characterization of the dissipative structure of doubly dissipative elastic waves. We now compare the dissipative structure of doubly dissipative elastic waves and elastic waves with friction or structural damping in [@Reissig2016; @ChenReissig2019SD]. For one thing, as $|\xi|\rightarrow0$, the dissipative structure of doubly dissipative elastic waves is the same as elastic waves with friction or structural damping $(-\Delta)^{\rho}u_t$ for $\rho\in[0,1/2)$, that is $\eta(|\xi|)\asymp|\xi|^{2-2\rho}$ for $|\xi|\rightarrow0$. For another, as $|\xi|\rightarrow\infty$, the dissipative structure of doubly dissipative elastic waves is the same as elastic waves with structural damping $(-\Delta)^{\theta}u_t$ for $\theta\in(1/2,1]$, that is $\eta(|\xi|)\asymp|\xi|^{2-2\theta}$ for $|\xi|\rightarrow\infty$. Now, we state our main result on energy estimates. \[Thm.EnergyEst\] Let us consider the Cauchy problem with $0\leq\rho<1/2<\theta\leq1$ and $U_0\in H^s\cap L^m$, where $s\geq0$ and $m\in[1,2]$. Then, the following estimates hold: $$\begin{aligned} \|U(t,\cdot)\|_{\dot{H}^s}\lesssim(1+t)^{-\frac{s}{2-2\rho}-\frac{2-m}{m(2-2\rho)}}\|U_0\big\|_{H^s\cap L^m}.\end{aligned}$$ According to Proposition \[Prop.INT\] and sharp pointwise estimate in Proposition \[Prop.PointwiseEst\], the energy estimates in Theorem \[Thm.EnergyEst\] are sharp for initial data $U_0\in H^s\cap L^m$, where $s\geq0$ and $m\in[1,2]$. We remark that the energy estimates for doubly dissipative elastic waves in Theorem \[Thm.EnergyEst\] are the same as damped elastic waves with damping term $(-\Delta)^{\rho}u_t$ for $\rho\in[0,1/2)$ in Theorems 7.2 and 7.3 in [@Reissig2016]. \[Remark.01\] From energy estimates in Theorem \[Thm.EnergyEst\], we observe that the decay rate is only determined by the damping term $(-\Delta)^{\rho}u_t$ with $\rho\in[0,1/2)$ in . For the other damping term $(-\Delta)^{\theta}u_t$ with $\theta\in(1/2,1]$, there is no any influence for the energy estimates. The main reason is that the decay rate for energy estimates of is mainly determined by dissipative structure for small frequencies. However, for the dissipative structure for small frequencies (see Proposition \[Prop.INT\]), the dominant influence of eigenvalues are determined by $|\xi|^{2-2\rho}$. Although the parameter $\theta$ in the damping term $(-\Delta)^{\theta}u_t$ has a great influence on the asymptotic behavior of eigenvalues for large frequencies, the solutions satisfies an exponential decay for large frequencies providing that we assume suitable regularity for initial data. To begin with, by using Proposition \[Prop.PointwiseEst\], we calculate $$\begin{aligned} \|W(t,\cdot)\|_{\dot{H}^s}&\lesssim\left\|\chi_{\operatorname{int}}(\xi)|\xi|^se^{-c|\xi|^{2-2\rho}t}W_0(\xi)\right\|_{L^2}+e^{-ct}\left\|\chi_{\text{bdd}}(\xi)|\xi|^sW_0(\xi)\right\|_{L^2}\\ &\quad+\left\|\chi_{\operatorname{ext}}(\xi)|\xi|^se^{-c|\xi|^{2-2\theta}t}W_0(\xi)\right\|_{L^2}\\ &\lesssim\left\|\chi_{\operatorname{int}}(\xi)|\xi|^se^{-c|\xi|^{2-2\rho}t}W_0(\xi)\right\|_{L^2}+e^{-ct}\|U_0\|_{H^s}.\end{aligned}$$ Next, we divide the proof into two cases. For the case when $m=2$ in Theorem \[Thm.EnergyEst\], we have $$\begin{aligned} \left\|\chi_{\operatorname{int}}(\xi)|\xi|^se^{-c|\xi|^{2-2\rho}t}W_0(\xi)\right\|_{L^2}&\lesssim\sup\limits_{\xi\in{\mathcal}{Z}_{\operatorname{int}}(\varepsilon)}\left||\xi|^se^{-c|\xi|^{2-2\rho}t}\right|\|W_0\|_{L^2}\\ &\lesssim(1+t)^{-\frac{s}{2-2\rho}}\|U_0\|_{L^2}.\end{aligned}$$ For the case when $m\in[1,2)$ in Theorem \[Thm.EnergyEst\], the applications of Hölder’s inequality and the Hausdorff-Young inequality yield $$\begin{aligned} \left\|\chi_{\operatorname{int}}(\xi)|\xi|^se^{-c|\xi|^{2-2\rho}t}W_0(\xi)\right\|_{L^2}&\lesssim\left\|\chi_{\operatorname{int}}(\xi)|\xi|^se^{-c|\xi|^{2-2\rho}t}\right\|_{L^{\frac{2m}{2-m}}}\|U_0\|_{L^m}\\ &\lesssim(1+t)^{-\frac{s}{2-2\rho}-\frac{2-m}{m(2-2\rho)}}\|U_0\|_{L^m}.\end{aligned}$$ Finally, by applying the Parseval-Plancherel theorem, we immediately complete the proof. Furthermore, we discuss energy estimates in a framework of weighted $L^1$ data. Before stating our result, we recall the Lemma 2.1 in the paper [@Ikehata2004]. \[Lem.IkehataWeightedL1\] Let $f\in L^{1,\gamma}$ with $\gamma\in(0,1]$. Then, the following estimate holds: $$\begin{aligned} \left|\hat{f}(\xi)\right|\leq C_{\gamma}|\xi|^{\gamma}\|f\|_{L^{1,\gamma}}+\left|\int_{{\mathbb}{R}^n}f(x)dx\right|, \end{aligned}$$ with a positive constant $C_{\gamma}>0$. \[Thm.EnergyWeighted.L1\] Let us consider the Cauchy problem with $0\leq\rho<1/2<\theta\leq1$ and $U_0\in H^s\cap L^{1,\gamma}$, where $s\geq0$ and $\gamma\in(0,1]$. Then, the following estimates hold: $$\begin{aligned} \|U(t,\cdot)\|_{\dot{H}^s}&\lesssim(1+t)^{-\frac{s+\gamma}{2-2\rho}-\frac{1}{2-2\rho}}\|U_0\|_{H^s\cap L^{1,\gamma}}+(1+t)^{-\frac{s}{2-2\rho}-\frac{1}{2-2\rho}}\left|\int_{{\mathbb}{R}^2}U_0(x)dx\right|.\end{aligned}$$ We remark that if we take initial data satisfying $\left|\int_{{\mathbb}{R}^2}U_0(x)dx\right|=0$ in Theorem \[Thm.EnergyWeighted.L1\], then we can observe that the decay rates given in Theorem \[Thm.EnergyEst\] when $m=1$ can be improved by $(1+t)^{-\frac{\gamma}{2-2\rho}}$ for $\gamma\in(0,1]$. To prove Theorem \[Thm.EnergyWeighted.L1\], we only need to modify the estimate for small frequencies. By using Lemma \[Lem.IkehataWeightedL1\], we have $$\begin{aligned} |W_0(\xi)|\lesssim |\xi|^{\gamma}\|U_0\|_{L^{1,\gamma}}+\left|\int_{{\mathbb}{R}^2}U_0(x)dx\right|.\end{aligned}$$ Then, we derive $$\begin{aligned} \left\|\chi_{\operatorname{int}}(\xi)|\xi|^se^{-c|\xi|^{2-2\rho}t}W_0(\xi)\right\|_{L^2}&\lesssim\left\|\chi_{\operatorname{int}}(\xi)|\xi|^{s+\gamma}e^{-c|\xi|^{2-2\rho}t}\right\|_{L^2}\|U_0\|_{L^{1,\gamma}}\\ &\quad+\left\|\chi_{\operatorname{int}}(\xi)|\xi|^{s}e^{-c|\xi|^{2-2\rho}t}\right\|_{L^2}\left|\int_{{\mathbb}{R}^2}U_0(x)dx\right|.\end{aligned}$$ Then, combining with the proof of Theorem \[Thm.EnergyEst\], we complete the proof. Diffusion phenomena {#Sec.AsymptoticProfiles} =================== Our main purpose in this section is to obtain diffusion phenomena for doubly dissipative elastic waves. According to Theorems \[Thm.EnergyEst\] and \[Thm.EnergyWeighted.L1\], we observe that the decay rate of energy estimates is determined by small frequencies (see Remark \[Remark.01\]). However, we may obtain an exponential decay estimates with suitable regularity on initial data for bounded frequencies and large frequencies. For this reason, we will interpret diffusion phenomena by the solutions localized in small frequency zone in this section. To do this, we first introduce the corresponding reference systems for the cases $\rho+\theta<1$, $\rho+\theta=1$ and $\rho+\theta>1$, respectively. Firstly, we introduce the matrices $$\begin{aligned} M_1:=\left( {\begin{array}{*{20}c} b^2 & 0 & 0 & 0 \\ 0 & a^2 & 0 & 0\\ 0 & 0 & -b^2 & 0 \\ 0 & 0 & 0 & -a^2\\ \end{array}} \right)\,\,\,\,\text{and}\,\,\,\,M_2:=\left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\\ \end{array}} \right).\end{aligned}$$ Motivated by the principle part of eigenvalues in Proposition \[Prop.INT\], we define the different reference systems between the following three cases. - In the case $\rho+\theta<1$, we define $\widetilde{U}=\widetilde{U}(t,x;\rho,\theta)$ is the solution to the following evolution system: $$\label{Ref.System01} \left\{ \begin{aligned} &\widetilde{U}_t+M_1(-\Delta)^{1-\rho}\widetilde{U}+M_2(-\Delta)^{\rho}\widetilde{U}+M_2(-\Delta)^{\theta}\widetilde{U}=0,&&x\in{\mathbb}{R}^2,\,\,t>0,\\ &\widetilde{U}(0,x)=T_1^{-1}U_0(x),&&x\in{\mathbb}{R}^2. \end{aligned} \right.$$ - In the case $\rho+\theta=1$, we define $\widetilde{U}=\widetilde{U}(t,x;\rho,\theta)$ is the solution to the following evolution system: $$\label{Ref.System02} \left\{ \begin{aligned} &\widetilde{U}_t+(M_1+M_2)(-\Delta)^{1-\rho}\widetilde{U}+M_2(-\Delta)^{\rho}\widetilde{U}=0,&&x\in{\mathbb}{R}^2,\,\,t>0,\\ &\widetilde{U}(0,x)=T_1^{-1}U_0(x),&&x\in{\mathbb}{R}^2. \end{aligned} \right.$$ - In the case $\rho+\theta>1$, we define $\widetilde{U}=\widetilde{U}(t,x;\rho,\theta)$ is the solution to the following evolution system: $$\label{Ref.System03} \left\{ \begin{aligned} &\widetilde{U}_t+M_1(-\Delta)^{1-\rho}\widetilde{U}+M_2(-\Delta)^{\rho}\widetilde{U}=0,&&x\in{\mathbb}{R}^2,\,\,t>0,\\ &\widetilde{U}(0,x)=T_1^{-1}U_0(x),&&x\in{\mathbb}{R}^2. \end{aligned} \right.$$ Here the matrix $T_1$ is defined in . Let us now give some explanation for these reference system. In the case when $\rho+\theta<1$, for the evolution system , we find that the reference system is made up of three different parabolic systems. We may interpret this new effect as *triple diffusion phenomena*. This effect is shown firstly in [@Chen2019TEP] for thermoelastic plate equations with structural damping. In this case, the damping term $(-\Delta)^{\theta}u_t$ with $\theta\in(1/2,1]$ in really has influence on the diffusion structure. But this effect does not appear in the other case $\rho+\theta\geq1$. However, we find that when $\rho+\theta\geq1$, the reference system is changed into and . Obviously, these reference systems are only made up of two different parabolic systems, whose structures are similar as reference system for elastic waves with damping term $(-\Delta)^{\rho}u$ for $\rho\in[0,1/2)$. We may interpret this effect as *double diffusion phenomena* (one may see the pioneering paper [@DabbiccoEbert2014]). From the above discussions, we observe a new threshold of diffusion structure for doubly dissipative elastic waves, that is $\rho+\theta=1$. In other words, the structure of the reference system will be changed with the parameters changing from $\rho+\theta<1$ to $\rho+\theta\geq1$. Let us begin to state our main theorems on diffusion phenomena. \[Thm.DiffusionPhenomena\] Let us consider the Cauchy problem with $0\leq \rho<1/2<\theta\leq1$ and $U_0\in L^m$ with $m\in[1,2]$. Then, the following refinement estimates hold: $$\begin{aligned} &\left\|\chi_{\operatorname{int}}(D)\left(U(t,\cdot)-T_1\widetilde{U}(t,\cdot;\rho,\theta)\right)\right\|_{\dot{H}^s}\lesssim(1+t)^{-\frac{s}{2-2\rho}-\frac{2-m}{m(2-2\rho)}-q(\rho,\theta)}\|U_0\|_{L^m},\end{aligned}$$ where the function $q=q(\rho,\theta)$ is defined by $$\label{Fun.q(rho,theta)} q(\rho,\theta):= \left\{ \begin{aligned} &\frac{2\theta-1}{2-2\rho},&&\text{if }\rho+\theta<1,\\ &\frac{1-2\rho}{2-2\rho},&&\text{if }\rho+\theta\geq1,\\ \end{aligned} \right.$$ the matrix $T_1$ is defined in . Here we only prove the case when $\rho+\theta<1$. For the other case when $\rho+\theta\geq1$, its proof is similar as the following discussion. Thus, we omit it. First of all, let us apply the partial Fourier transform with respect to spatial variable such that $\widetilde{W}(t,\xi;\rho,\theta)={\mathcal}{F}_{x\rightarrow\xi}(\widetilde{U}(t,x;\rho,\theta))$ to get $$\begin{aligned} \widetilde{W}(t,\xi;\rho,\theta)=\operatorname{diag}\left(e^{-\tilde{\lambda}_j(|\xi|)t}\right)_{j=1}^4T_1^{-1}W_0(\xi), \end{aligned}$$ where $$\begin{aligned} &\tilde{\lambda}_{1}(|\xi|)=b^2|\xi|^{2-2\rho},&&\tilde{\lambda}_{2}(|\xi|)=a^2|\xi|^{2-2\rho},\\ &\tilde{\lambda}_{3}(|\xi|)=|\xi|^{2\rho}+|\xi|^{2\theta}-b^2|\xi|^{2-2\rho},&& \tilde{\lambda}_{4}(|\xi|)=|\xi|^{2\rho}+|\xi|^{2\theta}-a^2|\xi|^{2-2\rho}. \end{aligned}$$ We remark that $\tilde{\lambda}_j(|\xi|)$ are the principle parts of eigenvalues $\lambda_j(|\xi|)$ for $j=1,\dots,4$ (one may recall the statement of Proposition \[Prop.INT\]).\ According to the representation of solutions for $\xi\in{\mathcal}{Z}_{\operatorname{int}}(\varepsilon)$ in Proposition \[Prop.INT\], we may obtain $$\begin{aligned} \chi_{\operatorname{int}}(\xi)|\xi|^s\left(W(t,\xi)-T_1\widetilde{W}(t,\xi;\rho,\theta)\right)=\chi_{\operatorname{int}}(\xi)|\xi|^s\left(J_1(t,|\xi|)+J_2(t,|\xi|)+J_3(t,|\xi|)\right)W_0(\xi),\end{aligned}$$ where $$\begin{aligned} J_1(t,|\xi|)&=T_1\operatorname{diag}\left(e^{-\lambda_j(|\xi|)t}-e^{-\tilde{\lambda}_j(|\xi|)t}\right)_{j=1}^4T_1^{-1},\\ J_2(t,|\xi|)&=T_1N_2(|\xi|)\operatorname{diag}\left(e^{-\lambda_j(|\xi|)t}\right)_{j=1}^4(I_{4\times4}+N_2(|\xi|))^{-1}T_1^{-1},\\ J_3(t,|\xi|)&=-T_1(I_{4\times4}+N_2(|\xi|))\operatorname{diag}\left(e^{-\lambda_j(|\xi|)t}\right)_{j=1}^4N_2(|\xi|)(I_{4\times 4}+N_2(|\xi|))^{-1}T_1^{-1}.\end{aligned}$$ Here the matrix $N_2(|\xi|)={\mathcal}{O}\left(|\xi|^{1-2\rho}\right)$ is defined in . In the above equation we used $$\begin{aligned} (I_{t\times4}+N_2(|\xi|))^{-1}=I_{4\times4}-N_2(|\xi|)(I_{4\times4}+N_2(|\xi|))^{-1}.\end{aligned}$$ We now begin to estimate $J_1(t,|\xi|)$ and $J_2(t,|\xi|)+J_3(t,|\xi|)$, respectively. By means value theorem, we know that $$\begin{aligned} \chi_{\operatorname{int}}(\xi)\left|e^{-\lambda_j(|\xi|)t}-e^{-\tilde{\lambda}_j(|\xi|)t}\right|\lesssim (1+t)\chi_{\operatorname{int}}(\xi)|\xi|^{1+2\theta-2\rho}e^{-\tilde{\lambda}_j(|\xi|)t}.\end{aligned}$$ Thus, $$\begin{aligned} \left\|\chi_{\operatorname{int}}(\xi)|\xi|^{s}J_1(t,\xi)W_0(\xi)\right\|_{L^2}&\lesssim(1+t)\left\|\chi_{\operatorname{int}}(\xi)|\xi|^{s+1+2\theta-2\rho}\operatorname{diag}\left(e^{-\tilde{\lambda}_j(|\xi|)t}\right)_{j=1}^4W_0(\xi)\right\|_{L^2}\\ &\lesssim(1+t)^{-\frac{s}{2-2\rho}-\frac{2-m}{m(2-2\rho)}-\frac{2\theta-1}{2-2\rho}}\|U_0\|_{L^m}.\end{aligned}$$ Due to the fact that $N_2(|\xi|)={\mathcal}{O}\left(|\xi|^{1-2\rho}\right)$, we have $$\begin{aligned} \left\|\chi_{\operatorname{int}}(\xi)|\xi|^{s}(J_2(t,\xi)+J_3(t,|\xi|))W_0(\xi)\right\|_{L^2}&\lesssim\left\|\chi_{\operatorname{int}}(\xi)|\xi|^{s+1-2\rho}\operatorname{diag}\left(e^{-\lambda_j(|\xi|)t}\right)_{j=1}^4W_0(\xi)\right\|_{L^2}\\ &\lesssim(1+t)^{-\frac{s}{2-2\rho}-\frac{2-m}{m(2-2\rho)}-\frac{1-2\rho}{2-2\rho}}\|U_0\|_{L^m}\end{aligned}$$ Summarizing the above estimates leads to $$\begin{aligned} \left\|\chi_{\operatorname{int}}(\xi)|\xi|^s\left(W(t,\xi)-T_1\widetilde{W}(t,\xi;\rho,\theta)\right)\right\|_{L^2}\lesssim(1+t)^{-\frac{s}{2-2\rho}-\frac{2-m}{m(2-2\rho)}-\frac{2\theta-1}{2-2\rho}}\|U_0\|_{L^m},\end{aligned}$$ where we used our condition $\rho+\theta<1$.\ Finally, applying the Parseval-Plancherel theorem, we complete the proof of the theorem. \[Thm.DiffusionPhenWeightL1\] Let us consider the Cauchy problem with $0\leq \rho<1/2<\theta\leq1$ and $U_0\in L^{1,\gamma}$ with $\gamma\in(0,1]$. Then, the following refinement estimates hold: $$\begin{aligned} &\left\|\chi_{\operatorname{int}}(D)\left(U(t,\cdot)-T_1\widetilde{U}(t,\cdot;\rho,\theta)\right)\right\|_{\dot{H}^s}\lesssim(1+t)^{-\frac{s+\gamma}{2-2\rho}-\frac{1}{2-2\rho}-q(\rho,\theta)}\|U_0\|_{L^{1,\gamma}},\end{aligned}$$ where the function $q=q(\rho,\theta)$ is defined in and the matrix $T_1$ is defined in . We may immediately compete the proof of this result by following the procedure from the proofs of Theorems \[Thm.EnergyWeighted.L1\] and \[Thm.DiffusionPhenomena\]. According to Theorems \[Thm.EnergyEst\], \[Thm.EnergyWeighted.L1\], \[Thm.DiffusionPhenomena\] and \[Thm.DiffusionPhenWeightL1\], the decay rate $(1+t)^{-q(\rho,\theta)}$ can be gained by subtracting the solutions $\widetilde{U}(t,x;\rho,\theta)$ for the reference systems , and . From the value of $q(\rho,\theta)$, we also find that the threshold for diffusion structure is $\rho+\theta=1$. Concluding remarks {#Sec.ConcludingRemarks} ================== Let us discuss about smoothing effect of solutions. We first introduce the Gevrey space $\Gamma^{\kappa}$ with $\kappa\in[1,\infty)$ (see [@Rodino1993]), where $$\begin{aligned} \Gamma^{\kappa}:=\left\{f\in L^2\,:\,\text{there exists a constant }c\text{ such that }\exp\left(c\langle\xi\rangle^{\frac{1}{\kappa}}\right){\mathcal}{F}(f)\in L^2\right\}.\end{aligned}$$ By using Proposition \[Prop.EXT\] with the same approach of [@Reissig2016], we immediately obtain the following results. Let us consider the Cauchy problem with $0\leq\rho<1/2<\theta<1$ and $U_0\in L^2$. Then, the solutions satisfy $|D|^sU(t,\cdot)\in\Gamma^{\frac{1}{2-2\theta}}$ with $s\geq0$. However, when $0\leq\rho<1/2<\theta=1$, the solutions do not belong to any Gevrey space. It is well-known that smoothing effect is mainly determined by asymptotic behavior of eigenvalues localized in large frequency zone (see Proposition \[Prop.EXT\]). For this reason, we may observe smoothing effect is only influenced by the damping term $(-\Delta)^{\theta}u_t$ with $\theta\in(1/2,1]$ in the Cauchy problem . From [@Reissig2016; @ChenReissig2019SD], we know the solution to elastic waves with friction $u_t$ does not have smoothing effect. However, in doubly dissipative elastic waves , the structural damping $(-\Delta)^{\theta}u_t$ with $\theta\in(1/2,1)$ brings Gevrey smoothing for the solutions even when $\rho=0$. In the present paper we focus on energy estimates with initial data taking from $H^s\cap L^m$ for $s\geq0$, $m\in[1,2]$ or from $H^{s}\cap L^{1,\gamma}$ for $\gamma\in(0,1]$. Here we restrict ourselves on estimating solutions in the $L^2$ norm. For estimating the solutions in the $L^q$ norm with $2\leq q\leq\infty$, by applying Lemma 4.2 in [@Chen2019TEP], one may obtain $L^p-L^q$ estimates with $1\leq p\leq 2\leq q\leq\infty$ and diffusion phenomena in a $L^p-L^q$ framework. 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--- abstract: 'We use direct numerical simulations to investigate the interaction between the temperature field of a fluid and the temperature of small particles suspended in the flow, employing both one and two-way thermal coupling, in a statistically stationary, isotropic turbulent flow. Using statistical analysis, we investigate this variegated interaction at the different scales of the flow. We find that the variance of the fluid temperature gradients decreases as the thermal response time of the suspended particles is increased. The probability density function (PDF) of the fluid temperature gradients scales with its variance, while the PDF of the rate of change of the particle temperature, whose variance is associated with the thermal dissipation due to the particles, does not scale in such a self-similar way. The modification of the fluid temperature field due to the particles is examined by computing the particle concentration and particle heat fluxes conditioned on the magnitude of the local fluid temperature gradient. These statistics highlight that the particles cluster on the fluid temperature fronts, and the important role played by the alignments of the particle velocity and the local fluid temperature gradient. The temperature structure functions, which characterize the temperature fluctuations across the scales of the flow, clearly show that the fluctuations of the fluid temperature increments are monotonically suppressed in the two-way coupled regime as the particle thermal response time is increased. Thermal caustics dominate the particle temperature increments at small scales, that is, particles that come into contact are likely to have very large differences in their temperature. This is caused by the nonlocal thermal dynamics of the particles, and the scaling exponents of the inertial particle temperature structure functions in the dissipation range reveal very strong multifractal behavior. Further insight is provided by the PDFs of the two-point temperature increments and by the flux of temperature increments across the scales. All together, these results reveal a number of non-trivial effects, with a number of important practical consequences.' author: - 'M. Carbone, A. D. Bragg' - 'M. Iovieno' bibliography: - 'JFM2018.bib' title: 'Multiscale fluid–particle thermal interaction in isotropic turbulence' --- Introduction ============ The interaction between inertial particles and scalar fields in turbulent flows plays a central role in many natural problems, ranging from cloud microphysics [@Pruppacher2010; @Grabowski2013] to the interactions between plankton and nutrients [@DeLillo2014], and dust particle flows in accretion disks [@Takeuchi2002]. In engineered systems, applications involve chemical reactors and combustion chambers, and more recently, microdispersed colloidal fluids where the enhanced thermal conductivity due to particle aggregations can give rise to non-trivial thermal behavior [@Prasher2006; @Momenifar2015], and which can be used in cooling devices for electronic equipment exposed to large heat fluxes [@Das2006]. In this work, we focus on the heat exchange between advected inertial particles and the fluid phase in a turbulent flow, with a parametric emphasis relevant to understanding particle-scalar interactions in cloud microphysics. Understanding the droplet growth in clouds requires to characterize the interaction between water droplets and the humidity and temperature fields. A major problem is to understand how the interaction between turbulence, heat exchange, condensational processes, and collisions can produce the rapid growth of water droplets that leads to rain initiation [@Pruppacher2010; @Grabowski2013]. While the study of the transport of scalar fields and particles in turbulent flows are well established research areas in both theoretical and applied fluid dynamics [@Kraichnan1994; @Taylor1922], the characterization of the interaction between scalars and particles in turbulent flows is a relatively new topic [@Bec2014], since the problem is hard to handle analytically, requires sophisticated experimental techniques, and is computationally demanding. When temperature differences inside the fluid are sufficiently small, the temperature field behaves almost like a passive scalar, that is, the fluid temperature is advected and diffused by the fluid motion but has negligible dynamical effect on the flow. Even in this regime, the statistical properties of the passive scalar field are significantly different from those of the underlying velocity field that advects it. Different regimes take place according to the Reynolds number and the ratio between momentum and scalar diffusivities [@Shraiman2000; @Warhaft2000; @Watanabe2004]. Experiments, numerical simulations and analytical models show that a passive scalar field is always more intermittent than the velocity field, and passive scalars in turbulence are characterized by strong anomalous scaling [@Holzer1994]. This is due to the formation of ramp–cliff structures in the scalar field [@Celani2000; @Watanabe2004]: large regions in which the scalar field is almost constant are separated by thin regions in which the scalar abruptly changes. The regions in which the scalar mildly changes are referred to as Lagrangian coherent structures. The thin regions with large scalar gradient, where the diffusion of the scalar takes place, are referred to as fronts. It has been shown that the large scale forcing influences the passive scalar statistics at small scales [@Gotoh2015]. In particular, a mean scalar gradient forcing preserves universality of the statistics while a large scale Gaussian forcing does not. However, the ramp-cliff structure was observed with different types of forcing, implying that this structure is universal to scalar fields in turbulence [@Watanabe2004; @Bec2014]. Moreover, recent measurements of atmospheric turbulence have shown that external boundary conditions, such as the magnitude and sign of the sensible heat flux, have a significant impact on the fluid temperature dynamics within the inertial range, while for the same scales the fluid velocity increments are essentially independent of these large-scale conditions [@zorzetto18]. When a turbulent flow is seeded with inertial particles, the particles can sample the surrounding flow in a non-uniform and correlated manner [@Toschi2009]. Particle inertia in a turbulent flow is measured through the Stokes number ${\text{\textit{St}}}\equiv\tau_p/\tau_\eta$, which compares the particle response time to the Kolmogorov time scale. A striking feature of inertial particle motion in turbulent flows is that they spontaneously cluster even in incompressible flows [@maxey87; @wang93; @Bec2007; @Ireland2016]. This clustering can take place across a wide range of scales [@Bec2007; @Bragg2015b; @Ireland2016], and the small-scale clustering is maximum when ${\text{\textit{St}}}={\textit{O}\left( 1 \right)}$. A variety of mechanisms has been proposed to explain this phenomena: when ${\text{\textit{St}}}\ll1$ the clustering is caused by particles being centrifuged out of regions of strong rotation [@maxey87; @chun05], while for ${\text{\textit{St}}}\geq {\textit{O}\left( 1 \right)}$, a non-local mechanism generates the clustering, whose effect is related to the particles memory of its interaction with the flow along its path-history [@gustavsson11b; @gustavsson16; @bragg14b; @bragg2015a; @Bragg2015b]. Note that recent results on the clustering of settling inertial particles in turbulence have corroborated this picture, showing that strong clustering can occur even in a parameter regime where the centrifuge effect cannot be invoked as the explanation for the clustering, but is caused by a non-local mechanism [@ireland16b]. When particles have finite thermal inertia, they will not be in thermal equilibrium with the fluid temperature field, and this can give rise to non-trivial thermal coupling between the fluid and particles in a turbulent flow. A thermal response time $\tau_\theta$ can be defined so that the particle thermal inertia is parameterized by the thermal Stokes number ${\text{\textit{St}}}_\theta \equiv \tau_\theta/\tau_\eta$ [@Zaichik2009]. Since both the fluid temperature and particle phase-space dynamics depend upon the fluid velocity field, there can exist non-trivial correlations between the fluid and particle temperatures even in the absence of thermal coupling. Indeed, it was show by [@Bec2014] that inertial particles preferentially cluster on the fronts of the scalar field. Associated with this is that the particles preferentially sample the fluid temperature field, and when combined with the strong intermittency of temperature fields in turbulent flows, that can cause particles to experience very large temperature fluctuations along their trajectories. Several works have considered aspects of the fluid-particle temperature coupling using numerical simulations. For example, [@Zonta2008] investigated a particle-laden channel flow, with a view to modeling the modification of heat transfer in micro–dispersed fluids. They considered both momentum and temperature two–way coupling and observed that, depending on the particle inertia, the heat flow at the wall can increase or decrease. [@Kuerten2011] considered a similar set-up with larger dispersed particles, and they observed a stronger modification of the fluid temperature statistics due to the particles. [@Zamansky2014; @Zamansky2016] considered turbulence induced by buoyancy, where the buoyancy was generated by heated particles. They observed that the resulting flow is driven by thermal plumes produced by the particles. As the particle inertia was increased, the inhomogeneity and the effect of the coupling were enhanced in agreement with the fact that inertial particles tend to cluster on the scalar fronts. [@Kumar2014] examined how the spatial distribution of droplets is affected by large scale inhomogeneities in the fluid temperature and supersaturation fields, considering the transition between homogeneous and inhomogeneous mixing. A similar flow configuration was also investigated by [@Gotzfried2017]. Each of these studies was primarily focused on the effect of the inertial particles on the large-scale statistics of the fluid temperature field. However, the results of [@Bec2014] imply that the effects of fluid-particle thermal coupling could be strong at the small scales, owing to the fact that they cluster on the fronts of the temperature field. Moreover, there is a need to understand and characterize the multiscale thermal properties of the particles themselves. In order to address these issues, we have conducted direct numerical simulations (DNS) to investigate the interaction between the scalar temperature field and the temperature of inertial particles suspended in the fluid, with one and two-way thermal coupling, in statistically stationary, isotropic turbulence. Using statistical analysis, we probe the multiscale aspects of the problem and consider the particular ways that the inertial particles contribute to the properties of the fluid temperature field in the two-way coupled regime. The paper is organized as follows. In section \[sec:simul\] we present the physical model used in the DNS, and present the parameters in the system. In section \[sec:diss\] the statistics of the fluid temperature and time derivative of the particle temperature are considered, which allow us to quantify the contributions to the thermal dissipation in the system from the fluid and particles. In section \[sec:fluct\] we consider the statistics of the fluid and particle temperature. In section \[sec:FATT\] we consider the heat flux due to the particle motion conditioned on the local fluid temperature gradients in order to obtain insight into the details of the thermal coupling. In section \[sec:SF\] we consider the structure functions of the fluid and particle temperature increments, along with their scaling exponents. In section \[sec:2pPDF\] we consider the probability density functions (PDFs) of the fluid and particle temperature increments, along with the PDFs of the fluxes of the fluid and particle temperature increments across the scales of the flow. Finally, concluding remarks are given in section \[sec:concl\]. The physical model {#sec:simul} ================== In this section we present the governing equations of the physical model which will be solved numerically to simulate the thermal coupling and behavior of a particle-laden turbulent flow. Fluid phase ----------- We consider a statistically stationary, homogeneous and isotropic turbulent flow, governed by the incompressible Navier-Stokes equations. The turbulent velocity field advects the fluid temperature field (assumed a passive scalar), together with the inertial particles. In this study, we account for two-way thermal coupling between the fluid and particles, but only one-way momentum coupling. Therefore, the governing equations for the fluid phase are $$\begin{aligned} \bnabla\bcdot{ \mathbf{u} } &=& 0 \label{NScontinuity},\\ \partial_t { \mathbf{u} } + { \mathbf{u} }\bcdot\bnabla{ \mathbf{u} } &=& -\frac{1}{\rho_0}\bnabla p + \nu\nabla^2 { \mathbf{u} } + { \mathbf{f} } \label{NSmomentum},\\ \partial_t T + { \mathbf{u} }\bcdot\bnabla T &=& \kappa \nabla^2 T - C_T + f_T \label{NSscalar}.\end{aligned}$$ Here ${ \mathbf{u} }\left({ \mathbf{x} },t\right)$ is the velocity of the fluid, $p\left({ \mathbf{x} },t\right)$ is the pressure, $\rho_0$ is the density of the fluid, $\nu$ is its kinematic viscosity, $T\left({ \mathbf{x} },t\right)$ is the temperature of the fluid and $\kappa$ is the thermal diffusivity. The ratio between the the momentum diffusivity $\nu$ and the thermal diffusivity $\kappa$ defines the Prandtl number $\Pran\equiv\nu/\kappa$. In this work, we consider $\Pran=1$, leaving further exploration of its effect on the system to future work. The ${ \mathbf{f} }$ and $f_T$ terms in equations and represent the external forcing, and $C_T$ is the thermal feedback of the particles on the fluid temperature field, that is, the heat exchanged per unit time and unit volume between the fluid and particles at position ${ \mathbf{x} }$. When the forcing is confined to sufficiently large scales, it is assumed that the details of the forcing do not influence the small-scale dynamics. Previous experimental evidence seems to confirm this [@Sreenivasan1996], leading to a universal behaviour of the small-scales. However, recent studies [@Gotoh2015] pointed out that this hypothesis of universality is partially violated by the advected scalar fields, whose inertial range statistics exhibit sensitivity to the details of the imposed forcing. Since we aim to characterize temperature and temperature gradient fluctuations in the dissipation range for different inertia of the suspended particles, we employ a forcing that imposes the same total dissipation rate for all the simulations. This produces results which can be meaningfully compared for different parameters of the suspended particles, since the response of the system to the same injected thermal power can be examined. Therefore, we employ a large scale forcing which imposes the average dissipation rate [@Kumar2014], that is $$\hat{{ \mathbf{f} }}({ \mathbf{k} }) = \varepsilon \frac{\hat{{ \mathbf{u} }}({ \mathbf{k} })} {\sum_{{ \mathbf{k} }\in\mathcal{K}_f} \left\Vert \hat{{ \mathbf{u} }}({ \mathbf{k} }) \right\Vert^2 }, \quad \hat{f}_T({ \mathbf{k} }) = \chi\frac{\hat{T}({ \mathbf{k} })}{\sum_{{ \mathbf{k} }\in\mathcal{K}_f} \left\vert \hat{T}({ \mathbf{k} }) \right\vert^2 },$$ in the wavenumber space (a hat indicates the Fourier transform and ${ \mathbf{k} }$ is the wavenumber). Here $\mathcal{K}_f$ is the set of forced wavenumbers while $\varepsilon$ and $\chi$ are the imposed dissipation rates of velocity and temperature variance, respectively. Since both the velocity and temperature statistics at large scales tend to be close to Gaussian, this forcing behaves similarly to a random Gaussian forcing. The value of the parameters relative to the fluid flow, employed in the simulations are in table \[tab:flow\]. ------------------------------------------- ---------------- ------------- Kinematic viscosity $\nu$ 0.005 Prandtl number $\Pran$ 1 Velocity fluctuations dissipation rate $\varepsilon$ 0.27 Temperature fluctuations dissipation rate $\chi$ $0.1$ Kolmogorov time scale $\tau_\eta$ 0.136 Kolmogorov length scale $\eta$ 0.0261 Taylor micro-scale $\lambda$ 0.498 Integral length scale $\ell$ 1.4 Root mean square velocity $u'$ 0.88 Kolmogorov velocity scale $u_\eta$ 0.192 Small scale temperature $T_\eta$ $0.117$ Taylor Reynolds number $\Rey_\lambda$ 88 Integral scale Reynolds number $\Rey_l$ 244 Forced wavenumber $k_f$ $\sqrt{2}$ Number of Fourier modes $N$ $128$ (3/2) Resolution $N\eta/2$ 1.67 ------------------------------------------- ---------------- ------------- : Flow parameters in dimensionless code units. The characteristic parameters of the fluid flow are defined from its energy spectrum $E{\left( k \right)}\equiv\int_{{\left\Vert { \mathbf{k} } \right\Vert}=k}\left\Vert { \mathbf{\hat{u}} }{\left( { \mathbf{k} } \right)} \right\Vert^2 {\text{\textrm{d}}}{ \mathbf{k} }/2$. The dissipation rate of turbulent kinetic energy is: $\varepsilon \equiv2\nu\int k^2 E{\left( k \right)} {\text{\textrm{d}}}k$. The Kolmogorov length $\eta\equiv\left(\nu^3/\varepsilon\right)^{1/4}$, time scale $\tau_\eta\equiv\left(\nu/\varepsilon\right)^{1/2}$ and velocity scale $u_\eta \equiv \eta/\tau_\eta$. The Taylor micro-scale is: $\lambda\equiv u'/\sqrt{\langle\left|\bnabla { \mathbf{u} }\right|^2\rangle}$. The root mean square velocity is $u'\equiv\sqrt{(2/3)\int E{\left( k \right)} {\text{\textrm{d}}}k}$ and the integral length scale $\ell \equiv \left.\upi\middle/\left(2u'^2\right)\right. \int E{\left( k \right)}/k {\text{\textrm{d}}}k$. Similarly, the spectrum, root mean square value and dissipation rate of the scalar field are: $E_T{\left( k \right)}\equiv\int_{{\left\Vert { \mathbf{k} } \right\Vert}=k}\left\vert \hat{T}{\left( { \mathbf{k} } \right)} \right\vert^2 {\text{\textrm{d}}}{ \mathbf{k} }/2$, $T' \equiv\sqrt{(1/2)\int E_{T}{\left( k \right)} {\text{\textrm{d}}}k}$, $\chi \equiv 2\kappa \int k^2 E_T{\left( k \right)} {\text{\textrm{d}}}k$. The small scale temperature is determined by the viscosity and dissipation rate: $T_\eta\equiv\sqrt{\chi\tau_\eta}$. Since the Prandtl number is unitary the small scales of the scalar and the velocity field are of the same order.[]{data-label="tab:flow"} Particle phase -------------- We consider rigid, point-like particles which are heavy with respect to the fluid, and small with respect to any scale of the flow. In particular, the particle density $\rho_p$ is much larger than the fluid density $\rho_p\gg\rho_0$, and the particle radius $r_p$ is much smaller than the Kolmogorov length scale $r_p \ll \eta$. With these assumptions (and neglecting gravity) the particle acceleration is described by the Stokes drag law. Analogously, the rate of change of the particle temperature is described by Newton’s law for the heat conduction $$\begin{aligned} {\frac{\textrm{d} { \mathbf{x} }_p}{\textrm{d} t}} &\equiv& { \mathbf{v} }_p, \label{eq:partx}\\ {\frac{\textrm{d} { \mathbf{v} }_p}{\textrm{d} t}} &=& \frac{ { \mathbf{u} } \left({ \mathbf{x} }_{p},t \right) - { \mathbf{v} }_p }{\tau_p}, \label{eq:partv}\\ {\frac{\textrm{d} \theta_p}{\textrm{d} t}} &=& \frac{ T \left( { \mathbf{x} }_{p} , t \right) - \theta_{p} }{\tau_\theta}. \label{eq:parttheta}\end{aligned}$$ Here $\tau_p \equiv\left.2\rho_pr_p^2\middle/\left(9\rho_0\nu\right)\right.$ is the particle momentum response time, $\tau_\theta \equiv \left. \rho_p c_p r_p^2\middle/\left(3\rho_0 c_0\kappa\right)\right.$ is the particle thermal response time, $c_p$ is the particle heat capacity, and $c_0$ is the fluid heat capacity at constant pressure. The Stokes number is defined as ${\text{\textit{St}}}\equiv \left.\tau_p\middle/\tau_\eta\right.$ ,and the thermal Stokes number is defined as ${\text{\textit{St}}}_\theta \equiv \left.\tau_\theta\middle/\tau_\eta\right.$, where $\tau_\eta$ is the Kolmogorov time scale. Our simulations focus on a dilute suspension regime with particle volume fraction $\phi=4\times 10^{-4}$. While this volume fraction is large enough for two-way momentum coupling between the particles and fluid to be important [e.g. @elghobashi91], we ignore this in the present study. The motivation is that including both two-way momentum and two-way thermal coupling introduces too many competing effects that would compound a thorough understanding of the problem. In this study we therefore ignore momentum coupling, but account for two-way thermal coupling, and in a follow up study we will include the effects of two-way momentum coupling. We consider nine values of ${\text{\textit{St}}}_\theta$ and three values of ${\text{\textit{St}}}$ in order to explore the behavior of the system over a range of parameter values. Since we are accounting for thermal coupling, each combination of ${\text{\textit{St}}}_\theta$ and ${\text{\textit{St}}}$ must be simulated separately, and when combined with the large number of particles in the flow domain, the set of simulations require considerable computational resources. Therefore, in the present study we restrict attention to $\Rey_\lambda =88$, but future explorations should consider larger $\Rey_\lambda$ in order to explore the behavior when there exists a well-defined inertial range in the flow. In order to obtain deeper insight into the role of the two-way thermal coupling, we perform simulations with (denoted by S1) and without (denoted by S2) the thermal coupling. The particle parameters employed in the simulations are in table \[tab:particles\]. Thermal coupling ---------------- In the two-way thermal coupling regime, the thermal energy contained in the fluid is finite with respect to the thermal energy of the particles, therefore, when heat flows from the fluid to the particle the fluid loses thermal energy at the particle position. Due to the point-mass approximation, the feedback from the particles on the fluid temperature field is a superposition of Dirac delta functions, centered on the particles. Hence the coupling term in equation is given by $$C_T \left({ \mathbf{x} },t\right) = \frac{4}{3}\upi\frac{\rho_p}{\rho_0}\frac{c_p}{c_0} r_p^3 \sum_{p=1}^{N_P}{\frac{\textrm{d} \theta_p}{\textrm{d} t}}\delta\left({ \mathbf{x} } - { \mathbf{x} }_p\right). \label{eq:CT}$$ --------------------------------- ------------------------------- ---------------------------------------------------- Particle phase volume fraction $\phi$ $0.0004$ Particle to fluid density ratio $\rho_p/\rho_0$ $1000$ Particle back reaction $C_T$ S1: included; S2: neglected. Stokes number ${\text{\textit{St}}}$ $0.5$; $1$; $3$. Thermal Stokes number ${\text{\textit{St}}}_\theta$ $0.2$; $0.5$; $1$; $1.5$; $2$; $3$; $4$; $5$; $6$. Number of particles $N_P$ $12500992$; $4419584$; $847872$. --------------------------------- ------------------------------- ---------------------------------------------------- : Particles parameters in dimensionless code units. The Stokes number is ${\text{\textit{St}}}\equiv\tau_p/\tau_\eta$ and thermal Stokes number ${\text{\textit{St}}}_\theta\equiv\tau_\theta/\tau_\eta$ and the particle response times are defined in the text. In the simulations, ${\text{\textit{St}}}_\theta$ is varied by varying the particle heat capacity. The different combinations of ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$ are simulated including the two-way thermal thermal coupling (simulations S1) and neglecting it (simulations S2).[]{data-label="tab:particles"} ![(*a*) Three-dimensional energy spectrum of the fluid velocity field (open squares) and temperature field (open circles). The temperature field is computed without any feedback from the particles on the fluid flow (simulations S2). (*b*) Second order longitudinal structure functions of the particle velocity for various Stokes numbers.[]{data-label="fig:flowstat"}]({figure1-crop}.pdf){width="\textwidth"} Numerical method ---------------- We perform direct numerical simulation of incompressible, statistically steady and isotropic turbulence on a tri-periodic cubic domain. Equations , , and are solved by means of the pseudo-spectral Fourier method for the spatial discretization. The $3/2$ rule is employed for dealiasing [@Canuto1988], so that the maximum resolved wavenumber is $k_{\textrm{max}}=N/2$. The required Fourier transforms are executed in parallel using the P3DFFT library [@Pekurovsky2012]. Forcing is applied to a single scale, that is to all wavevectors satisfying $\left\Vert { \mathbf{k} } \right\Vert^2 = k_f$, with $k_f=2$, and the equations for the fluid velocity and temperature Fourier coefficients are evolved in time by means of a second order Runge-Kutta exponential integrator [@Hochbruck2010]. This method has been preferred to the standard integrating factor because of its higher accuracy and, above all, because of its consistency. Indeed, in order to obtain an accurate representation of small scale temperature fluctuations, it is critical that the numerical solution conserves thermal energy. The same time integration scheme is used to solve particle equations , and , thus providing overall consistency, since the system formed by fluid and particles is evolved in time as a whole. The fluid velocity and temperature are interpolated at the particle position by means of fourth order B-spline interpolation. The interpolation is implemented as a backward Non Uniform Fourier Transform with B-spline basis: the fluid field is projected onto the B-spline basis in Fourier space through a deconvolution, than transformed into the physical space by means of a inverse Fast Fourier Transform (FFT). A convolution provides the interpolated field at particle position [@Beylkin1995]. Since B-splines have a compact support in physical space and deconvolution in Fourier space reduces to a division, this provide an efficient way to obtain high order interpolation. This guarantees smooth and accurate interpolation and its efficient implementation is suitable for pseudo-spectral methods [@vanHinsberg2012]. Moreover, the same method is used to obtain the spectral representation of the coupling term . The coupling term has to be projected on the Cartesian grid used to represent the fields. This is performed by means of the forward Nonuniform Fast Fourier Transform (NUFFT) with B-spline basis [@Beylkin1995]. Briefly, the algorithm works as follows. The convolution of the distribution $C_T{\left( { \mathbf{x} },t \right)}$ with the B-spline polynomial basis $B{\left( { \mathbf{x} } \right)}$ is computed in physical space, so that it can be effectively represented on the Cartesian grid $$\widetilde{C}\left({ \mathbf{x} },t \right) = \int C_T{\left( { \mathbf{x} },t \right)} B{\left( { \mathbf{x} }-{ \mathbf{y} } \right)}{\text{\textrm{d}}}{ \mathbf{y} }.$$ Then, the regularized field $\widetilde{C}$ is transformed to Fourier space and the convolution with the B-spline basis is efficiently removed: $$\widehat{{C}}{\left( { \mathbf{k} },t \right)} = \frac{\widehat{\widetilde{C}}{\left( { \mathbf{k} },t \right)}}{\widehat{B}{\left( { \mathbf{k} } \right)}}.$$ This algorithm allows an efficient and accurate spectral representation of the particle back-reaction [@Carbone2018]. Indeed, the NUFFT satisfies the constraints for interpolation schemes [@Sundaram1996]: the backward and forward transformations are symmetric and the non locality, introduced in physical space due to the convolution, is removed in Fourier space. For these reasons this technique is preferred to shape regularization functions, [@Maxey1997]. Characterization of the thermal dissipation rate {#sec:diss} ================================================ In the flow under consideration, the total dissipation rate of the temperature field $\chi$ is constant due to the forcing term $f_T$. The total dissipation has a contribution from the fluid and particle phases and is given by [@Sundaram1996] $$\chi = \kappa {\left\langle \left\Vert \bnabla T \right\Vert^2 \right\rangle} + \frac{\phi}{\tau_\theta}\frac{\rho_p c_p}{\rho_0 c_0} {\left\langle \left( T {\left( { \mathbf{x} }_p,t \right)} - \theta_p \right)^2 \right\rangle}.\label{eq:dissbal0}$$ We indicate with $\chi_f$ the dissipation due to the fluid temperature gradient and with $\chi_p$ the dissipation due to the particles, the two terms in the right hand side of equation , so that $\chi=\chi_f+\chi_p$. Note that both contributions to the dissipation rate are proportional to the kinematic thermal conductivity of the fluid since $\tau_\theta\propto 1/\kappa$, and hence both the dissipation mechanisms are due to molecular diffusivity. A characteristic length of the dissipation due to the particles can be defined as $$\eta_p \equiv \frac{r_p}{\sqrt{3 \phi }}\label{eq:disslen}$$ and using this, the balance of the dissipation of the temperature fluctuations can be written as $$\chi = \kappa \left[ {\left\langle \left\Vert \bnabla T \right\Vert^2 \right\rangle} + {\left\langle \left(\frac{ T {\left( { \mathbf{x} }_p,t \right)} - \theta_p}{\eta_p}\right)^2 \right\rangle} \right].\label{eq:dissbal}$$ In these simulations the volume fraction $\phi$ is constant, so the characteristic length of the dissipation due to the particles is proportional to the particle radius. The portion of temperature fluctuations dissipated by the two different mechanisms depends on the statistics of the differences between the particle and local fluid temperatures. In the limit ${\text{\textit{St}}}_\theta\to 0$ we have $T {\left( { \mathbf{x} }_p,t \right)}=\theta_p$, such that all of the dissipation is associated with the fluid. In the general case, the statistics of $T {\left( { \mathbf{x} }_p,t \right)}-\theta_p$ depend not only on ${\text{\textit{St}}}_\theta$, but also implicitly upon ${\text{\textit{St}}}$, with the statistics of $T {\left( { \mathbf{x} }_p,t \right)}$ depending on the spatial clustering of the particles. This coupling between the particle momentum and temperature dynamics can lead to non-trivial effects of particle inertia on $\chi_p$. Thermal dissipation due to the temperature gradients ---------------------------------------------------- Since the flow is isotropic, $\chi_f$ is given by $$\chi_f = 3\kappa{\left\langle {\left( \partial_x T \right)}^2 \right\rangle}$$ We consider fixed Reynolds number and $\Pran=1$, thus $\kappa$ is the same in all the presented simulations, and so $\langle(\partial_x T)^2\rangle$ fully characterizes $\chi_f$. Moreover, given the expected structure of the field $\partial_x T$, it is instructive to consider its full Probability Density Function (PDF), in addition to its moments in order to know how different regions of the flow contribute to the average dissipation rate $\chi_f$. ![PDF of the fluid temperature gradient $\partial_x T$ from simulations S1, for ${\text{\textit{St}}}=1$ (*a*) and ${\text{\textit{St}}}=3$ (*b*), and for various ${\text{\textit{St}}}_\theta$. (*c*) Dissipation rate $\chi_f$ of the fluid temperature fluctuations, for different ${\text{\textit{St}}}$ as a function of ${\text{\textit{St}}}_\theta$. (*d*) Kurtosis of the fluid temperature gradient PDF.[]{data-label="fig:PDFgrT"}]({figure2-crop}.pdf){width="\textwidth"} Figures \[fig:PDFgrT\](a-b) show the normalized PDFs of $\partial_x T$ for ${\text{\textit{St}}}=1$ and ${\text{\textit{St}}}=3$ respectively, and for various ${\text{\textit{St}}}_\theta$, where the PDFs are normalized using the standard deviation of the distribution, $\sigma_{\partial_x T}$. The distribution is almost symmetric and it displays elongated exponential tails. The largest temperature gradients exceed the standard deviation by an order of magnitude [@Overholt1996]. Remarkably, the shape of the PDF shows a very weak dependence on ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$, such that the PDF shape scales with $\sigma_{\partial_x T}$. The variance of the fluid temperature gradient is proportional to the actual dissipation rate of the temperature fluctuation (the proportionality factor being $3\kappa$, fixed in our simulations). In contrast to the PDF shape, the suspended particles have a strong impact on $\chi_f$, as shown in figure \[fig:PDFgrT\]. As ${\text{\textit{St}}}_\theta$ is increased, $\chi_f$ decreases. However, this is mainly due to the fact that as ${\text{\textit{St}}}_\theta$ is increased, $\chi_p$ increases, and so $\chi_f$ must decrease since $\chi=\chi_f+\chi_p$ is fixed. The influence of the Stokes number on $\chi_p$ is very small in the range of parameters considered. The kurtosis of the fluid temperature gradients is shown in figure \[fig:PDFgrT\](d), as a function of ${\text{\textit{St}}}_\theta$ and for various ${\text{\textit{St}}}$. The kurtosis is approximately constant, and much larger than the value of a Gaussian distribution. The behavior of the kurtosis confirms that the fluid temperature gradient PDF is approximately self-similar. Thermal dissipation due to the particle dynamics ------------------------------------------------ The dissipation rate due to the particles, $\chi_p$, depends on the difference between the particle temperature and the fluid temperature at the particle position $$\chi_p = \kappa{\left\langle \left(\frac{ T {\left( { \mathbf{x} }_p,t \right)} - \theta_p}{\eta_p}\right)^2 \right\rangle}$$ For notational simplicity, we define $\varphi_p \equiv \left.\left(T {\left( { \mathbf{x} }_p,t \right)} - \theta_p\right)\middle/\eta_p\right.$. When $\varphi_p$ is normalized by its standard deviation, we can relate this to the rate of change of the particle temperature using equation $$\frac{ \dot{\theta}_p }{ \sigma_{ \dot{\theta}_p } } = \frac{\varphi_p}{\sigma_{\varphi_p}}.$$ ![PDF of $\dot{\theta}_p$ for ${\text{\textit{St}}}=1$ (*a-b*) and ${\text{\textit{St}}}=3$ (*c-d*), and for various ${\text{\textit{St}}}_\theta$. Plots (*a-c*) are from simulations S1, in which the two-way thermal coupling is considered, while plots (*b-d*) are from simulations S2, in which the two-way coupling is neglected. (*e*) Dissipation rate $\chi_p$ of the temperature fluctuations due to the particles, for different ${\text{\textit{St}}}$ as a function of ${\text{\textit{St}}}_\theta$. (*f*) Kurtosis of the PDF of $\dot{\theta}_p$.[]{data-label="fig:PDFdeT"}]({figure3-crop}.pdf){width="\textwidth"} The normalized PDF of $\dot{\theta}_p$ for ${\text{\textit{St}}}=1$ and ${\text{\textit{St}}}=3$, and for various ${\text{\textit{St}}}_\theta$ is shown in figure \[fig:PDFdeT\]. Figure \[fig:PDFdeT\](a) shows the normalized PDF of $\dot{\theta}_p$, for ${\text{\textit{St}}}=1$ for the set of simulations S1, in which the two-way thermal coupling is taken to account. Figure \[fig:PDFdeT\](b) shows the corresponding results for simulations S2, in which the two-way thermal coupling is neglected. The normalized PDF of $\dot{\theta}_p$ for ${\text{\textit{St}}}=3$, with and without the two-way thermal coupling, is shown in figures \[fig:PDFdeT\](c-d). ![(a) Variance of the particle temperature rate of change as a function of the thermal Stokes number for different Stokes numbers. The dotted lines represent the expected asymptotic behaviour for ${\text{\textit{St}}}_\theta\ll1$ and ${\text{\textit{St}}}_\theta\gg1$. (b) Normalized PDF of the particle temperature rate of change, $\dot{\theta_p}$ at ${\text{\textit{St}}}_\theta=1$ for various Stokes number, ${\text{\textit{St}}}<0.5$. The dotted line shows a Gaussian PDF for reference. Results obtained neglecting the particle thermal feedback.[]{data-label="fig:compare"}]({figure4-crop}.pdf){width="\textwidth"} In contrast to the fluid temperature gradient PDFs, the shape of the PDF of $\dot{\theta}_p$ is not self-similar with respect to its variance. As ${\text{\textit{St}}}_\theta$ is increased, the normalized PDF becomes narrower. This is due to the fact that as ${\text{\textit{St}}}_\theta$ is increased, the particles respond more slowly to changes in the fluid temperature field, analogous to the “filtering” effect for inertial particle velocities in turbulence [@salazar12a; @Ireland2016]. The PDF shapes are mildly affected by ${\text{\textit{St}}}$, and for larger ${\text{\textit{St}}}_\theta$, extreme fluid temperature-particle temperature differences are suppressed when the two-way thermal coupling is neglected. The variance of $\dot{\theta}_p$ is proportional to the particle dissipation rate $\chi_p$, and the results for this are shown in figure \[fig:PDFdeT\](e), for various ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$, and for simulations S1 and S2. The results show that as ${\text{\textit{St}}}_\theta$ is increased, $\chi_p$ increases. This is mainly because as ${\text{\textit{St}}}_\theta$ is increased, the thermal memory of the particle increases, and the particle temperature depends strongly on its encounter with the fluid temperature field along its trajectory history for times up to ${\textit{O}\left( \tau_\theta \right)}$ in the past. As a result, the particle temperature can differ strongly from the local fluid temperature. The results also show that $\chi_p$ is dramatically suppressed when two-way thermal coupling is accounted for. One reason for this is that as shown earlier, two-way thermal coupling leads to a suppression in the fluid temperature gradients. As these gradients are suppressed, the fluid temperature along the particle trajectory history differs less from the local fluid temperature than it would have in the absence of two-way thermal coupling, and as a result $\chi_p$ is decreased. The results for kurtosis of $\dot{\theta}_p$, as a function of ${\text{\textit{St}}}_\theta$ and for various ${\text{\textit{St}}}$ are shown in figure \[fig:PDFdeT\](f). The results show that the kurtosis decreases with increasing ${\text{\textit{St}}}_\theta$. This is mainly due to the filtering effect mentioned earlier, wherein as ${\text{\textit{St}}}_\theta$ is increased, the particles are less able to respond to rapid fluctuations in the fluid temperature along their trajectory. Further, the kurtosis is typically larger when the two-way thermal coupling is taken into account (simulations S1), and is maximum for ${\text{\textit{St}}}=1$. This is due to the particle clustering on the fronts of the fluid temperature field, as will be discussed in section \[sec:FATT\]. Our results for the PDF of $\dot{\theta}_p$ and its moments differ somewhat from those in [@Bec2014]. This is in part due to the difference in the forcing methods employed by [@Bec2014] and that in our study. The solution of may be written as [@Bec2014] $${\left\langle \dot{\theta_p}^2 \right\rangle} = \frac{1}{2\tau_\theta^3}\int_0^\infty {\left\langle \Big( \delta_t T_p(t)\Big)^2 \right\rangle}\exp{\left( -\frac{t}{\tau_\theta} \right)}{\text{\textrm{d}}}t, \label{eq:solthetalag}$$ where $\delta_t T_p(t)\equiv T{\left( { \mathbf{x} }_p{\left( t \right)},t \right)}-T{\left( { \mathbf{x} }_p{\left( 0 \right)},0 \right)}$. In the regime ${\text{\textit{St}}}_\theta\ll1$, the exponential in decays very fast in time so that the main contribution to the integral comes from $\delta_t T_p$ for infinitesimal $t$, with $\delta_t T_p \sim t^n$ for $t\to 0$. Substituting $\delta_t T_p \sim t^n$ into we obtain the leading order behavior$${\left\langle \dot{\theta_p}^2 \right\rangle} \sim \frac{1}{2\tau_\theta^3}\int_0^\infty t^{2n} \exp{\left( -\frac{t}{\tau_\theta} \right)}{\text{\textrm{d}}}t \sim {\text{\textit{St}}}_\theta^{2n-2},\;{\text{\textit{St}}}_\theta\ll 1.$$ [@Bec2014] used a white in time forcing for the fluid scalar field, giving $n=1/2$, and yielding $\langle\dot{\theta_p}^2\rangle \sim {\text{\textit{St}}}_\theta^{-1}$ for ${\text{\textit{St}}}_\theta\ll1$. However, the forcing scheme that we have employed generates a field $T({ \mathbf{x} },t)$ that evolves smoothly in time, so $n=1$ and $\langle\dot{\theta_p}^2\rangle\sim$ constant for ${\text{\textit{St}}}_\theta\ll1$. For ${\text{\textit{St}}}_\theta\gg 1$, the integral in is dominated by uncorrelated temperature increments, $\delta_t T \sim t^0$, such that $\langle\dot{\theta_p}^2\rangle \sim {\text{\textit{St}}}_\theta^{-2}$. The comparison between figure \[fig:compare\](a) and figure 5 of [@Bec2014] highlights the different asymptotic behavior of $\sigma_{\dot{\theta_p}}^2\equiv \langle\dot{\theta_p}^2\rangle$ for ${\text{\textit{St}}}_\theta\ll 1$, but the same behavior $\langle\dot{\theta_p}^2\rangle \sim {\text{\textit{St}}}_\theta^{-2}$ for ${\text{\textit{St}}}_\theta\gg 1$. Further, as expected, our DNS data approaches these asymptotic regimes for both the cases with and without two-way thermal coupling. Another difference is that in the results of [@Bec2014], the tails of the PDFs of $\dot{\theta}_p$ for ${\text{\textit{St}}}_\theta=1$ become heavier as ${\text{\textit{St}}}$ is increased, whereas our results in figure \[fig:PDFdeT\] show that while the kurtosis of these PDFs increases from ${\text{\textit{St}}}=0.5$ to ${\text{\textit{St}}}=1$, it then decreases from ${\text{\textit{St}}}=1$ to ${\text{\textit{St}}}=3$. In order to examine this further, we performed simulations (without two-way thermal coupling) for ${\text{\textit{St}}}_\theta=1$ and ${\text{\textit{St}}}\leq 0.4$. The results are shown in figure \[fig:compare\](b), and in this regime we do in fact observe that the tails of the PDFs of $\dot{\theta}_p$ become increasingly wider as ${\text{\textit{St}}}$ is increased. Taken together with the results in figure \[fig:PDFdeT\], this implies that in our simulations, the tails of the PDFs of $\dot{\theta}_p$ become increasingly wider as ${\text{\textit{St}}}$ is increased until ${\text{\textit{St}}}\approx 1$, where this behavior then saturates, and upon further increase of ${\text{\textit{St}}}$ the tails start to narrow. This non-monotonic behavior is due to the particle clustering in the fronts of the temperature field, which is strongest for ${\text{\textit{St}}}\approx 1$ (see §\[sec:FATT\]). While the results in [@Bec2014] over the range ${\text{\textit{St}}}\leq 3.7$ do not show the tails of the PDFs of $\dot{\theta}_p$ becoming narrower, their results clearly show that the widening of the tails saturates (see inset of figure 5 in [@Bec2014]). It is possible that if they had considered larger ${\text{\textit{St}}}$, they would have also began to observe a narrowing of the tails as ${\text{\textit{St}}}$ was further increased. Possible reasons why the widening of the tails saturates at a lower value of ${\text{\textit{St}}}$ in our DNS than it does in theirs include is the effect of Reynolds number ($\Rey_\lambda=315$ in their DNS, whereas in our DNS $\Rey_\lambda=88$), and differences in the scalar forcing method. Characterization of the temperature fluctuations {#sec:fluct} ================================================ This section consists of a short overview of the one-point temperature statistics. Note that due to the large scale forcing used in the DNS, the one-point statistics of the flow are affected by the forcing method employed. Fluid temperature fluctuations ------------------------------ ![PDF of the fluid temperature for ${\text{\textit{St}}}=1$ (*a*) and ${\text{\textit{St}}}=3$ (*b*), and for various ${\text{\textit{St}}}_\theta$. (*c*) Variance of the fluid temperature fluctuations for different ${\text{\textit{St}}}$ as a function of ${\text{\textit{St}}}_\theta$. (*d*) Kurtosis of the fluid temperature PDF. These results are from simulations S1 in which the two-way thermal coupling is considered.[]{data-label="fig:PDFT"}]({figure5_106-crop}.pdf){width="\textwidth"} Figures \[fig:PDFT\](a-b) show the normalized one-point PDF of the fluid temperature for ${\text{\textit{St}}}=1$ and ${\text{\textit{St}}}=3$, respectively, and for various ${\text{\textit{St}}}_\theta$. The PDFs are normalized with the standard deviation of the distribution $\sigma_{T}$. The PDFs are almost Gaussian for low ${\text{\textit{St}}}_\theta$, while the tails become wider as ${\text{\textit{St}}}_\theta$ is increased. However, we are unable to explain the cause of this enhanced non-Gaussianity. The temperature PDFs are also not symmetric, and display a bump in the right tail. This behavior was also reported by [@Overholt1996] for the case without particles, and it appears to be a low Reynolds number effect that is also dependent on the forcing method employed. The effect of ${\text{\textit{St}}}$ on $\sigma_T$ is striking, whereas we saw earlier in figure \[fig:PDFgrT\](c) that $\chi_f$ only weakly depends on ${\text{\textit{St}}}$. To explain the dependence upon the Stokes number we note that the energy balance can be rewritten as $$\chi = \kappa \left[ {\left\langle \left\Vert \bnabla T \right\Vert^2 \right\rangle} + \frac{2}{3}\frac{\phi}{\tau_\eta}\frac{\rho_p}{\rho_0}\frac{1}{{\text{\textit{St}}}}{\left\langle \left( T {\left( { \mathbf{x} }_p,t \right)} - \theta_p\right)^2 \right\rangle} \right]\label{eq:balSt}.$$ The factor $\phi\rho_p/\left(\rho_0\tau_\eta\right)$ is constant in our simulations. Therefore, since our DNS data suggest that $\chi_f$ is a function of ${\text{\textit{St}}}_\theta $ only (see figure \[fig:PDFgrT\](c)), from and we obtain $${\left\langle T{\left( { \mathbf{x} }_p,t \right)}^2 \right\rangle} - {\left\langle \theta_p^2 \right\rangle} \propto {\text{\textit{St}}}f{\left( {\text{\textit{St}}}_\theta \right)}.$$ The kurtosis of the fluid temperature fluctuation is shown in figure \[fig:PDFT\](d), as a function of ${\text{\textit{St}}}_\theta$ and for various ${\text{\textit{St}}}$. For small ${\text{\textit{St}}}_\theta$, the kurtosis of the fluid temperature fluctuation is close to the value for a Gaussian PDF, namely $3$. However, as ${\text{\textit{St}}}_\theta$ is increased, the kurtosis increases. Furthermore, the kurtosis decreases with increasing ${\text{\textit{St}}}$ for the range considered in our simulations. The explanation of these trends in the kurtosis is unclear. Particle temperature fluctuations --------------------------------- ![PDF of the particle temperature for ${\text{\textit{St}}}=1$ (*a-b*) and ${\text{\textit{St}}}=3$ (*c-d*), for various ${\text{\textit{St}}}_\theta$. Plots (*a-c*) are from simulations S1, in which the two-way thermal coupling is considered, while plots (*b-d*) are from simulations S2, in which the two-way coupling is neglected. (*e*) Variance of the particle temperature fluctuations for different ${\text{\textit{St}}}$ numbers as a function of ${\text{\textit{St}}}_\theta$. (*f*) Kurtosis of the particle temperature distribution.[]{data-label="fig:PDFtheta"}]({figure6_106-crop}.pdf){width="\textwidth"} Figures \[fig:PDFtheta\](a-b) show the normalized one-point PDF of the particle temperature with ${\text{\textit{St}}}=1$, for various ${\text{\textit{St}}}_\theta$, and for simulations S1 and S2. Figures \[fig:PDFtheta\](c-d) show the corresponding results for ${\text{\textit{St}}}=3$, and the PDFs are normalized by their standard deviations. When the two-way thermal coupling is accounted for, the tails of the particle temperature distribution tend to become wider as ${\text{\textit{St}}}_\theta$ is increased. On the other hand, when the two-way coupling is neglected, the PDF of the particle temperature is very close to Gaussian, and its shape is not sensitive to either ${\text{\textit{St}}}$ or ${\text{\textit{St}}}_\theta$. The variance of the particle temperature fluctuations monotonically decrease with increasing ${\text{\textit{St}}}_\theta$, as shown in figure \[fig:PDFtheta\](e). The results also show a strong dependence on ${\text{\textit{St}}}$, but most interestingly, the dependence on ${\text{\textit{St}}}$ is the opposite for the cases with and without two-way coupling. To understand this we note that using the formal solution to the equation for $\dot{\theta}_p(t)$ (ignoring initial conditions) we may construct the result $$\Big\langle \theta^2_p(t)\Big\rangle=\frac{1}{\tau_\theta^2}\int^t_0\int^t_0 \Big\langle T({ \mathbf{x} }_p(s),s)T({ \mathbf{x} }_p(s'),s')\Big\rangle e^{-(2t-s-s')/\tau_\theta}\,ds\,ds'.$$ If we now substitute into this the exponential approximation $$\langle T({ \mathbf{x} }_p(s),s)T({ \mathbf{x} }_p(s'),s')\rangle \approx \langle T^2({ \mathbf{x} }_p(t),t)\rangle \exp[-|s-s'|/\tau_T],$$ where $\tau_T$ is the timescale of $T({ \mathbf{x} }_p(t),t)$, then we obtain $$\Big\langle \theta^2_p(t)\Big\rangle= \frac{\langle T^2({ \mathbf{x} }_p(t),t)\rangle}{1+\tau_\theta/\tau_T} .$$ This result reveals that the particle temperature variance is influenced by ${\text{\textit{St}}}$ in two ways. First, $\langle T^2({ \mathbf{x} }_p(t),t)\rangle$ depends upon the spatial clustering of the inertial particles, and this depends essentially upon ${\text{\textit{St}}}$. Second, the timescale $\tau_T$ is the timescale of the fluid temperature field measured along the inertial particle trajectories, and hence depends upon ${\text{\textit{St}}}$. For isotropic turbulence, this timescale is expected to decrease as ${\text{\textit{St}}}$ is increased, which would lead to $\langle \theta^2_p(t)\rangle$ decreasing as ${\text{\textit{St}}}$ increases, which is the behavior observed in figure \[fig:PDFtheta\](e). In the presence of two-way coupling, however, $\langle T^2({ \mathbf{x} },t)\rangle$ increases with increasing ${\text{\textit{St}}}$, as shown earlier. In the two-way coupled regime this increase in $\langle T^2({ \mathbf{x} },t)\rangle$ leads to an increase in $\langle T^2({ \mathbf{x} }_p(t),t)\rangle$ that dominates over the decrease of $\tau_T$ with increasing ${\text{\textit{St}}}$, and as a result $\langle \theta^2_p(t)\rangle$ increases with increasing ${\text{\textit{St}}}$. The kurtosis of the particle temperature increases with increasing ${\text{\textit{St}}}_\theta$ when the two-way thermal coupling is accounted for, as shown in figure \[fig:PDFtheta\](f) (simulations S1, filled symbols). Conversely, the kurtosis of the particle temperature remains constant as ${\text{\textit{St}}}_\theta$ is increased when the two-way thermal coupling is ignored (simulations S2, open symbols). Statistics conditioned on the local fluid temperature gradients {#sec:FATT} =============================================================== In this section we consider additional quantities to obtain deeper insight into the one-point particle to fluid heat flux. In particular, we explore the relationship between this heat flux and the local fluid temperature gradients. Particle clustering on the temperature fronts --------------------------------------------- ![(*a*) Radial distribution function (RDF) as a function of the separation $r/\eta$ for various ${\text{\textit{St}}}$. (*b*) Particle number density conditioned on the magnitude of the fluid temperature gradient at the particle position, for various ${\text{\textit{St}}}$. These results are from simulations S2, in which the two-way thermal coupling is neglected.[]{data-label="fig:FATTconc"}]({figure7-crop}.pdf){width="\textwidth"} It is well known that inertial particles in turbulence form clusters [@Bec2007], which may be quantified using the radial distribution function (RDF). As shown in figure \[fig:FATTconc\](a), the particle number density in our simulations at small separations is a order of magnitude larger than the mean density when ${\text{\textit{St}}}=\textit{O}{\left( 1 \right)}$. [@Bec2014] showed that inertial particles also exhibit a tendency to preferentially cluster in the fluid temperature fronts where the temperature gradients are large. To demonstrate this, they measured the temperature dissipation rate at the particle positions and showed that this was higher than the Eulerian dissipation rate of the fluid temperature fluctuations. Alternatively, we may quantify this tendency for inertial particles to cluster in the fluid temperature fronts by computing the particle number density conditioned on the magnitude of the fluid temperature gradient $$n_P{\left( \left\Vert\bnabla T\right\Vert \right)} = \frac{\sum_p\int_V \delta{\left( { \mathbf{x} }-{ \mathbf{x} }_p \right)}{\text{\textrm{d}}}{ \mathbf{x} }}{N_P},\; V=\left\{ { \mathbf{x} }: \left\Vert\bnabla T{\left( { \mathbf{x} } \right)}\right\Vert = \left\Vert\bnabla T\right\Vert\right\}.$$ Defining $\|\bnabla T\|_{rms}$ as the rms value of $\|\bnabla T\|$, small values of $\|\bnabla T\|/\|{ \mathbf{\bnabla} } T\|_{rms}$ may be interpreted as corresponding to the large scales, and are associated with the Lagrangian coherent structures in which the temperature field is almost constant. Large values of $\|\bnabla T\|/\|\bnabla T\|_{rms}$ may be interpreted as corresponding to the small scales, and are associated with fronts in the fluid temperature field. The results for $n_P$ are shown in figure \[fig:FATTconc\](b), corresponding to simulations without two-way thermal coupling (the results show only a weak dependence on ${\text{\textit{St}}}_\theta$ when the two-way coupling is included). For fluid particles, $n_P$ decays almost exponentially with increasing $\|\bnabla T\|$. For values of ${\text{\textit{St}}}$ at which the maximum particle clustering takes place, $n_P$ is an order of magnitude larger than the value for fluid particles in regions of strong temperature gradients. These results therefore support the conclusions of [@Bec2014] that inertial particles preferentially cluster in the fronts of the fluid temperature field where $\|\bnabla T\|/\|\bnabla T\|_{rms}$ is large. Particle motion across the temperature fronts --------------------------------------------- To obtain further insight into the thermal coupling between the particles and fluid we consider the properties of the particle heat flux conditioned on $\|\bnabla T\|$. In particular, we consider the following quantity $$q_n {\left( \left\Vert \bnabla T \right\Vert \right)} \equiv {{\left( T{\left( { \mathbf{x} }_p \right)}-\theta_p \right)}^n{ \mathbf{v} }_p\bcdot { \mathbf{n} }_T{\left( { \mathbf{x} }_p \right)}}\Big\vert_{\left\Vert \bnabla T \right\Vert},$$ where $\mathbf{n}_T$ is the normalized temperature gradient $$\mathbf{n}_T{\left( { \mathbf{x} }_p \right)} \equiv \frac{\bnabla T{\left( { \mathbf{x} }_p \right)}}{\left\Vert \bnabla T{\left( { \mathbf{x} }_p \right)} \right\Vert}.$$ ![(*a*) Results for ${\left\langle \left\vert q_0(\|\bnabla T\|)\right\vert \right\rangle}/u_\eta$, for various ${\text{\textit{St}}}$. (*b*) Results for $\langle|\cos \alpha_p|\rangle$ as a function of $\|\bnabla T\|$, for various ${\text{\textit{St}}}$. These results are from simulations S2, in which the two-way thermal coupling is neglected.[]{data-label="fig:FATTq0"}]({figure8-crop}.pdf){width="\textwidth"} The statistics of $q_n$ provide a way to quantify the relationship between the particle heat flux and the local temperature gradients in the fluid. Understanding this relationship is key to understanding how the particles modify the properties of the fluid temperature and temperature gradient fields. The efficiency with which the particles cross the fronts in the fluid temperature field is quantified by ${\left\langle \left\vert q_0\right\vert \right\rangle}$, and our results for this quantity are shown in figure \[fig:FATTq0\](a). The curves are normalized with the Kolmogorov velocity scale $u_\eta$. The results show that as ${\text{\textit{St}}}$ is increased, the particles move across the fronts with increasingly large velocities. This behavior is non-trivial since it is known that the kinetic energy of an inertial particle decreases with increasing ${\text{\textit{St}}}$ [@Zaichik2009; @Ireland2016]. It is also important to consider whether the reduction of ${\left\langle \left\vert q_0\right\vert \right\rangle}$ as $\|\bnabla T\|$ increases is due to the reduction of the norm of the particle velocity or to the lack of alignment between the particle velocity and the fluid temperature gradient at the particle position. Figure \[fig:FATTq0\](b) displays the average of the absolute value of the cosine of the angle between the particle velocity and temperature gradient $$\cos\alpha_p \equiv \frac{{ \mathbf{v} }_p}{\left\Vert { \mathbf{v} }_p \right\Vert} \bcdot \frac{\bnabla T {\left( { \mathbf{x} }_p \right)}}{\left\Vert \bnabla T {\left( { \mathbf{x} }_p \right)}\right\Vert},$$ conditioned on $\|{ \mathbf{\bnabla} }T\|$. The results show that as $\|\bnabla T\|$ is increased, the particle motion becomes misaligned with the local fluid temperature gradient. This then shows that the reduction of ${\left\langle \left\vert q_0\right\vert \right\rangle}$ as $\|\bnabla T\|$ increases is due to non-trivial statistical geometry in the system. The results also show that as ${\text{\textit{St}}}$ is increased, the cosine of the angle between the fluid temperature gradient and the particle velocity becomes almost independent of $\|\bnabla T\|$, and ${\left\langle \left\vert\cos\alpha_p\right\vert \right\rangle}\approx 1/2$, the value corresponding to $\cos{\left( \alpha_p \right)}$ being a uniform random variable. This shows that as ${\text{\textit{St}}}$ is increased, the correlation between the direction of the particle velocity and the local fluid velocity gradient vanishes. Heat flux due to the particle motion across the fronts ------------------------------------------------------ ![Results for ${\left\langle q_1{\left( {\left\Vert \bnabla T \right\Vert} \right)} \right\rangle}/{\left( u_\eta T_\eta \right)}$ for ${\text{\textit{St}}}=0.5$ (*a-b*), ${\text{\textit{St}}}=1$ (*c-d*) and ${\text{\textit{St}}}=3$ (*e-f*), and for various ${\text{\textit{St}}}_\theta$. Plots (*a-c-e*) are from simulations S1, in which the two-way thermal coupling is considered, while plots (*b-d-f*) are from simulations S2, in which the two-way coupling is neglected.[]{data-label="fig:FATTq1"}]({figure9-crop}.pdf){width="\textwidth"} We now turn to consider the quantity ${\left\langle q_1 \right\rangle}$. When the particle moves from a cold to a warm region of the fluid, the component of the particle velocity along the temperature gradient is positive, ${ \mathbf{v} }_p\bcdot { \mathbf{n} }_T{\left( { \mathbf{x} }_p \right)} > 0$. If the particle is also cooler than the local fluid so that $T{\left( { \mathbf{x} }_p \right)}-\theta_p>0$, then as it moves into the region where the fluid is warmer, $q_1>0$ meaning that the particle will absorb heat from the fluid, and will therefore tend to reduce the local fluid temperature gradient. When the particle moves from a warm to a cold region of the flow, if $T{\left( { \mathbf{x} }_p \right)}-\theta_p<0$ then $q_1$ is also positive, so that again the particle will act to reduce the local temperature gradient in the fluid. Therefore, $q_1>0$ indicates that the action of the inertial particles is to smooth out the fluid temperature field, reducing the magnitude of its temperature gradients, and $q_1<0$ implies the particles enhance the temperature gradients. The results for ${\left\langle q_1 \right\rangle}$ are shown in figure \[fig:FATTq1\] for various ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$, including (simulations S1) and neglecting (simulations S2) the two-way thermal coupling. On average we observe ${\left\langle q_1 \right\rangle}\ge 0$, such that the particles tend to make the fluid temperature field more uniform. The results show that ${\left\langle q_1 \right\rangle}$ tends to zero as $\|\bnabla T\|\to0$. This indicates that the particles spend enough time in the Lagrangian coherent structures to adjust to the temperature of the fluid. However, ${\left\langle q_1 \right\rangle}$ increases significantly as $\|\bnabla T\|$ increases, suggesting that inertial particles can carry large temperature differences across the fronts. In the limit ${\text{\textit{St}}}_\theta\to 0$, ${\left\langle q_1 \right\rangle}\to0$ reflecting the thermal equilibrium between the particles and the fluid. As ${\text{\textit{St}}}_\theta$ is increased, the heat-flux becomes finite, however, if ${\text{\textit{St}}}_\theta$ is too large, the particle temperature decorrelates from the fluid temperature and the heat exchange is not effective. Hence, ${\left\langle q_1 \right\rangle}$ can saturate with increasing ${\text{\textit{St}}}_\theta$. The results show that ${\left\langle q_1 \right\rangle}$ increases with increasing ${\text{\textit{St}}}$, associated with the decoupling of ${ \mathbf{v} }_p$ and ${ \mathbf{n} }_T{\left( { \mathbf{x} }_p \right)}$ discussed earlier. Finally, the results also show that two-way thermal coupling reduces ${\left\langle q_1 \right\rangle}$. This is simply a reflection of the fact that since the particles tend to smooth out the fluid temperature gradients, the disequilibrium between the particle and local fluid temperature is reduced, which in turn reduces the heat flux due to the particles. Temperature structure functions {#sec:SF} =============================== We now turn to consider two-point quantities in order to understand how the two-way thermal coupling affects the system at the small scales. Fluid temperature structure functions ------------------------------------- The $n$-th order structure function of the fluid temperature field is defined as $$S^n_T{\left( r \right)} \equiv {\left\langle \left\vert \Delta T(r,t)\right\vert^n \right\rangle}$$ where $\Delta T(r,t)$ it the difference in the temperature field at two points separated by the distance $r$ (the “temperature increment”). The results for $S^2_T$, with different ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$ are shown in figure \[fig:SFT\]. ![Results for $S^2_T$ for different ${\text{\textit{St}}}_\theta$, for ${\text{\textit{St}}}=0.5$ (*a*), ${\text{\textit{St}}}=1$ (*b*) and ${\text{\textit{St}}}=3$ (*c*). (*d*) Scaling exponents of the fluid temperature structure functions at small separation, $r\le2\eta$, at ${\text{\textit{St}}}=1$. The data is from simulations S1 in which the two-way thermal coupling is considered.[]{data-label="fig:SFT"}]({figure10-crop}.pdf){width="\textwidth"} The results show that $S^2_T$ decreases monotonically with increasing ${\text{\textit{St}}}_\theta$ at all scales when the two-way thermal coupling is taken to account. In the dissipation range, $S^2_T$ is directly connected to the dissipation rate, and is suppressed in the same way for the three different ${\text{\textit{St}}}$ considered. Conversely, the suppression of the large scale fluctuations is stronger as ${\text{\textit{St}}}$ is reduced, at least for the range of ${\text{\textit{St}}}$ considered here. The scaling exponents of the structure functions of the temperature field $$\zeta^n_T \equiv \frac{{\text{\textrm{d}}}\log S^n_T {\left( r \right)}}{{\text{\textrm{d}}}\log r}$$ are shown in figure \[fig:SFT\](d) for $r\le2\eta$. The results show that the fluid temperature field remains smooth (to within numerical uncertainty) even when suspended particles are suspended in the flow. This is not trivial since the contribution from the particle to fluid coupling term $C_T \left({ \mathbf{x} }+{ \mathbf{r} },t\right)-C_T \left({ \mathbf{x} },t\right)$ need not be a smooth function of $r$. Particle temperature structure functions ---------------------------------------- The $n$-th order structure function of the particle temperature $\theta_p{\left( t \right)}$ is defined as $$S^n_{\theta}{\left( r \right)} \equiv {\left\langle \left\vert \Delta \theta_{p}\right\vert^n \right\rangle}_{ r}$$ where $\Delta \theta_{p}(t)$ is the difference in the temperature of the two particles, and the brackets denote an ensemble average, conditioned on the two particles having separation $r$. The results for $S^2_{\theta}$ for different ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$, with and without two-way thermal coupling, are shown in figure \[fig:SFtheta\]. ![Results for $S^2_\theta$ for different ${\text{\textit{St}}}_\theta$, for ${\text{\textit{St}}}=0.5$ (*a-b*), ${\text{\textit{St}}}=1$ (*c-d*) and ${\text{\textit{St}}}=3$ (*e-f*). Plots (*a-c-e*) are from simulations S1, in which the two-way thermal coupling is considered, while plots (*b-d-f*) are from simulations S2, in which the two-way coupling is neglected.[]{data-label="fig:SFtheta"}]({figure11-crop}.pdf){width="\textwidth"} The results show that $S^2_{\theta}$ depends on ${\text{\textit{St}}}_\theta$ in much the same way as the inertial particle relative velocity structure functions depend on ${\text{\textit{St}}}$ [@Ireland2016]. This is not surprising since the equation governing $\dot{\theta}_p$ is structurally identical to the equation governing the particle acceleration. However, important differences are that $\dot{\theta}_p$ depends on both ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$, and also that the fluid temperature field is structurally different from the fluid velocity field, with the temperature field exhibiting the well-known ramp-cliff structure. To obtain further insight into the behavior of $S^2_{\theta}$ and $S^n_{\theta}$ in general, we note that the formal solution for $\Delta \theta_{p}(t)$ is given by (ignoring initial conditions) $$\Delta\theta_p{\left( t \right)} =\frac{1}{\tau_\theta} \int_{0}^t \Delta T{\left( { \mathbf{x} }_p{\left( s \right)},{ \mathbf{r} }_p{\left( s \right)},s \right)} \exp{\left( -\frac{t-s}{\tau_\theta} \right)} {\text{\textrm{d}}}s, \label{eq:soltheta}$$ where $\Delta T{\left( { \mathbf{x} }_p{\left( s \right)},{ \mathbf{r} }_p{\left( s \right)},s \right)}$ is the difference in the fluid temperature at the two particle positions ${ \mathbf{x} }_p{\left( s \right)}$ and ${ \mathbf{x} }_p{\left( s \right)}+{ \mathbf{r} }_p{\left( s \right)}$. Equation shows that $\Delta\theta_p{\left( t \right)}$ depends upon $\Delta T$ along the path-history of the particles, and $\Delta\theta_p{\left( t \right)}$ is therefore a non-local quantity. The role of the path-history increases as ${\text{\textit{St}}}_\theta$ is increased since the exponential kernel in the convolution integral decays more slowly as $\tau_\theta$ is increased. Since the statistics of $\Delta T$ increase with increasing separation, particle-pairs at small separations are able to be influenced by larger values of $\Delta T$ along their path-history, such that $\Delta\theta_p{\left( t \right)}$ can significantly exceed the local fluid temperature increment $\Delta T{\left( { \mathbf{x} }_p{\left( t \right)},{ \mathbf{r} }_p{\left( t \right)},t \right)}$. This then causes $S^2_{\theta}$ to increase with increasing ${\text{\textit{St}}}_\theta$, as shown in figure \[fig:SFtheta\]. This effect is directly analogous to the phenomena of caustics that occur in the relative velocity distributions of inertial particles at the small scales of turbulence [@Wilkinson2005], and which occur because the inertial particle relative velocities depend non-locally on the fluid velocity increments experienced along their trajectory history [@bragg14c]. In analogy, we may therefore refer to the effect as “thermal caustics”, and they may be of particular importance for particle-laden turbulent flows where particles in close proximity thermally interact. The results in figure \[fig:SFtheta\] also reveal a strong effect of ${\text{\textit{St}}}$, and one way that ${\text{\textit{St}}}$ affects these results is through the spatial clustering and preferential sampling of the fluid temperature field by the inertial particles. There is, however, another mechanism through which ${\text{\textit{St}}}$ can affect $S^2_{\theta}$. In particular, since, due to caustics, the relative velocity of the particles increases with increasing ${\text{\textit{St}}}$ at the small scales, then the values of $\Delta T{\left( { \mathbf{x} }_p{\left( s \right)},{ \mathbf{r} }_p{\left( s \right)},s \right)}$ that may contribute to $\Delta\theta_p{\left( t \right)}$ become larger. This follows since if their relative velocities are larger, then over the time span $t-s\leq {\textit{O}\left( \tau_\eta \right)}$ the particle-pair can come from even larger scales where (statistically) $\Delta T{\left( { \mathbf{x} }_p{\left( s \right)},{ \mathbf{r} }_p{\left( s \right)},s \right)}$ is bigger. This effect would cause $S^2_{\theta}$ to increase with ${\text{\textit{St}}}$ for a given ${\text{\textit{St}}}_\theta$, further enhancing the thermal caustics, which is exactly what is observed in figure \[fig:SFtheta\]. The results also show that the thermal caustics are stronger for ${\text{\textit{St}}}_\theta\geq {\textit{O}\left( 1 \right)}$ when the two-way thermal coupling is ignored. This is mainly due to the reduction in the fluid temperature gradients due to the two-way thermal coupling described earlier, noting that in the limit of vanishing fluid temperature gradients, the thermal caustics necessarily disappear. At larger scales where the statistics of $\Delta T$ vary more weakly with $r$, the non-local effect weakens, the thermal caustics disappear, and a filtering mechanism takes over which causes $S^2_{\theta}$ to decrease with increasing ${\text{\textit{St}}}_\theta$. This filtering effect is directly analogous to that dominating the large-scale velocities of inertial particles in isotropic turbulence, and is associated with the sluggish response of the particles to the large scale flow fluctuations due to their inertia [@Ireland2016]. ![Scaling exponent of the structure functions of the particle temperature at small separation, $r\le2\eta$, for various thermal Stokes numbers ${\text{\textit{St}}}_\theta$, at ${\text{\textit{St}}}=0.5$ (*a*) and ${\text{\textit{St}}}=1$ (*b*).[]{data-label="fig:scalexpSFtheta"}]({figure12-crop}.pdf){width="\textwidth"} In the dissipation range our results show that $S^n_{\theta}$ behave as power laws, and the associated scaling exponents $\zeta^n_\theta$ are shown in figure \[fig:scalexpSFtheta\]. To reduce statistical noise, we estimate $\zeta^n_\theta$ by fitting the data for $S^n_{\theta}$ over the range $r\leq 2\eta$. Over this range, $S^n_{\theta}$ do not strictly behave as power laws, and hence the exponents measured are understood as average exponents. The results in figure \[fig:scalexpSFtheta\] reveal that particle temperature increments exhibit a strong multifractal behaviour. This multifractility is due to the non-local thermal dynamics of the particles and the formation of thermal caustics, described earlier. In particular, there exists a finite probability to find inertial particle-pairs that are very close but have large temperature differences because they experienced very different fluid temperatures along their trajectory histories. As with the thermal caustics, the multifractility is enhanced as ${\text{\textit{St}}}$ is increased. Most interestingly, the results for $\zeta^n_\theta$ are only weakly affected by the two-way thermal coupling, despite the fact that we observed a significant effect of the coupling on $S^2_{\theta}$. This suggests that the two-way coupling affects the strength of the thermal caustics, but only weakly affects the scaling of the structure functions in the dissipation range. Mixed structure functions ------------------------- We turn to consider the behaviour of the flux of the temperature increments across the scales of the flow, which is associated with the mixed structure functions $$S_Q (r)\equiv {\left\langle \left(\Delta T(r,t)\right)^2 \Delta u_\parallel(r,t) \right\rangle}$$ where $\Delta u_\parallel$ is the longitudinal relative velocity difference. The results for $S_Q$, for different ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$ are shown in figure \[fig:mixedSFT\]. Just as we observed for the fluid temperature structure functions, $-S_Q$ decreases monotonically with increasing ${\text{\textit{St}}}_\theta$, as was also observed for the fluid temperature dissipation rate $\chi_f$. ![Second order mixed structure functions of the fluid temperature field, for different thermal Stokes numbers of the suspended particles, at ${\text{\textit{St}}}=0.5$ (*a*) and ${\text{\textit{St}}}=1$ (*b*). The data refer to the set of simulations S1, with thermal particle back-reaction included.[]{data-label="fig:mixedSFT"}]({figure13-crop}.pdf){width="\textwidth"} ![Second order mixed structure functions of the particle temperature, for different thermal Stokes numbers, at ${\text{\textit{St}}}=0.5$ (*a-b*) and ${\text{\textit{St}}}=1$ (*c-d*). The plots on the left (*a-c*) refer to the set of simulations S1, in which the thermal particle back-reaction is included. The plots on the right (*b-d*) refer to the set of simulations S2, in which the thermal particle back-reaction is neglected.[]{data-label="fig:mixedSFtheta"}]({figure14-crop}.pdf){width="\textwidth"} To consider the flux of the particle temperature increments, we begin by considering the exact equation that can be constructed for $S^n_\theta$ using PDF transport equations. In particular, if we introduce the PDF $\mathcal{P}({ \mathbf{r} },\Delta\theta,t)\equiv\langle \delta({ \mathbf{r} }_p(t)-{ \mathbf{r} })\delta(\Delta\theta_p(t)-\Delta\theta)\rangle$ and the associated marginal PDF $\varrho({ \mathbf{r} },t)\equiv \int\mathcal{P}\,d\Delta\theta$, where ${ \mathbf{r} }$ and $\Delta\theta$ are time-independent phase-space coordinates, then we may derive for a statistically stationary system the result (see [@bragg14b; @bragg2015a] for details on how to derive such results) $$\Big\langle [\Delta\theta_p(t)]^2\Big\rangle_{ \mathbf{r} }=\Big\langle \Delta T({ \mathbf{x} }_p(t){ \mathbf{r} }_p(t),t)\Delta\theta_p(t)\Big\rangle_{ \mathbf{r} }-\frac{\tau_\theta}{2\varrho}\frac{\partial}{\partial{ \mathbf{r} }}\bcdot\varrho \Big\langle [\Delta\theta_p(t)]^2{ \mathbf{w} }_p(t)\Big\rangle_{ \mathbf{r} },$$ where ${ \mathbf{w} }_p(t)\equiv \partial_t{ \mathbf{r} }_p(t)$. The first term on the right-hand side is the local contribution that remains when there exist no fluxes across the scales, and this term determines the behavior of $\langle [\Delta\theta_p(t)]^2\rangle_{ \mathbf{r} }$ at the large scales of homogeneous turbulence where the statistics are independent of ${ \mathbf{r} }$. The second term on the right-hand side is the non-local contribution that arises for ${\text{\textit{St}}}_\theta>0$, and it is this term that is responsible for the thermal caustics discussed earlier. It depends on the spatial clustering of the particles through $\varrho$ (which is proportional to the RDF), and the flux $\langle [\Delta\theta_p(t)]^2{ \mathbf{w} }_p(t)\rangle_{ \mathbf{r} }$ which, for an isotropic system, is determined by the longitudinal component $$S_{Q_p} (r)\equiv \frac{{ \mathbf{r} }}{r}\bcdot\Big\langle [\Delta\theta_p(t)]^2{ \mathbf{w} }_p(t)\Big\rangle_{r}.$$ The results for $S_{Q_p}$ from our simulations are shown in figure \[fig:mixedSFtheta\], and they show that without two-way coupling, $-S_{Q_p}$ monotonically increases with increasing ${\text{\textit{St}}}_\theta$ at the smallest scales. However, with two-way coupling, $-S_{Q_p}$ is maximum for intermediate values of ${\text{\textit{St}}}_\theta$, and this occurs because as shown earlier, as ${\text{\textit{St}}}_\theta$ is increased, the fluid temperature fluctuations are suppressed across the scales. Distribution of the temperature increments and fluxes {#sec:2pPDF} ===================================================== In this section we look at the distribution of the fluid and particle temperature increments in the dissipation range. ![Probability density function in normal form of the fluid temperature increments at small separations, $r=10\eta$, at ${\text{\textit{St}}}=1$ (*a*) and ${\text{\textit{St}}}=3$ (*b*). The data refer to the set of simulations S1, with thermal feedback included. (*c*) Standard deviation of the fluid temperature increments at small separation. (*d*) Kurtosis of the distribution of the fluid temperature increments at small separation.[]{data-label="fig:PDFdT"}]({figure15-crop}.pdf){width="\textwidth"} Temperature increments in the dissipation range ----------------------------------------------- The normalized PDF of the fluid temperature increments at separations $r=10\eta$ are shown in figure \[fig:PDFdT\] (for the fluid temperature field, we do not consider the PDFs of the velocity increments for $r\leq {\textit{O}\left( \eta \right)}$ since these are essentially identical to the PDFs of the fluid temperature gradients that were considered earlier). Just as we observed earlier for the PDFs of the fluid temperature gradients, the results in figure \[fig:PDFdT\] show that at larger separations the PDFs of the fluid temperature increments are also self similar and approximately collapse when scaled by their standard deviation. The standard deviation and kurtosis of the PDF, also shown in \[fig:PDFdT\], show that while the kurtosis is almost independent of ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$, the variance decreases with increasing ${\text{\textit{St}}}_\theta$, and increases with increasing ${\text{\textit{St}}}$. This latter result differs significantly from the behavior of the variance of the fluid temperature gradients which were almost independent of ${\text{\textit{St}}}$. ![Probability density function in normal form of the particle temperature increments at small separations, $r\le2\eta$, at ${\text{\textit{St}}}=1$ (*a-b*) and ${\text{\textit{St}}}=3$ (*c-d*). The plots on the left (*a-c*) refer to the set of simulations S1, in which the thermal particle back-reaction is included. The plots on the right (*b-d*) refer to the set of simulations S2, in which the thermal particle back-reaction is neglected. (*e*) Standard deviation of the particle temperature increments at small separation. (*f*) Kurtosis of the distribution of the particle temperature increments at small separation.[]{data-label="fig:PDFdtheta"}]({figure16-crop}.pdf){width="\textwidth"} The PDF of the particle temperature increments, along with its variance and kurtosis are shown in figure \[fig:PDFdtheta\]. The results show that while the variance of the PDF monotonically increases with increasing ${\text{\textit{St}}}_\theta$, the kurtosis can increase slightly with increasing ${\text{\textit{St}}}_\theta$ when ${\text{\textit{St}}}_\theta$ is below some threshold, after which the kurtosis monotonically decreases with increasing ${\text{\textit{St}}}_\theta$. However, across the parameter range studied, the PDFs are strongly non-Gaussian, with a maximum kurtosis value of $\approx 13$. The kurtosis values are also strongly and non-monotonically dependent on ${\text{\textit{St}}}$, with the largest values tending to occur for ${\text{\textit{St}}}=1$. This may be due to the clustering of the particles in the fronts of the temperature field, leading to large particle temperature differences. It may also be due to the non-local mechanisms described earlier since although the non-local effects can enhance non-Gaussianity in certain regimes, in the regime where the behavior is entirely non-local (e.g. for ${\text{\textit{St}}}\gg1$), the behavior becomes ballistic and $\Delta\theta_p(t)$ is governed by a central limit theorem and the PDF of $\Delta\theta_p(t)$ approaches a Gaussian distribution. Flux of temperature increments in the dissipation range ------------------------------------------------------- We finally turn to consider the PDFs of the fluid temperature flux $Q=\left(\Delta T(r,t)\right)^2 \Delta u_\parallel(r,t)$ and particle temperature flux $Q=[\Delta\theta_p(t)]^2{w}_\parallel(t)$, where ${w}_\parallel(t)$ is the parallel component of the particle-pair relative velocity. ![Probability density function in normal form of the flux of fluid temperature increments at small separations, $r\le2\eta$, at ${\text{\textit{St}}}=0.5$ (*a*) and ${\text{\textit{St}}}=1$ (*b*). The data refer to the set of simulations S1, with thermal feedback included.[]{data-label="fig:PDFfluxdT"}]({figure17-crop}.pdf){width="\textwidth"} ![Probability density function in normal form of the flux of particle temperature increments at small separations, $r\le2\eta$, at ${\text{\textit{St}}}=0.5$ (*a-b*), ${\text{\textit{St}}}=1$ (*c-d*), ${\text{\textit{St}}}=3$ (*e-f*). The plots on the left (*a-c-e*) refer to the set of simulations S1, in which the thermal particle back-reaction is included. The plots on the right (*b-d-f*) refer to the set of simulations S2, in which the thermal particle back-reaction is neglected.[]{data-label="fig:PDFfluxdtheta"}]({figure18-crop}.pdf){width="\textwidth"} The PDF of the fluid temperature flux is plotted in normal form for $r\le2\eta$ in figure \[fig:PDFfluxdT\]. These normalized PDFs collapse onto each other for all ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$ values considered. Thus, the fluid temperature flux simply scales with its variance in the dissipation range, and the variance of the flux is modulated by the particles but the shape of the distribution is not affected by the particle dynamics. The PDF are strongly negatively skewed and have a negative mean value, associated with the mean flux of thermal fluctuations from large to small scales in the flow. The PDF of the particle temperature flux is plotted in normal form for $r\le2\eta$ in figure \[fig:PDFfluxdtheta\]. The PDF of the particle temperature flux across the scales is not self-similar with respect to its variance. Furthermore, the PDF becomes more symmetric as ${\text{\textit{St}}}_\theta$ is increased. This is associated with the increasingly non-local thermal dynamics of the particles, which allows the particle-pairs to traverse many scales of the flow with minimal changes in their temperature difference. Conclusions {#sec:concl} =========== Using direct numerical simulations, we have investigated the interaction between the scalar temperature field and the temperature of inertial particles suspended in the fluid, with one and two-way thermal coupling, in statistically stationary, isotropic turbulence. We found that the shape of the probability density function (PDF) of the fluid temperature gradients is not affected by the presence of the particles when two-way thermal coupling is considered, and scales with its variance. On the other hand, the variance of the fluid temperature gradients decreases with increasing ${\text{\textit{St}}}_\theta$, while ${\text{\textit{St}}}$ plays a negligible role. The PDF of the rate of change of the particle temperature, whose variance is associated with the thermal dissipation due to the particles, does not scale in a self-similar way with respect to its variance, and its kurtosis decreases with increasing ${\text{\textit{St}}}_\theta$. The particle temperature PDFs and their moments exhibit qualitatively different dependencies on ${\text{\textit{St}}}$ for the case with and without two-way thermal coupling. To obtain further insight into the fluid-particle thermal coupling, we computed the number density of particles conditioned on the magnitude of the local fluid temperature. In agreement with [@Bec2014], we observed that the particles cluster in the fronts of the temperature field. We also computed quantities related to moments of the particle heat flux conditioned on the magnitude of the local fluid temperature. These results showed how the particles tend to decrease the fluid temperature gradients, and that it is associated with the statistical alignments of the particle velocity and the local fluid temperature gradient field. The two-point temperature statistics were then examined to understand the properties of the temperature fluctuations across the scales of the flow. By computing the structure functions, we observed that the fluctuations of the fluid temperature increments are monotonically suppressed as ${\text{\textit{St}}}_\theta$ increases in the two-way coupled regime. The structure functions of the particle temperatures revealed the dominance of thermal caustics at the small scales, wherein the particle temperature differences at small separations rapidly increase as ${\text{\textit{St}}}_\theta$ and ${\text{\textit{St}}}$ are increased. This allows particles to come into contact with very large temperature differences, which has a number of important practical implications. The scaling exponents of the inertial particle temperature structure functions in the dissipation range revealed strongly multifractal behavior. PDFs of the fluid temperature increments at different separations were found to scale in a self-similar way with their variance, just as was found for the temperature gradients. However, PDFs of the particle temperature increments do not exhibit this self-similarity, and their non-Gaussianity is much stronger than that for the fluid. Finally, the flux of fluid temperature increments across the scales was found to decrease monotonically with increasing ${\text{\textit{St}}}_\theta$. The PDFs of this flux are strongly negatively skewed and have a negative mean value, indicating that the flux is predominately from the large to the smallest scales of the flow. In the two-way coupled regime, the presence of the inertial particles does not change the shape of the PDF. The PDF of the flux of particle temperature increments in the dissipation range becomes more and more symmetric as ${\text{\textit{St}}}_\theta$ is increased, associated with the increasingly non-local thermal dynamics of the particles. The results presented have revealed a number of non-trivial effects and behavior of the particle temperature statistics. In future work it will be important to consider the role of gravitation settling and coupling with water vapor fields, both of which are important for the cloud droplet problem. Moreover, it will be interesting to include the two-way momentum coupling and to consider the non-dilute regime. Acknowledgments =============== This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562 [@xsede]. Specifically, the Comet cluster was used under allocation CTS170009. The authors also acknowledge the computational resources provided by LaPalma Supercomputer at the Instituto de Astrofísica de Canarias through the Red Española de Supercomputación (project FI-2018-1-0044).

--- abstract: 'We introduce pretty clean modules, extending the notion of clean modules by Dress, and show that pretty clean modules are sequentially Cohen-Macaulay. We also extend a theorem of Dress on shellable simplicial complexes to multicomplexes.' address: - 'Jürgen Herzog, Fachbereich Mathematik und Informatik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany' - 'Dorin Popescu, Institute of Mathematics “Simion Stoilow”, University of Bucharest, P.O.Box 1-764, Bucharest 014700, Romania' author: - Jürgen Herzog and Dorin Popescu title: Finite filtrations of modules and shellable multicomplexes --- [^1] Introduction {#introduction .unnumbered} ============ Let $R$ be a Noetherian ring, and $M$ a finitely generated $R$-module. A basic fact in commutative algebra (see [@M Theorem 6.4]) says that there exists a finite filtration $$\mathcal{F}\: 0=M_0\subset M_1\subset \cdots \subset M_{r-1} \subset M_r=M$$ with cyclic quotients $M_i/M_{i-1}\iso R/P_i$ and $P_i\in \Supp(M)$. We call any such filtration of $M$ a [*prime filtration*]{}. The set of prime ideals $\{P_1,\ldots, P_r\}$ which define the cyclic quotients of $\mathcal{F}$ will be denoted by $\Supp(\mathcal{F})$. Another basic fact [@M Theorem 6.5] says that $$\Ass(M)\subset \Supp(\mathcal{F})\subset \Supp(M).$$ Let $\Min(M)$ denote the set of minimal prime ideals. Dress [@D] calls a prime filtration ${\mathcal F}$ of $M$ [*clean*]{}, if $\Supp(\mathcal{F})\subset\Min(M)$. The module $M$ is called [*clean*]{}, if $M$ admits a clean filtration. It is clear that for a clean filtration $\mathcal F$ of $M$ one has $$\Min(M)=\Ass(M)=\Supp({\mathcal F}).$$ Cleanness is the algebraic counterpart of shellability for simplicial complexes. Indeed, let $\Delta$ be a simplicial complex and $K$ a field. Dress [@D] showed that $\Delta$ is (non-pure) shellable in the sense of Björner and Wachs [@BW], if and only if the Stanley-Reisner ring $K[\Delta]$ is clean. On the other hand Stanley [@St] showed that if $\Delta $ is shellable, then $K[\Delta]$ is sequentially Cohen-Macaulay. In this paper we show more generally that any clean module over a Cohen-Macaulay ring which admits a canonical module is sequentially Cohen-Macaulay if all factors in the clean filtration are Cohen-Macaulay. In fact, we prove this result (Theorem \[sequentially\]) for an even larger class of modules which we call pretty clean. These modules are defined by the property that they have a prime filtration as above, and such that for all $i<j$ for which $P_i \subset P_j$ it follows that $P_i=P_j$. We now describe the content of this paper in more detail. In Section 1 we recall the concept of dimension filtrations introduced by Schenzel [@Sc], and note (Proposition \[characterization\]) that the dimension filtration of a module is characterized by the associated prime ideals of its factors. In the next section we discuss some basic properties of sequentially Cohen-Macaulay modules. Such modules were introduced by Schenzel [@Sc] and Stanley [@St]. It was Schenzel who observed that a module is sequentially Cohen-Macaulay if and only the non-zero factors of the dimension filtration are Cohen-Macaulay. The following section is devoted to introduce clean and pretty clean modules. We show that a pretty clean filtration $\mathcal F$ of a module $M$ satisfies $\supp({\mathcal F})=\Ass(M)$, and we give an example of a module $M$ which admits a prime filtration ${\mathcal F}$ with $\supp({\mathcal F})=\Ass(M)$ but which is not pretty clean. We also observe that that all pretty clean filtrations of a module have the same length. In Section 4 we show (Theorem \[sequentially\]) that under the mild assumptions, mentioned above, pretty clean modules are sequentially Cohen-Macaulay, and we show in Corollary \[interesting\] that under the same assumptions a module is pretty clean if and only if the factors in its dimension filtration are all clean. In Section 5 we give an interesting class of pretty clean rings, namely of rings whose defining ideal is of Borel type. This generalizes a result in [@HPV] where it is shown that such rings are sequentially Cohen-Macaulay. In the following section we consider graded and multigraded pretty clean rings and modules. Of particular interest is the case that $R=S/I$ where $S$ is a polynomial ring and $I\subset S$ a monomial ideal. Using a result of Nagel and Römer [@NR Theorem 3.1] we show that in this case the length of each multigraded pretty clean filtrations of $S/I$ is equals to the arithmetic degree of $S/I$. In [@St1] Stanley conjectured that the depth of $S/I$ is a lower bound for the ‘size’ of the summands in any Stanley decomposition of $S/I$. We show in Theorem \[stanley1\] that Stanley’s conjecture holds if $R$ is a multigraded pretty clean ring. In Section 7 we show that for a given prime filtration $\mathcal{F}\: 0=M_0\subset M_1\subset \cdots \subset M_{r-1} \subset M_r=M$ of $M$ with factors $M_i/M_{i-1}=R/P_i$ there exists irreducible submodules $P_j$-primary submodules $N_j$ of $M$ such that $M_i=\Sect_{j>i}^rN_j$ for $i=0,\ldots, r$. It turns out, as demonstrated in the next and the following sections, that this presentation of the modules $M_i$ is the algebraic interpretation of shellability for clean and pretty clean filtrations. This becomes obvious in the next section where we recall the theorem of Dress and show that the shelling numbers of a simplicial complex can be recovered from the graded clean filtration, see Proposition \[shelling numbers\]. In Section 9 we introduce multicomplexes. These are subsets $\Gamma\subset \NN^n_\infty$ which are closed under limits of sequence $a_i\in \Gamma$ with $a_i\leq a_{i+1}$ (componentwise), and have the property that whenever $a\in \Gamma$ and $b\leq a$ (componentwise), then $b\in \Gamma$. Here $\NN_\infty=\NN\union \{\infty\}$. We show that if $\Gamma$ is a multicomplex and $a\in \Gamma$, then there exists a maximal element $m\in \Gamma$ with $a\leq m$. Here we need that $\Gamma$ is closed with respect to limits of non-decreasing sequences. Then we define the facets of $\Gamma$ to be those elements $a\in\Gamma$ with the property that if $a\leq m$ and $m$ is maximal in $\Gamma$, then the infinite part of $a$ coincides with the infinite part of $m$, which means that the $i$th component of $a$ is infinite if and only if the $i$th component of $m$ is infinite. We show that each multicomplex has only a finite number of facets. Multicomplexes in $\NN^n_\infty$ correspond to monomial ideals in $S=K[x_1,\ldots,x_n]$. The monomial ideal $I$ defined by a multicomplex $\Gamma$ is the ideal spanned by all monomials whose exponents belong to $\NN^n\setminus \Gamma$. Our definition of the facets of $\Gamma$ is partly justified by the fact, shown in Lemma \[pairs\], that there is a bijection between the set of facets of $\Gamma$ and the standard pairs of $I$ as defined by Sturmfels, Trung and Vogel in [@STV]. However the main justification of the definition is given by Proposition \[multiprimary\] where we show that a pretty clean filtration of $S/I$ determines uniquely the facets of $\Gamma$. This result finally leads us to the definition of shellable multicomplexes. In Proposition \[extend\] we show that our definition of shellable multicomplexes extends the corresponding notion known for simplicial complexes. However the main result of the final section is Theorem \[multi2\] which asserts that for a monomial ideal $I$ the ring $S/I$ is multigraded pretty clean if and only if the corresponding multicomplex is shellable. The dimension filtration ======================== Let $M$ be an $R$-module of dimension $d$. In [@Sc] Schenzel introduced the [*dimension filtration*]{} $${\mathcal F}\: 0\subset D_0(M)\subset D_1(M)\subset \cdots \subset D_{d-1}(M)\subset D_d(M)=M$$ of $M$, which is defined by the property that $D_i(M)$ is the largest submodule of $M$ with $\dim D_i(M)\leq i$ for $i=0,\ldots,d$. It is convenient to set $D_{-1}(M)=(0)$. For all $i$ we set $\Ass^i(M)=\{P\in\Ass(M)\: \dim R/P=i\}$. The following characterization of a dimension filtration will be useful for us: \[characterization\] Let ${\mathcal F}\: 0\subset M_0\subset M_1\subset \cdots \subset M_{d-1}\subset M_d=M$ be a filtration of $M$. The following conditions are equivalent: 1. $\Ass(M_i/M_{i-1})=\Ass^{i}(M)$ for all $i$; 2. ${\mathcal F}$ is the dimension filtration of $M$. That the dimension filtration satisfies condition (a) has been shown by Schenzel in [@Sc Corollary 2.3 (c)]. For the converse we show that if ${\mathcal F}$ satisfies condition (a), then it is uniquely determined. Since the dimension filtration satisfies this condition, it follows then that ${\mathcal F}$ must be the dimension filtration of $M$. The integers $i$ for which $M_i=M_{i-1}$ are exactly those for which $\Ass^{i}(M)=\emptyset$, and hence this set is uniquely determined. Thus it remains to show, if $M_i\neq M_{i+1}$, then $M_{i}$ is uniquely determined. To this end, consider the multiplicatively closed set $$S=R\setminus\Union_{P\in\Ass(M),\atop \dim R/P\geq i+1}P,$$ and let $U$ be the kernel of the natural map $M\to M_S$. We claim that $M_i=U$. This will imply the uniqueness of the filtration. We first notice that $(M_j/M_{j-1})_S=0$ for $j\leq i$. Indeed, if $(M_j/M_{j-1})_S\neq 0$, then $PR_S\in\Ass_{R_S}(M_j/M_{j-1})_S$ for some $P\in \Ass_R(M_j/M_{j-1})$. By (a), $\dim R/P\leq i$, and hence $P\sect S\neq\emptyset$, a contradiction. We conclude that $(M_i)_S=0$, and hence $M_i\subset U$. Condition (a) implies that $$\Ass(M/M_{i})\subset \Union_{P\in\Ass(M),\atop \dim R/P\geq i+1}P.$$ Therefore all elements of $S$ are non-zerodivisors on $M/M_{i}$, and hence the natural map $M/M_i\to (M/M_i)_S$ is injective. This implies that $U\subset M_i$. It follows from condition (a) of Proposition \[characterization\] that if $M_i/M_{i-1}\neq 0$, then $M_i/M_{i-1}$ is equidimensional of dimension $i$ and has no embedded prime ideals. The arguments in the proof of the previous proposition yield the following description of the dimension filtration. \[description\] Let $\Sect_{i=1}^nQ_i$ be a primary decomposition of $(0)$ in $M$, where $Q_i$ is $P_i$-primary. Then $$D_i(M)=\Sect_{\dim R/P_j\geq i+1}Q_j,$$ for $i=1,\ldots, \dim M$. Sequentially Cohen-Macaulay modules =================================== Let $(R,\mm)$ be a local Noetherian ring, or a standard graded $K$-algebra with graded maximal ideal $\mm$. All modules considered will be finitely generated, and graded if $R$ is graded. The following definition is due to Stanley [@St Section II, 3.9], and Schenzel [@Sc]. \[stanley\] *Let $M$ be a finitely generated (graded) $R$-module. A finite filtration $$0=M_0\subset M_1\subset M_2\subset\ldots\subset M_r=M$$ of $M$ by (graded) submodules of $M$ is called a [*CM-filtration*]{}, if each quotient $M_i/M_{i-1}$ is Cohen-Macaulay (CM for short), and $$\dim (M_1/M_0)<\dim(M_2/M_1)<\ldots <\dim(M_r/M_{r-1}).$$* The module $M$ is called [*sequentially Cohen-Macaulay*]{} if $M$ admits a CM-filtration. We recall a few basic facts whose proof in the graded case can be found in [@HS], but which are proved word by word in the same way in the local case. \[exti\] Let $R$ be Cohen-Macaulay of dimension $n$ with canonical module $\omega_R$. Suppose that $M$ is sequentially CM with a CM-filtration as in [*\[stanley\]*]{}, and assume further that $d_i=\dim M_i/M_{i-1}$ for $i=1,\ldots, r$. Then 1. $\Ext^{n-d_i}_R(M,\omega_R)\iso \Ext^{n-d_i}_R(M_i/M_{i-1},\omega_R)$; 2. $\Ext^{n-d_i}_R(M,\omega_R)$ is CM of dimension $d_i$ for $i=1,\ldots,r$; 3. $\Ext^j_R(M,\omega_R)=0$ if $j\not\in\{n-d_1,\ldots,n-d_r\}$; 4. $\Ext^{n-d_i}_R(\Ext^{n-d_i}_R(M,\omega_R),\omega_R)\iso M_i/M_{i-1}$ for $i=1,\ldots,r$. \[Ass\] With the assumptions and notation introduced in Proposition \[exti\] we have $$\Ass(\Ext^{n-d_i}_R(M,\omega_R))=\Ass(M_i/M_{i-1}).$$ Let ${\bf x}=x_1,\ldots, x_{n-d_i}$ be a maximal regular sequence in $\Ann(M_i/M_{i-1})$, and set $S=R/({\bf x})$. Then (a) implies that $\Ext^{n-d_i}_R(M,\omega_R)\iso \Hom_S(M_i/M_{i-1},\omega_S)$, and that $M_i/M_{i-1}$ may be viewed a maximal CM module over $S$. It follows that $$\Ass(\Ext^{n-d_i}_R(M,\omega_R))=\Supp(M_i/M_{i-1})\sect\Ass(\omega_S)=\Supp(M_i/M_{i-1})\sect \Min(S).$$ Since $M_i/M_{i-1}$ is a maximal CM module over $S$, we have $$\Ass(M_i/M_{i-1})=\Min(M_i/M_{i-1})=\Supp(M_i/M_{i-1})\sect \Min(S).$$ This proves the assertion. It follows from Proposition \[exti\] that if $M$ is sequentially CM, then the non-zero modules among the $\Ext_R^{n-i}(M,\omega_R)$ are CM of dimension $i$. Peskine noticed that this property characterizes sequentially CM modules. Indeed one has \[peskine\] The following two conditions are equivalent: 1. $M$ is sequentially CM; 2. for all $i$, the modules $\Ext_R^{n-i}(M,\omega_R)$ are either $0$ or CM of dimension $i$. We conclude this section with a result due to Schenzel [@Sc Corollary 2.3]. \[faridi\] Let $M$ be sequentially CM with a CM-filtration as above. Then $\Ass(M_i/M_{i-1})=\Ass^{d_i}(M)$ for all $i$. In particular, $\Ass(M)=\Union_i\Ass(M_i/M_{i-1})$. Since $M_i/M_{i-1}$ is CM of dimension $d_i$, it follows that $\dim R/P=d_i$ for all $P\in \Ass(M_i/M_{i-1})$. Therefore it suffices to show that $\Ass(M)=\Union_i\Ass(M_i/M_{i-1})$. Using the fact that for an exact sequence $0\to U\to V\to W\to 0$ of $R$-modules one has that $\Ass(V)\subset \Ass(U)\union \Ass(W)$, one easily concludes by induction on the length $r$ of the filtration, that $\Ass(M)\subset \Union_i\Ass(M_i/M_{i-1})$. Conversely, let $P\in \Ass(M_i/M_{i-1})$. Then $(M_j/M_{j-1})_P=0$ for all $j<i$, since $\dim M_j/M_{j-1}<\dim R/P$. This implies that $(M_{i-1})_P=0$, so that $(M_i/M_{i-1})_P=(M_i)_P$. Thus $PR_P\in \Ass_{R_P}(M_i)_P$, and hence $P\in \Ass(M_i)\subset \Ass(M)$. Combining Proposition \[faridi\] with Corollary \[Ass\] we obtain \[need\] Let $M$ be sequentially CM, then $\Ass(\Ext_R^{n-i}(M,\omega_R))=\Ass^i(M)$ for all $i$. The following characterization of sequentially Cohen-Macaulay modules, due to Schenzel [@Sc Proposition 4.3], is a consequence of Proposition \[characterization\] and Proposition \[faridi\]. \[comparison\] A module $M$ is sequentially CM, if and only if the factors in the dimension filtration of $M$ are either 0 or CM. Clean and pretty clean modules ============================== Let $R$ be a Noetherian ring, and $M$ a finitely generated $R$-module. Recall from the introduction that according to Dress [@D] a prime filtration $\mathcal F$ of $M$ is called [*clean*]{} if $\Supp({\mathcal F})=\Min(M)$, and that $M$ itself is called [*clean*]{} if $M$ admits a clean filtration. \[character\] Let $\mathcal F$ be a prime filtration of $M$. The following conditions are equivalent: 1. $\mathcal F$ is a clean filtration of $M$; 2. For all $P, Q\in \Supp({\mathcal F})$ with $P\subset Q$ one has $P=Q$. (a)(b) is trivial. Conversely suppose ${\mathcal F}: 0=M_0\subset M_1\subset\cdots\subset M_r=M$ with $M_i/M_{i-1}=R/P_i$ and let $P\in \Supp({\mathcal F})$. Since there are no non-trivial inclusions between the prime ideals in $\Supp({\mathcal F})$ it follows that $M_P$ has a filtration $(0)=(M_0)_P\subset (M_1)_P\subset \cdots \subset (M_r)_P=M_P$ such that $$(M_i)_P/(M_{i-1})_P = \left\{ \begin{array}{lll} R_P/PR_P, & \text{if} & P=P_i,\\ 0, & \mbox{if} & P\neq P_i. \end{array} \right.$$ Hence we see that $\Ass_{R_P}(M_P)=\{PR_P\}$, and so $P\in \Ass(M)$. It follows that $\Supp({\mathcal F})=\Ass(M)$. Applying again assumption (b), we conclude that $\Ass(M)=\Min(M)$. \[induced\] Let $0=M_0\subset M_1\subset \ldots \subset M_{r-1}\subset M_r=M$ be a clean filtration of $M$. Then for all $i=0,\ldots,r$ $$0=M_i/M_i\subset M_{i+1}/M_i\subset\ldots \subset M_{r-1}/M_i\subset M_r/M_i,$$ and $$0=M_0\subset M_1\subset \ldots\subset M_{i-1}\subset M_i$$ are clean filtrations. In particular, $M_i$ and $M/M_i$ are clean. A weakening of condition (b) of Lemma \[character\] leads to \[pretty\] *A prime filtration ${\mathcal F}\: 0=M_0\subset M_1\subset \ldots \subset M_{r-1}\subset M_r=M$ of $M$ with $M_i/M_{i-1}=R/P_i$ is called [*pretty clean*]{}, if for all $i<j$ for which $P_i\subset P_j$ it follows that $P_i=P_j$.* In other words, a proper inclusion $P_i\subset P_j$ is only possible if $i>j$. The module $M$ is called [*pretty clean*]{}, if it has a pretty clean filtration. A ring is called pretty clean if it is a pretty clean module, viewed as a module over itself. [*Let ${\mathcal F}\: 0=M_0\subset M_1\subset \ldots \subset M_{r-1}\subset M_r=M$ be a pretty clean filtration of $M$. It follows immediately from the definition that for all $i$ the filtrations $$0=M_i/M_i\subset M_{i+1}/M_i\subset\ldots \subset M_{r-1}/M_i\subset M_r/M_i,$$ and $$0=M_0\subset M_1\subset \ldots\subset M_{i-1}\subset M_i$$ are pretty clean.* ]{} \[important\] Let ${\mathcal F}\: 0=M_0\subset M_1\subset \ldots \subset M_{r-1}\subset M_r=M$ be a pretty clean filtration of $M$. Then $P_i\in \Ass(M_i)$ for all $i$. We use the same argument as in the proof of Lemma \[character\]: set $P=P_i$. Then $0=(M_0)_P\subset (M_1)_P\subset \ldots \subset (M_{i-1})_P\subset (M_{i})_P$ is a finite filtration of the $R_P$-module $(M_i)_P$. Let $j\leq i$. Since ${\mathcal F}$ is pretty clean we get $$(M_j)_P/(M_{j-1})_P = \left\{ \begin{array}{lll} R_P/PR_P, & \text{if} & P_j=P,\\ 0, & \mbox{if} & P_j\neq P. \end{array} \right.$$ This implies that $PR_P\in \Ass_{R_p}((M_i)_P)$. Therefore $P\in \Ass(M_i)$. \[nice\] Let ${\mathcal F}$ be a pretty clean filtration of $M$. Then $\Supp({\mathcal F})=\Ass(M)$. For all $i$ we have $P_i\in \Ass(M_i)\subset \Ass(M)$. Therefore $\Supp({\mathcal F})\subset \Ass(M)$. The other inclusion holds for any prime filtration. \[equal\] Let $M$ be a pretty clean module. The following conditions are equivalent: 1. $M$ is clean; 2. $\Ass(M)=\Min(M)$. *Let $S=K[x,y]$ be the polynomial ring over the field $K$, $I\subset S$ the ideal $I=(x^2,xy)$ and $R=S/I$. Then $R$ is pretty clean but not clean. Indeed, $0\subset (x)\subset R$ is a pretty clean filtration of $R$ with $(x)=R/(x,y)$, so that $P_1=(x,y)$ and $P_2=(x)$. $R$ is not clean since $\Ass(R)\neq \Min(R)$.* Note $R$ has a different prime filtration, namely, ${\mathcal G}: 0\subset (y)\subset (x,y)\subset R$ with factors $(y)=R/(x)$ and $(x,y)/(y)=R/(x,y)$. Hence this filtration is not pretty clean, even though $\Supp({\mathcal G})=\Ass(M)$. On the other hand, in the next section we give an example of a module which admits a prime filtration ${\mathcal F}$ with $\supp({\mathcal F})=\Ass(M)$, but which is not pretty clean. We conclude this section by showing that all pretty clean filtrations have the same length. For $\pp\in\Spec(R)$ the number $$\mult_M(\pp)=\length (H^0_\pp(M_\pp)),$$ is called the [*length multiplicity*]{} of $\pp$ with respect to $M$. Obviously, one has $\mult_M(\pp)>0$, if and only if $\pp\in \Ass(M)$. Localizing a pretty clean filtration of $M$ we immediately get \[allthesame\] Let $M$ be a pretty clean module. Then all pretty clean filtrations of $M$ have the same length, namely their common length equals $\sum_{\pp\in\Ass(M)}\mult_M(\pp)$. Assume now that $(R,\mm)$ is local. Recall that the [*arithmetic degree*]{} of $M$ is defined to be $\sum_\pp\mult_M(\pp)\deg(R/\pp)$ where $\deg(R/\pp)$ is the multiplicity of the associated graded ring of $R/\pp$. The preceding lemma shows that the length of a pretty clean filtration is bounded above by the arithmetic degree of the module, and equals the arithmetic degree if and only if $\deg R/\pp=1$ for all $\pp\in\Ass(M)$. Pretty clean modules are sequentially Cohen-Macaulay ==================================================== The purpose of this section is to show \[sequentially\] Let $R$ be a local or standard graded CM ring admitting a canonical module $\omega_R$, and let $M$ be an $R$-module with pretty clean filtration ${\mathcal F}$ such that $R/P$ is Cohen-Macaulay for all $P\in \Supp({\mathcal F})$. Furthermore suppose that $M$ is graded if $R$ is graded. Then $M$ is sequentially Cohen-Macaulay. Moreover, if $\dim R/P=\dim M$ for all $P\in\Supp({\mathcal F})$, then $M$ is clean and Cohen-Macaulay. Let $n=\dim R$. We may assume that $R$ is local. In the graded case the arguments are the same. For all $i$ we show: the module $\Ext_R^{n-i}(M,\omega_R)$ is either $0$ or Cohen-Macaulay of dimension $i$. We show this by induction on the length $r$ of the pretty clean filtration ${\mathcal F}\: 0=M_0\subset M_1\subset\cdots \subset M_{r-1}\subset M_r=M$ of $M$. Since, as we already noticed, the module $U=M_{r-1}$ is pretty clean with a pretty clean filtration of length $r-1$, we may assume by induction that $U$ is sequentially Cohen-Macaulay. Let $M/U=R/P$. By hypothesis, $R/P$ is Cohen-Macaulay, say of dimension $d$. The short exact sequence $$0\To U\To M\To R/P\To 0$$ gives rise to the long exact sequence $$\cdots \Ext_R^{n-i-1}(U,\omega_R)\to \Ext_R^{n-i}(R/P,\omega_R)\to \Ext_R^{n-i}(M,\omega_R)\to\Ext_R^{n-i}(U,\omega_R)\to\cdots,$$ Since $$\Ext_R^{n-i}(R/P,\omega_R)= \left\{ \begin{array}{lll} \omega_{R/P}, & \text{if} & i=d\,\\ 0, & \mbox{if} & i\neq d, \end{array} \right.$$ it follows that $\Ext_R^{n-i}(M,\omega_R)\iso \Ext_R^{n-i}(U,\omega_R)$ for all $i\neq d,d+1$. Thus for such $i$ we have $\Ext_R^{n-i}(M,\omega_R)$ is Cohen-Macaulay of dimension $i$ if not the zero module. Moreover we have the exact sequence $$\begin{aligned} 0&\to &\Ext_R^{n-d-1}(M,\omega_R)\to \Ext_R^{n-d-1}(U,\omega_R)\to \Ext_R^{n-d}(R/P,\omega_R)\\ &\to& \Ext_R^{n-d}(M,\omega_R)\to \Ext_R^{n-d}(U,\omega_R)\to 0.\end{aligned}$$ Suppose the map $\Ext_R^{n-d-1}(U,\omega_R)\to \Ext_R^{n-d}(R/P,\omega_R)\iso\omega_{R/P}$ is not the zero map. Then its image $C\subset \omega_{R/P}$ is not zero. Since $R/P$ is domain, $\omega_{R/P}$ may be identified with an ideal in $R/P$, see [@BH Proposition 3.3.18]. Hence also $C$ may be identified with an ideal in $R/P$. Again using that $R/P$ is a domain, we conclude that $CR_P\neq 0$. It follows that $\Ext_R^{n-d-1}(U,\omega_R)_P\neq 0$, and so the set $$\Ass_{R_P}(\Ext_R^{n-d-1}(U,\omega_R)_P)=\{QR_P\: Q\in \Ass_R(\Ext_R^{n-d-1}(U,\omega_R)),\; Q\subset P\}$$ is not empty. Thus there exists $Q\in \Ass_R(\Ext_R^{n-d-1}(U,\omega_R))$ with $Q\subset P$. By Corollary \[need\] we know that $\Ass_R(\Ext_R^{n-d-1}(U,\omega_R))\subset \Ass_R^{d+1} (U)$. Therefore, since $\dim R/P=d$, the inclusion $Q\subset P$ must be proper. But this contradicts the fact that ${\mathcal F}$ is a pretty clean filtration of $M$. It follows now that $$\Ext_R^{n-d-1}(M,\omega_R)\iso \Ext_R^{n-d-1}(U,\omega_R),$$ and that the sequence $$\begin{aligned} \label{exact} 0\To\omega_{R/P}\to \Ext_R^{n-d}(M,\omega_R)\To \Ext_R^{n-d}(U,\omega_R)\To 0\end{aligned}$$ is exact. Using the induction hypothesis we conclude that $\Ext_R^{n-d-1}(M,\omega_R)$ is either Cohen-Macaulay of dimension $d+1$ or the zero module, and that $\Ext_R^{n-d}(M,\omega_R)$ is Cohen-Macaulay of dimension $d$. If $\dim R/P=\dim M$ for all $P\in \Supp({\mathcal F})$, then the pretty clean filtration ${\mathcal F}$ is necessarily clean, and $M$ is unmixed. Since any unmixed sequentially Cohen-Macaulay module is Cohen-Macaulay, all assertions are proved. As a consequence of the previous theorem we get \[interesting\] Let $M$ be an $R$-module. If the non-zero factors of the dimension filtration of $M$ are clean, then $M$ is pretty clean. Conversely assume that $R$ is a local or standard graded CM ring with canonical module $\omega_R$, and that $M$ admits a pretty clean filtration ${\mathcal F}$ such that $R/P$ is CM for all $P\in \Supp({\mathcal F})$. Furthermore assume that $M$ is graded if $R$ is graded. Then the non-zero factors of the dimension filtration of $M$ are clean. Suppose all factors $D_i(M)/D_{i-1}(M)$ in the dimension filtration of $M$ are clean. Then it is obvious that the dimension filtration can be refined to yield a pretty clean filtration of $M$. We prove the second statement of the corollary by induction on the length $r$ of the filtration $\mathcal F$. The claim is obvious if $r=1$. Now let $r>1$, and set $U=M_{r-1}$. We obtain the exact sequence $0\to U\to M\to R/P\to 0$ with $P\in \Spec(R)$. Let $d=\dim R/P$. Then, as we have seen in the proof of Theorem \[sequentially\], one has $ \Ext_R^{n-i}(M,\omega_R)\iso \Ext_R^{n-i}(U,\omega_R)$ for all $i\neq d$, as well as the exact sequence $$0\To\omega_{R/P}\to \Ext_R^{n-d}(M,\omega_R)\To \Ext_R^{n-d}(U,\omega_R)\To 0.$$ Since $M$ is sequentially CM by the previous theorem, these isomorphisms together with Proposition \[exti\](d) and Corollary \[comparison\] imply that $$D_i(M)/D_{i-1}(M)\iso D_i(U)/D_{i-1}(U)$$ for $i\neq d$. Hence, since the factors $D_i(U)/D_{i-1}(U)$ are clean by induction hypothesis, the same is true for the factors $D_i(M)/D_{i-1}(M)$ with $i\neq d$. Applying the functor $\Ext_R^{n-d}(-,\omega_R)$ to the above exact sequence and using Proposition \[exti\](d) again we obtain the exact sequence $$0\To D_d(U)/D_{d-1}(U)\To D_d(M)/D_{d-1}(M)\To R/P\to 0.$$ Since all modules in this exact sequence are of dimension $d$, and since $D_d(U)/D_{d-1}(U)$ is clean, it follows that $D_d(M)/D_{d-1}(M)$ is clean as well. \[conclusion\] Let $S=K[x_1,\ldots,x_n]$ be a the polynomial ring and $I\subset S$ a monomial ideal. Then the following conditions are equivalent: 1. $S/I$ is pretty clean; 2. $S/I$ is sequentially CM, and the non-zero factors in the dimension filtration of $S/I$ are clean; 3. the non-zero factors in the dimension filtration of $S/I$ are clean. (a)(b): Since the associated prime ideals of $S/I$ are all generated by subsets of $\{x_1,\ldots,x_n\}$, all hypotheses Theorem \[sequentially\] and Corollary \[interesting\] are satisfied, so that the assertions follow. (b)(c) is trivial. (c)(a): The refinement of the dimension filtration by the clean filtrations of the non-zero factors gives us the desired pretty clean filtration of $S/I$. *Let $S=K[x,z,u,v]$, and consider the ideals $L=(u,v,z)$, $Q_1=(x,z^2)$, $Q_2=(x,v^2,z^3)$ and $I=L\sect Q_1\sect Q_2$. We claim that the module $M=L/I$ is not pretty clean, but that $M$ has a prime filtration $\mathcal F$ with $\Supp({\mathcal F})=\Ass(M)$.* Note that $(L\sect Q_1)\sect (L\sect Q_2)$ modulo $I$ is an irredundant primary decomposition of $(0)$ in $M$. Hence since $\emptyset \neq \Ass(L/L\sect Q_i)\subset \Ass(S/Q_i)=\{P_i\}$ with $P_1=(x,z)$ and $P_2=(x,v,z)$ we see that $\Ass(M)=\{P_1,P_2\}$. It follows from Corollary \[description\] that $D_1(M)=(L\sect Q_1)/I$ and that $D_2(M)=M$. We show that $D_2(M)/D_1(M)=L/L\sect Q_1$ is not clean. Indeed, suppose $L/L\sect Q_1$ is clean. Then, since $\Ass(L/L\sect Q_1)=\{P_1\}$, this module has a filtration with all factors isomorphic to $S/P_1$, and the number of these factors equals the length of the $S_{P_1}$-module $(L/L\sect Q_1)_{P_1}=S_{P_1}/Q_1S_{P_1}$. This length is obviously $2$. On the other hand, since $L/L\sect Q_1$ is generated by 3 elements, it cannot have a filtration with two factors, both of them being cyclic. Knowing now that $D_2(M)/D_1(M)$ is not clean, we conclude from Corollary \[interesting\] that $M$ is not pretty clean. Finally we construct a prime filtration ${\mathcal F}$ of $M$ with $\Supp({\mathcal F})=\Ass(M)$. The filtration $\mathcal F$ will be the following refinement of the dimension filtration. Denote by $\bar{a}$ the residue class of an element $a\in L$ in $L/L\sect Q_1=D_2(M)/D_1(M)$. Then $(0)\subset (\bar{z})\subset(\bar{z},\bar{v})\subset (\bar{z},\bar{v},\bar{u})=D_2(M)/D_1(M)$ is a filtration of $D_2(M)/D_1(M)$ with $(\bar{z})=S/P_1$, $(\bar{z},\bar{v})/(\bar{z})=S/P_1$ and $(\bar{z},\bar{v},\bar{u})/(\bar{z},\bar{v})=S/P_2$. Furthermore, denote the residue class of an element $a\in S$ in $S/I$ by $\tilde{a}$. Then $D_1(M)=L\sect Q_1/I$ is generated by $\tilde{z}^2$, and so $D_1(M)\iso S/(x,z,v^2)$. It is clear that this filtration can be further refined so that all factors are isomorphic to $S/P_2$. Classes of pretty clean rings ============================= Let $S=K[x_1,\ldots, x_n]$ be the polynomial ring over a field $K$. In this section we present a class of monomial ideals for which $S/I$ is pretty clean. Quite generally we have \[total\] Let $I\subset S$ be a monomial ideal, and suppose that $\Ass(S/I)$ is totally ordered by inclusion. Then $S/I$ is pretty clean. Let $\Ass(S/I)=\{P_1,\ldots,P_r\}$ and suppose that $P_1\supset P_2\supset \cdots \supset P_r$, and set $d_i=\dim S/P_i$ for $i=1,\ldots, r$. The ideal $I$ can be written as an intersection $I=\Sect_{i=1}^rQ_i$ where each $Q_i$ is a $P_i$-primary monomial ideal. There exist subsets $J_i\subset [n]$ such that $P_i$ is generated by $x_j$ with $j\in J_i$. It follows from our assumption that $J_1\supset J_2\supset \cdots \supset J_r$. Set $U_i=\Sect_{j>i}Q_j$. Then according to Corollary \[description\] we have $U_i/I=D_{d_i}(S/I)$. By Corollary \[conclusion\](c) it suffices to show that $U_i/Q_i\sect U_i$ is clean for all $i$. We have $\emptyset \neq \Ass(U_i/Q_i\sect U_i)\subset \Ass(S/Q_i)=\{P_i\}$. Let $S'$ be the polynomial ring over $K$ in the variables $x_j$ with $j\in J_i$, and set $P_i'=P_i\sect S'$. Then $P_i'$ is the graded maximal ideal of $S'$ and $P_i=P_i'S$. Similarly, since $J_k\subset J_i$ for $k\geq i$, we have $Q_k=Q_k'S$ and $U_k=U_k'S$ where $Q_k'=Q_k\sect S'$ and $U_k'=U_k\sect S'$. The $S'$-module $U'_i/Q'_i\sect U'_i$ is a clean since it is of finite length. By base change, $U_i/Q_i\sect U_i\iso (U'_i/Q'_i\sect U'_i)\tensor_{S'}S$ is a clean $S$-module. In Gröbner basis theory, Borel fixed ideals play an important role as they are just the generic initial ideals of graded ideals in a polynomial ring. By a theorem of Bayer and Stillman (see [@Ei Proposition 15.24]) a Borel fixed ideal $I\subset S=K[x_1,\ldots,x_n]$ is a monomial ideal with the property that $$\begin{aligned} \label{borel} I:x_j^\infty=I:(x_1,\ldots,x_j)^\infty\end{aligned}$$ for all $j=1,\ldots,n$. In [@HPV], any monomial ideal satisfying condition (\[borel\]) is called an ideal of [*Borel type*]{}, and it is shown that $S/I$ is sequentially Cohen-Macaulay if $I$ is of Borel type. Here we show the following slightly stronger \[cleanborel\] Let $I\subset S$ be an ideal of Borel type. Then $S/I$ is pretty clean. Let $P\in \Ass(S/I)$, and let $j$ be the largest integer such that $x_j\in P$. There exists a monomial $u\in S$ such that $(I,u)/I\iso S/P$. Since $x_ju\in I$ it follows that $u\in I:x_j^{\infty}$, and hence $u\in I:(x_1,\ldots,x_j)^{\infty}$. Therefore $u(x_1,\ldots, x_j)^k\subset I$ for some integer $k>0$,and hence $(x_1,\ldots, x_j)^k\subset P$. Since $P$ is prime ideal we conclude that $(x_1,\ldots, x_j)\subset P$. By the definition of $j$, it follows then that $P=(x_1,\ldots, x_j)$. Thus the associated prime ideal of $S/I$ are totally ordered and the assertion follows from Proposition \[total\]. Graded pretty clean modules =========================== Let $K$ be a field and $R$ a standard graded $K$-algebra, and let $M$ be a graded $R$-module. A prime filtration of $M$ $${\mathcal F}\: (0)=M_0\subset M_1\subset M_{r-1}\subset M_r=M.$$ is called [*graded*]{}, if all $M_i$ of $M$ are graded submodules of $M$, and if there are graded isomorphisms $M_i/M_{i-1}\iso R/P_i(-a_i)$ with some $a_i\in\ZZ$ and some graded prime ideals $P_i$. The module $M$ is called a [*graded (pretty) clean module*]{}, if it admits a (pretty) clean filtration which is a graded prime filtration. Similarly we define multigraded filtrations and multigraded (pretty) clean modules. We denote by $(N)_i$ the $i$th graded component of a graded $R$-module $N$, and by $$\Hilb(N)=\sum_i\dim_K(N)_it^i\in\ZZ[t,t^{-1}]$$ its Hilbert-series. By the additivity of the Hilbert-series, one obtains for a module with a graded prime filtration as above the Hilbert-series $$\Hilb(M)=\sum_{i=1}^r\Hilb(R/P_i)t^{a_i}.$$ We now consider a more specific case \[monomial\] Let $S=K[x_1,\ldots, x_n]$ be the polynomial ring, and $I\subset S$ a monomial ideal. Assume that $S/I$ is a graded pretty clean ring whose graded pretty clean filtration has the factors $M_j/M_{j-1}\iso S/P_j(-a_j)$ for $j=1,\ldots, r$, $a_j\in \NN$ and $P_j\in\Ass(S/I)$. For all $k$ and $i$ set $$h_{ki}=|\{j\: a_j=k,\; \dim S/P_j=i\}|.$$ Then $$\Hilb(S/I)=\sum_iH_i(t) \quad \text{with}\quad H_i(t)=\frac{Q_i(t)}{(1-t)^i} \quad\text{where} \quad Q_i(t)=\sum_kh_{ki}t^k.$$ We have $$\begin{aligned} \Hilb(S/I)&=&\sum_i \sum_{j\atop \dim S/P_j=i}\Hilb(S/P_j)t^{a_j}\\ &=& \sum_i(\sum_{j\atop \dim S/P_j=i}t^{a_j})/(1-t)^i.\end{aligned}$$ The last equality holds, since all associated prime ideals of $S/I$ are generated by subsets of the variables. Finally the desired formula follows, if we combine in the sum $\sum_{j,\; \dim S/P_j=i}t^{a_j}$ all powers of $t$ with the same exponent. The attentive reader will notice the similarity of formula \[monomial\] with the formula of McMullen and Walkup for shellable simplicial complexes, see [@BH Corollary 5.1.14]. The precise relationship will become apparent in Section 8 where the numbers $a_j$ are interpreted as shelling numbers. We now derive similar formulas for the modules $\Ext^i_S(M,\omega_S)$ when $M$ is a graded pretty clean module. Suppose $\dim S=n$. Using the graded version of the exact sequence (\[exact\]) in the proof of Theorem \[sequentially\], and induction on the length of the pretty clean filtration it follows easily that $$\Hilb(\Ext_S^i(M,\omega_S))=\sum_{j\atop \dim S/P_j=n-i}\Hilb(\omega_{S/P_j})t^{-a_j} \quad\text{for}\quad i=0,\ldots, \dim M.$$ In particular we have \[hilbert\] With the assumptions and notation of [*\[monomial\]*]{}, one has $$\Hilb(\Ext_S^i(S/I,S(-n)))=(\sum_k h_{k,n-i}t^{n-i-k})/(1-t)^{n-i}= (-1)^{n-i}H_{n-i}(t^{-1}).$$ The first equality follows from the fact that $\omega_{S/P_j}=S/P_j(-(n-i))$ if $\dim S/P_j=n-i$, so that $\Hilb(\omega_{S/P_j})=t^{n-i}/(1-t)^{n-i}$. To obtain the second equality, we divide numerator and denominator of $(\sum_k h_{k,n-i}t^{n-i-k})/(1-t)^{n-i}$ by $t^{n-i}$ and get $$(\sum_k h_{k,n-i}t^{n-i-k})/(1-t)^{n-i}=(\sum_k h_{k,n-i}t^{-k})/(t^{-1}-1)^{n-i}=(-1)^{n-i}H_{n-i}(t^{-1}).$$ Let $S=K[x_1,\ldots, x_n]$ and $M$ be a graded $S$-module. We set $$b_j=\min\{k\: \Ext_S^j(M, S(-n))_k\neq 0\}.$$ Then the regularity of $M$ is given by $$\reg(M)=\max\{n-j-b_j\: j=0,\ldots, \},$$ cf. [@Ei Section 20.5]. \[regularity\] Let $S=K[x_1,\ldots, x_n]$ be the polynomial ring, and $I\subset S$ a monomial ideal. Assume that $S/I$ is a graded pretty clean ring with filtration as in [*\[monomial\]*]{}. Then 1. $\reg(S/I)=\max\{k\: h_{ki}\neq 0 \ \ \mbox{for some i}\}=\max\{a_j\: j=0,\ldots r\}$; 2. $\Hilb(D_i(S/I)/D_{i-1}(S/I))=H_i(t)$ for all $i$. \(a) the first equality follows immediately from Proposition \[hilbert\] and the definition of $\reg(S/I)$. The second equality results from the definition of the numbers $h_{ki}$. \(b) By Proposition \[exti\] we have $$D_i(S/I)/D_{i-1}(S/I)\iso\Ext_S^{n-i}(\Ext_S^{n-i}(S/I,\omega_S),\omega_S).$$ Thus the assertion follows from Proposition \[hilbert\] and [@BH Theorem 4.4.5(a)]. We denote by $e(M)$ the multiplicity of a graded module. \[independent\] Let $i$ and $k$ be integers. Then the number of factors $S/P(-k)$ in a graded pretty clean filtration of $S/I$ satisfying $\dim S/P=i$ is independent of the chosen filtration. In particular, all graded pretty clean filtrations of $S/I$ have the same length, namely $\sum_{i=0}^ne(\Ext_S^i(S/I,S))$, and this number equals the arithmetic degree of $S/I$. The number in question equals $h_{ki}$, the $k$-th coefficient of the $h$-vector of $D_i(S/I)/D_{i-1}(S/I)$. Hence this number only depends on $S/I$. Moreover, it follows that the length of a graded pretty clean filtration of $S/I$ equals $\sum_{i=0}^nQ_i(1)$. As a consequence of Proposition \[hilbert\] and [@BH Proposition 4.1.9] we have that $Q_i(1)=e(\Ext_S^i(S/I,S))$. In [@NR Theorem 3.11] Nagel and Römer have shown that the arithmetic degree of a sequentially Cohen-Macaulay $R$-module $M$ equals the number $\sum_{i=0}^ne(\Ext_R^i(M,\omega_R))$. Since by Theorem \[sequentially\], $S/I$ is sequentially CM, all assertions follow. We would like to remark that the fact that the length of all pretty clean filtrations of $S/I$ have length equal to the arithmetic degree of $S/I$ also follows from Lemma \[allthesame\]. Suppose that $I\subset S$ is a monomial ideal, and that $\mathcal F$ is a multigraded prime filtration of $S/I$ with factors $(S/P_i)(-a_i)$, $i=1,\ldots,r$, where $a_i\in \NN^n$. Then this filtration decomposes $S/I$ as a multigraded $K$-vectorspace, that is, we have $$S/I\iso \Dirsum_{i=1}^rS/P_i(-a_i).$$ Each module $M_i$ in the filtration $\mathcal F$ is of the form $I_i/I$ where $I_i$ is a monomial ideal. The monomials not belonging to $I_i$ form a $K$-basis of $(S/I)/M_i=S/I_i$, and so $S/I=(S/I)/M_i\dirsum M_i$ decomposes naturally as a $K$-vectorspace. Identifying $S/P_i(-a_i)=M_i/M_{i-1}\subset S/I_{i-1}$ with its image in $S$, we get $S/P_i(-a_i)=u_iK[Z_i]$ where $u_i=\prod_{j=1}^nx_j^{a_i(j)}$ and $Z_i=\{x_j\: j\not\in P_j\}$. Thus $$S/I= \Dirsum_{i=1}^ru_iK[Z_i].$$ Any decomposition of $S/I$ as a direct sum of $K$-vectorspaces of the form $uK[Z]$ where $Z$ is a subset of $X=\{x_1,\ldots,x_n\}$ and $u$ is a monomial of $K[X]$ is called a [*Stanley decomposition*]{}. Stanley decompositions have been studied in various combinatorial and algebraic contexts, see [@A], [@HT], and [@MS]. Not all Stanley decompositions arise from prime filtrations, see [@MS]. Stanley [@St1] conjectured that there always exists a Stanley decomposition $S/I= \Dirsum_{i=1}^ru_iK[Z_i]$ such that $|Z_i|\geq \depth S/I$. In [@A] Apel studied cases in which Stanley’s conjecture holds. We conclude this section by showing \[stanley1\] Let $I\subset S$ a monomial ideal, and suppose that $S/I$ is a multigraded pretty clean ring. Then Stanley’s conjecture holds for $S/I$. Stanley’s conjecture follows if we can show that there exist a multigraded prime filtration $\mathcal F$ of $S/I$ with factors $S/P_i(-a_i)$ such that $\depth S/P_i\geq \depth S/I$. Since $S/I$ is multigraded pretty clean, it follows from Corollary \[conclusion\] that all nonzero factors $D_i(S/I)/D_{i-1}(S/I)$ of the dimension filtration are clean. Moreover, since $S/I$ is sequentially Cohen-Macaulay, it follows from Proposition \[exti\] that $\depth S/I=t$ where $t=\min\{i\: D_i(S/I)/D_{i-1}(S/I)\neq 0\}$. Since $D_i(S/I)/D_{i-1}(S/I)$ is clean, we obtain a pretty clean filtration of $S/I$ as a refinement of the dimension filtration by the clean filtrations of the factors $D_i(S/I)/D_{i-1}(S/I)$. Thus in this prime filtration each factor $S/P$ belongs to $\Ass(D_i(S/I)/D_{i-1}(S/I))$ for some $i$. It follows that $\depth S/P\geq t$, as desired. Prime filtrations and primary decompositions ============================================ In this section we give another characterization of pretty clean modules in terms of primary decompositions. \[primary\] Let $M$ be an $R$-module, and suppose $M$ admits the prime filtration ${\mathcal F}: (0)=M_0\subset M_1\subset \cdots \subset M_{r-1}\subset M_r=M$ with $M_i/M_{i-1}\iso R/P_i$ for all $i$. Then for $j=1,\ldots, r$ there exist irreducible $P_j$-primary submodules $N_j$ of $M$ such that $M_i=\Sect_{j=i}^rN_j$ for $i=0,\ldots, r$. In the proof of this result we shall need the following \[complement\] Let $U\subset V\subset M$ be submodules of $M$ such that $V/U\iso R/P$ for some $P\in \Spec(R)$. Then there exists an irreducible submodule $W$ of $M$ such that $U=V\sect W$. By Noetherian induction there exists a maximal submodule $W$ of $M$ such that $U=V\sect W$. We claim that $W$ is an irreducible submodule of $M$. Indeed, suppose that $W=W_1\sect W_2$. Then $U=(V\sect W_1)\sect(V\sect W_2)$ is a decomposition of $U$ in $V$. However, $U$ is irreducible in $V$ since $V/U\iso R/P$. It follows that $V\sect W_1=U$ or $V\sect W_2=U$. Since $W$ was chosen to be maximal with this intersection property, we see that $W=W_1$ or $W=W_2$. Thus $W$ is irreducible, as desired. (a)(b): Let ${\mathcal F}$ be a prime filtration as given in (a). We show by decreasing induction on $i<r$ that for $j=i+1,\ldots, r$ there exist irreducible $P_j$-primary submodules $N_j$ of $M$ such that $M_{i}=\Sect_{j={i+1}}^rN_j$. For $i=r$ we may choose $N_r=M_{r-1}$, since $M/M_{r-1}\iso R/P_r$. Now let $1<i<r$, and assume that $M_{i}=\Sect_{j=i+1}^rN_j$ where $N_j$ is an irreducible $P_j$-primary submodule of $M$ for $j=i+1,\cdots, r$. Since $M_{i}/M_{i-1}\iso R/P_{i}$, it follows by Lemma \[complement\] that there exists an irreducible submodule $N_{i}$ of $M$ such that $M_{i-1}=M_{i}\sect N_{i}$. Since $R/P_{i}\iso M_{i}/M_{i-1}=M_{i}/M_{i}\sect N_{i}\subset M/N_{i}$, it follows that $\{P_{i}\}=\Ass(M_{i}/M_{i-1})\subset \Ass(M/N_{i})$. However $\Ass(M/N_{i})$ has only one element, therefore $\Ass(M/N_{i})=\{P_{i}\}$. Clean filtrations and shellings =============================== In this section we recall the main result of the paper of Dress [@D] (see also [@Si]), and provide some extra information. Let $\Delta$ be a simplicial complex on the vertex set $[n]=\{1,\ldots,n\}$. Recall that $\Delta$ is [*shellable*]{}, if the facets of $\Delta$ can be given a linear order $F_1,\ldots, F_m$ such that for all $i,j$, $1\leq i<j\leq m$, there exists some $v\in F_i\setminus F_j$ and some $k<i$ with $F_i\setminus F_k=\{v\}$. Note that we do [*not*]{} insist that $\Delta$ is pure, that is, that all facets of $\Delta$ have the same dimension. Sometimes such a shelling is called a [*non-pure shelling*]{}. Let $K$ be a field. The [*Stanley-Reisner ring*]{} of $K[\Delta]$ of $\Delta$ is the factor ring of $S=K[x_1,\ldots, x_n]$ modulo the ideal $I_\Delta$ generated by all squarefree monomials $x_{i_1}x_{i_2}\cdots x_{i_k}$ such that $\{i_1,\ldots, i_k\}$ is not a face of $\Delta$. One has \[dress\] The simplicial complex $\Delta$ is shellable if and only if $K[\Delta]$ is a clean ring. For a subset of faces $G_1,\ldots, G_r$ of $\Delta$ we denote by $\langle G_1,\ldots, G_r\rangle$, the smallest subcomplex of $\Delta$ containing the faces $G_1,\ldots, G_r$. With this notation, the shellability of $\Delta$ can also be characterized as follows: $\Delta$ is shellable if and only if the facets of $\Delta$ can be ordered $F_1,\ldots, F_r$ such that for $i=2,\ldots,m$ the facets of $\langle F_1,\ldots, F_{i-1}\rangle\sect \langle F_i\rangle$ are maximal proper faces of $\langle F_i\rangle$. For $i\geq 2$ we denote by $a_i$ the number of facets of $\langle F_1,\ldots, F_{i-1}\rangle\sect \langle F_i\rangle$, and set $a_1=0$. We call the $a_1,\ldots, a_r$ the sequence of [*shelling numbers*]{} of the given shelling of $\Delta$. Set $P_{F_i}=(\{x_j\}_{j\not\in F_i})$. Then $I_\Delta=\Sect_{i=1}^rP_{F_i}$. Therefore, if $F_1,\ldots, F_r$ is a shelling of $\Delta$, then for $i=2,\ldots,r$ we have $$\Sect_{j=1}^{i-1}P_{F_j}+P_{F_i} =P_{F_i}+(f_i).$$ Here $f_i=\prod_k x_k$, where the product is taken over those $k\in F_i$ such that $F_i\setminus\{k\}$ is a facet of $\langle F_1,\ldots, F_{i-1}\rangle\sect \langle F_i\rangle$. In particular it follows that $\deg f_i$ equals the $i$th shelling number $a_i$. We obtain the following isomorphisms of graded $S$-modules $$\begin{aligned} (\Sect_{j=1}^{i-1}P_{F_j})/(\Sect_{j=1}^{i}P_{F_j})&\iso &(\Sect_{j=1}^{i-1}P_{F_j}+P_{F_i})/P_{F_i} = (P_{F_i}+(f_i))/P_{F_i}\\ &\iso & (f_i)/(f_i)P_{F_i}\iso S/P_{F_i}(-a_i).\end{aligned}$$ The isomorphism $(P_{F_i}+(f_i))/P_{F_i}\iso (f_i)/(f_i)P_{F_i}$ results from the fact that $(f_i)\sect P_{F_i}=(f_i)P_{F_i}$ since the set of variables dividing $f_i$ and the set of variables generating $P_{F_i}$ have no element in common. Thus we have shown \[shelling numbers\] Let $\Delta$ be a shellable simplicial complex with shelling $F_1,\ldots, F_r$ and shelling numbers $a_1,\ldots, a_r$. Then $(0)=M_0\subset M_1\subset\cdots\cdots M_{r-1}\subset M_r=K[\Delta]$ with $$M_i=\Sect_{j=1}^{r-i}P_{F_j}\quad \text{and}\quad M_i/M_{i-1}\iso S/P_{F_{r-i+1}}(-a_{r-i+1})$$ is a clean filtration of $S/I_{\Delta}$. Multicomplexes ============== The aim of this and the next section is to extend the result of Dress to multicomplexes. Stanley [@St] calls a subset $\Gamma\subset \NN^n$ a multicomplex if for all $a\in \Gamma$ and all $b\in\NN^n$ with $b\leq a$, it follows that $b\in\Gamma$. The elements of $\Gamma$ are called [*faces*]{}. What are the facets of $\Gamma$? We define on $\NN^n$ the partial order given by $$(a(1),\ldots, a(n))\leq (b(1),\ldots,b(n))\quad \text{if}\quad a(i)\leq b(i)\quad \text{for all}\quad i.$$ An element $m\in \Gamma$ is called maximal if there exists no $a\in \Gamma$ with $a> m$. We denote by ${\mathcal M}(\Gamma)$ the set of maximal elements of $\Gamma$. One would expect that ${\mathcal M}(\Gamma)$ is the set of facets of $\Gamma$. However ${\mathcal M}(\Gamma)$ may be the empty set, for example for $\Gamma=\NN^n$. To remedy this defect we will consider “closed" subsets $\Gamma$ in $\NN^n_\infty$, where $\NN_\infty=\NN\union \{\infty\}$. Let $a\in\Gamma$. Then $$\ip a=\{i\: a(i)=\infty\}$$ is called the [*infinite part*]{} of $a$. We first notice that \[finitem\] Let $\Gamma\subset \NN^n_\infty$. Then ${\mathcal M}(\Gamma)$ is finite. Let $F\subset [n]$, and set $\Gamma_F=\{a\in \Gamma\: \ip a =F\}.$ It is clear that if $a\in \Gamma_F$ is maximal in $\Gamma$ then $a$ is maximal in $\Gamma_F$. Since there are only finitely many subsets $F$ of $[n]$, it suffices to show that $\Gamma_F$ has only finitely many maximal elements. Let $[n]\setminus F=\{i_1,\ldots, i_k\}$ with $i_1<i_2<\cdots <i_k$. For each $a\in\Gamma_F$ we let $a'\in \NN^k$ be the integer vector with $a'(j)=a(i_j)$ for $j=1,\ldots,k$. Now if $a$ and $b$ are two maximal elements in $\Gamma_F$ with $a\neq b$, then $a'$ and $b'$ are incomparable vectors, that is, $a'\not\leq b'$ and $b'\not\leq a'$. This implies that the set of monomials $\{x^{a'}\: a\in \Gamma_F,\; \text{$a$ maximal}\}$ is a minimal set of generators of the monomial ideal they generate in $K[x_1,\ldots,x_k]$. Hence this set is finite. Thus the set of maximal elements $\Gamma_F$ is finite for all $F\subset [n]$, and ${\mathcal M}(\Gamma)$ is finite. We say that a sequence of natural numbers $a(i)$ has limit $\lim a(i)=\infty$, if for all integers $b$ there exists an integer $j$ such that $a(i)\geq b$ for all $i\geq j$. Of course any non-decreasing sequence in $\NN$ has a limit – either it is eventually constant, and this constant is its limit, or the limit is $\infty$. As usual we set $a\leq \infty$ for all $a\in \NN$. and extend the partial order on $\NN^n$ naturally to $\NN^n_\infty $. By what we just said it follows that any sequence $a_i$, $i=1,2,\ldots$ of elements in $\NN^n_\infty$ with $a_i\leq a_{i+1}$ has a limit – the limit being taken componentwise. Let $\Gamma\subset \NN_{\infty}^n$. The set $\bar{\Gamma}$ of all $a\in\NN^n_\infty$ which are limits of ascending sequences in $\Gamma$ is called [*the closure*]{} of $\Gamma§$. It is clear that $\Gamma\subset \bar{\Gamma}$ and that $\bar{\bar{\Gamma}}=\bar{\Gamma}$. *A subset $\Gamma\subset \NN^n_\infty$ is called a [*multicomplex*]{} if* 1. for all $a\in\Gamma$ and all $b\in\NN^n_\infty$ with $b\leq a$ it follows that $b\in\Gamma$; 2. $\Gamma=\bar{\Gamma}$. The elements of a multicomplex are called [*faces*]{}. The next result shows that each face of a multicomplex is bounded by a face in ${\mathcal M}(\Gamma)$. \[start\] Let $\Gamma\subset \NN^n_\infty$ be a set satisfying property $(1)$ of multicomplexes. Then the following conditions are equivalent: 1. $\Gamma=\bar{\Gamma}$; 2. for each $a\in\Gamma$ there exists $m\in{\mathcal M}(\Gamma)$ with $a\leq m$. (a)(b): We proceed by induction on $n-|\ip a|$. If $n-|\ip a|=0$, then $a(i)=\infty$ for all $i$, and hence $a\in {\mathcal M}(\Gamma)$. Suppose now that $n-|\ip a|>0$ and that there is no $m\in {\mathcal M}(\Gamma)$ with $a\leq m$. Then there exists a strictly ascending sequence $a=a_1<a_2<\ldots$ in $\Gamma$. Since $\Gamma=\Bar{\Gamma}$ it follows that $b=\lim a_i\in \Gamma$. Obviously one has $n-|\ip b|<n-|\ip a|$. Hence by induction hypothesis, there exists $m\in{\mathcal M}(\Gamma)$ with $b\leq m$, and thus $a<m$. (b)(a): Let $a_i$, $i=1,2,\ldots$ be an ascending sequence in $\Gamma$. By assumption, there exist $m_i\in{\mathcal M}(\Gamma)$ with $a_i\leq m_i$. Since ${\mathcal M}(\Gamma)$ is finite (see Lemma \[finitem\]), there exists $i_0$ such that $m_i=m_{i_0}$ for all $i\geq i_0$. It follows that $a_i\leq m_{i_0}$ for all $i$. Hence $\lim a_i\leq m_{i_0}$. In particular, $\lim a_i\in\Gamma$. Combining Lemma \[start\] with Lemma \[finitem\] we get \[m\] Let $\Gamma\subset \NN^n_\infty$. Then $\Gamma$ is a multicomplex if and only if there exist finitely many elements $m_1,\ldots, m_r\in\NN^n_\infty$ such that $$\Gamma=\{a\in\NN^n_\infty\: a\leq m_i \text{ for some }i=1,\ldots, r\}.$$ We have \[closure\] Suppose $\Gamma\subset \NN^n$ satisfies property $(1)$ of multicomplexes, then so does $\bar{\Gamma}$. The statement is clear if $a\in\Gamma$. Suppose now that $a\in\bar{\Gamma}$, and let $a_i\in \Gamma$ be a non-descending sequence with $\lim a_i=a$. Let $b_i(j)=\min\{a_i(j),b(j)\}$ for $j=1,\ldots,n$. Then $b_i=(b_i(1),\ldots,b_i(n))\leq a_i$ for all $i$, and hence $b_i\in \Gamma$ for all $i$. Moreover, $b=\lim b_i$ and so $b\in\bar{\Gamma}$. The lemma shows that if $\Gamma\subset \NN^n$ is a multicomplex in the sense of Stanley, then $\bar{\Gamma}\subset \NN^n_\infty$ is a multicomplex in our sense. Moreover $\bar{\Gamma}\sect\NN^n=\Gamma$. Thus the assignment $\Gamma\mapsto \bar{\Gamma}$ establishes a bijection between these different concepts of multicomplexes. In the following we will use the term multicomplex only in our sense, that is, we will always assume that $\Gamma=\bar{\Gamma}$. Note that $\Delta(\Gamma)=\{\ip a\: a\in\bar{\Gamma}\}$ is a simplicial complex on the vertex set $[n]=\{1,\ldots,n\}$. It is called the simplicial complex associated to the multicomplex $\Gamma$. The number $\dim a=|\ip a|-1$ is called the [*dimension of $a$*]{}. The [*dimension of $\Gamma$*]{} is defined to be $$\dim \Gamma= \max\{\dim a\: a\in \Gamma\}.$$ Obviously one has $\dim \Gamma =\dim \Delta(\Gamma)$. An element $a\in \Gamma$ is called a [*facet*]{} of $\Gamma$ if for all $m\in {\mathcal M}(\Gamma)$ with $a\leq m$ one has $\ip a=\ip m$. The set of facets of $\Gamma $ will be denoted by $\mathcal{F}(\Gamma)$. It is clear that $\mathcal{M}(\Gamma)\subset \mathcal{F}(\Gamma)$. The facets in $\mathcal{M}(\Gamma)$ are called [*maximal facets*]{}. Consider for example the multicomplex $\Gamma\in\NN^2_\infty$ with faces $$\{a\: a\leq (0,\infty )\; \text{or}\; a\leq (2,0)\}.$$ Then ${\mathcal M}(\Gamma)=\{(0, \infty), (2,0)\}$ and ${\mathcal F}(\Gamma)=\{(0,\infty), (2,0), (1,0)\}$. Besides its facets, $\Gamma$ admits the infinitely many faces $(0,i)$ with $i\in \NN$. \[finite\] Each multicomplex has a finite number of facets. Let $\Gamma$ be the given multicomplex. Given $m\in{\mathcal M}(\Gamma)$. By \[finitem\] it remains to show that the set $$\{a\in \Gamma\: a\leq m\; \text{and}\; \ip a=\ip m\}$$ is finite. But this is obviously the case since for each $i\not\in \ip m$ there are only $m(i)+1$ numbers $j\in \NN$ with $j\leq m(i)$. \[intersection\] An arbitrary intersection and a finite union of multicomplexes is again a multicomplex. Let $(\Gamma_i)_{i\in I}$ be a family of multicomplexes, and set $\Gamma=\Sect_{i\in I}\Gamma_i$. If $a\in \Gamma$ and $b\leq a$, then obviously $b\in \Gamma$. Thus it remains to show that $\Gamma=\bar{\Gamma}$. Let $a_j$, $j=1,2,\ldots$ be an ascending sequence in $\Gamma$. Since $\Gamma_i=\bar{\Gamma}_i$ for all $i\in I$, it follows that $\lim a_j\in \Gamma_i$ for all $i$, and hence $\lim a_i\in \Gamma$, as desired. On the other hand, suppose $J=\{1,\ldots,k\}$ and let $\Gamma=\Union_{i=1}^k\Gamma_i$. Then $\Gamma$ satisfies obviously condition (1) of a multicomplex. By Lemma \[finitem\] the sets ${\mathcal M}(\Gamma_i)$ are finite, and $\Union_{i=1}^k\Gamma_i$ is the set of all $a\in\NN^n_\infty$ for which there exists $j\in J$ and $m\in {\mathcal M}(\Gamma_j)$ such that $a\leq m$. Thus it follows from Corollary \[m\] the $\Gamma$ is a multicomplex. \[closure\] Let $A\subset \NN^n_\infty$ be an arbitrary subset of $\NN^n_\infty$. Then there exists a unique smallest multicomplex $\Gamma(A)$ containing $A$. Let $\Gamma$ be a multicomplex, and let $I(\Gamma)$ be the $K$-subspace in $S=K[x_1,\ldots,x_n]$ spanned by all monomials $x^a$ such that $a\not\in \Gamma$. Note that if $a\in \NN^n$ and $b\in \NN^n\setminus\Gamma$, then $a+b\in \NN^n\setminus \Gamma$, that is, if $x^a\in I(\Gamma)$ then $x^ax^b\in I(\Gamma)$ for all $x^b\in S$. In other words, $I(\Gamma)$ is a monomial ideal. In particular, the monomials $x^a$ with $a\in \Gamma$ form a $K$-basis of $S/I(\Gamma)$. For example for the above multicomplex $\Gamma=\{a\: a\leq (0,\infty )\; \text{or}\; a\leq (2,0)\}$ in $\NN^2_\infty$ we have $I(\Gamma)=(x_1^3,x_1x_2)$. Conversely, given an arbitrary monomial ideal $I\subset S$, there is a unique multicomplex $\Gamma$ with $I=I(\Gamma)$. Indeed, let $A =\{a\in \NN^n\: x^a\not\in I\}$; then $\Gamma=\Gamma(A)$. The monomial ideal of a multicomplex behaves with respect to intersections and unions of multicomplexes as follows: \[ideal\] Let $\Gamma_j$, $j\in J$ be a family of multicomplexes. Then 1. $I(\Sect_{j\in J}\Gamma_j)=\sum_{j\in J} I(\Gamma_j)$, 2. if $J$ is finite, then $I(\Union_{j\in J}\Gamma_j)=\Sect_{j\in J}I(\Gamma_j)$. Next we describe the relationship between simplicial complexes and multicomplexes. Let $\Delta$ be a simplicial complex on the vertex set $[n]$. To each facet $F\in \Delta$ we associate the element $a_F\in\NN^n_\infty$ with $$a_F(i)=\left\{ \begin{array}{lll} \infty, & \text{if} & i\in F\,\\ 0, & \mbox{if} & i\not\in F, \end{array} \right.$$ Then $\{a_F\: F\in \Delta\}$ is the set of facets of a multicomplex $\Gamma(\Delta)$, and $I(\Gamma(\Delta))=I_\Delta$, where $I_\Delta$ is the Stanley-Reisner ideal of $\Delta$. Moreover one has $\dim \Gamma=\dim \Delta(\Gamma)$. For a multicomplex $\Gamma$ and $a\in \Gamma$ we let $P_a$ be the prime ideal generated by all $x_i$ with $i\not\in \ip a$. Thus $P_a$ is generated by all $x_i$ with $a(i)\in \NN$. \[irred\] Let $\Gamma$ be a multicomplex. The following statements are equivalent: 1. $\Gamma$ has just one maximal facet $a$; 2. $I(\Gamma)$ is an irreducible ideal. If the equivalent conditions hold, then $I(\Gamma)$ is generated by $\{x_i^{a(i)+1}\: i\in [n]\setminus \ip a\}$. In particular, $I(\Gamma)$ is a $P_a$-primary ideal. If $a$ is the unique maximal facet of $\Gamma$ then $$I(\Gamma)=(x^b\: b\in\NN^n,\; b(i)>a(i)\; \text{for some $i$}) =(x_i^{a(i)+1}\: i\in [n]\setminus \ip a).$$ Conversely, if $I(\Gamma)$ is irreducible, then according to [@Vi Theorem 5.1.16] there exists a subset $A\subset \{1,\ldots,n\}$ and for each $i\in A$ an integer $a_i>0$ such that $I(\Gamma)=(x_i^{a_i}:i\in A, a_i>0)$. Set $a(i)=a_i-1$ for $i\in A$ and $a(i)=\infty$ for $i\not \in A$. Then $a$ is the unique facet of $\Gamma$. \[irrprime\] Let $\Gamma\subset \NN_{\infty}^n$ be a multicomplex with just one facet $a$. Then $I(\Gamma)=P_a$. Suppose $a(i)\neq 0$ for some $i\not\in \ip a$. Then $a-e_i$ is a facet, different from $a$. Here $e_i$ is the canonical $i$th unique vector. Thus we see that $a(i)\in \{0,\infty\}$ for $i=1,\ldots,n$, so that $I(\Gamma)=I(\Gamma(a))=P_a$. The next result describes how the maximal facets of a multicomplex $\Gamma$ are related to the irreducible components of $I(\Gamma)$. \[irrdec\] Let $\Gamma\subset \NN_{\infty}^n$ be a multicomplex, and $a_1,\ldots,a_r$ its maximal facets. Then $$I(\Gamma)=\Sect_{j=1}^rI(\Gamma(a_j))$$ is the unique irredundant irreducible decomposition of $I(\Gamma)$ in $S=K[x_1,\ldots,x_n]$. Conversely, let $I\subset S$ be a monomial ideal, $I=\Sect_{j=1}^rI_j$ the unique irredundant irreducible decomposition of $I$ in $S$, and let $\Gamma$ be the multicomplex with $I(\Gamma)=I$. Then $\Gamma$ has $r$ maximal facets $a_1,\ldots, a_r$ which can be labelled such that $$I(\Gamma(a_j)) =I_j\quad \text{for}\quad j=1,\ldots, r.$$ Since $\Gamma=\Union_{i=1}^r\Gamma(a_i)$, it follows from Lemma \[ideal\] that $I(\Gamma)=\Sect_{j=1}^rI(\Gamma (a_j))$. That each $I(\Gamma(a_i))$ is irreducible, we have seen in Lemma \[irred\]. Conversely, let $I=\Sect_{j=1}^rI_j$ be the unique irredundant irreducible decomposition of $I$, and let $\Gamma_j$ be the unique multicomplex with $I(\Gamma_j)=I_j$. By Lemma \[irred\], each $\Gamma_j$ has exactly one maximal facet, say $a_j$. Hence $\Gamma_j=\Gamma(a_j)$ for $j=1,\ldots, r$. Let $\Gamma$ be the unique multicomplex with $I(\Gamma)=I$. Then since $I(\Gamma)=\Sect_{i=1}^rI((\Gamma(a_j))$, it follows from Lemma \[ideal\] that $I(\Gamma)=I(\Gamma( a_1,\ldots,a_r))$, and hence that $\Gamma=\Gamma(a_1,\ldots,a_r)$. Each of the $a_j$ is a maximal facet of $\Gamma$, because if there would be an inclusion among them, then there would also be an inclusion among the $I_j$, contradicting the minimality of the decomposition. \[dimension\] Let $\Gamma$ be a multicomplex. Then $\dim S/I(\Gamma)=\dim \Gamma+1$. By the preceding proposition it suffices to prove the assertion in case that $\Gamma$ has just one maximal facet, say $a$. Suppose that $\dim \Gamma=d-1$. We may, then assume that $a(i)=\infty$ for $i\geq n-d+1$. Then $I(\Gamma)=(x_1^{a(1)+1},\ldots,x_{n-d}^{a(n-d)+1})$, and $\dim S/I(\Gamma)=d$. Finally we will show that the facets of a multicomplex $\Gamma$ correspond to the standard pairs of $I=I(\Gamma)$ introduced by Sturmfels, Trung and Vogel [@STV]: let $u$ be a monomial of $S=K[x_1,\ldots, x_n]$. Then we set $\supp(u)=\{x_i\: x_i\text{ divides } u\}$. A pair $(u,Z)$ where $u$ is a monomial and $Z$ is a subset of the set of variables $X=\{x_1,\ldots, x_n\}$ is called [*admissible*]{} if no $x_i\in Z$ divides $u$, that is, if $\supp(u)\sect Z=\emptyset$. The set of admissible pairs is partially ordered as follows: $$(u,Z)\leq(u',Z')\quad\iff\quad \text{$u$ divides $u'$}\quad \text{and}\quad \supp(u'/u)\union Z'\subset Z.$$ An admissible pair $(u,Z)$ is called [*standard*]{} with respect to $I$, if $u K[Z]\sect I=\{0\}$, and $(u, Z)$ is minimal with this property. The set of standard pairs with respect to $I$ is denoted by $\std(I)$. For a monomial $u\in S$, with $u=\prod_{i=1}^nx_i^{a_i}$ we set $\log u= (a_1,\ldots, a_n)$, and for a subset $Z\subset X$ we let $c(Z)\in \NN_\infty^n$ the element with $$c(Z)(i) = \left\{ \begin{array}{lll} \infty, & \text{if} & x_i\in Z,\\ 0, & \mbox{if} & x_i\not\in Z. \end{array} \right.$$ With this notation we have \[pairs\] Let $I\subset S$ be a monomial ideal, and $\Gamma$ the multicomplex associated with $I$. Then the standard pairs with respect to $I$ correspond bijectively to the facets of $\Gamma$. The bijection is established by the following assignment: $$\std(I)\To {\mathcal F}(\Gamma), \quad (u,Z)\mapsto \log u+c(Z).$$ Let $\mathcal A$ be the set of admissible pairs. Since $\supp u\sect Z=\emptyset$ for $(u,Z)\in \mathcal A$ it follows that the map $${\mathcal A}\To \NN^n_\infty,\quad (u,Z)\mapsto \log u+c(Z)$$ is injective. Moreover, for each $(u,Z)\in \mathcal A$ we have $$uK[Z]\sect I=\{0\}\iff \log u+c(Z)\in \Gamma.$$ Now let $(u,Z)\in\std(I)$, and set $a=\log u+c(Z)$. Let $m\in {\mathcal M}(\Gamma)$ with $a\leq m$. Suppose that $\ip a \neq \ip m$. Then there exists $i$ such that $a(i)<m(i)=\infty$. Let $v=u/x_i^{a(i)}$ and $W=Z\union \{x_i\}$. Then $(v,W)<(u,Z)$ and $v\cdot K[W]\sect I=\{0\}$, a contradiction. Therefore, $a\in{\mathcal F}(\Gamma)$. Conversely let $a\in {\mathcal F}(\Gamma)$. Set $u=\prod_{i\not\in\ip(a)}x_i^{a(i)}$ and $Z=\{x_i\: i\in\ip a\}$. Then $(u,Z)\in\mathcal A$ and $a=\log u+c(Z)$. Since $a\in\Gamma$ it follows that $u\cdot K[Z]\sect I=\{0\}$. Suppose that $(u,Z)$ is not minimal with this property. Then there exists $(v,W)\in \mathcal A$ with $v\cdot K[W]\sect I=\{\ 0\}$ and $(v,W)<(u,Z)$, and we have 1. $b=\log v+c(W)\in \Gamma$; 2. $v$ divides $u$; 3. $\supp(u/v)\union Z\subset W$. The properties (2) and (3) imply that $a(i)=b(i)$ for all $i$ such that $b(i)<\infty$. Thus $a\leq b$, and $a=b$ if and only if $\ip a=\ip b$. However since $a\neq b$, we have $\ip a\neq \ip b$. By property (1) there exists $m\in {\mathcal M}(\Gamma)$ with $b\leq m$. Then $a\leq m$ and $\ip b\subset \ip m$. In particular, $\ip a \neq \ip m$. It follows that $a\not \in {\mathcal F}(\Gamma)$, a contradiction. Pretty clean filtrations and shellable multicomplexes ===================================================== In this section we introduce shellable multicomplexes and show how this concept is related to clean filtrations. Our concept of shellability is a translation of Corollary \[primary1\] into the language of multicomplexes. In that corollary we characterized pretty clean filtrations in terms of primary decompositions. Here we need a refined multigraded version of this result. \[multiprimary\] Let $S=K[x_1,\ldots, x_n]$ be the polynomial ring, and $I\subset S$ a monomial ideal. The following conditions are equivalent: 1. $S/I$ admits a multigraded prime filtration ${\mathcal F}: (0)=M_0\subset M_1\subset \cdots \subset M_{r-1}\subset M_r=S/I$ such that $M_i/M_{i-1}\iso S/P_i(-a_i)$ for all $i$; 2. there exists a chain of monomial ideals $I=I_0\subset I_1\subset \cdots \subset I_r=S$ and monomials $u_i$ of multidegree $a_i$ such that $I_i=(I_{i-1},u_i)$ and $I_{i-1}:u_i=P_i$; If the equivalent conditions hold, then there exist irreducible monomial ideals $J_1,\ldots J_r$ such that $I_i=\Sect_{j=i+1}^r J_j$ for $i=0,\ldots, r$. Moreover, if the prime filtration is pretty clean, then this set of irreducible ideals $\{J_1,\ldots, J_r\}$ is uniquely determined. In fact, this set corresponds bijectively to the set of facets of the multicomplex associated with $I$. The statements (a) and (b) are obviously equivalent, while the existence of of the irreducible ideals $J_i$ is just the multigraded version of Proposition \[primary\]. Now we assume that the prime filtration $\mathcal F$ is pretty clean. Since $J_i$ is an irreducible monomial ideal, it follows that $J_i=\Gamma(a_i)$ for some $a_i\in\NN^n_\infty$, see Lemma \[irred\]. We claim that ${\mathcal A}=\{a_1,\ldots, a_r\}$ is the set of facets of the unique multicomplex $\Gamma$ with $I=I(\Gamma)$. We first show that all $a_j$ are facets of $\Gamma$. Note that ${\mathcal M}(\Gamma)\subset \mathcal A$. Indeed, by Proposition \[irrdec\] we have that $$I(\Gamma)=\Sect _{a\in {\mathcal M}(\Gamma)}I(\Gamma(a))$$ is the unique irredundant decomposition of $I(\Gamma)$ into irreducible ideals. Since from any redundant such decomposition, like the decomposition $I=\Sect_{j=1}^rJ_j$, we obtain an irredundant by omitting redundant components we obtain the desired inclusion. We also see that for each $J_j$ there exists a maximal facet $a$ of $\Gamma$ such that $I(\Gamma(a))\subset J_j$, that is, for each $a_j\in \mathcal A$ there exists a maximal facet $a$ of $\Gamma$ such that $a_j\leq a$. We claim that $\ip a_j=\ip a$, in other words, that $P_a= P_j$. In fact, since $a\in \mathcal A$ as we have just seen, there exists an integer $i$ such that $a=a_i$, and hence $I(\Gamma(a))=J_i$ is $P_i$-primary, and $P_i\subset P_j$. Suppose that $P_i\neq P_j$. Then, since $\mathcal F$ is pretty clean, we conclude that $i>j$. It follows that $\Sect_{t>j}J_t=\Sect_{t\geq j}J_t$, contradicting (b). Thus we have shown that all elements of $\mathcal A$ are facets of $\Gamma$. Next we prove that $r=|{\mathcal F}(\Gamma)|$. This then implies that ${\mathcal A}={\mathcal F}(\Gamma)$, and that the elements of $\mathcal A$ are pairwise distinct. We know from Corollary \[independent\] that $r$ equals the arithmetic degree of $S/I$. On the other hand we have shown in Lemma \[pairs\] that the facets of $\Gamma$ correspond to the standard pairs of $I$. In [@STV Lemma 3.3] it is shown that the number of standard pairs of $I$ is equal to the arithmetic degree of $S/I$ as well. Thus $|{\mathcal F}(\Gamma)|=r$, as desired. In Section 6 we have considered the Stanley decomposition of $S/I$ into subspaces of the form $uK[Z]$ where $u$ is a monomial in the variables $X=\{x_1,\ldots, x_n\}$ and $Z\subset X$. We call $S\subset \NN^n_\infty$ a [*Stanley set*]{} if there exists $a\in \NN^n$ and $m\in \NN^n_\infty$ with $m(i)\in\{0,\infty\}$ such that $S=a+S^*$, where $S^*=\Gamma(m)$ . The [*dimension of $S$*]{} is defined to be $\dim\Gamma(m)$. Obviously Stanley sets correspond to subspaces of the form $uK[Z]$. *A multicomplex $\Gamma$ is [*shellable*]{} if the facets of $\Gamma$ can be ordered $a_1,\ldots, a_r$ such that* 1. $S_i= \Gamma(a_i)\setminus\Gamma(a_1,\ldots, a_{i-1})$ is a Stanley set for $i=1,\ldots,r$, and 2. whenever $S_i^*\subset S_j^*$, then $S_i^*=S_j^*$ or $i>j$. Any order of the facets satisfying (1) and (2) is called a [*shelling*]{} of $\Gamma$ The next result shows that our definition of shellability of multicomplexes extends the classical concept of shellability of simplicial complexes. \[extend\] Let $\Delta$ be a simplicial complex with facets $F_1,\ldots, F_r$, and $\Gamma$ be the multicomplex with facets $a_{F_1},\ldots, a_{F_m}$. Then $F_1,\ldots, F_m$ is a shelling of $\Delta$ if and only if $a_{F_1},\ldots, a_{F_r}$ is a shelling of $\Gamma$. We denote by $e_i$ the $i$th standard unit vector in $\NN^n$, and set $\Gamma_i=\Gamma(a_{F_i})$. Then $$\begin{aligned} \Gamma(a_{F_i})\setminus\Gamma(a_{F_1},\ldots,a_{F_{i-1}})&=&\Sect_{j=1}^{i-1}(\Gamma(a_{F_i})\setminus\Gamma(a_{F_j}))\\ &=&\Sect_{j=1}^{i-1}(\Union_{k\in F_i\setminus F_j}(e_k+\Gamma_i))\end{aligned}$$ We notice that $$(e_k+\Gamma_i)\sect (e_l+\Gamma_i)=\left\{ \begin{array}{lll} e_k+\Gamma_i, & \text{if} & k=l,\\ e_k+e_l+\Gamma_i, & \mbox{if} & k\neq l, \end{array} \right.$$ Thus $$\Gamma(a_{F_i})\setminus\Gamma(a_{F_1},\ldots,a_{F_{i-1}})=\Union_{L\in\mathcal L}(e_L+\Gamma_i),$$ where $${\mathcal L}=\{\{k_1,\ldots, k_{i-1}\}\: k_j\in F_i\setminus F_j \text{ for } j=1,\ldots,{i-1}\}$$ and where $e_L=\sum_{j\in L}e_{j}$ for each $L\in\mathcal L$. The union $$\Union_{L\in\mathcal L}(e_L+\Gamma_i)$$ is a Stanley set if and only if there exists $L\in\mathcal L$ such that $e_{L'}+\Gamma_i\subset e_L+\Gamma_i$ for all $L'\in \mathcal L$, and this is the case if and only if there exists $L\in\mathcal L$ such that $L\subset L'$ for all $L'\in \mathcal L$. We claim that the last condition is equivalent to the condition that all facets of $\langle F_i\rangle\sect \langle F_1,\ldots, F_{i-1}\rangle$ are maximal proper subfaces of $\langle F_i\rangle$. Suppose first that there is a set $L_0\in\mathcal L$ which is minimal under inclusion. We may assume that $L_0=[m]$. Let $k\in [m]$ and assume that all sets $F_i\setminus F_j$ which contain $k$ have more than one element. Then for each such set we can pick $k_j\in F_i\setminus F_j$ with $k_j\neq k$, and hence there exists $L\in\mathcal L$ which does not contain $k$, a contradiction, since $k\in L_0\subset L$. Thus for each $k\in L_0$ there exists an integer $j_k\in [i-1]$ such that $F_i\setminus F_{j_k}=\{k\}$. Now let $j\in [i-1]$ be arbitrary. If $|F_i\setminus F_j|=1$, then by definition of the sets $L$, the set $F_i\setminus F_j$ is a subset of each $L$, and in particular of $L_0$. Thus we see that the subfaces of $\langle F_i\rangle\sect \langle F_1,\ldots, F_{i-1}\rangle$ of codimension $1$ are exactly the faces $F_i\setminus\{k\}$ for $k=1,\ldots, m$. Suppose now there exists $j\in [i-1]$ for which $F_i\sect F_j$ is not contained in any of these codimension 1 subfaces of $F_i$ (in which case not all facets of $\langle F_i\rangle\sect \langle F_1,\ldots, F_{i-1}\rangle$ would be maximal proper subfaces of $\langle F_i\rangle$.). Then $k\not\in F_i\setminus F_j$ for $k=1,\ldots, m$, and hence $(F_i\setminus F_j)\sect L_0=\emptyset$. This a contradiction, since any $L\subset \mathcal L$ contains an element of $F_i\setminus F_j$. Conversely, suppose that all facets of $\langle F_i\rangle\sect \langle F_1,\ldots, F_{i-1}\rangle$ are maximal proper subfaces of $\langle F_i\rangle$. Then there exist $j_1,\ldots, j_m\in [i-1]$ such that $|F_i\setminus F_{j_k}|=1$, and for any $j\in [i-1]$ there exists $k\in [m]$ such that $F_i\setminus F_{j_k}\subset F_i\setminus F_j$. For simplicity we may assume that $F_i\setminus F_{j_k}=\{k\}$ for $k=1,\ldots,m$. Then obviously $L_0\in\mathcal L$ and $L_0\subset L$ for any other $L\in\mathcal L$. [*Condition (2) in the definition of shellability is superfluous in case $\Gamma$ is the multicomplex corresponding to a simplicial complex, because in this case the sets $S_i^*$ correspond to the minimal prime ideals of $I(\Gamma)$, and hence there is no inclusion among them.*]{} As an extension of the theorem of Dress we now show \[multi2\] The multicomplex $\Gamma$ is shellable if and only if $S/I(\Gamma)$ is a multigraded pretty clean ring. Let $a_1,\ldots, a_r$ be the facets of $\Gamma$, and let $J_{j}=\Gamma(a_j)$ for $j=1,\ldots,r$. Then $J_j$ is an irreducible monomial ideal, and $I(\Gamma(a_1,\ldots, a_{i}))=\Sect_{j=1}^{i}J_j$. We set $I_i=\Sect_{j=1}^{r-i+1}J_j$ and $M_i=I_i/I$ for $i=0,\ldots, r$, $I=I(\Gamma)$. Then ${\mathcal F}\: (0)=M_0\subset M_1\subset \cdots\subset M_r=S/I$ is a multigraded filtration of $S/I$. Since $\Gamma(a_i)\setminus \Gamma(a_1,\ldots, a_{i-1})=\Gamma(a_1,\ldots, a_{i})\setminus \Gamma(a_1,\ldots, a_{i-1})$, we see that $b\in \Gamma(a_i)\setminus \Gamma(a_1,\ldots, a_{i-1})$ if and only $x^b\in \Sect_{j=1}^{i-1}iJ_j\setminus \Sect_{j=1}^{i}J_j$. In other words, the monomials $x^b$ with $b\in \Gamma(a_i)\setminus \Gamma(a_1,\ldots, a_{i-1})$ form $K$-basis of the factor module $I_i/I_{i-1}=M_{i}/M_{i-1}$. The discussion at the end of Section 6 shows that $\mathcal F$ is a prime filtration if and only if $M_i/M_{i-1}$ as monomial vectorspace is isomorphic to $uK[Z]$ for some monomial $u\in S$ and some subset $Z\subset \{x_1,\ldots,x_n\}$. Consequently, $\mathcal F$ is a prime filtration if and only $\Gamma(a_i)\setminus \Gamma(a_1,\ldots, a_{i-1})$ is a Stanley set for all $i=1,\ldots,r$. Hence the theorem follows from Proposition \[multiprimary\]. Let $K$ be field, and let $S=K[x_1,\ldots,x_n]$ be the polynomial ring. We call a multicomplex $\Gamma\subset \NN^n_\infty$ [*Cohen-Macaulay*]{} or [*sequentially Cohen-Macaulay over $K$*]{} if $S/I(\Gamma)$ has the corresponding property. $\Gamma$ is simply called Cohen-Macaulay, or sequentially Cohen-Macaulay, if $S/I(\Gamma)$ has the corresponding property over any field. \[properties\] Let $\Gamma$ be a shellable multicomplex. Then $\Gamma$ is sequentially Cohen-Macaulay. If moreover, all facets of $\Gamma$ have the same dimension, then $\Gamma$ is Cohen-Macaulay. Theorem \[multi2\] implies that $S/I(\Gamma)$ is pretty clean. Hence the assertions follow from Theorem \[sequentially\]. \[criterion\] A multicomplex $\Gamma$ is shellable if and only if there exists an order $a_1,\ldots, a_r$ of the facets such that for $i=1,\ldots,r$ the sets $S_i=\Gamma(a_i)\setminus\Gamma(a_1,\ldots, a_{i-1})$ are Stanley sets with $\dim S_1\geq \dim S_2\geq \ldots\geq \dim S_r$. Suppose the conditions of the corollary are satisfied, and that $S_i^*\subset S_j^*$ for some $i<j$. Then, since $\dim S_i \geq S_j$, it follows that $S_i^*=S_j^*$. Thus $\Gamma$ is shellable. Conversely, suppose hat $\Gamma$ is shellable. Then $S/I$ is pretty clean. Thus by Corollary \[conclusion\] the non-zero factors of the dimension filtration are clean. Refining the dimension filtration by the clean filtrations of the factors we obtain a pretty clean filtration with $\dim S_1\geq \dim S_2\geq \ldots\geq \dim S_r$. [1]{} J. Apel, On a conjecture of R.P.Stanley; Part I-Monomial Ideals, J.  of Alg.  Comb. [**17**]{}, (2003), 36–59. A. Björner, M.  Wachs, Shellable nonpure complexes and posets. I. Trans. Amer. Math. Soc. [**349**]{}(10), (1997), 3945–3975. W. Bruns, J. Herzog, [*Cohen-Macaulay rings*]{}, Revised Edition, Cambridge, 1996. A. Dress, A new algebraic criterion for shellability, Beitrage zur Alg. und Geom., [**340**]{}(1), (1993),45–55. D.  Eisenbud, [*Commutative algebra, with a view toward geometry*]{}, Graduate Texts Math. Springer, 1995. J. Herzog, D. Popescu, M. Vladoiu, On the Ext-modules of ideals of Borel type, Contemporary Math. [**331**]{} (2003), 171-186. J. Herzog, E. Sbarra, Sequentially Cohen-Macaulay modules and local cohomology, in [*Arithmetic and Geometry*]{}, Proceed. of Intern. Coll. on Alg., 327-340. S. Hoşten, R. R. Thomas, Standard pairs and group relaxations in integer programming, J. Pure Appl. Alg.  [**139**]{} (1999), 133–157. D. Maclagan, G. Smith, Uniform bounds on multigraded regularity, J. Alg. Geom. [**14**]{} (2005), 137–164. H. Matsumura, [*Commutative Ring Theory*]{}, Cambridge, 1986. U. Nagel, T. Römer, Extended degree functions and monomial modules, Preprint 2004. K. Pardue, [*Nonstandard Borel fixed ideals*]{}, Dissertation, Brandeis University, 1994. P. Schenzel, On the dimension filtration and Cohen-Macaulay filtered modules, Proceed. of the Ferrara meeting in honour of Mario Fiorentini, ed. F. Van Oystaeyen, Marcel Dekker, New-York, 1999. R. S. Simon, Combinatorial Properties of “Cleanness", J. of Alg. [**167**]{} (1994), 361-388. R. P. Stanley, [*Combinatorics and Commutative Algebra*]{}, Birkhäuser, 1983. R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. [**68**]{} (1982), 175–193. B. Sturmfels, N. V. Trung, W. Vogel, Bounds on Degrees of Projective Schemes, Math. Ann. [**302**]{} (1995), 417–432. R. H.  Villarreal, [*Monomial Algebras*]{}, Dekker, New York, NY, 2001. [^1]: The second author was mainly supported by Marie Curie Intra-European Fellowships MEIF-CT-2003-501046 and partially supported by CNCSIS and the Ceres programs 4-147/2004 and 4-131/2004 of the Romanian Ministery of Education and Research

--- abstract: 'We evaluate the elastic scattering cross section of vector dark matter with nucleon based on the method of effective field theory. The dark matter is assumed to behave as a vector particle under the Lorentz transformation and to interact with colored particles including quarks in the Standard Model. After formulating general formulae for the scattering cross sections, we apply them to the case of the first Kaluza-Klein photon dark matter in the minimal universal extra dimension model. The resultant cross sections are found to be larger than those calculated in previous literature.' address: - '$^1$ Department of Physics, Nagoya University, Nagoya 464-8602, Japan' - '$^2$ Department of Physics, University of Tokyo, Tokyo 113-0033, Japan' author: - 'Natsumi Nagata$^{1, 2}$' title: 'A calculation for vector dark matter direct detection[^1]' --- Introduction ============ The existence of dark matter (DM) has been established by cosmological observations [@Komatsu:2010fb]. One of the most attractive candidates is what we call Weakly Interacting Massive Particles (WIMPs), which are stable particles with masses of the electroweak scale and weakly interact with ordinary matters. This interactions enable us to search for WIMP DM by using the scattering signal of DM with nuclei on the earth. Such kind of experiments are called the direct detection experiments of WIMP DM. For the past years, a lot of efforts have been dedicated to the direct detection of WIMP DM, and their sensitivities have been extremely improving. The XENON100 Collaboration, for example, gives a severe constraint on the spin-independent (SI) elastic scattering cross section of WIMP DM with nucleon $\sigma^{\rm SI}_N$ ($\sigma^{\rm SI}_N < 2.0\times 10^{-45}~{\rm cm}^2$ for WIMPs with a mass of 55 GeV$/c^2$) [@Aprile:2012nq]. Moreover, ton-scale detectors for the direct detection experiments are now planned and expected to have significantly improved sensitivities. In order to study the nature of DM based on these experiments, we need to evaluate the WIMP-nucleon elastic scattering cross section precisely. In this work, we assume the WIMP DM to be a vector particle, and evaluate its cross section scattering off a nucleon. Several candidates for vector DM have been proposed in various models, and there have been a lot of previous work computing the scattering cross sections [@Cheng:2002ej; @Servant:2002hb; @Birkedal:2006fz]. However, we found that in the calculations some of the leading contributions to the scattering cross section are not evaluated correctly, or in some cases completely neglected. Taking such situation into account, we study the way of evaluating the cross section systematically by using the method of effective field theory. Direct detection of vector dark matter ====================================== In this section we discuss the way of evaluating the elastic scattering cross section of vector DM with nucleon. First, we write down the effective interactions of vector DM ($B_\mu$) with light quarks and gluon [@Hisano:2010yh]: $$\mathcal{L}^{\mathrm{eff}}=\sum_{q=u,d,s}\mathcal{L}^{\mathrm{eff}}_q +\mathcal{L}^{\mathrm{eff}}_G,$$ with $$\begin{aligned} \mathcal{L}^{\mathrm{eff}}_q &=& f_q m_q B^{\mu}B_{\mu}\bar{q}q+ \frac{d_q}{M} \epsilon_{\mu\nu\rho\sigma}B^{\mu}i\partial^{\nu}B^{\rho} \bar{q}\gamma^{\sigma}\gamma^{5}q+\frac{g_q}{M^2} B^{\rho}i\partial^{\mu}i\partial^{\nu}B_{\rho}\mathcal{O}^q_{\mu\nu}, \label{eff_lagq} \\ \mathcal{L}^{\mathrm{eff}}_G&=&f_G B^{\rho}B_{\rho}G^{a\mu\nu}G^a_{\mu\nu}, \label{eff_lagG}\end{aligned}$$ where $m_q$ are the masses of light quarks, $M$ is the DM mass, and $\epsilon^{\mu\nu\rho\sigma}$ is the totally antisymmetric tensor defined as $\epsilon^{0123}=+1$. The covariant derivative is defined as $D_\mu\equiv\partial_\mu+i g_sA^a_\mu T_a$, with $g_s$, $T_a$ and $A^a_\mu$ being the strong coupling constant, the SU(3)$_C$ generators, and the gluon fields, respectively. The gluon field strength tensor is denoted by $G^a_{\mu\nu}$, and $\mathcal{O}^q_{\mu\nu}\equiv\frac12 \bar{q} i \left(D_{\mu}\gamma_{\nu} + D_{\nu}\gamma_{\mu} -\frac{1}{2}g_{\mu\nu}{{\ooalign{\hfil/\hfil\crcr$D$}}} \right) q $ are the twist-2 operators of light quarks. When we write down the effective Lagrangian, we consider the fact that the scattering process is non-relativistic. The coefficients of the operators are to be determined by integrating out the heavy particles in high energy theory. The second term in Eq. (\[eff\_lagq\]) gives rise to the spin-dependent (SD) interaction, while the other terms yield the spin-independent (SI) interactions. We focus on the SI interactions hereafter, because the experimental constraint is much severe for the SI interactions, rather than for the SD interactions. In order to obtain the effective coupling of the vector DM with nucleon induced by the effective Lagrangian, we need to evaluate the nucleon matrix elements of the quark and gluon operators in Eqs.(\[eff\_lagq\]) and (\[eff\_lagG\]). First, the nucleon matrix elements of the scalar-type quark operators are parametrized as $$f_{Tq}\equiv \langle N \vert m_q \bar{q} q \vert N\rangle/m_N~,$$ with $\vert N\rangle$ and $m_N$ the one-particle state and the mass of nucleon, respectively. The parameters are called the mass fractions and their values are obtained from the lattice simulations [@Young:2009zb; @:2012sa]. Second, for the quark twist-2 operators, we can use the parton distribution functions (PDFs): $$\begin{aligned} \langle N(p)\vert {\cal O}_{\mu\nu}^q \vert N(p) \rangle &=&\frac{1}{m_N} (p_{\mu}p_{\nu}-\frac{1}{4}m^2_N g_{\mu\nu})\ (q(2)+\bar{q}(2)) \ ,\end{aligned}$$ where $q(2)$ and $\bar{q}(2)$ are the second moments of PDFs of quark $q(x)$ and anti-quark $\bar{q}(x)$, respectively, which are defined as $q(2)+ \bar{q}(2) =\int^{1}_{0} dx ~x~ [q(x)+\bar{q}(x)]$. These values are obtained from Ref. [@Pumplin:2002vw]. Finally, the matrix element of gluon field strength tensor can be evaluated by using the trace anomaly of the energy-momentum tensor in QCD [@Shifman:1978zn]. The resultant expression is given as $$\langle N\vert G^a_{\mu\nu}G^{a\mu\nu}\vert N\rangle =-\frac{8\pi}{9\alpha_s} m_N f_{TG}$$ with $f_{TG}\equiv 1-\sum_{q=u,d,s}f_{Tq}$. Note that the right hand side of the expression is divided by the strong coupling constant, $\alpha_s$. For this reason, although the gluon contribution is induced by higher loop diagrams, it can be comparable to the quark contributions [@Hisano:2010ct]. Briefly speaking, the enhancement comes from the large gluon contribution to the mass of nucleon. As a result, the SI effective coupling of vector DM with nucleon, $f_N$, is given as $$\begin{aligned} f_N/m_N&=&\sum_{q=u,d,s} f_q f_{Tq} +\sum_{q=u,d,s,c,b} \frac{3}{4} \left(q(2)+\bar{q}(2)\right)g_q -\frac{8\pi}{9\alpha_s}f_{TG} f_G ~. \label{f}\end{aligned}$$ Using the effective coupling, we eventually obtain the SI scattering cross section of DM with nucleon: $$\sigma^{\rm (SI)}_{N}= \frac{1}{\pi}\biggl(\frac{m_N}{M+m_N}\biggr)^2~\vert f_N\vert ^2~.$$ Now, all we have to do reduces to evaluate the coefficients of the effective operators by integrating out the heavy fields in the high-energy theories. For example, we take the case where the interaction Lagrangian of the vector DM has a generic form as $$\begin{aligned} \mathcal{L}= \bar{\psi}_2 ~(a_{\psi_2\psi_1} \gamma^{\mu}+b_{\psi_2\psi_1} \gamma^{\mu}\gamma_5)\psi_1 B_{\mu} + \mathrm{h.c.}~, \label{eq:simpleL}\end{aligned}$$ where $\psi_1$ and $\psi_2$ are colored fermions with masses $m_1$ and $m_2$ ($m_1<m_2$), respectively. ![Tree-level diagrams of exchanging colored fermion $\psi_2$ to generate interaction of vector dark matter with light quarks.[]{data-label="fig:tree_general"}](tree_general.eps){height="4cm"} In this case, the vector DM is scattered by light quarks at tree-level. The relevant interaction Lagrangian is given by taking $\psi_1=q$ in Eq. (\[eq:simpleL\]), and the corresponding diagrams are shown in Fig. \[fig:tree\_general\]. After integrating out the heavy particle $\psi_2$, we obtain $$\begin{aligned} f_q&=&\frac{a^2_{\psi_2 q}-b^2_{\psi_2 q}}{m_q}\frac{m_{2}}{m^2_{2}-M^2} -(a^2_{\psi_2 q}+b^2_{\psi_2 q}) \frac{m^2_{2}}{2(m^2_{2}-M^2)^2}, \label{tree_general_1} \\ g_q&=&-\frac{2M^2(a^2_{\psi_2 q}+b^2_{\psi_2 q})}{(m^2_{2}-M^2)^2}. \label{tree_general_3}\end{aligned}$$ One can easily find that the effective couplings obtained here are enhanced when the vector DM and the heavy colored fermion are degenerate in mass [@Hisano:2011um]. ![One-loop contributions to scalar-type effective coupling with gluon. []{data-label="fig:loop_general"}](loop_general.eps){height="4cm"} The effective coupling of the vector DM with gluon is induced by 1-loop diagrams illustrated in Fig. \[fig:loop\_general\]. In those processes, all the particles $\psi_1$ and $\psi_2$ which couple with $B_{\mu}$ run in the loop. The resultant expressions are somewhat complicated, and thus we just quote Ref. [@Hisano:2010yh] for their complete formulae as well as their derivation. Application and Results ======================= Next, we deal with a particular model for vector DM as an application. We carry out the calculation for the first Kaluza-Klein (KK) photon DM [@Cheng:2002ej; @Servant:2002hb] in the minimal universal extra dimension (MUED) model [@Appelquist:2000nn; @Cheng:2002iz]. In this model, an extra dimension is compactified on an $S^1/\mathbb{Z}_2$ orbifold with the compactification radius $R$, and all of the Standard Model (SM) particles propagate in the dimension. The lightest KK-odd particle (LKP) is prevented from decaying to the SM particles, so it becomes DM. This model has just three undetermined parameters: the radius of the extra dimension $R$, the mass of Higgs boson $m_h$, and the cutoff scale $\Lambda$. In general, extra dimensional models give rise to the degenerate mass spectrum at tree-level, which is broken by radiative corrections. By evaluating the radiative corrections [@Cheng:2002iz], one finds that the first KK photon $B^{(1)}$ is the lightest KK-odd particle, thus, becomes the DM in the Universe. Moreover, since the mass difference is induced by radiative corrections, the mass degeneracy is tight for a small cut-off scale. In such a case, although it is difficult to probe the MUED model at the LHC because of the soft QCD jets, the direct detection rate of dark matter is expected to be enhanced [@Hisano:2011um; @Arrenberg:2008wy], as we shall see soon later[^2]. ![Tree-level diagrams for the elastic scattering of the KK photon DM $B^{(1)}$ with light quarks: (a) Higgs boson exchange contribution, and (b) KK quark exchange contributions.[]{data-label="fig:tree"}](tree.eps){height="3.5cm"} Now we evaluate the SI scattering cross section of the KK photon DM with nucleon. The effective interaction of the KK photon DM $B^{(1)}$ with light quarks is induced by the tree-level diagrams shown in Fig. \[fig:tree\]. Here, $h^0$ and $q^{(1)}$ are the Higgs boson and the first KK quark, respectively. ![One-loop diagrams for the effective interaction of $B^{(1)}$ with gluon.[]{data-label="fig:qloop"}](1loop.eps){height="4cm"} Also, there are one-loop diagrams we should evaluate for the gluon contribution. They are illustrated in Fig. \[fig:qloop\]. In these diagrams, all of the KK quarks run in the loop. Taking all the contributions into account, we obtain the effective interactions. Their expressions are given in Ref. [@Hisano:2010yh]. ![Each contribution in effective coupling $f_N/m_N$ given in Eq. (\[f\]). Here we set $m_h=125~{\rm GeV}$ and $(m_{\rm 1st}-M)/M=0.1$.[]{data-label="fig:fN_10"}](fN.eps){height="7cm"} ![Spin-independent cross section with proton for $m_h = 125$ GeV. Each line corresponds to $\Lambda = 5/R$, $20/R$, and $50/R$, respectively. []{data-label="mhfixed"}](sigmasi.eps){height="7.5cm"} In Fig. \[fig:fN\_10\], we plot each contribution to the SI effective coupling of DM with a proton. The solid line shows the contribution of the Higgs boson exchanging diagrams, the dashed line indicates the twist-2 type contribution, the dash-dotted line corresponds to the scalar-type contribution (except for the Higgs-exchanging contribution), and the dotted line represents the gluon contribution (except for the Higgs-exchanging contribution). Here we set the Higgs boson mass[^3] equal to 125 GeV, and the mass difference between the DM and the first KK quark to be 10 % by hand. We find that all of the contributions have the same sign thus they are constructive. The twist-2 contribution is dominant when $M\lesssim 800~{\rm GeV}$, while that of Higgs boson exchanging process becomes dominant above it. Moreover, although the tree-level quark contributions are dominant, it is found that the gluon contribution is not negligible at all. By using the effective couplings obtained above, we evaluate the SI scattering cross sections. In Fig. \[mhfixed\], we plot the SI cross section of KK photon DM with a proton as a function of the DM mass. Here again, the Higgs boson mass is set to be 125 GeV. We take the cut-off scale $\Lambda=5/R,~20/R, ~50/R$ from top to bottom. We find that the scattering cross sections reduce as the cut-off scale is taken to be large, as expected. As a result, we obtain the SI scattering cross section which is larger than those obtained by the previous calculations. Note that the DM mass with which the thermal relic abundance is preferred by the WMAP observation [@Komatsu:2010fb] is around 1300 GeV [@Belanger:2010yx]. It corresponds to $\sigma^{\rm SI}_N=$(2.5-5)$\times 10^{-47}$ cm$^{2}$, so the direct detection experiments with ton-scale detectors might be able to probe the DM in the future. Conclusion ========== We calculate the spin independent elastic scattering cross sections of vector dark matter with nucleon based on the effective field theory. It is found that the interaction of dark matter with gluon as well as quarks yields sizable contribution to the scattering cross section, though the gluon contribution is induced at loop level. The scattering cross section of the first Kaluza-Klein photon dark matter in the MUED model turns out to be larger than those obtained by the previous calculations. The author would like to thank Junji Hisano, Koji Ishiwata, and Masato Yamanaka for collaboration. This work is supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. References {#references .unnumbered} ========== [99]{} J. Hisano, K. Ishiwata, N. Nagata and M. Yamanaka, Prog. Theor. Phys.  [**126**]{}, 435 (2011) . E. Komatsu [*et al.*]{} \[WMAP Collaboration\], Astrophys. J. Suppl.  [**192**]{}, 18 (2011) . E. Aprile [*et al.*]{} \[XENON100 Collaboration\], arXiv:1207.5988 \[astro-ph.CO\]. H. C. P. Cheng, J. L. Feng and K. T. Matchev, Phys. Rev. Lett.  [**89**]{}, 211301 (2002). G. Servant and T. M. P. Tait, New J. Phys.  [**4**]{}, 99 (2002). A. Birkedal, A. Noble, M. Perelstein and A. Spray, Phys. Rev.  D [**74**]{}, 035002 (2006). R. D. Young and A. W. Thomas, Phys. Rev. D [**81**]{}, 014503 (2010) . H. Ohki , [*et al.*]{} \[JLQCD Collaboration\], arXiv:1208.4185 \[hep-lat\]. J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. M. Nadolsky and W. K. Tung, JHEP [**0207**]{}, 012 (2002). M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Phys. Lett. B [**78**]{}, 443 (1978). 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[^2]: Indirect DM searches also might be a powerful alternative in such a case [@Asano:2011ik; @Garny:2012eb]. [^3]: Recent searches for the Standard Model Higgs boson at the LHC indicate its mass to be around 125 GeV [@:2012gk; @:2012gu].

--- abstract: 'We investigate the occurrence of anomalous diffusive transport associated with acoustic wave fields propagating through highly-scattering periodic media. Previous studies had correlated the occurrence of anomalous diffusion to either the random properties of the scattering medium or to the presence of localized disorder. In this study, we show that anomalous diffusive transport can occur also in perfectly periodic media and in the absence of disorder. The analysis of the fundamental physical mechanism leading to this unexpected behavior is performed via a combination of deterministic, stochastic, and fractional-order models in order to capture the different elements contributing to this phenomenon. Results indicate that this anomalous transport can indeed occur in perfectly periodic media when the dispersion behavior is characterized by anisotropic (partial) bandgaps. In selected frequency ranges, the propagation of acoustic waves not only becomes diffusive but its intensity distribution acquires a distinctive L[é]{}vy $\alpha$-stable profile having pronounced heavy-tails. In these ranges, the acoustic transport in the medium occurs according to a hybrid transport mechanism which is simultaneously propagating and anomalously diffusive. We show that such behavior is well captured by a fractional diffusive transport model whose order can be obtained by the analysis of the heavy tails.' author: - Salvatore Buonocore - Mihir Sen - Fabio Semperlotti bibliography: - 'ref.bib' title: 'Occurrence of anomalous diffusion and non-local response in highly-scattering acoustic periodic media' --- Introduction {#Introduction} ============ In recent years, several theoretical and experimental studies have shown that field transport processes in non-homogeneous and complex media can occur according to either hybrid or anomalous mechanisms. Some examples of these physical mechanisms include anomalous diffusive transport (such as non-Fourier [@povstenko2013fractional; @borino2011non; @ezzat2010thermoelectric], or non-Fickian diffusion [@benson2000application; @benson2001fractional; @cushman2000fractional; @fomin2005effect] with heavy-tailed distribution) or hybrid wave transport (characterized by simultaneous propagation and diffusion [@mainardi1996fractional; @mainardi1996fundamental; @mainardi1994special; @mainardi2010fractional; @chen2003modified; @chen2004fractional]). Simultaneous hybrid and anomalous transport has also been observed, particularly in wave propagation problems involving random scattering media. Electromagnetic waves traveling through a scattering material[@yamilov2014position] such as fog [@belin2008display] or murky water [@zevallos2005time] are relevant examples of practical problems where such transport process can arise. A distinctive feature of anomalous transport is the occurrence of heavy-tailed distributions of the representative field quantities [@benson2001fractional]. In this case, the diffusion process does not follow a classical Gaussian distribution but instead is characterized by a high-probability of occurrence of the events associated with large variance (i.e. those described by the “heavy” tails). This behavior is typically not accounted for in traditional field transport theories based on integer order differential or integral models. Purely numerical methods, such as Monte Carlo or finite element simulations[@huang1991optical; @ishimaru2012imaging; @mosk2012controlling; @sebbah2012waves; @gibson2005recent], can capture this response but are very computationally intensive and do not provide any additional insight in the physical mechanisms generating the macroscopic dynamic behavior. The ability to accurately predict the anomalous response and to retrieve information hidden in diffused fields remains a challenging and extremely important topic in many applications. Acoustical and optical imaging, non-intrusive monitoring of engineering and biomedical materials are just a few examples of practical problems in which the ability to carefully predict the field distribution is of paramount importance to achieve accurate and physically meaningful solutions. Nevertheless, in most classical approaches, information contained in the heavy tails is typically discarded because it cannot be properly captured and interpreted by integer-order transport models. Hybrid and anomalous diffusive transport mechanisms are pervasive also in acoustics. This type of transport can arise when acoustic fields propagate in a highly scattering medium such as a urban environment [@albert2010effect; @remillieux2012experimental], a forest [@aylor1972noise; @tarrero2008sound], a stratified fluid (e.g. the ocean) [@baggeroer1993overview; @dowling2015acoustic; @casasanta2012fractional], or a porous medium [@benson2001fractional; @schumer2001eulerian; @fellah2003measuring; @fellah2000transient]. From a general perspective, classical diffusion of wave fields occurs within a range where the wavelength is comparable to the size of the scatterers, the so-called Mie scattering regime. Any deviation from classical diffusion, being either sub-diffusion [@metzler2000random; @goychuk2012fractional] (typically linked to Anderson localization) or super-diffusion (typically linked to L[é]{}vy-flights) [@barthelemy2008levy; @bertolotti2010multiple], still arises within the same regime. The two dominant factors are either the relation between the transport mean free path and the wavelength, or the statistical distributions of the scattering paths in presence of disorder. When a wave field interacts with scattering elements, it undergoes a variety of physical phenomena including reflection, refraction, diffraction, and absorption that significantly alter its initial characteristics. Depending on the quantity, distribution, and properties of the scatterers the momentum vector of an initially coherent wave can become quickly randomized. For most processes, the Central Limit Theorem (CLT) guarantees that the distribution of macroscopic observable quantities (e.g. the field intensity) converges to a Gaussian profile in full agreement with the predictions from classical Fourier diffusion. At the same time, the transition to a macroscopic diffusion behavior leads to an inevitable coexistence of diffusive and wave-like processes at the meso- and macro-scales. There are numerous physical processes in nature whose *basin of attraction* is given by the normal (Gaussian) distribution. On the other hand, when the distribution of characteristic step-length has infinite variance, the diffusion process no longer follows the standard diffusion theory, but rather acquires an anomalous behavior with a basin of attraction given by the so-called $\alpha$-stable L[é]{}vy distribution. In the latter case, the unbounded value of the variance of the step-length distribution is due to the non-negligible probability of existence of steps whose lengths greatly differ from the mean value; these are usually denoted as L[é]{}vy flights. The distinctive feature of the $\alpha$-stable L[é]{}vy distributions is the occurrence of heavy tails having a power-law decay of the form $p(l) \sim l^{-(\alpha+1)}$. This characteristic suggests that transport phenomena evolving according to L[é]{}vy statistics are dominated by infrequent but very long steps, and therefore their dynamics are profoundly different from those predicted by the random (Brownian) motion. Many of the complex hybrid transport mechanisms mentioned above fall in this category, and therefore cannot be successfully described in the framework of classical diffusion theory. In addition, these complex transport mechanisms are typically not amenable to closed-form analytical solutions therefore requiring either fully numerical or statistical approaches to predict the field quantities under various input conditions. Typical modeling approaches rely on random walk statistical models [@metzler2000random; @bouchaud1990anomalous] or on semi-empirical corrections to the fundamental diffusive transport equation via renormalization theory [@asatryan2003diffusion; @cobus2016anderson]. These approaches imply a considerable computational cost and do not provide physical insight in the operating mechanisms of the anomalous response. A few studies have also indicated that, for this type of processes, the macroscopic governing equation describing the evolution of the wave field intensity could be described by a generalization of the classical diffusion equation using fractional derivatives [@bertolotti2010multiple; @metzler2000random; @bertolotti2007light]. To-date, the occurrence of anomalous diffusion of wave fields has been connected and observed only in random and disordered media [@barthelemy2008levy; @burresi2012weak; @bouchaud1990anomalous; @asatryan2003diffusion; @cobus2016anderson]. In this study, we show theoretical and numerical evidence that anomalous behavior can occur even in presence of perfectly periodic media and in absence of disorder. We present this analysis in the context of diffusive transport of acoustic waves although the results could be generalized to other wave fields. In particular, we investigate the specific case of propagation of acoustic waves in a medium with identical and periodically distributed hard scatterers. We develop a theoretical framework for multiple scattering in super-diffusive periodic media. We first show, by full field numerical simulations, that under certain conditions, acoustic waves propagating through a periodic medium are subject to anomalous diffusion. Then, we propose an approach based on a combination of deterministic and stochastic methodologies to explore the physical origin of this unexpected behavior. Ultimately, we show that fractional order models can predict, more accurately and effectively, the resulting anomalous field quantities. More important, we will show that the analysis of the heavy tails provide a reliable means to extract the equivalent fractional order of the medium. Anomalous diffusion in acoustic periodic media: overview of the method ====================================================================== We consider the generic problem of an acoustic bulk medium made of periodically-distributed cylindrical hard scatterers in air (Fig. \[Fig\_1\]). We assume a monopole-like acoustic source, located in the center of the lattice, which emits at a selected harmonic frequency chosen within the scattering regime. The main objective is to characterize the propagation of acoustic waves in such medium based on different regimes of dispersion. As previously anticipated, in selected regimes the propagation of acoustic waves will exhibit anomalous diffusive transport properties. The reminder of this study will be dedicated to investigating the causes leading to the occurrence of such phenomenon. In order to identify the fundamental mechanisms at the origin of this behavior, we have designed a multi-folded approach capable of characterizing the different processes contributing to the macrosopic anomalous response. The approach consists of the following components. First, we investigate via full-field numerical simulations the propagation of acoustic waves in either a 1D or a 2D periodic scattering medium. The numerical results will allow making important observations on the different propagation mechanisms occurring in the two systems and on the corresponding diffusive processes. Then, the radiative transfer theory will be applied to interpret the evolution of the wave intensity distribution and analyze the nature of the diffusive phenomena in the context of a renormalization approach. In order to identify the physical mechanism at the origin of the anomalous diffusion, a multiple scattering analysis based on the multipole expansion method will be applied in order to characterize the interaction between different scatterers. In particular, this approach was intended to identify and quantify possible long-range interactions between pairs of scatterers. Based on the results of the multiple scattering analysis, a Monte Carlo model is used to confirm that the anomalous transport is in fact originated by the long-range interactions between different directions of propagation in the lattice. Finally, we show that the behavior of the lattice can be effectively described in a homogenized sense, by a fractional continuum diffusion model whose fractional order can be identified by fitting an $\alpha$-stable distribution to the heavy tails of the wave intensity. This approach can be seen as an equivalent *fractional homogenization* of the medium. Of particular interest is the fact that the fractional (homogenized) model allows a closed-form analytical solution the agrees very well with the numerical predictions. Scattering and diffusive transport {#Overview} ================================== From a general perspective, it is possible to identify four different wave propagation regimes in scattering media which are classified based on the relative ratio of quantities such as the transport mean free path $l_t$, the wavelength of the propagating field $\lambda$, and the characteristic size of the scattering domain $L$. The four regimes are: 1. The *homogenized regime*: it occurs when the wavelength of the incident wave field is much larger than the typical characteristic size $d$ of the scatterer, that is $\lambda \gg d$. 2. The *diffusive regime*: it occurs when the wavelength $\lambda$ satisfies the relation $\lambda/2\pi \ll l_t \ll L $. In this regime the wave intensity can be approximated by the diffusion equation. 3. The *anomalously diffusive regime*: it occurs when the interference of waves causes the reduction of the transport mean free path $l_{t}$ and consequently a renormalization of the macroscopic diffusion constant $D$. In this regime, the transport mean free path varies according to the size of the cluster and to the degree of disorder. 4. The *localization regime*: it occurs in the range $\lambda/2\pi \geq l_{t}$ and corresponds to a diffusion constant $D$ tending to 0. As mentioned in the classification above, there are regimes in which the intensity of the wave field can be properly described by the *diffusion approximation*, that is it varies in space as prescribed by the field evolution in a diffusion equation. In particular, when the incident wave has a wavelength smaller than the length-scale characterizing the material and/or of the geometric variations of the physical medium, the wave field undergoes multiple scattering with a consequent randomization of its phase and direction of propagation. In order to characterize this phenomenon a statistical description based on random walk models is typically employed. These models rely on phenomenological quantities such as the scattering $l_s$ and the transport $l_t$ mean free paths. From a physical perspective, $l_s$ represents the average distance between two successive scattering events, while $l_t$ is the mean distance after which the wave field loses memory of its initial direction and becomes randomized [@van1999multiple]. When the filling factor $f$ (which describes the density of scatterers) is low, $l_{s}$ and $l_{t}$ are defined as [@ishimaru1978wave]: $$\begin{aligned} \label{ls_lt} l_{s} &=& \dfrac{1}{\rho\sigma_{t}}\\ l_{t} &=& \dfrac{l_{s}}{1-\langle\cos{\theta}\rangle} \nonumber \label{eq:lt}\end{aligned}$$ where $\sigma_{t}$ is the total scattering cross section, $\langle\cos{\theta}\rangle$ is the *anisotropy factor* and $\rho$ is the scatterers concentration. Note that the relations in Eq. (\[ls\_lt\]) are valid only for low filling factors, approximately in the range $f \leq 0.1$. For increased values of the filling factor, the scattering cross section $\sigma_t$ needs to be rescaled. The rescaling factor in the range $0.1 \leq f \leq 0.6$ is given by $\sigma_t \rightarrow \sigma_t (1-f)$, while higher filling factors require a more elaborated rescaling procedure [@ishimaru1978wave]. The scattering cross section plays a crucial role in the characterization of multiple scattering phenomena and in two dimensions takes the form: $$\begin{aligned} \label{sigma_t} \begin{split} \sigma_t = \int_{2\pi}\sigma_d(\theta) d\theta . \end{split}\end{aligned}$$ The integrand $\sigma_d$ is the differential scattering cross section defined as: $$\begin{aligned} \label{diff_scatt_cross_sect} \begin{split} \sigma_d(\theta)= \lim_{R \rightarrow \infty} R\left [ (S_s(\theta))/S_i\right ]. \end{split} \end{aligned}$$ . In Eq. (\[diff\_scatt\_cross\_sect\]), the term $S_s$ is the scattered power flux density at a distance $R$ from the scatterer in the direction $\mathbf{\hat{o}}$ caused by an incident power flux density $S_i$. The azimuthal angle $\theta$ is the angle between the incident ($\mathbf{\hat{i}}$) and the scattered wave fields ($\mathbf{\hat{o}}$). The *scattering phase function* is obtained by normalizing the differential scattering cross section with respect to $\sigma_t$: $$\begin{aligned} \label{} p\mathbf{( \hat{o},\hat{i})}=p(\cos\theta) = \frac{\sigma_d(\theta)}{\sigma_{t}} \end{aligned}$$ and represents the probability that a wave field impinging on the scatterer from a given direction will be scattered by an angle $\theta$. The mean value of the previous probability distribution defines the *anisotropy factor*: $$\begin{aligned} \label{} \langle \cos \theta \rangle = \int_{2\pi} p(\cos \theta) \cos(\theta) d \theta. \end{aligned}$$ This factor varies between 0 and 1, and it accounts for the existence of preferential scattering directions. For $\langle\cos{\theta}\rangle=0$ all the scattering directions have the same probability and the scattering is isotropic. As $\langle\cos{\theta}\rangle$ approaches 1, the forward scattering becomes the most probable event. These quantities will be used in the following analyses in order to identify the different scattering regimes. One-dimensional medium ====================== Consider a one-dimensional bulk scattering medium composed of $N$ hard cylindrical scatterers equally distributed in an air background (Fig. \[Fig\_3\]). This system can be interpreted by all means as a classical 1D acoustic metamaterial. The radius of the individual scatterer is $a = 0.2 d$, where $d$ indicates the distance between two neighboring cylinders. The filling fraction for this particular cluster is $f = \pi a^2/d^2 \approx 0.1257$. The waveguide is excited by a monochromatic acoustic monopole $S$ that replaces the center cylinder. The response of the system is obtained numerically by means of a commercial finite element software (Comsol Multiphysics) and using symmetric boundary conditions on the top and bottom edges and perfectly matched layers (PML) on the left and right edges. The frequency of excitation is selected in the first bandgap (see Fig. \[Fig\_10\] for the general dispersion properties of this waveguide) and has a non-dimensional value $\Omega = 0.0831$. In this excitation regime the diffusive behavior is expected. Remember that, in the absence of disorder or trapping mechanisms and in the range of excitation frequencies where the diffusion approximation holds, the variance of the step-length distribution characterizing the multiple scattering process of the acoustic field is expected to be finite and, if the steps are independent (by virtue of the Central Limit Theorem) the limit distribution should be the Normal distribution as predicted by the standard diffusion model. The resulting normalized magnitude of the acoustic pressure field generated in the waveguide is shown in Fig. \[Fig\_4\](a) in terms of a contour plot and in Fig. \[Fig\_4\](b),(c) in terms of the intensity profile along the mid-line of the waveguide as defined later in §\[Modell\]. From Fig. \[Fig\_4\](b),(c) a characteristic exponential decay of the type $e^{-x/l_s}$ (consistent with the Beer-Lambert law) is very well identifiable. This trend represents the decay of the coherent part of the intensity and corresponds to the squared absolute value of the Green’s function solution. In more general terms, this result shows a solution which is perfectly consistent with the classical diffusion behavior. This is a well expected result and it is reported here only for comparison with the results that will be presented below. Two-dimensional medium ====================== Radial lattice -------------- The immediate extension of the previous scenario to a two-dimensional system corresponds to a radial distribution of equally distributed hard scatterers. As in the 1D case, the system is excited by a monochromatic monopole source located in the center of the 2D lattice at point $S$. The source is monochromatic and it is actuated at the non-dimensional frequency $\Omega = 0.0831$, that belongs to the first bandgap. The normalized magnitude of the acoustic pressure field for this system is numerically calculated and shown in Fig. \[Fig\_6\](a). Fig. \[Fig\_6\](b) provides a closeup view of the field around the source (the area within the black dashed line). The acoustic intensity profile along the $x$-axis direction shows an exponential decay as illustrated by Fig. \[Fig\_7\]. All radial directions (not shown) exhibit an identical response as expected due the azimuthal symmetry of the system. As in the 1D case, this linear decay of the intensity distribution was expected and confirms that, in systems with a high degree of symmetry, a classical diffusion behavior should be recovered. From a practical perspective, this radial lattice could be seen as a radial arrangement of 1D waveguides. Rectangular lattice {#rect_lattice} ------------------- The dynamic behavior of the lattice changes quite drastically when the axial-symmetry is removed. Consider the square lattice of scatterers schematically illustrated in Fig. \[Fig\_1\]. Assume each scatterer having an individual radius of $a = 0.2 d$, where $d$ is the distance between two neighboring cylinders. The filling fraction for this periodic cluster is $f = \pi a^2/d^2 \approx 0.1257$. ### Dispersion analysis {#DispersionRelation} In order to understand the dynamic behavior of this lattice and interpret the results that will follow, we start analyzing the fundamental dispersion structure of the square lattice. The dispersion was calculated using finite element analysis and the band structure is plotted along the irreducible part of the first Brillouin zone, as shown in Fig. \[Fig\_10\]. The results highlight the existence of anisotropy in terms of directions of propagation. These directions are connected to the existence of a partial bandgap in the $\Gamma-X$ direction between the non-dimensional frequencies $\Omega = 0.0824$ and $\Omega = 0.1103$. When the system is excited at a frequency within the bandgap, the propagation acquires an anisotropic distribution (see § \[Forced\] and Fig. \[Fig\_8\]), because propagation can only occur in the $\Gamma-M$ direction. This is not an unexpected result and, in fact, it is fully consistent with the propagation behavior expected in square periodic lattice. However, we will show that these dispersion characteristics play a key role in the occurrence of anomalous behavior. ### Forced response {#Forced} The forced response of the lattice was also numerically evaluated. In this case, the lattice is excited by a monochromatic acoustic monopole $S$ that replaces the center cylinder. As for the previous two lattices, the total acoustic pressure field is calculated numerically using the finite element method and reported in Fig. \[Fig\_8\]. More specifically, Fig. \[Fig\_8\](a) presents the response to an excitation outside the first bandgap, while Fig. \[Fig\_8\](b) reports the case just inside the first bandgap. Note that due to symmetry considerations, only a quarter of the domain was solved. As the acoustic wave fronts propagate through the medium in the radial directions and interact with the scattering particles, the rays are scattered in multiple directions. In both cases it is evident that the propagation is strongly anisotropic and occurs mostly along the diagonal directions of the lattice. The response of the medium is shown in Fig. \[Fig\_8\] in terms of the normalized magnitude of the acoustic pressure distribution. Contrarily to what observed for the radial lattice, in this case the intensity distribution does not decay linearly. This behavior is very evident by performing a numerical fit of the simulation data, as shown in Fig. \[Fig\_9\]. These results suggest the occurrence of an unexpected mechanism of diffusion despite the lattice periodicity. This is a remarkable departure compared with available results in the literature that, to-date, have highlighted the occurrence of anomalous diffusion only in connection with random distributions of geometric or material properties. Radiative transport approach {#Modell} ============================ The results presented above illustrated that in case of anisotropic propagation a departure from the classical diffusive behavior is observed. In this section, we use a traditional radiative transport approach with renormalization to show that this observed behavior can be mapped to anomalous diffusion. We investigate the presence of anomalous diffusion for wavelength ranges in the passband and in the bandgap. As already pointed out, within the regime $\lambda/2 \pi<l_{t}<L$ the diffusion approximation applies and the spatial evolution of the wave amplitude can be predicted by a diffusion equation for the wave intensity. Starting from a cluster of particles, as schematically illustrated in Fig. \[Fig\_11\], and applying the diffusion approximation the 2D diffusion equation for harmonic excitation and lossless scatterers is given by: $$\begin{aligned} \label{Diffusion equation} \begin{split} \nabla^2 I =-\frac{P_0}{\pi l_t}\delta(\vec{r}-\vec{r}_s) \end{split}\end{aligned}$$ where $I$ is the intensity of the acoustic wavefield, $P_0$ is the total emitted acoustic power, $\vec{r}$ and $\vec{r}_s$ are the position vectors of the source $S$ and of a generic point $P$, respectively. The average acoustic intensity of a monochromatic monopole source can be obtained as $\langle I \rangle = ||0.5*Re(p \cdot v')||$, where $p$ is the pressure field, and $v'$ is the complex conjugate of the velocity field. The diffusion equation Eq. (\[Diffusion equation\]) requires the following boundary conditions at the edge of the domain to be solved: $$\begin{aligned} \begin{split} I\mathbf{(r_s)}-\frac{\pi l_t}{4}\frac{\partial }{\partial n} I\mathbf{(r_s)} = 0 \end{split}\end{aligned}$$ where $\mathbf{\hat{n}}$ is the unit inward normal. These boundary conditions are obtained by the requirement of zero inward flux at the boundaries[@ishimaru1978wave]. The numerical value of this boundary condition on the intensity was obtained by the previous finite element model. By enforcing this boundary condition, Eq. (\[Diffusion equation\]) can be solved analytically: $$\begin{aligned} \begin{split} I = -\frac{P_0}{2\pi^2 l_t}ln\frac{|\vec{r}-\vec{r}_s|}{L}+ I_{0} \end{split} \label{Solution}\end{aligned}$$ where $I_{0}$ is the value of the intensity at the boundary of the cluster of scatterers and $L$ is the size of the computational domain. In order to be able to solve Eq. (\[Solution\]), we need to estimate the parameters $l_t$ and $l_s$ and characterize the specific regime of propagation. To achieve this result, we first plot $\langle l_{s}\rangle$ and $\langle l_{t}\rangle$ versus the wavelength $\lambda$ as shown in Fig. \[Fig\_12\]. These curves were numerically determined using the model presented in §\[rect\_lattice\] and the Eqs. (\[ls\_lt\]). The transport mean free path $\langle l_{t}\rangle$ is always expected to be greater than $\langle l_{s}\rangle$ and to converge asymptotically to $\langle l_{s}\rangle$ for large wavelengths. In fact, for long wavelengths the wavefield is marginally affected by the presence of the scatterers. In the short wavelength limit, $l_s/d$ tends to 1 because the wave fronts are highly directional (this is the range of validity of ray acoustics approximation) and **$l_s$** is approximately given by the average distance between two neighboring scatterers. Fig. \[Fig\_13\] shows a detailed view of the previous curves in the frequency range corresponding to the first bandgap and within the diffusive regime. The labels $A$ and $B$ indicate the non-dimensional wavelengths corresponding to the excitation conditions analyzed in the following sections. Note that these curves provide the foundation to investigate the different regimes of propagation and to implement the renormalization approach. ### Renormalization and anomalous diffusion {#Numerical_1} Fig. \[Fig\_14\] shows the acoustic intensity distribution $I$ along the $x$ axis for the two excitation conditions identified by the labels A and B. The red circles show the numerical solution obtained by the FE model and provide a one-dimensional section of the data in Fig. \[Fig\_8\] along the $x$-axis. The continuous blue line is the analytical solution of the diffusion equation Eq. (\[Solution\]) after having rescaled the transport mean free path. In particular, for excitation wavelengths in the first passband the value $\langle l_{t}\rangle/d\approx 1.211$ was rescaled to $\langle l_{t}\rangle/d\approx 0.32 \pm 0.02$ (label $A^*$ in Fig. \[Fig\_13\]), while for the first bandgap the value $\langle l_{t}\rangle/d\approx 1.42$ was rescaled to $\langle l_{t}\rangle/d\approx 0.48 \pm 0.02$ (label $B^*$ in Fig.  \[Fig\_13\]). These results show that, in order to be able to predict the numerical data by using the diffusion approximation, a renormalization of the transport mean free path (and consequently of the diffusion coefficient) must take place. The renormalization requires smaller values of the transport parameters which is a clear indication of superdiffusive anomalous behavior. Causes of anomalous diffusion {#NumericalViewPer} ============================= In the previous sections we showed the occurrence of anomalous diffusion of acoustic waves in perfectly periodic square lattices and suggested that the possible origin of this mechanism is linked to the anisotropy of the dispersion properties (i.e. to the anisotropy of the bandgaps). In this section we will present theoretical and numerical models with the intent of uncovering the physical mechanism leading to this unexpected propagation modality. It is anticipated that the occurrence of anomalous diffusion will be connected to the existence of long range interactions between different directions of propagation governed by either bandpass or stopband behavior. We will use a combination of both deterministic and stochastic methods in order to quantify the long-range interactions and to demostrate that they are at the origin of the macroscopic anomalous diffusion mechanism. More specifically, we will use a scattering matrix approach to quantify the interaction between different scatterers in different regimes. Then, we will use a discrete random walk diffusion model (which uses probability density functions obtained from the scattering model) to show that, under these assumptions, the anomalous diffusion process matches well with the numerically predicted behavior. The scattering matrix --------------------- In order to evaluate and quantify the strength of the interaction between different scatterers in the lattice, we use a multiple scattering approach based on the multipole expansion method. According to this method, after applying the Jacobi’s expansion and the Graf’s addition theorem, the general solution of the wave field can be expressed as: $$\begin{aligned} \begin{split} \label{Eq.206} p(\vec{r}_m)=\sum_{j=-\infty}^{\infty} (e^{i\vec{k}\cdot \vec{P}_m} e^{ij(\pi /2-\psi_0)}J_j(\vec{r}_m)+A_j^m H_j(\vec{r}_m))+\\ \sum_{n=1,n\neq m}^{N}\sum_{q=-\infty}^{\infty}A_n^q\sum_{j=-\infty}^{\infty}H_{q-j}(kR_{nm})e^{i(q-j)\Phi_{nm}}J_j(\vec{r}_m) \end{split}\end{aligned}$$ where $\vec{k}$ represents the wave vector, $\psi_0$ is the angle of the impinging wave field with respect to the $x$-axis, $\vec{P}_m$ is the position vector of the scatterer’s center $O_m$ with respect to the origin $O$ of the system of reference, $\vec{r}_m$ is the position vector of a generic point $P$ with respect to the scatterer’s center $O_m$, $R_{nm} = \left |\vec{P}_m - \vec{P}_n \right |$, $J_q(\cdot)$ are the Bessel functions of the first kind, $H_q(\cdot)$ are the Hankel functions of the first kind. To determine the unknown amplitude coefficients $A_m^q$ the boundary conditions at the surface of the $m$th cylinder must be enforced. The result is a linear set of equations as follows[@linton2005multiple; @martin2006multiple; @kafesaki1999multiple]: $$\begin{aligned} \begin{split} \label{Eq.207} A_m^p+Z_p\sum_{n=1,n\neq m}^{N}\sum_{q=-\infty}^{\infty}A_n^qH_{q-p}(kR_{nm})e^{i(q-p)\Phi_{nm}}= \\ -Z_p e^{i\vec{k}\cdot\vec{P}_m} e^{ip(\pi/2-\psi_0)},\\ \quad m=1,...,N ,\quad p=0,+1,-1,... \end{split}\end{aligned}$$ where $Z_p =J'_p(\cdot)/ H'_p(\cdot)$ specifies the Neumann boundary conditions on the surface of the cylinders. The unknown amplitude coefficients $A_m^q$ can be determined by solving the infinite system of algebraic equations with inner sum truncated at some positive integer $|q| =Q$. The information about the relative energy exchange between the scatterers can be obtained by rearranging Eq. (\[Eq.207\]) in matrix form as: $$\begin{aligned} \begin{split} \label{Eq.209} (I-TS)f=Ta \end{split}\end{aligned}$$ where $I$ is the unit matrix, $T$ is the block diagonal impedance matrix, the vector $f$ represents the unknown expansions of scattered waves, and the vector $a$ stands for the expansion vector of incident waves on all the scattering cylinders. Finally the matrix $S$ is the so called combined translation matrix that can be expressed as follows: $$\begin{aligned} \begin{split} \label{Eq.210} S=\begin{bmatrix} 0 & L_{12} &...& L_{1N}\\ L_{21} & 0 &...&L_{2N}\\ \vdots & \vdots & \ddots & \vdots \\ L_{N1}& L_{N2} & ... & 0 \end{bmatrix} \end{split}\end{aligned}$$ where the matrix $L_{nm}$ is defined as follows: $$\begin{aligned} \begin{split} \label{Eq.211} L_{nm}(q,p)=H_{q-p}(kR_{nm})e^{i(q-p)\Phi_{nm}}. \end{split}\end{aligned}$$ The matrix $L_{nm}$ represents the translation matrix between the $n$th and the $m$th cylinder, representing therefore the incident wave on the $n$th cylinder caused by the scattered wave off the $m$th cylinder. The elements of the translation matrix can be obtained from the addition theorem of cylindrical harmonics also known as Graf’s theorem. The generic term $S_{mn}$ quantifies the portion of the acoustic intensity scattered by the cylinder $m$ capable of reaching the cylinder $n$. Equivalently, it represents the fraction of the acoustic intensity reaching the cylinder $n$ due to the wave scattered by the cylinder $m$. This approach was applied to model both the 1D and the 2D waveguides. The normalized scattering coefficients for the 1D waveguide are shown in matrix form in Fig. \[Fig\_16\]. Each block of this matrix has a size $\bar{Q} = 2*Q+1$, where $Q$ is the total number of spherical harmonics used in the multipole series expansion. The total size of the matrix is $N*\bar{Q}$, where $N$ is the total number of cylinders in the cluster. The main diagonal represents the coefficient $S_{mm}$, that is the scattering of a given cylinder $m$ towards itself, and therefore these terms are all zero. The analysis of Fig. \[Fig\_16\] shows, as expected, that in a 1D waveguide the scatterers only interact with their closest neighbors. In other terms, there is no evidence of long-range interaction in 1D periodic waveguides. This is not surprising because we had already found from the full-field simulations in Fig. \[Fig\_4\](a) that the diffusive transport was following a purely Gaussian distribution (hence dominated by nearest neighbor interactions). In a similar way, the analysis can be repeated for the 2D waveguide with square lattice structure. The resulting scattering coefficients are shown in Fig. \[Fig\_17\]. Contrarily to the 1D example, this scattering matrix has the appearance of a tridiagonal matrix that highlights the substantial interactions between distant neighbors. In other terms, the rectangular lattice show strong evidence of long-range interactions. These results provide a first important observation concerning the cause of anomalous diffusion in periodic rectangular lattice, that is the anisotropy of the dispersion bands gives rise to long-range interactions that ultimately alter the diffusion process. Discrete random walk models: approximate acoustic intensity ----------------------------------------------------------- The previous analysis is not yet sufficient to provide conclusive evidence that the long-range interactions due to the bandgap anisotropy are the main cause of the anomalous wave diffusion. In order to identify this further logical link, we developed a discrete random walk (DRW) model capable of simulating the diffusion process resulting from the multiple scattering of the acoustic waves. The interaction between the different elements of a DRW model is typically represented by probability density functions (*pdf*). In the following, the *pdf*s are synthesized based on the coefficients of the scattering matrix. The model can then be numerically solved in order to predict the approximate acoustic intensity resulting from the scattered field. ### 1D discrete random walk model The DRW model for a 1D waveguide is composed of a series of boxes (see Fig. \[Fig\_18\]), each one representing a scatterer. This model can be seen as the direct discrete equivalent of the 1D waveguide in Fig. \[Fig\_3\]. The dots in each box represent the different acoustic rays impinging on a given scatterer and being refracted towards different (scattering) elements. This model follows a ray acoustic approximation which is a reasonable assumption in the range of wavelength we have been considering. To simulate the monopole acoustic source located at the center of the waveguide, the center box (labeled $i$) contains a source term that serves as an omni-directional source of rays. In the 1D model, the rays emitted from the center box can be scattered both to the left and to the right according to the associated *pdf* synthesized based on the elements of the scattering matrix. At every time increment, the rays “jump” into another box following a Markovian process and a *pdf* proportional to the coefficients extracted from the scattering matrix. The equilibrium condition needed to solve the DRW model and simulate the evolution of the acoustic intensity upon scattering is given by imposing the conservation of rays: $$\begin{aligned} \label{eqn:Conservation number of particles} \begin{split} n_{i,j+1}=n_{i,j}+\sum_{k=1}^{N_L} n_{k,j}P(i-k)+\sum_{k=1}^{N_R}n_{k,j}P(k-i)- \\ \sum_{k=1}^{N_L}n_{i,j}P(i-k)-\sum_{k=1}^{N_R}n_{i,j}P(k-i) + B_i \end{split}\end{aligned}$$ where $i$ is the box index, $j$ is the time index, $n(i,j)$ is the number of rays at time $i$ entering the box $j$ (i.e. impinging on the scatterer $j$), $B_i$ is the source term, and $N_L$ and $N_R$ represent the number of boxes on the left and right side, respectively. The previous equation can be rearranged as follows: $$\begin{aligned} \label{eqn:Conservation number of particles 2} \begin{split} n_{i,j+1}=n_{i,j}+\sum_{k=1}^{N_L}(n_{k,j}-n_{i,j})P(i-k)- \\ \sum_{k=1}^{N_R}(n_{k,j}-n_{i,j})P(k-i)+ B_i. \end{split}\end{aligned}$$ The comparison between the intensity distributions obtained with the FE model and by the equivalent 1D DRW model is given in Fig. \[Fig\_19\]. The direct comparison of the results shows a very good agreement between the two models. Note that the DRW is a diffusive model therefore the comparison between the intensity distributions is meaningful only in the tail region. As expected the tails evolve according to a Gaussian distribution. The comparison with the 1D waveguide was provided to illustrate the validity of the proposed approach and to confirm that, under the given assumptions, the results from the DRW converge to the full-field simulations. ### 2D discrete random walk model The same approach illustrated above for the 1D waveguide can be applied to the analysis of the 2D square lattice. In this case, the DRW model is composed of a 2D distribution of boxes simulating the scatterers. The interactions between different boxes are again expressed in terms of *pdf*s that are synthesized based on the scattering coefficients obtained from the 2D multipole expansion model (Fig. \[Fig\_17\]). The equilibrium condition for the 2D DRW model is given by: $$\begin{aligned} \label{eqn:Conservation number of particles 2D} \begin{split} n_{i,h,j+1}=n_{i,h,j}+ \\ \sum_{k_h=1}^{N_D}\sum_{k_i=1}^{N_L}(n_{k_i,k_h,j}-n_{i,h,j})P(i-k_i,h-k_h)-\\ \sum_{k_h=1}^{N_U}\sum_{k_i=1}^{N_R}(n_{k_i,k_h,j}-n_{i,h,j})P(k_i-i,k_h-h)+ B_{ih} \end{split}\end{aligned}$$ where $i$ and $h$ are the box indices, $j$ is the time index, $n(i,h,j)$ is the number of particles at time $j$ in the box $(i,h)$, $B_{ih}$ is the source term and $N_L $, $N_R$, $N_U$ and $N_D$ represent the number of boxes on the left, right, up and down sides, respectively. The comparison between the intensity distributions obtained by numerical FE simulations and by equivalent 2D DRW model is reported in Fig. \[Fig\_20\]. Also in this case, the DRW model is in very good agreement with the FE simulations and, most important, is perfectly capable of capturing the anomalous (power-law) decay of the tails of the distribution. This result provides the conclusive proof that the anomalous behavior observed in the square lattice is in fact the result of long-range (L[é]{}vy flights) interactions due to scattering events occurring along different directions of propagation that are characterized by anisotropic dispersion. $\alpha$-stable distributions and fractional diffusion equation {#fractional diffusion} =============================================================== The renormalization criterion used in section §\[Numerical\_1\] to determine the existence of the anomalous diffusion regime is theoretically well-grounded but it does not allow a convenient approach to classify the anomalous regime. This classification typically requires the analysis of the time scales involved in the evolution of the moments of the distribution [@metzler2000random]. Here we suggest a different approach that, not only provides a more direct classification based on the available data, but opens new routes for an analytical treatment of the resulting diffusion problem. The intensity distributions reported in Fig. \[Fig\_14\] suggest a power-law behavior of the tails. Recent studies [@mainardi1996fractional; @mainardi1995fractional; @benson2000application; @benson2001fractional] have shown that, for physical phenomena exhibiting this characteristic distribution of the field variables, the governing equations are generalizations to the fractional order of the classical diffusion equation. Power-law distributions, associated with infinite variance random variables (the so called L[é]{}vy flights), are in the domain of attraction of $\alpha$-stable random variables also called L[é]{}vy stable densities (their properties are summarized in Appendix \[Appendix\]). On the other hand finite-variance random variables are in the Normal domain of attraction that is a subset of L[é]{}vy stable densities. This suggests that the trend of the tails carries information about the $\alpha$-stable order of the underlying distribution. In order to show that this situation occurs also in the present case, we performed numerical fits of the acoustic intensity profiles (Fig. \[Fig\_14\]) using $\alpha$-stable distributions. The four parameters defining the $\alpha$-stable distributions were obtained by numerically solving a nonlinear optimization problem. The most important parameter is the characteristic exponent $\alpha$ (also called the index of stability) that is also connected to the slope of the tails. For the square lattice distribution, the values of $\alpha$ determined with the optimization procedure are $\alpha = 0.89$ and $\alpha = 0.57$ for the passband and bandgap excitation wavelengths, respectively. In order to show that the order of the $\alpha$-stable distribution effectively describes the anomalous diffusive dynamics of the system, we use a generalized fractional diffusion equation [@mainardi2007fundamental]: $$\begin{aligned} \label{Fractional diffusion} \begin{split} _{x}\textrm{D}_{\theta}^{\alpha}u(x,t)=_{t}\textrm{D}_{*}^{\beta}u(x,t) \quad x \in \mathbb{R} , \quad t \in \mathbb{R}^+ \end{split}\end{aligned}$$ where $\alpha$, $\theta$, $\beta$ are real parameters always restricted as follows: $$\begin{aligned} \label{Restrictions} \begin{split} 0< \alpha \leq2, \quad |\theta|\leq min\left \{ \alpha,2-\alpha \right \}, 0 <\beta \leq 2. \end{split}\end{aligned}$$ In Eq. (\[Fractional diffusion\]), $u = u(x,t)$ is the field variable, $_{x}\textrm{D}_{\theta}^{\alpha}$ is the *Riesz-Feller* space fractional derivative of order $\alpha$, and $_{t}\textrm{D}_{*}^{\beta}$ is the Caputo time-fractional derivative of order $\beta$. The fractional operator in this equation exhibits a non-local behavior which makes it ideally suited to model dynamical systems dominated by long-range interactions. Mainardi [@mainardi2007fundamental] reported the Green’s function for a Cauchy problem based on the space-time fractional diffusion equation. The self-similar nature of the solution allows the application of a similarity method that separates the solution into a space dependent (the reduced Green’s function $K$) and a time dependent term. In our system, we use a harmonic (constant amplitude) source and we analyze the steady state response, that is we consider a self-similar problem. In other terms, the reduced Green’s function $K$ proposed by Mainardi coincides with the normalized solution of the forced fractional diffusion equation governing our problem: $$\begin{aligned} \label{Reduced Green function} \begin{split} \textrm{K}_{\alpha,\beta}^{\alpha}(x) = \frac{1}{\pi x}\sum_{n=1}^{\infty}\frac{\Gamma (1+\alpha n)}{\Gamma(1+\beta n)}\sin\left [ \frac{n\pi}{2}(\theta-\alpha) \right ](-x^{-\alpha})^n. \end{split}\end{aligned}$$ Note that this solution is valid in the case $\alpha<\beta$. In our case, $\beta=1$ to model a space fractional diffusion equation. Fig. \[Fig\_22\] shows the comparison between the normalized acoustic intensity from the FE numerical data and the result from the reduced Green’s function $K$ (Eq. (\[Reduced Green function\])) calculated for the order $\alpha$ obtained by the previous $\alpha$-stable fits. The above results clearly show that the fractional diffusion equation is able to capture the heavy-tailed behavior of the intensity distribution with good accuracy. They also confirm that the use of $\alpha$-stable fits provides a reliable approach to classify the anomalous behavior and to extract the corresponding fractional order of the operator. The above numerical results provide also further confirmation that the observed dynamic behavior from the full field numerical simulations is in fact dominated by anomalous diffusion. These results are particularly relevant if seen in a perspective of developing predictive capabilities for transport processes in highly inhomogeneous systems. As an example, fractional models would provide an excellent framework for the solution of inverse problems in imaging and remote sensing through highly scattering media. The ability to properly capture a mixed transport behavior, such as partially propagating and diffusive, would allow extracting more information from the measured response therefore improving the sensitivity and resolution of these approaches. From a broader perspective, this methodology has general applicability and could be extended to a variety of applications involving wave-like field transport such as those mentioned in the introduction. Conclusions {#Conclusion} =========== In this paper, we investigated the scattering behavior of sound waves in a perfectly periodic acoustic medium composed of a square lattice of hard cylinders in air. From a general perspective, the most remarkable result consists in the observation of the occurrence of anomalous hybrid transport in perfectly periodic lattice structures without disorder or random properties. This result is particularly relevant because the anomalous response of a scattering system was previously observed only in systems with either stochastic material or geometric properties. By using a combination of theoretical and numerical models, both deterministic and stochastic, it was determined that the existence of long-range interactions associated with the anisotropy of the dispersion bands was the driving factor leading to the occurrence of the anomalous transport behavior. The resulting diffused intensity fields were characterized by heavy-tails with marked asymptotic power-law decay, that were well described by $\alpha$-stable distributions. It was also shown that the $\alpha$-stable nature of the dynamic response provided a reliable approach for the classification and characterization of the non-local effect via the intrinsic parameters of $\alpha$-stable distributions. Observing that $\alpha$-stable distributions represent the fundamental kernel for the solutions of fractional continuum models, we showed that a space fractional diffusion equation having the order predicted by the $\alpha$-stable fit of the acoustic intensity was capable of capturing very accurately the characteristic features of the anomalous transport process. From a general perspective, this approach can be interpreted as a fractional order homogenization of the periodic medium which is capable of mapping the complex inhomogeneous system to a (fractional) governing equation that still accepts an analytical solution. This latter characteristic is particularly remarkable if seen from a practical application perspective because it could open the way to accurate and non-iterative inverse problems that play a critical role in remote sensing, imaging, and material design, just to name a few. Another key observation concerns the strong deviation of the tails of the acoustic intensity from the Gaussian distribution which highlights that much information is still contained in the tails. This aspect is particularly relevant for imaging and sensing in scattering media because traditional analytical methodologies typically assume a Gaussian distribution of the measured intensity field hence leading to two main drawbacks: 1) the loss of important information about the internal structure of the medium which is contained in the tails, and 2) the lack of a proper model capable of extracting and interpreting this information from measured data. Acknowledgments {#acknowledgments .unnumbered} =============== The authors gratefully acknowledge the financial support of the National Science Foundation under the grants DCSD CAREER $\#1621909$ and of the Air Force Office of Scientific Research under Grant No. YIP FA9550-15-1-0133. Appendix A {#Appendix} ========== This appendix summarizes some basic properties of the $\alpha$-stable distributions that have been used to analyze and interpret the simulation data in the paper. The family of $\alpha$-stable distributions are defined by the Fourier transform of their characteristics functions $\psi(w)$ that can be written in the explicit form as[@herranz2004alpha; @benson2002fractional]: $$\begin{aligned} \label{alpha-stab0} \psi(w)=\exp\left \{ i\mu w-\gamma\left | w \right |^\alpha B_{w,\alpha} \right \} \\ B_{w,\alpha}=\left\{\begin{matrix} \left [1+i\beta \operatorname{sgn}(w) \tan\frac{\alpha \pi}{2} \right ] \quad \alpha\neq1 \nonumber \\ \left [1+i\beta \operatorname{sgn}(w) \frac{2}{\pi} \log\left | w \right | \right ] \quad \alpha=1 \end{matrix}\right.\end{aligned}$$ where $0<\alpha\leq 2$, $-1\leq\beta\leq 1$, $\gamma>0$, and $-\infty<\mu<\infty$. The parameters $\alpha$, $\beta$, $\gamma$ and $\mu$ uniquely and completely identify the stable distribution. 1. The parameter $\alpha$ is the *characteristic exponent*, or the *stability parameter*, and it defines the degree of impulsiveness of the distribution. As $\alpha$ decreases the level of impulsiveness of the distribution increases. For $\alpha=2$ we recover the Gaussian distribution. A particular case is obtained for $\alpha=1$ and $\beta=0$ that corresponds to the Cauchy distribution. For $\alpha \notin (0,2]$ the inverse Fourier transform $\psi(w)$ is not positive-definite and hence is not a proper probability density function. 2. The parameter $\beta$ is the *symmetry*, or *skewness parameter*, and determines the skewness of the distribution. Symmetric distributions have $\beta=0$, whereas $\beta=1$ and $\beta=-1$ correspond to completely skewed distributions. 3. The parameter $\gamma$ is the *scale parameter*. It is a measure of the spread of the samples from a distribution around the mean. 4. The parameter $\mu$ is the *location parameter* and corresponds to a shift in the $x$-axis of the pdf. For a symmetric $(\beta=0)$ distribution, $\mu$ is the mean when $1<\alpha\leq 2$ and the median when $0<\alpha\leq 1$. The characteristic functions described in Eq. (\[alpha-stab0\]) are equivalent to a probability density function and do not have analytical solutions except for few special cases. The main feature of these characteristic functions is the presence of heavy-tails when compared to a Gaussian distribution. The probability density functions with tails heavier than Gaussian are also denoted as *impulsive*. An impulsive process is characterized by the presence of large values that significantly deviates from the mean value of the distribution with non-negligible probability. In this sense the $\alpha$-stable distribution represents a generalization of the Gaussian distribution that allows to model impulsive processes by using only four parameters instead of an infinite number of moments. The possibility of describing the distribution of particles in anomalous diffusion phenomena by using $\alpha$-stable distributions has numerous advantages: 1) many methods exist to perform statistical inference on $\alpha$-stable environments [@nikias1995signal; @janicki1993simulation], 2) these distributions are simple because they are completely characterized by only four parameters, 3) the use of $\alpha$-stable distributions finds a theoretical justification in the fact that they satisfy the generalized central limit theorem which states that the limit distribution on infinitely many [i.i.d.]{} random variables, is a stable distribution, 4) they include the Gaussian distribution as a particular case for a specific set of parameters. These distributions are stable since the output of a linear system in response to $\alpha$-stable inputs is again $\alpha$-stable.

--- abstract: | We identify complete fragments of the Simple Theory of Types with Infinity ($\mathrm{TSTI}$) and Quine’s $\mathrm{NF}$ set theory. We show that $\mathrm{TSTI}$ decides every sentence $\phi$ in the language of type theory that is in one of the following forms: - $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s_1} \cdots \exists y_l^{s_l} \theta$ where the superscripts denote the types of the variables, $s_1 > \ldots > s_l$ and $\theta$ is quantifier-free, - $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s} \cdots \exists y_l^{s} \theta$ where the superscripts denote the types of the variables and $\theta$ is quantifier-free. This shows that $\mathrm{NF}$ decides every stratified sentence $\phi$ in the language of set theory that is in one of the following forms: - $\phi= \forall x_1 \cdots \forall x_k \exists y_1 \cdots \exists y_l \theta$ where $\theta$ is quantifier-free and $\phi$ admits a stratification that assigns distinct values to all of the variable $y_1, \ldots, y_l$, - $\phi= \forall x_1 \cdots \forall x_k \exists y_1 \cdots \exists y_l \theta$ where $\theta$ is quantifier-free and $\phi$ admits a stratification that assigns the same value to all of the variables $y_1, \ldots, y_l$. author: - Anuj Dawar - Thomas Forster - Zachiri McKenzie bibliography: - 'decidablefragementsoftst24.bib' title: 'Decidable fragments of the Simple Theory of Types with Infinity and $\mathrm{NF}$ [^1]' --- Introduction ============ Roland Hinnion showed in his thesis [@hin75] that [*Every consistent $\exists^*$ sentence in the language of set theory is a theorem of $\mathrm{NF}$*]{} or, equivalently: [*Every finite binary structure can be embedded in every model of $\mathrm{NF}$*]{}. Both these formulations invite generalisations. On the one hand we find results like [*every countable binary structure can be embedded in every model of $\mathrm{NF}$*]{} (this is theorem 4 of [@for87]) and on the other we can ask about the status of sentences with more quantifiers: $\forall^*\exists^*$ sentences in the first instance; it is the second that will be our concern here.\ \ It is elementary to check that $\mathrm{NF}$ does not decide all $\forall^*\exists^*$ sentences, since the existence of Quine atoms ($x = \{x\}$) is consistent with, and independent of, $\mathrm{NF}$. However ‘$(\forall x)(x \not= \{x\})$’ is not stratified, and this invites the conjecture that (i) $\mathrm{NF}$ decides all stratified $\forall^*\exists^*$ sentences and that (ii) all unstratified $\forall^*\exists^*$ sentences can be proved both relatively consistent and independent by means of Rieger-Bernays permutation methods. It’s with limb (i) of this conjecture that we are concerned here.\ \ The foregoing is all about $\mathrm{NF}$; the connection with the Simple Theory of Types with Infinity ($\mathrm{TSTI}$) arises because of work of Ernst Specker [@spe62] and [@spe53]: $\mathrm{NF}$ decides all stratified $\forall^*\exists^*$ sentences of the language of set theory if and only if $\mathrm{TSTI} + \mathrm{Ambiguity}$ decides all $\forall^*\exists^*$ sentences of the language of type theory. > [**Conjecture**]{}: All models of $\mathrm{TSTI}$ agree on all $\forall^*\exists^*$ sentences. It is towards a proof of this conjecture that our efforts in this paper are directed.\ \ Observe that [*there is a total order of $V$*]{} is consistent with and independent of $\mathrm{TST}$ and it can be said with three blocks of quantifiers: $$(\exists O)[(\forall x y \in O)(x \subseteq y \lor y \subseteq x) \land (\forall u v)( u \not= v \to (\exists x \in O)( u \in x \iff v \not\in x))]$$ making it $\exists^1\forall^6\exists^1$. Background and definitions {#Sec:Background} ========================== The Simple Theory of Types is the simplification of the Ramified Theory of Types, the underlying system of [@rw08], that was independently discovered by Frank Ramsey and Leon Chwistek. Following [@mat01] we use $\mathrm{TSTI}$ and $\mathrm{TST}$ to abbreviate the Simple Theory of Types with and without an axiom of infinity respectively. These theories are naturally axiomatised in a many-sorted language with sorts for each $n \in \mathbb{N}$. We use $\mathcal{L}_{\mathrm{TST}}$ to denote the $\mathbb{N}$-sorted language endowed with binary relation symbols $\in_n$ for each sort $n \in \mathbb{N}$. There are variables $x^n, y^n, z^n, \ldots$ for each sort $n \in \mathbb{N}$ and well-formed $\mathcal{L}_{\mathrm{TST}}$-formulae are built-up inductively from atomic formulae in the form $x^n \in_n y^{n+1}$ and $x^n = y^n$ using the connectives quantifiers of first-order logic. We refer to sorts of $\mathcal{L}_{\mathrm{TST}}$ as types. We will attempt to stick to the convention of denoting $\mathcal{L}_{\mathrm{TST}}$-structures using calligraphy letters ($\mathcal{M}, \mathcal{N}, \ldots$). A $\mathcal{L}_{\mathrm{TST}}$-structure $\mathcal{M}$ consists of domains $M_n$ for each type $n \in \mathbb{N}$ and interpretations of the relations $\in_n^{\mathcal{M}} \subseteq M_n \times M_{n+1}$ for each type $n \in \mathbb{N}$; we write $\mathcal{M}= \langle M_0, M_1, \ldots, \in_0^{\mathcal{M}}, \in_1^{\mathcal{M}}, \ldots \rangle$. If is an $\mathcal{L}_{\mathrm{TST}}$-structure then we call the elements of $M_0$ atoms. We use $\mathrm{TST}$ to denote the $\mathcal{L}_{\mathrm{TST}}$-theory with axioms - (Extensionality) for all $n \in \mathbb{N}$, $$\forall x^{n+1} \forall y^{n+1} (x^{n+1}= y^{n+1} \iff \forall z^n(z^{n} \in_n x^{n+1} \iff z^n \in y^{n+1})),$$ - (Comprehension) for all $n \in \mathcal{N}$ and for all well-formed $\mathcal{L}_{\mathrm{TST}}$-formulae $\phi(x^n, \vec{z})$, $$\forall \vec{z} \exists y^{n+1} \forall x^n (x^n \in_n y^{n+1} \iff \phi(x^n, \vec{z})).$$ Comprehension ensures that every successor type is closed under the set-theoretic operations: union ($\cup$), intersection ($\cap$), difference ($\backslash$) and symmetric difference ($\triangle$). For all $n \in \mathbb{N}$, we use $\emptyset^{n+1}$ to denote the point at type $n+1$ which contains no points from type $n$ and we use $V^{n+1}$ to denote the point at type $n+1$ that contains every point from type $n$. The Wiener-Kuratowski ordered pair allows us to code ordered pairs in the form $\langle x, y \rangle$ as objects in $\mathrm{TST}$ which have type two higher than the type of $x$ and $y$. Functions, as usual, are thought of as collections of ordered pairs. This means that a function $f: X \longrightarrow Y$ will be coded by an object in $\mathrm{TST}$ that has type two higher than the type of $X$ and $Y$. The theory $\mathrm{TSTI}$ is obtained from $\mathrm{TST}$ by asserting the existence of a Dedekind infinite collection at type $1$. We use $\mathrm{TSTI}$ to denote the $\mathcal{L}_{\mathrm{TST}}$-theory obtained from $\mathrm{TST}$ by adding the axiom $$\exists x^1 \exists f^3(f^3: x^1 \longrightarrow x^1 \textrm{ is injective but not surjective}).$$ Let $X$ be a set. If the $\mathcal{L}_{\mathrm{TST}}$-structure $\mathcal{M}= \langle M_0, M_1, \ldots, \in_0, \in_1, \ldots \rangle$ is defined by $M_n= \mathcal{P}^n(X)$ and $\in_n^\mathcal{M}= \in \upharpoonright \mathcal{P}^n(X) \times \mathcal{P}^{n+1}(X)$ for all $n \in \mathbb{N}$, then $\mathcal{M} \models \mathrm{TST}$. If $m \in \mathbb{N}$ and $|X|= m$ then $\mathcal{M}$ is the unique, up to isomorphism, model of $\mathrm{TST}$ with exactly $m$ atoms and we say that $\mathcal{M}$ is finitely generated by $m$ atoms. Alternatively, if $X$ is Dedekind infinite then $\mathcal{M} \models \mathrm{TSTI}$. This shows that $\mathrm{ZFC}$ proves the consistency of $\mathrm{TSTI}$. In fact, in [@mat01] it is shown that $\mathrm{TSTI}$ is equiconsistent with Mac Lane Set Theory.\ \ We say that an $\mathcal{L}^\prime$-theory $T$ decides an $\mathcal{L}^\prime$-sentence $\phi$ if and only if $T \vdash \phi$ or $T \vdash \neg \phi$. The Completeness Theorem implies that $T$ decides $\phi$ if and only if $\phi$ holds in all $\mathcal{L}^\prime$-structures $\mathcal{M} \models T$, or $\neg \phi$ holds in all $\mathcal{L}^\prime$-structures $\mathcal{M} \models T$. We say that a $\mathcal{L}_{\mathrm{TST}}$-sentence $\phi$ is $\exists^* \forall^*$ if and only if\ where $\theta$ is quantifier-free. We say that an $\mathcal{L}_{\mathrm{TST}}$-sentence $\phi$ is $\forall^* \exists^*$ if and only if\ where $\theta$ is quantifier-free. We will show that $\mathrm{TSTI}$ decides a significant fragment of the $\forall^* \exists^*$ sentences (and thus it also decides the $\exists^* \forall^*$ sentences that are logically equivalent to the negation of these $\forall^* \exists^*$ sentences). We achieve this result by showing that every sentence or negation of a sentence in this fragment that is true in some model of $\mathrm{TSTI}$ is true in all models of $\mathrm{TST}$ that are finitely generated by sufficiently many atoms. We say that an $\mathcal{L}_{\mathrm{TST}}$-sentence $\phi$ has the finitely generated model property if and only if, if there exists an $\mathcal{N} \models \mathrm{TSTI}+\phi$ then there exists a $k \in \mathbb{N}$ such that for all $m \geq k$, if $\mathcal{M} \models \mathrm{TST}$ is finitely generated by $m$ atoms then $\mathcal{M} \models \phi$. Note that if $\Gamma$ is class of $\mathcal{L}_{\mathrm{TST}}$-sentences that have the finitely generated model property and $\Gamma$ is closed under negations then $\mathrm{TST}$ decides every sentence in $\Gamma$.\ \ In [@qui37] Willard van Orman Quine introduces a set theory by identifying a syntactic condition on formulae in the single sorted language of set theory that captures the restricted comprehension available in $\mathrm{TST}$. This set theory has been dubbed ‘New Foundations’ ($\mathrm{NF}$) after the title of [@qui37]. We will use $\mathcal{L}$ to denote the language of set theory — the language of first-order logic endowed with a binary relation symbol $\in$ whose intended interpretation is membership. Before giving the axioms of $\mathrm{NF}$ we first recall Quine’s definition of a stratified formulae. If $\phi$ is an $\mathcal{L}$-formula then we use $\mathbf{Var}(\phi)$ to denote the set of variables (both free and bound) which appear in $\phi$. Let $\phi(x_1, \ldots, x_n)$ be an $\mathcal{L}$-formula. We say that $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$ is a stratification of $\phi$ if and only if - if ‘$x \in y$’ is a subformula of $\phi$ then $\sigma(\textrm{`}y\textrm{'})= \sigma(\textrm{`}x\textrm{'})+1$, - if ‘$x = y$’ is a subformula of $\phi$ then $\sigma(\textrm{`}y\textrm{'})= \sigma(\textrm{`}x\textrm{'})$. If there exists a stratification of $\phi$ then we say that $\phi$ is stratified. Let $\phi$ be an $\mathcal{L}$-formula. Note that $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$ is a stratification of $\phi$ if and only if the formula obtained by decorating every variable appearing in $\phi$ with the type given by $\sigma$ yields a well-formed $\mathcal{L}_{\mathrm{TST}}$-formula. Conversely, let $\theta$ be a well-formed $\mathcal{L}_{\mathrm{TST}}$-formula and let $\phi$ an $\mathcal{L}$-formula obtained for $\theta$ by deleting the types from the variables appearing in $\theta$ while ensuring (by relabeling variables) that no two distinct variables in $\theta$ become the same variable in $\phi$. Then the $\mathcal{L}$-formula $\phi$ is stratified and the function which sends a variable in $\phi$ to the type index of the corresponding variable in $\theta$ is a stratification. Let $\phi$ be an $\mathcal{L}$-formula with stratification $\sigma: \mathbf{Var}(\phi)$. We use $\phi^{(\sigma)}$ to denote the $\mathcal{L}_{\mathrm{TST}}$-formula obtained by assigning each variable ‘$x$’ appearing $\phi$ the type $\sigma(\textrm{`}x\textrm{'})$. $\mathrm{NF}$ is the $\mathcal{L}$-theory with the axiom of extensionality and comprehension for all stratified $\mathcal{L}$-formulae. We use $\mathrm{NF}$ to denote the $\mathcal{L}$-theory with axioms - (Extensionality) $\forall x \forall y (x=y \iff \forall z(z \in x \iff z \in y))$, - (Stratified Comprehension) for all stratified $\phi(x, \vec{z})$, $$\forall \vec{z} \exists y \forall x (x \in y \iff \phi(x, \vec{z})).$$ We direct the interested reader to [@for95] for detailed treatment of $\mathrm{NF}$. One interesting feature of $\mathrm{NF}$ is that it refutes the Axiom of Choice and so proves the Axiom of Infinity (see [@spe53]). There is a strong connection between the theories $\mathrm{NF}$ and $\mathrm{TSTI}$. [@spe62] shows that models of $\mathrm{NF}$ can be obtained from models of $\mathrm{TSTI}$ plus the scheme $\phi \iff \phi^+$, for all $\mathcal{L}_{\mathrm{TST}}$-sentences $\phi$, where $\phi^+$ is obtained from $\phi$ by incrementing the types of all the variables appearing in $\phi$. Conversely, let $\mathcal{M}= \langle M, \in^{\mathcal{M}} \rangle$ be an $\mathcal{L}$-structure with $\mathcal{M} \models \mathrm{NF}$. The $\mathcal{L}_{\mathrm{TST}}$-structure $\mathcal{N}= \langle N_0, N_1, \ldots, \in_0^{\mathcal{N}}, \in_1^{\mathcal{N}}, \ldots \rangle$ defined by $N_n= M$ and $\in_n^{\mathcal{N}}= \in^{\mathcal{M}}$ is such that $\mathcal{N} \models \mathrm{TSTI}$. Moreover, if $\phi$ is an $\mathcal{L}$-sentence with stratification $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$ and $\mathcal{M} \models \phi$ then $\mathcal{N} \models \phi^{(\sigma)}$. This immediately shows that a decidable fragment of $\mathrm{TSTI}$ yields a decidable fragment of $\mathrm{NF}$. \[Th:DecidableFragmentsOfNF\] Let $\phi$ be an $\mathcal{L}$-sentence with stratification $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$. If $\mathrm{TSTI}$ decides $\phi^{(\sigma)}$ then $\mathrm{NF}$ decides $\phi$. $\exists^* \forall^*$ sentences have the finitely generated model property ========================================================================== In this section we prove that all $\exists^* \forall^*$ sentences have the finitely generated model property. This result follows from the fact that if $\mathcal{N}$ is a model of $\mathrm{TSTI}$, $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{N}$ with $r_1 \leq \ldots \leq r_k$ and $\mathcal{M}$ is a model of $\mathrm{TST}$ that is finitely generated by sufficiently many atoms then there is an embedding of $\mathcal{M}$ into $\mathcal{N}$ with $a_1^{r_1}, \ldots, a_k^{r_k}$ in the range. Given $k \in \mathbb{N}$ we define the function $\mathbf{G}_k: \mathbb{N} \longrightarrow \mathbb{N}$ by recursion $$\mathbf{G}_k(0)= k \textrm{ and } \mathbf{G}_k(n+1)= \binom{\mathbf{G}_k(n)}{2}+ k.$$ \[Th:EmeddingProperty\] Let $\mathcal{N} \models \mathrm{TSTI}$ and let with $r_1 \leq \ldots \leq r_k$. If $\mathcal{M} \models \mathrm{TST}$ is finitely generated by at least $\mathbf{G}_k(r_k)$ atoms then there exists a sequence $\langle f_n \mid n \in \mathbb{N} \rangle$ such that for all $n \in \mathbb{N}$, - $f_n: M_n \longrightarrow N_n$ is injective, - for all $x \in M_n$ and for all $y \in M_{n+1}$, $$\mathcal{M} \models x \in_n y \textrm{ if and only if } \mathcal{N} \models f_n(x) \in_n f_{n+1}(y),$$ - $$a_1^{r_1}, \ldots, a_k^{r_k} \in \bigcup_{m \in \mathbb{N}} \mathrm{rng}(f_m).$$ Let $\mathcal{N}= \langle N_0, N_1, \ldots, \in_0^{\mathcal{N}}, \in_1^{\mathcal{N}}, \ldots \rangle$ be such that $\mathcal{N} \models \mathrm{TSTI}$ and let $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{N}$ with $r_1 \leq \ldots \leq r_k$. Let $\mathcal{M}= \langle M_0, M_1, \ldots, \in_0^{\mathcal{M}}, \in_1^{\mathcal{M}}, \ldots \rangle$ be such that $\mathcal{M} \models \mathrm{TST}$ is finitely generated and $|M_0| \geq \mathbf{G}_k(r_k)$. We begin by defining $C \subseteq \mathcal{N}$ such that $|C \cap N_0| \leq \mathbf{G}_k(r_k)$ and for any two points $x \neq y$ in $C$ that are not atoms, there exists a point $z$ in $C$ which $\mathcal{N}$ believes is in the symmetric difference of $x$ and $y$. Define $C_0= \{a_1^{r_1}, \ldots, a_k^{r_k} \} \subseteq \mathcal{N}$. Note that $|C_0 \cap N_{r_k}| \leq \mathbf{G}_k(0)= k$ and for all $0\leq m < r_k$, $|C_0 \cap N_m| \leq k$. For $0 < n \leq r_k$ we recursively define $C_n \subseteq \mathcal{N}$ which satisfies - $|C_n \cap N_{r_k-n}| \leq \mathbf{G}_k(n)$, - for all $0 \leq m < r_k-n$, $|C_n \cap N_m| \leq k$. Suppose that $n < r_k$ and $C_n \subseteq \mathcal{N}$ has been defined and satisfies (I) and (II). For all $y, z \in N_{r_k-n}$ with $y \neq z$, let $\gamma_{\{y, z\}} \in N_{r_k-(n+1)}$ be such that $$\mathcal{N} \models \gamma_{\{y, z\}} \in_{r_k-(n+1)} y \triangle z.$$ Define $$C_{n+1}= C_n \cup \{ \gamma_{\{y, z\}} \mid \{y, z\} \in [N_{r_k-n} \cap C_n]^2 \}.$$ It follows from (I) and (II) that $$|C_{n+1} \cap N_{r_k-(n+1)}| \leq |C_n \cap N_{r_k-(n+1)}|+\binom{|C_n \cap N_{r_k-n}|}{2} \leq k + \binom{\mathbf{G}_k(n)}{2}= \mathbf{G}_k(n+1)$$ and for all $0 \leq m < r_k-(n+1)$, $|C_{n+1} \cap N_m| \leq k$. Now, let $C= C_{r_k}$. This recursion ensures that $|C \cap N_0| \leq \mathbf{G}_k(r_k)$.\ We now turn to defining the family of maps $\langle f_n \mid n \in \mathbb{N} \rangle$ which embed $\mathcal{M}$ into $\mathcal{N}$. We define the sequence $\langle f_n \mid n \in \mathbb{N} \rangle$ by induction. Let $C^\prime= C \cap N_0$. Let $f_0: M_0 \longrightarrow N_0$ be an injection such that $C^\prime \subseteq \mathrm{rng}(f_0)$. Suppose that $\langle f_0, \ldots, f_n \rangle$ has been defined such that - for all $0 \leq j \leq n$, $f_j: M_j \longrightarrow N_j$ is injective, - for all $0 \leq j < n$, for all $x \in M_j$ and for all $y \in M_{j+1}$, $$\mathcal{M} \models x \in_j y \textrm{ if and only if } \mathcal{N} \models f_j(x) \in_j f_{j+1}(y),$$ - for all $0 \leq j \leq n$, $C \cap N_j \subseteq \mathrm{rng}(f_j)$. If $0 \leq j \leq n$ and $x \in M_{j+1}$ then we use $f_j``x$ to denote the point in $N_{j+1}$ such that $\mathcal{N} \models f_j``x= \{ f_j(y) \mid \mathcal{M} \models y \in_j x \}$. Note that, since $\mathcal{M}$ is finitely generated, for all $x \in M_{j+1}$, $f_j``x$ exists in $\mathcal{N}$. We define $f_{n+1}: M_{n+1} \longrightarrow N_{n+1}$ by $$f_{n+1}(x)= \left\{ \begin{array}{ll} \gamma & \textrm{if } \gamma \in C \cap N_{n+1} \textrm{ and } \mathcal{N} \models f_n``x= \gamma \cap f_n``(V^{n+1})^\mathcal{M}\\ f_n``x & \textrm{otherwise} \end{array} \right.$$ We first need to show that the map $f_{n+1}$ is well-defined. Suppose that $\xi_1, \xi_2 \in C \cap N_{n+1}$ with $\xi_1 \neq \xi_2$ and $x \in M_{n+1}$ are such that $$\mathcal{N} \models f_n``x= \xi_1 \cap f_n``(V^{n+1})^\mathcal{M} \textrm{ and } \mathcal{N} \models f_n``x= \xi_2 \cap f_n``(V^{n+1})^\mathcal{M}.$$ Now, there is a $\gamma \in C \cap N_n$ such that $\mathcal{N} \models \gamma \in_n \xi_1 \triangle \xi_2$. By (III’), $\gamma \in \mathrm{rng}(f_n)$, which is a contradiction. Therefore $f_{n+1}$ is well-defined.\ The fact that $f_n$ is injective ensures that $f_{n+1}$ is injective.\ We now turn to showing that the sequence $\langle f_0, \ldots, f_{n+1} \rangle$ satisfies (II’). Let $x \in M_n$ and let $y \in M_{n+1}$. There are two cases. Firstly, suppose that $f_{n+1}(y)= \gamma \in C$. Therefore $\mathcal{N} \models f_n``y= \gamma \cap f``(V^{n+1})^\mathcal{M}$. If $\mathcal{M} \models x \in_n y$ then $\mathcal{N}\models f_n(x) \in_n f_n``y$ and so $\mathcal{N} \models f_n(x) \in_n f_{n+1}(y)$. Conversely, if $\mathcal{N} \models f_n(x) \in_n \gamma$ then $\mathcal{N} \models f_n(x) \in_n f_n``y$ and so $\mathcal{M} \models x \in_n y$. The second case is when $f_{n+1}(y)= f_n``y$. In this case it is clear that $$\mathcal{M} \models x \in_n y \textrm{ if and only if } \mathcal{N} \models f_n(x) \in_n f_{n+1}(y).$$ This shows that the sequence $\langle f_0, \ldots, f_{n+1} \rangle$ satisfies (II’).\ This concludes the induction step of the construction and shows that we can construct a sequence $\langle f_n \mid n \in \mathbb{N} \rangle$ that satisfies (i)-(iii). This embedding property allows us to show that every $\exists^* \forall^*$ sentence has the finitely generated model property. \[Th:ExistentialUniversalSentencesHaveFinitelyGeneratedModelProperty\] Let $\phi= \exists x_1^{r_1} \cdots \exists x_k^{r_k} \forall y_1^{s_1} \cdots \forall y_l^{s_l} \theta$ where $r_1 \leq \ldots \leq r_k$ and $\theta$ is quantifier-free. If $\mathcal{N} \models \mathrm{TSTI}+\phi$ and $\mathcal{M} \models \mathrm{TST}$ is finitely generated by at least $\mathbf{G}_k(r_k)$ atoms then $\mathcal{M} \models \phi$. Let $\mathcal{N}= \langle N_0, N_1, \ldots, \in_0^{\mathcal{N}}, \in_1^{\mathcal{N}}, \ldots \rangle$ be such that $\mathcal{N} \models \mathrm{TSTI}+\phi$. Let $\mathcal{M}= \langle M_0, M_1, \ldots, \in_0^{\mathcal{M}}, \in_1^{\mathcal{M}}, \ldots \rangle$ be such that $\mathcal{M} \models \mathrm{TST}$ and $\mathcal{M}$ is finitely generated by at least $\mathbf{G}_k(r_k)$ atoms. Let $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{N}$ be such that $$\mathcal{N} \models \forall y_1^{s_1} \cdots \forall y_l^{s_l} \theta[a_1^{r_1}, \ldots, a_k^{r_k}].$$ Using Lemma \[Th:EmeddingProperty\] we can find a sequence $\langle f_n \mid n \in \mathbb{N} \rangle$ such that - $f_n: M_n \longrightarrow N_n$ is injective, - for all $x \in M_n$ and for all $y \in M_{n+1}$, $$\mathcal{M} \models x \in_n y \textrm{ if and only if } \mathcal{N} \models f_n(x) \in f_{n+1}(y),$$ - $$a_1^{r_1}, \ldots, a_k^{r_k} \in \bigcup_{m \in \mathbb{N}} \mathrm{rng}(f_m).$$ Let $b_1^{r_1}, \ldots, b_k^{r_k} \in \mathcal{M}$ be such that for all $1 \leq j \leq k$, $f_{r_j}(b_j^{r_j})= a_j^{r_j}$. Let $c_1^{s_1}, \ldots, c_l^{s_l} \in \mathcal{M}$. Since $\mathcal{N} \models \theta[a_1^{r_1}, \ldots, a_k^{r_k}, f_{s_1}(c_1^{s_1}), \ldots, f_{s_l}(c_l^{s_l})]$, it follows that $$\mathcal{M} \models \theta[b_1^{r_1}, \ldots, b_k^{r_k}, c_1^{s_1}, \ldots, c_l^{s_l}].$$ Therefore $$\mathcal{M} \models \forall y_1^{s_1} \cdots \forall y_l^{s_l}\theta[b_1^{r_1}, \ldots, b_k^{r_k}],$$ which proves the theorem. Decidable fragments of the $\forall^* \exists^*$ sentences ========================================================== In this section we will show that $\mathrm{TSTI}$ decides every $\forall^* \exists^*$ sentence $\phi$ that is in one of the following forms: - $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s_1} \cdots \exists y_l^{s_l} \theta$ where $s_1 > \ldots > s_l$ and $\theta$ is quantifier-free, - $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s} \cdots \exists y_l^{s} \theta$ where $\theta$ is quantifier-free. By applying Theorem \[Th:DecidableFragmentsOfNF\] it then follows that $\mathrm{NF}$ decides every stratified $\mathcal{L}$-sentence $\phi$ that is in one of the following forms: - $\phi= \forall x_1 \cdots \forall x_k \exists y_1 \cdots \exists y_l \theta$ where $\theta$ is quantifier-free and $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$ is a stratification of $\phi$ that assigns distinct values to all of the variables $y_1, \ldots, y_l$, - $\phi= \forall x_1 \cdots \forall x_k \exists y_1 \cdots \exists y_l \theta$ where $\theta$ is quantifier-free and $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$ is a stratification of $\phi$ that assigns the same value to all of the variables $y_1, \ldots, y_l$. Throughout this section we will fix $k, l \in \mathbb{N}$ and a sequence $r_1 \leq \ldots \leq r_k$ that will represent the types of the universally quantified variables in a $\forall^* \exists^*$ sentence. Let $k^\prime$ be the number of distinct elements in the list $r_1, \ldots, r_k$. Let $K_1, \ldots, K_{k^\prime}$ be the multiplicities of the elements in the list $r_1, \ldots, r_k$, so $k= \sum_{1 \leq i \leq k^\prime} K_i$, and let $K= \max\{ K_1, \ldots, K_{k^\prime}, l\}$. We also fix structures $\mathcal{N}= \langle N_0, N_1, \ldots, \in_0^{\mathcal{N}}, \in_1^{\mathcal{N}}, \ldots \rangle$ with $\mathcal{N} \models \mathrm{TSTI}$ and $\mathcal{M}= \langle M_0, M_1, \ldots, \in_0^{\mathcal{M}}, \in_1^{\mathcal{M}}, \ldots \rangle$ with $\mathcal{M} \models \mathrm{TST}$ finitely generated by at least $(2^K)^{k^\prime+2}$ atoms. Let $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{M}$.\ \ Our approach will be to define colour classes $\mathcal{C}_{i, j}$, the elements of which we will call colours, and functions $c_{i, j}^{\mathcal{M}}: M_i \longrightarrow \mathcal{C}_{i, j}$ and $c_{i, j}^{\mathcal{N}}: N_i \longrightarrow \mathcal{C}_{i, j}$, which we will call colourings, for all $i \in \mathbb{N}$ and for all $0 \leq j \leq k^\prime$. For all $0 < j \leq k^\prime$, the colourings $c_{i, j}^{\mathcal{M}}$ will be defined using the elements $a_1^{r_1}, \ldots, a_{j^\prime}^{r_{j^\prime}}$ where $j^\prime= \sum_{1 \leq m \leq j} K_m$, and in the process of defining the colourings $c_{i, j}^{\mathcal{N}}$ we will construct corresponding elements $b_1^{r_1}, \ldots, b_{j^\prime}^{r_{j^\prime}} \in \mathcal{N}$. The colourings will be designed with the following properties: - For a fixed colour $\alpha$ in some $\mathcal{C}_{i, j}$, the property of being an element of $\mathcal{N}$ that is given colour $\alpha$ by $c_{i, j}^{\mathcal{N}}$ will be definable by an $\mathcal{L}_{\mathrm{TST}}$-formula, $\Phi_{i, j, \alpha}$, with parameters over $\mathcal{N}$. - The colour given to an element $x$ in $\mathcal{M}$ (or $\mathcal{N}$) by the colouring $c_{i, j}^{\mathcal{M}}$ (respectively $c_{i, j}^{\mathcal{N}}$) will tell us which quantifier-free $\mathcal{L}_{\mathrm{TST}}$-formulae with parameters $a_1^{r_1}, \ldots, a_{j^\prime}^{r_{j^\prime}}$ (respectively $b_1^{r_1}, \ldots, b_{j^\prime}^{r_{j^\prime}}$), where $j^\prime= \sum_{1 \leq m \leq j} K_m$, are satisfied by $x$ in $\mathcal{M}$ (respectively $\mathcal{N}$). - For every colour $\beta$ in $\mathcal{C}_{i, j}$, the colour given to an element $x$ in $\mathcal{M}$ (or $\mathcal{N}$) by the colouring $c_{i+1, j}^{\mathcal{M}}$ (respectively $c_{i+1, j}^{\mathcal{N}}$) will tell us whether or not there is an element $y$ in $\mathcal{M}$ (respectively $\mathcal{N}$) such that $\mathcal{M} \models y \in_i x$ (respectively $\mathcal{N} \models y \in_i x$) and $y$ is given colour $\beta$ by $c_{i, j}^{\mathcal{M}}$ (respectively $c_{i, j}^{\mathcal{N}}$). - For every colour $\beta$ in $\mathcal{C}_{i, j}$, the colour given to an element $x$ in $\mathcal{M}$ (or $\mathcal{N}$) by the colouring $c_{i+1, j}^{\mathcal{M}}$ (respectively $c_{i+1, j}^{\mathcal{N}}$) will tell us whether or not there is an element $y$ in $\mathcal{M}$ (respectively $\mathcal{N}$) such that $\mathcal{M} \models y \notin_i x$ (respectively $\mathcal{N} \models y \notin_i x$) and $y$ is given colour $\beta$ by $c_{i, j}^{\mathcal{M}}$ (respectively $c_{i, j}^{\mathcal{N}}$). Note that since $\mathcal{M}$ is finitely generated, the analogue of condition (i) automatically holds for $\mathcal{M}$.\ \ Before defining the colour classes $\mathcal{C}_{i, j}$ and the colourings $c_{i,j}^{\mathcal{M}}$ and $c_{i, j}^{\mathcal{N}}$ we first introduce the following definitions: Let $m \in \mathbb{N}$. We say that a colour $\alpha \in \mathcal{C}_{i, j}$ is $m$-special with respect to a colouring $f: X \longrightarrow \mathcal{C}_{i, j}$ if and only if $$|\{ x \in X \mid f(x)= \alpha\}|= m.$$ If $\alpha \in \mathcal{C}_{i, j}$ is $0$-special then we say that $\alpha$ is forbidden. Let $m \in \mathbb{N}$. We say that a colour $\alpha \in \mathcal{C}_{i, j}$ is $m$-abundant with respect to a colouring $f: X \longrightarrow \mathcal{C}_{i, j}$ if and only if $$|\{ x \in X \mid f(x)= \alpha\}|\geq m.$$ Let $J \in \mathbb{N}$. We say that colourings $f: X \longrightarrow \mathcal{C}_{i, j}$ and $g: Y \longrightarrow \mathcal{C}_{i, j}$ are $J$-similar if and only if for all $0 \leq m < J$ and for all $\alpha \in \mathcal{C}_{i, j}$, $$\alpha \textrm{ is } m \textrm{-special w.r.t. } f \textrm{ if and only if } \alpha \textrm{ is } m \textrm{-special w.r.t. } g.$$ The colour classes $\mathcal{C}_{i, j}$ and colourings $c_{i, j}^\mathcal{M}$ and $c_{i, j}^\mathcal{N}$ for all $i \in \mathbb{N}$ and for all $0 \leq j \leq k^\prime$ will be defined by a two-dimensional recursion. At each stage of the construction we will ensure that $c_{i, j}^\mathcal{M}$ and $c_{i, j}^\mathcal{N}$ are $(2^K)^{k^\prime-j+2}$-similar.\ \ Let $\mathcal{C}_{0, 0}= \{0\}$. Define $c_{0, 0}^{\mathcal{M}}: M_0 \longrightarrow \mathcal{C}_{0, 0}$ by $$c_{0, 0}^{\mathcal{M}}(x)= 0 \textrm{ for all } x \in M_0.$$ Define $c_{0, 0}^{\mathcal{N}}: N_0 \longrightarrow \mathcal{C}_{0, 0}$ by $$c_{0, 0}^{\mathcal{N}}(x)= 0 \textrm{ for all } x \in N_0.$$ Let $\Phi_{0, 0, 0}(x^0)$ be the $\mathcal{L}_{\mathrm{TST}}$-formula $x^0=x^0$. Note that for all $x \in N_0$, $$\mathcal{N} \models \Phi_{0, 0, 0}[x] \textrm{ if and only if } c_{0, 0}^\mathcal{N}(x)= 0.$$ The colourings $c_{0, 0}^{\mathcal{M}}$ and $c_{0, 0}^{\mathcal{N}}$ are $(2^K)^{k^\prime+2}$-similar. This follows immediately from the fact that $|M_0| \geq (2^K)^{k^\prime+2}$. We now turn to defining the colour classes $\mathcal{C}_{i, 0}$ and colourings $c_{i, 0}^{\mathcal{M}}: M_i \longrightarrow \mathcal{C}_{i, 0}$ and $c_{i, 0}^{\mathcal{N}}: N_i \longrightarrow \mathcal{C}_{i, 0}$ for all $i \in \mathbb{N}$. Suppose that we have defined the colour class $\mathcal{C}_{n, 0}$ with a canonical ordering, colourings $c_{n, 0}^{\mathcal{M}}: M_n \longrightarrow \mathcal{C}_{n, 0}$ and $c_{n, 0}^{\mathcal{N}}: N_i \longrightarrow \mathcal{C}_{n, 0}$ and $\mathcal{L}_{\mathrm{TST}}$-formulae $\Phi_{n, 0, \alpha}(x^n)$ for all $\alpha \in \mathcal{C}_{n, 0}$ with the following properties: - $c_{n, 0}^{\mathcal{M}}$ and $c_{n, 0}^{\mathcal{N}}$ are $(2^K)^{k^\prime+2}$-similar, - for all $\alpha \in \mathcal{C}_{n, 0}$ and for all $x \in N_n$, $$\mathcal{N} \models \Phi_{n, 0, \alpha}[x] \textrm{ if and only if } c_{n, 0}^{\mathcal{N}}(x)= \alpha.$$ Let $\mathcal{C}_{n, 0}= \{ \alpha_1, \ldots, \alpha_q \}$ be the enumeration obtained from the canonical ordering. Define $\mathcal{C}_{n+1, 0}= 2^{2 \cdot q}$ — the set of all 0-1 sequences of length $2 \cdot q$. Define $c_{n+1, 0}^{\mathcal{M}}: M_{n+1} \longrightarrow \mathcal{C}_{n+1, 0}$ such that for all $x \in M_{n+1}$, $$c_{n+1, 0}^{\mathcal{M}}(x)= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$ $$\textrm{where } f_i= \left\{ \begin{array}{ll} 0 & \textrm{if for all } y \in M_n, \textrm{ if } c_{n, 0}^{\mathcal{M}}(y)= \alpha_i \textrm{ then } \mathcal{M}\models y \notin_n x\\ 1 & \textrm{if there exists } y \in M_n, \textrm{ s.t. } c_{n, 0}^{\mathcal{M}}(y)= \alpha_i \textrm{ and } \mathcal{M} \models y \in_n x \end{array} \right.$$ $$\textrm{and } g_i= \left\{ \begin{array}{ll} 0 & \textrm{if for all } y \in M_n, \textrm{ if } c_{n, 0}^{\mathcal{M}}(y)= \alpha_i \textrm{ then } \mathcal{M} \models y \in_n x\\ 1 & \textrm{if there exists } y \in M_n \textrm{ s.t. } c_{n, 0}^{\mathcal{M}}(y)= \alpha_i \textrm{ and } \mathcal{M} \models y \notin_n x \end{array} \right.$$ Using this definition we get $\mathcal{C}_{1, 0}= \{ \langle 0, 0 \rangle, \langle 1, 0 \rangle, \langle 0, 1 \rangle, \langle 1, 1 \rangle \}$. There are no $x \in M_1$ which are given the colour $\langle 0, 0 \rangle$ by $c_{1, 0}^{\mathcal{M}}$. The only point in $M_1$ which is given the colour $\langle 1, 0 \rangle$ by $c_{1, 0}^{\mathcal{M}}$ is $(V^1)^{\mathcal{M}}$. Similarly, the only point in $M_1$ which is given the colour $\langle 0, 1 \rangle$ by $c_{1, 0}^{\mathcal{M}}$ is $(\emptyset^1)^{\mathcal{M}}$. Every other point in $M_1$ is given the colour $\langle 1, 1 \rangle$ by $c_{1, 0}^{\mathcal{M}}$. We define the colouring $c_{n+1, 0}^{\mathcal{N}}: N_{n+1} \longrightarrow \mathcal{C}_{n+1, 0}$ identically. Define $c_{n+1, 0}^{\mathcal{N}}: N_{n+1} \longrightarrow \mathcal{C}_{n+1, 0}$ such that for all $x \in N_{n+1}$, $$c_{n+1, 0}^{\mathcal{N}}(x)= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$ $$\textrm{where } f_i= \left\{ \begin{array}{ll} 0 & \textrm{if for all } y \in N_n, \textrm{ if } c_{n, 0}^{\mathcal{N}}(y)= \alpha_i \textrm{ then } \mathcal{N} \models y \notin_n x\\ 1 & \textrm{if there exists } y \in N_n, \textrm{ s.t. } c_{n, 0}^{\mathcal{N}}(y)= \alpha_i \textrm{ and } \mathcal{N} \models y \in_n x \end{array} \right.$$ $$\textrm{and } g_i= \left\{ \begin{array}{ll} 0 & \textrm{if for all } y \in N_n, \textrm{ if } c_{n, 0}^{\mathcal{N}}(y)= \alpha_i \textrm{ then } \mathcal{N} \models y \in_n x\\ 1 & \textrm{if there exists } y \in N_n \textrm{ s.t. } c_{n, 0}^{\mathcal{N}}(y)= \alpha_i \textrm{ and } \mathcal{N} \models y \notin_n x \end{array} \right.$$ We first show that there are $\mathcal{L}_{\mathrm{TST}}$-formulae $\Phi_{n+1, 0, \beta}$, for all $\beta \in \mathcal{C}_{n+1, 0}$, that satisfy condition (II) above for the colouring $c_{n+1, 0}^{\mathcal{N}}$. \[Th:LiftedColouringDefinable\] For all $\beta \in \mathcal{C}_{n+1, 0}$, there is an $\mathcal{L}_{\mathrm{TST}}$-formula $\Phi_{n+1, 0, \beta}(x^{n+1})$ such that for all $x \in N_{n+1}$, $$\mathcal{N} \models \Phi_{n+1, 0, \beta}[x] \textrm{ if and only if } c_{n+1, 0}^{\mathcal{N}}(x)= \beta.$$ For all $1 \leq i \leq q$, let $\Phi_{n, 0, \alpha_i}(x^{n})$ be such that for all $x \in N_n$, $$\mathcal{N} \models \Phi_{n, 0, \alpha_i}[x] \textrm{ if and only if } c_{n, 0}^{\mathcal{N}}(x)= \alpha_i.$$ Let $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle \in \mathcal{C}_{n+1, 0}$. For all $1 \leq i \leq q$ and $j \in \{ 0, 1 \}$ define the $\mathcal{L}_{\mathrm{TST}}$-formula $\Theta_{i, j}^\beta(x^{n+1})$ by: $$\Theta_{i, 0}^\beta(x^{n+1}) \textrm{ is } \left\{ \begin{array}{ll} \forall y^n(\Phi_{n, 0, \alpha_i}(y^n) \Rightarrow y^n \notin x^{n+1}) & \textrm{if } f_i= 0\\ \exists y^n( y^n \in x^{n+1} \land \Phi_{n, 0, \alpha_i}(y^n)) & \textrm{if } f_i= 1 \end{array}\right.$$ $$\Theta_{i, 1}^\beta(x^{n+1}) \textrm{ is } \left\{ \begin{array}{ll} \forall y^n(\Phi_{n, 0, \alpha_i}(y^n) \Rightarrow y^n \in x^{n+1}) & \textrm{if } g_i= 0\\ \exists y^n( y^n \notin x^{n+1} \land \Phi_{n, 0, \alpha_i}(y^n)) & \textrm{if } g_i= 1 \end{array}\right.$$ Define $\Phi_{n+1, 0, \beta}(x^{n+1})$ to be the $\mathcal{L}_{\mathrm{TST}}$-formula $$\bigwedge_{1 \leq i \leq q} \bigwedge_{j \in \{0, 1\}} \Theta_{i, j}^\beta(x^{n+1}).$$ It follows from the definition of $c_{n+1, 0}^{\mathcal{N}}$ that for all $x \in N_{n+1}$, $$\mathcal{N} \models \Phi_{n+1, 0, \beta}[x] \textrm{ if and only if } c_{n+1, 0}^{\mathcal{N}}(x)= \beta.$$ We now turn to showing that $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ are $(2^K)^{k^\prime+2}$-similar. In order to prove this we introduce the following sets: $$\mathrm{FOR}_n= \{ i \in [q] \mid \alpha_i \textrm{ is forbidden w.r.t. } c_{n, 0}^{\mathcal{M}} \textrm{ and } c_{n, 0}^{\mathcal{N}} \},$$ $$m\textrm{-}\mathrm{SPC}_n= \{ i \in [q] \mid \alpha_i \textrm{ is } m\textrm{-special w.r.t. } c_{n, 0}^{\mathcal{M}} \textrm{ and } c_{n, 0}^{\mathcal{N}} \} \textrm{ for } 1 \leq m < (2^K)^{k^\prime+2},$$ $$\mathrm{ABN}_n= \{ i \in [q] \mid \alpha_i \textrm{ is } (2^K)^{k^\prime+2}\textrm{-abundant w.r.t. } c_{n, 0}^{\mathcal{M}} \textrm{ and } c_{n, 0}^{\mathcal{N}} \}.$$ We classify the colours in $\mathcal{C}_{n+1, 0}$ which are forbidden, $1$-special and abundant with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$. \[Th:ClassifyForbiddenBase\] Let $\beta \in \mathcal{C}_{n+1, 0}$ with $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$. The colour $\beta$ is forbidden with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ if and only if either - there exists an $i \in [q]$ with $i \notin \mathrm{FOR}_n$ such that $f_i= g_i= 0$ OR, - there exists an $i \in 1\textrm{-}\mathrm{SPC}_n$ such that $f_i= g_i= 1$ OR, - there exists an $i \in \mathrm{FOR}_n$ such that $f_i= 1$ or $g_i=1$. It is clear that if any of the conditions (i)-(iii) hold then the colour $\beta$ is forbidden. Conversely, suppose that none of the conditions (i)-(iii) hold. We need to show that $\beta$ is not forbidden with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$. We first construct a point in $\mathcal{N}$ that is given colour $\beta$ by $c_{n+1, 0}^{\mathcal{N}}$. For all $1 \leq i \leq q$, let $\Phi_{n, 0, \alpha_i}(x^{n})$ be such that for all $x \in N_n$, $$\mathcal{N} \models \Phi_{n, 0, \alpha_i}[x] \textrm{ if and only if } c_{n, 0}^{\mathcal{N}}(x)= \alpha_i.$$ Let $\Theta_1(x^n)$ be the $\mathcal{L}_{\mathrm{TST}}$-formula $$\bigvee_{g_i=0} \Phi_{n, 0, \alpha_i}(x^n).$$ We work inside $\mathcal{N}$. Let $X_1= \{ x^n \mid \Theta_1(x^n) \}$. Note that comprehension ensures that $X_1$ exists. Let $$B= \mathrm{ABN}_n \cup \bigcup_{2 \leq m < (2^K)^{k^\prime+2}} m \textrm{-}\mathrm{SPC}_n$$ and let $A= \{ i \in B \mid f_i=g_i=1 \}$. Let $\Theta_2(x^n)$ be the $\mathcal{L}_{\mathrm{TST}}$-formula $$\bigvee_{i \in A} \Phi_{n, 0, \alpha_i}(x^n).$$ Let $X_2= \{ x^n \mid \Theta_2(x^n) \}$. Again, comprehension ensures that $X_2$ exists. For all $i \in A$, let $x_i \in N_n$ be such that $c_{n, 0}^{\mathcal{N}}(x_i)= \alpha_i$. Now, let $X= X_1 \cup (X_2 \backslash \{ x_i \mid i \in A \})$. Comprehension guarantees that $X$ exists in $\mathcal{N}$ and our construction ensures that $c_{n+1, 0}^{\mathcal{N}}(X)= \beta$. An identical construction shows that if none of the conditions (i)-(iii) hold then there is a point $X$ in $\mathcal{M}$ such that $c_{n+1, 0}^{\mathcal{M}}(X)= \beta$. \[Th:ClassifyOneSpecialBase\] Let $\beta \in \mathcal{C}_{n+1, 0}$ with $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$. The colour $\beta$ is $1$-special with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ if and only if $\beta$ is not forbidden with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ and for all $i \in [q]$ with $i \notin \mathrm{FOR}_n$, $f_i= 0$ or $g_i= 0$. Suppose $\beta$ is not forbidden with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ and for all $i \in [q]$ with $i \notin \mathrm{FOR}_n$, $f_i= 0$ or $g_i= 0$. If $x$ is a point that is given colour $\beta$ by $c_{n+1, 0}^{\mathcal{M}}$ or $c_{n+1, 0}^{\mathcal{N}}$ then $x$ is completely determined in $\mathcal{M}$ or $\mathcal{N}$ respectively. Therefore $\beta$ is $1$-special.\ Conversely, suppose that $\beta$ is not forbidden and there exists an $i \in [q]$ with $i \notin \mathrm{FOR}_n$ such that $f_i= g_i= 1$. We will show that $\beta$ is not $1$-special with respect to $c_{n+1, 0}^{\mathcal{M}}$ or $c_{n+1, 0}^{\mathcal{N}}$. We first construct two distinct points of $\mathcal{N}$ that are given colour $\beta$ by $c_{n+1, 0}^{\mathcal{N}}$. For all $1 \leq i \leq q$, let $\Phi_{n, 0, \alpha_i}(x^{n})$ be such that for all $x \in N_n$, $$\mathcal{N} \models \Phi_{n, 0, \alpha_i}[x] \textrm{ if and only if } c_{n, 0}^{\mathcal{N}}(x)= \alpha_i.$$ We work inside $\mathcal{N}$. Let $A= \{i \in [q] \mid f_i= g_i= 1 \}$. Since $\beta$ is not forbidden, for all $i \in A$, we can find $x_i, y_i \in N_n$ such that $c_{n, 0}^{\mathcal{N}}(x_i)= c_{n, 0}^{\mathcal{N}}(y_i)= \alpha_i$ and $x_i \neq y_i$. Let $\Theta_1(x^n)$ be the $\mathcal{L}_{\mathrm{TST}}$-formula $$\bigvee_{g_i=0} \Phi_{n, 0, \alpha_i}(x^n).$$ Let $\Theta_2(x^n)$ be the $\mathcal{L}_{\mathrm{TST}}$-formula $$\bigvee_{i \in A} \Phi_{n, 0, \alpha_i}(x^n).$$ Let $X_1= \{ x^n \mid \Theta_1(x^n) \}$ and let $X_2= \{ x^n \mid \Theta_2(x^n) \}$. Comprehension guarantees that both $X_1$ and $X_2$ exist. Let $X= X_1 \cup (X_2 \backslash \{ x_i \mid i \in A \})$ and let $Y= X_1 \cup (X_2 \backslash \{ y_i \mid i \in A \})$. Now, this construction ensures that $c_{n+1, 0}^{\mathcal{N}}(X)= c_{n+1, 0}^{\mathcal{N}}(Y)= \beta$ and $X \neq Y$. Therefore $\beta$ is not $1$-special with respect to $c_{n+1, 0}^{\mathcal{N}}$. An identical construction shows that $\beta$ is not $1$-special with respect to $c_{n+1, 0}^{\mathcal{M}}$. \[Th:ClassifyAbundantBase\] Let $\beta \in \mathcal{C}_{n+1, 0}$ with $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$. If $\beta$ is not forbidden with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ and there exists an $i \in \mathrm{ABN}_n$ such that $f_i= g_i= 1$ then $\beta$ is $(2^K)^{k^\prime+2}$-abundant with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$. Suppose that $\beta$ is not forbidden with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ and there exists an $i \in \mathrm{ABN}_n$ such that $f_i= g_i= 1$. We first construct $(2^K)^{k^\prime+2}$ distinct points in $\mathcal{N}$ that are given colour $\beta$ by $c_{n+1, 0}^{\mathcal{N}}$. For all $1 \leq i \leq q$, let $\Phi_{n, 0, \alpha_i}(x^{n})$ be such that for all $x \in N_n$, $$\mathcal{N} \models \Phi_{n, 0, \alpha_i}[x] \textrm{ if and only if } c_{n, 0}^{\mathcal{N}}(x)= \alpha_i.$$ We work inside $\mathcal{N}$. Let $u \in \mathrm{ABN}_n$ be such that $f_u= g_u= 1$. Let $A= \{ i \in [q] \mid f_i= g_i= 1 \}$. For all $i \in A$ with $i \neq u$, let $x_i \in N_n$ be such that $c_{n, 0}^{\mathcal{N}}(x_i)= \alpha_i$. Let $y_1, \ldots, y_{(2^K)^{k^\prime+2}} \in N_n$ be such that for all $1 \leq v \leq (2^K)^{k^\prime+2}$, $c_{n, 0}^{\mathcal{N}}(y_v)= \alpha_u$ and for all $1 \leq v_1 < v_2 \leq (2^K)^{k^\prime+2}$, $y_{v_1} \neq y_{v_2}$. Let $\Theta_1(x^n)$ be the $\mathcal{L}_{\mathrm{TST}}$-formula $$\bigvee_{g_i=0} \Phi_{n, 0, \alpha_i}(x^n).$$ Let $\Theta_2(x^n)$ be the $\mathcal{L}_{\mathrm{TST}}$-formula $$\bigvee_{i \in A} \Phi_{n, 0, \alpha_i}(x^n).$$ Let $X_1= \{ x^n \mid \Theta_1(x^n) \}$ and let $X_2= \{ x^n \mid \Theta_2(x^n) \}$. Comprehension guarantees that $X_1$ and $X_2$ exist. For all $1 \leq v \leq (2^K)^{k^\prime+2}$, let $$Y_v= X_1 \cup (X_2 \backslash (\{ x_i \mid i \in A \land i \neq u \} \cup \{y_v\})).$$ This construction ensures that for all $1 \leq v_1 < v_2 \leq (2^K)^{k^\prime+2}$, $Y_{v_1} \neq Y_{v_2}$ and for all $1 \leq v \leq (2^K)^{k^\prime+2}$, $c_{n+1, 0}^{\mathcal{N}}(Y_v)= \beta$. Therefore $\beta$ is $(2^K)^{k^\prime+2}$-abundant with respect to $c_{n+1, 0}^{\mathcal{N}}$. An identical construction shows that $\beta$ is $(2^K)^{k^\prime+2}$-abundant with respect to $c_{n+1, 0}^{\mathcal{M}}$. This allows us to show that the colourings $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ are $(2^K)^{k^\prime+2}$-similar. \[Th:BaseColouringsSimilar\] The colourings $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ are $(2^K)^{k^\prime+2}$-similar. Lemma \[Th:ClassifyForbiddenBase\] shows that for all $\beta \in \mathcal{C}_{n+1,0}$, $$\beta \textrm{ is forbidden w.r.t. } c_{n+1, 0}^{\mathcal{M}} \textrm{ if and only if } \beta \textrm{ is forbidden w.r.t. } c_{n+1, 0}^{\mathcal{N}}.$$ Lemma \[Th:ClassifyOneSpecialBase\] shows that for all $\beta \in \mathcal{C}_{n+1,0}$, $$\beta \textrm{ is } 1\textrm{-special w.r.t. } c_{n+1, 0}^{\mathcal{M}} \textrm{ if and only if } \beta \textrm{ is } 1\textrm{-special w.r.t. } c_{n+1, 0}^{\mathcal{N}}.$$ Let $\beta \in \mathcal{C}_{n+1, 0}$ with $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$. Lemma \[Th:ClassifyAbundantBase\] shows that if $\beta$ is not forbidden with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ and there is an $i \in \mathrm{ABN}_n$ such that $f_i= g_i= 1$ then $\beta$ is $(2^K)^{k^\prime+2}$-abundant with respect to both $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$. The remaining case is if $\beta$ is not forbidden or $1$-special and for all $i \in \mathrm{ABN}_n$, $f_i= 0$ or $g_i= 0$. Let $$B= \bigcup_{2 \leq m < (2^K)^{k^\prime+2}} m \textrm{-}\mathrm{SPC}.$$ In this case the number of $x \in M_{n+1}$ ($\in N_{n+1}$) with colour $\beta$ is completely determined by the number of $y \in M_{n}$ ($\in N_n$ respectively) with colour $\alpha_i$ such that $i \in B$ and $f_i= g_i= 1$. Therefore, the colourings $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ are $(2^K)^{k^\prime+2}$-similar. Therefore, by induction, for all $i \in \mathbb{N}$, the colourings $c_{i, 0}^\mathcal{M}: M_i \longrightarrow \mathcal{C}_{i, 0}$ and $c_{i, 0}^\mathcal{N}: N_i \longrightarrow \mathcal{C}_{i, 0}$ are $(2^K)^{k^\prime+2}$-similar.\ \ We now turn to defining the colour classes $\mathcal{C}_{i, j}$, and the colourings $c_{i, j}^\mathcal{M}: M_i \longrightarrow \mathcal{C}_{i, j}$ and $c_{i, j}^\mathcal{N}: N_i \longrightarrow \mathcal{C}_{i, j}$ for $1 \leq j \leq k^\prime$ and $i \in \mathbb{N}$. Let $0 \leq n < k^\prime$. Suppose that the colour classes $\mathcal{C}_{i, n}$ have been defined for all $i \in \mathbb{N}$ and that each of these colour classes has a canonical ordering. Let $j^\prime= \sum_{1 \leq m \leq n} K_m$ and suppose that $b_1^{r_1}, \ldots, b_{j^\prime}^{r_{j^\prime}} \in \mathcal{N}$ have been chosen. Moreover, suppose that for all $i \in \mathbb{N}$ and for all $\alpha \in \mathcal{C}_{i, n}$, the colourings $c_{i, n}^\mathcal{M}: M_i \longrightarrow \mathcal{C}_{i, n}$ and $c_{i, n}^\mathcal{N}: N_i \longrightarrow \mathcal{C}_{i, n}$, and the $\mathcal{L}_{\mathrm{TST}}$-formulae $\Phi_{i, n, \alpha}(x^i, \vec{z})$ have been defined with the following properties - $c_{i, n}^\mathcal{M}$ and $c_{i, n}^\mathcal{N}$ are $(2^K)^{k^\prime-n+2}$-similar, - for all $x \in N_i$, $$\mathcal{N} \models \Phi_{i, n, \alpha}[x, b_1^{r_1}, \ldots, b_{j^\prime}^{r_{j^\prime}}] \textrm{ if and only if } c_{i, n}^\mathcal{N}(x)= \alpha.$$ Observe that $r_{j^\prime+1}= \ldots = r_{j^\prime+K_{n+1}}$ and let $r= r_{j^\prime+1}$. We will define the colour classes $\mathcal{C}_{i, n+1}$ and colourings $c_{i, n+1}^\mathcal{M}: M_i \longrightarrow \mathcal{C}_{i, n+1}$ and $c_{i, n+1}^\mathcal{N}: N_i \longrightarrow \mathcal{C}_{i, n+1}$ such that for all $i \in \mathbb{N}$, $c_{i, n+1}^\mathcal{M}$ and $c_{i, n+1}^\mathcal{N}$ are $(2^K)^{k^\prime-n+1}$-similar and the colouring $c_{i, n+1}^\mathcal{N}$ is definable in $\mathcal{N}$. In the process of achieving this goal we will identify points $b_{j^\prime+1}^{r}, \ldots, b_{j^\prime+K_{n+1}}^{r} \in N_{r}$.\ \ For all $0 \leq i < r-1$, define $$\mathcal{C}_{i, n+1}= \mathcal{C}_{i, n},$$ $$c_{i, n+1}^\mathcal{M}= c_{i, n}^\mathcal{M},$$ $$c_{i, n+1}^\mathcal{N}= c_{i, n}^\mathcal{N}.$$ We now define the colour class $\mathcal{C}_{r-1, n+1}$, and the colourings $c_{r-1, n+1}^\mathcal{M}: M_{r-1} \longrightarrow \mathcal{C}_{r-1, n+1}$ and $c_{r-1, n+1}^\mathcal{N}: N_{r-1} \longrightarrow \mathcal{C}_{r-1, n+1}$. Let $\mathcal{C}_{r-2, n+1}= \mathcal{C}_{r-2, n}= \{ \alpha_1, \ldots, \alpha_q \}$ be obtained from the canonical ordering. Consider $a_{j^\prime+1}^{r}, \ldots, a_{j^\prime+K_{n+1}}^{r} \in M_{r}$ and use $\bar{a}_1, \ldots, \bar{a}_{K_{n+1}}$ to denote this sequence of elements. Define $\mathcal{C}_{r-1, n+1}= 2^{K_{n+1}} \times \mathcal{C}_{r-1, n}$ — the set of all 0-1 sequences of length $K_{n+1}+2\cdot q$. Define $c_{r-1, n+1}^{\mathcal{M}}: M_{r-1} \longrightarrow \mathcal{C}_{r-1, n+1}$ such that for all $x \in M_{r-1}$, $$c_{r-1, n+1}^{\mathcal{M}}(x)= \langle F_1, \ldots, F_{K_{n+1}}, f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$ $$\textrm{where } c_{r-1, n}^{\mathcal{M}}(x)= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$ $$\textrm{and } F_p= \left\{ \begin{array}{ll} 0 & \textrm{if } \mathcal{M} \models x \notin_{r-1} \bar{a}_p\\ 1 & \textrm{if } \mathcal{M} \models x \in_{r-1} \bar{a}_p \end{array}\right. \textrm{ for all } 1 \leq p \leq K_{n+1}.$$ \[Th:ColourRefinement\] There exists $\bar{b}_1, \ldots, \bar{b}_{K_{n+1}} \in N_{r}$ such that $c_{r-1, n+1}^{\mathcal{M}}$ and the colouring $c_{r-1, n+1}^{\mathcal{N}}: N_{r-1} \longrightarrow \mathcal{C}_{r-1, n+1}$, defined such that for all $x \in N_{r-1}$, $$c_{r-1, n+1}^{\mathcal{N}}(x)= \langle F_1, \ldots, F_{K_{n+1}}, f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$ $$\label{Eq:RefinedNColouring} \textrm{where } c_{r_{j^\prime+1}-1, n}^{\mathcal{N}}(x)= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$ $$\textrm{and } F_p= \left\{ \begin{array}{ll} 0 & \textrm{if } \mathcal{N} \models x \notin_{r-1} \bar{b}_p\\ 1 & \textrm{if } \mathcal{N} \models x \in_{r-1} \bar{b}_p \end{array}\right. \textrm{ for all } 1 \leq p \leq K_{n+1},$$ are $(2^K)^{k^\prime-n+1}$-similar. Let $\mathcal{C}_{r-1, n}= \{ \alpha_1, \ldots, \alpha_{q^\prime} \}$ be obtained from the canonical ordering. For all $1 \leq i \leq q^\prime$ and for all $\sigma \in 2^{K_{n+1}}$ define $X_\sigma^i \subseteq M_{r-1}$ by $$X_\sigma^i= \{ x \in M_{r-1} \mid (c_{r-1, n}^{\mathcal{M}}(x)= \alpha_i) \land (\forall v \in K_{n+1})(\sigma(v)=1 \iff x \in \bar{a}_v)\}.$$ Note that for all $1 \leq i \leq q^\prime$, the sets $\langle X_\sigma^i \mid \sigma \in 2^{K_{n+1}} \rangle$ partition the elements of $M_{r-1}$ that are given colour $\alpha_i$ by $c_{r-1, n}^{\mathcal{M}}$ into $2^{K_{n+1}}$ pieces. For each $1 \leq i \leq q^\prime$ choose a sequence $\langle Z_\sigma^i \mid \sigma \in 2^{K_{n+1}} \rangle$ such that for all $\sigma \in 2^{K_{n+1}}$, - $Z_\sigma^i \in N_{r}$, - for all $z \in N_{r-1}$ with $\mathcal{N} \models z \in_{r-1} Z_\sigma^i$, $c_{r-1, n}^{\mathcal{N}}(z)= \alpha_i$, - if $|X_\sigma^i| < (2^K)^{k^\prime-n+1}$ then $|\{z \in \mathcal{N} \mid \mathcal{N} \models z \in_{r-1} Z_\sigma^i \}|= |X_\sigma^i|$, - if $|X_\sigma^i| \geq (2^K)^{k^\prime-n+1}$ then $|\{z \in \mathcal{N} \mid \mathcal{N} \models z \in_{r-1} Z_\sigma^i \}| \geq (2^K)^{k^\prime-n+1}$. To see that we can make this choice we work inside $\mathcal{N}$. For all $1 \leq i \leq q^\prime$, let $\Phi_{r-1, n, \alpha_i}(x^{r-1}, \vec{z})$ be such that for all $x \in N_{r-1}$, $$\mathcal{N} \models \Phi_{r-1, n, \alpha_i}[x, b_1^{r_1}, \ldots, b_{j^\prime}^{r_{j^\prime}}] \textrm{ if and only if } c_{r-1, n}^\mathcal{N}(x)= \alpha_i.$$ For all $1 \leq i \leq q^\prime$, let $W_i= \{ x^{r-1} \mid \Phi_{r-1, n, \alpha_i}(x^{r-1}, b_1^{r_1}, \ldots, b_{j^\prime}^{r_{j^\prime}}) \}$. Comprehension ensures that the $W_i$s exist. For all $1 \leq i \leq q^\prime$ and for all $\sigma \in 2^{K_{n+1}}$, $Z_\sigma^i$ can be chosen to be a finite or cofinite subset of $W_i$. Moreover, the fact that $c_{r-1, n}^\mathcal{M}$ and $c_{r-1, n}^\mathcal{N}$ are $(2^K)^{k^\prime-n+2}$-similar ensures that for all $1 \leq i \leq q^\prime$ we can choose the sequence $\langle Z_\sigma^i \mid \sigma \in 2^{K_{n+1}} \rangle$ to satisfy condition (iii) above.\ Now, for all $1 \leq p \leq K_{n+1}$, let $\bar{b}_p \in N_{r}$ be such that $$\mathcal{N} \models \bar{b}_p= \bigcup_{1 \leq i \leq q^\prime} \bigcup_{^{\sigma \in 2^{K_{n+1}}}_{\textrm{s.t. } \sigma(p)=1}} Z_\sigma^i.$$ This construction ensures that the colourings $c_{r-1, n+1}^{\mathcal{M}}$ and $c_{r-1, n+1}^{\mathcal{N}}$ define by (\[Eq:RefinedNColouring\]) are $(2^K)^{k^\prime-n+1}$-similar. Let $b_{j^\prime+1}^{r}, \ldots, b_{j^\prime+K_{n+1}}^{r} \in \mathcal{N}$ be the points $\bar{b}_1, \ldots, \bar{b}_{K_{n+1}}$ produced in the proof of Lemma \[Th:ColourRefinement\] and let $c_{r-1, n+1}^{\mathcal{N}}$ be defined by (\[Eq:RefinedNColouring\]). Therefore $c_{r-1, n+1}^{\mathcal{M}}$ and $c_{r-1, n+1}^{\mathcal{N}}$ are $(2^K)^{k^\prime-n+1}$-similar. We can immediately observe that the colouring $c_{r-1, n+1}^{\mathcal{N}}$ is definable in $\mathcal{N}$ by an $\mathcal{L}_{\mathrm{TST}}$-formula using parameters $b_{1}^{r_{1}}, \ldots, b_{j^\prime+K_{n+1}}^{r_{j^\prime+K_{n+1}}}$. \[Th:ColourRefinmentDefinable\] For all $\alpha \in \mathcal{C}_{r-1, n+1}$, there exists an $\mathcal{L}_{\mathrm{TST}}$-formula $\Phi_{r-1, n+1, \alpha}(x^{r-1}, \vec{z})$ such that for all $x \in N_{r-1}$, $$\mathcal{N} \models \Phi_{r-1, n+1, \alpha}[x, b_{1}^{r_{1}}, \ldots, b_{j^\prime+K_{n+1}}^{r_{j^\prime+K_{n+1}}}] \textrm{ if and only if } c_{r-1, n+1}^{\mathcal{N}}(x)= \alpha.$$ Let $t= \sum_{1 \leq m \leq n+1} K_m$. Lemma \[Th:ColourRefinement\] and Lemma \[Th:ColourRefinmentDefinable\] show that we can define colourings $c_{r-1, n+1}^{\mathcal{M}}$ and $c_{r-1, n+1}^{\mathcal{N}}$, and $\mathcal{L}_{\mathrm{TST}}$-formulae $\Phi_{r-1, n+1, \alpha}(x^{r-1}, \vec{z})$ for all $\alpha \in \mathcal{C}_{r-1, n+1}$ which satisfy the following properties: - $c_{r-1, n+1}^{\mathcal{M}}$ and $c_{r-1, n+1}^{\mathcal{N}}$ are $(2^K)^{k^\prime-n+1}$-similar, - for all $x \in N_{r-1}$, $$\mathcal{N} \models \Phi_{r-1, n+1, \alpha}[x, b_1^{r_1}, \ldots, b_t^{r_t}] \textrm{ if and only if } c_{r-1, n+1}^{\mathcal{N}}(x)= \alpha.$$ We now turn to defining the colour classes $\mathcal{C}_{i, n+1}$, and the colourings $c_{i, n+1}^\mathcal{M}: M_i \longrightarrow \mathcal{C}_{i, n+1}$ and $c_{i, n+1}^\mathcal{N}: N_i \longrightarrow \mathcal{C}_{i, n+1}$ for all $i \geq r$. Let $i \geq r-1$. Suppose that the colour class $\mathcal{C}_{i, n+1}$ has been defined with a canonical ordering. Suppose, also, that the colourings $c_{i, n+1}^{\mathcal{M}}: M_i \longrightarrow \mathcal{C}_{i, n+1}$ and $c_{i, n+1}^{\mathcal{N}}: N_i \longrightarrow \mathcal{C}_{i, n+1}$, and the $\mathcal{L}_{\mathrm{TST}}$-formulae $\Phi_{i, n+1, \alpha}(x^{i}, \vec{z})$ have been defined and satisfy: - $c_{i, n+1}^{\mathcal{M}}$ and $c_{i, n+1}^{\mathcal{N}}$ are $(2^K)^{k^\prime-n+1}$-similar, - for all $x \in N_i$, $$\mathcal{N} \models \Phi_{i, n+1, \alpha}[x, b_1^{r_1}, \ldots, b_t^{r_t}] \textrm{ if and only if } c_{i, n+1}^{\mathcal{N}}(x)= \alpha.$$ We ‘lift’ the colour class $\mathcal{C}_{i, n+1}$ and the colourings $c_{i, n+1}^{\mathcal{M}}$ and $c_{i, n+1}^{\mathcal{N}}$ in the same way that we ‘lifted’ the colour classes $\mathcal{C}_{i, 0}$ and the colourings $c_{i, 0}^{\mathcal{M}}$ and $c_{i, 0}^{\mathcal{N}}$ above. Let $\mathcal{C}_{i, n+1}= \{ \alpha_1, \ldots, \alpha_q \}$ be obtained from the canonical ordering. Define $\mathcal{C}_{i+1, n+1}= 2^{2 \cdot q}$— the set of all 0-1 sequence of length $2 \cdot q$. Define $c_{i+1, n+1}^{\mathcal{M}}: M_{i+1} \longrightarrow \mathcal{C}_{i+1, n+1}$ such that for all $x \in M_{i+1}$, $$c_{i+1, n+1}^{\mathcal{M}}(x)= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$ $$\textrm{where } f_p = \left\{ \begin{array}{ll} 0 & \textrm{if for all } y \in M_i, \textrm{ if } c_{i,n+1}^{\mathcal{M}}(y)= \alpha_p \textrm{ then } \mathcal{M} \models y \notin_i x\\ 1 & \textrm{if there exists } y \in M_i \textrm{ such that } c_{i, n+1}^{\mathcal{M}}(y)= \alpha_p \textrm{ and } \mathcal{M} \models y \in_i x \end{array}\right.$$ $$\textrm{and } g_p = \left\{ \begin{array}{ll} 0 & \textrm{if for all } y \in M_i, \textrm{ if } c_{i, n+1}^{\mathcal{M}}(y)= \alpha_p \textrm{ then } \mathcal{M} \models y \in_i x\\ 1 & \textrm{if there exists } y \in M_i \textrm{ such that } c_{i, n+1}^{\mathcal{M}}(y)= \alpha_p \textrm{ and } \mathcal{M} \models y \notin_i x \end{array}\right.$$ Again, we define $c_{i+1, n+1}^{\mathcal{N}}$ identically. Define $c_{i+1, n+1}^{\mathcal{N}}: N_{i+1} \longrightarrow \mathcal{C}_{i+1, n+1}$ such that for all $x \in N_{i+1}$, $$c_{i+1, n+1}^{\mathcal{N}}(x)= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$ $$\textrm{where } f_p= \left\{ \begin{array}{ll} 0 & \textrm{if for all } y \in N_i, \textrm{ if } c_{i, n+1}^{\mathcal{N}}(y)= \alpha_p \textrm{ then } \mathcal{N} \models y \notin_i x\\ 1 & \textrm{if there exists } y \in N_i \textrm{ such that } c_{i, n+1}^{\mathcal{N}}(y)= \alpha_p \textrm{ and } \mathcal{N} \models y \in_i x \end{array}\right.$$ $$\textrm{and } g_p= \left\{ \begin{array}{ll} 0 & \textrm{if for all } y \in N_i, \textrm{ if } c_{i, n+1}^{\mathcal{N}}(y)= \alpha_p \textrm{ then } \mathcal{N} \models y \in_i x\\ 1 & \textrm{if there exists } y \in N_i \textrm{ such that } c_{i, n+1}^{\mathcal{N}}(y)= \alpha_p \textrm{ and } \mathcal{N} \models y \notin_i x \end{array}\right.$$ We first observe that there exists $\mathcal{L}_{\mathrm{TST}}$-formulae $\Phi_{i+1, n+1, \beta}(x^{i+1}, \vec{z})$ for each $\beta \in \mathcal{C}_{i+1, n+1}$ which witness the fact that the colouring $c_{i+1, n+1}^{\mathcal{N}}$ satisfies condition (II”’). For all $\beta \in \mathcal{C}_{i+1, n+1}$, there is an $\mathcal{L}_{\mathrm{TST}}$-formula $\Phi_{i+1, n+1, \beta}(x^{i+1}, \vec{z})$ such that for all $x \in N_{i+1}$, $$\mathcal{N} \models \Phi_{i+1, n+1, \beta}[x, b_1^{r_1}, \ldots, b_t^{r_t}] \textrm{ if and only if } c_{i+1, n+1}^{\mathcal{N}}(x)= \beta.$$ Identical to the proof Lemma \[Th:LiftedColouringDefinable\] using the fact that $c_{i, n+1}^{\mathcal{N}}$ satisfies condition (II”’). We now turn to showing that $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$ are $(2^K)^{k^\prime-n+1}$-similar. To do this we prove analogues of Lemmata \[Th:ClassifyForbiddenBase\], \[Th:ClassifyOneSpecialBase\] and \[Th:ClassifyAbundantBase\]. $$\mathrm{FOR}_i^{n+1}= \{ v \in [q] \mid \alpha_v \textrm{ is forbidden w.r.t. } c_{i, n+1}^{\mathcal{M}} \textrm{ and } c_{i, n+1}^{\mathcal{N}} \},$$ $$m\textrm{-}\mathrm{SPC}_i^{n+1}= \{ v \in [q] \mid \alpha_v \textrm{ is } m \textrm{-special w.r.t. } c_{i, n+1}^{\mathcal{M}} \textrm{ and } c_{i, n+1}^{\mathcal{N}} \} \textrm{ for } 1 \leq m < (2^K)^{k^\prime-n+1},$$ $$\mathrm{ABN}_i^{n+1}= \{ v \in [q] \mid \alpha_v \textrm{ is } (2^K)^{k^\prime-n+1} \textrm{-abundant w.r.t. } c_{i, n+1}^{\mathcal{M}} \textrm{ and } c_{i, n+1}^{\mathcal{N}} \}.$$ \[Th:ClassifyForbiddenRefined\] Let $\beta \in \mathcal{C}_{i+1, n+1}$ with $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$. The colour $\beta$ is forbidden with respect to $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$ if and only if either - there exists a $v \in [q]$ with $v \notin \mathrm{FOR}_i^{n+1}$ such that $f_v= g_v= 0$ OR, - there exists a $v \in 1\textrm{-}\mathrm{SPC}_i^{n+1}$ such that $f_v= g_v= 1$ OR, - there exists a $v \in \mathrm{FOR}_i^{n+1}$ such $f_v= 1$ or $g_v=1$. Identical to the proof of Lemma \[Th:ClassifyForbiddenBase\]. \[Th:ClassifyOneSpecialRefined\] Let $\beta \in \mathcal{C}_{i+1, n+1}$ with $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$. The colour $\beta$ is $1$-special with respect to $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$ if and only if $\beta$ is not forbidden with respect to $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$ and for all $v \in [q]$ with $v \notin \mathrm{FOR}_i^{n+1}$, $f_v=0$ or $g_v=0$. Identical to the proof of Lemma \[Th:ClassifyOneSpecialBase\]. \[Th:ClassifyAbundantRefined\] Let $\beta \in \mathcal{C}_{i+1, n+1}$ with $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$. If $\beta$ is not forbidden with respect to $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$ and there exists a $v \in \mathrm{ABN}_i^{n+1}$ with $f_v= g_v= 1$ then $\beta$ is $(2^K)^{k^\prime-n+1}$-abundant with respect to $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$. Identical to the proof of Lemma \[Th:ClassifyAbundantBase\]. These results allow us to show that $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$ are $(2^K)^{k^\prime-n+1}$-similar. \[Th:RefinedColouringsSimilar\] The colourings $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$ are $(2^K)^{k^\prime-n+1}$-similar. Identical to the proof of Lemma \[Th:BaseColouringsSimilar\] using Lemmata \[Th:ClassifyForbiddenRefined\], \[Th:ClassifyOneSpecialRefined\] and \[Th:ClassifyAbundantRefined\]. This recursion allows us to define the colour classes $\mathcal{C}_{n, k^\prime}$ and colourings $c_{n, k^\prime}^{\mathcal{M}}$ and $c_{n, k^\prime}^{\mathcal{N}}$ for all $n \in \mathbb{N}$, and elements $b_1^{r_1}, \ldots, b_1^{r_k} \in \mathcal{N}$. The above arguments show that for all $n \in \mathbb{N}$, $c_{n, k^\prime}^{\mathcal{M}}$ and $c_{n, k^\prime}^{\mathcal{N}}$ are $2^K$-similar. We have constructed the colourings $c_{n, k^\prime}^{\mathcal{M}}$ and $c_{n, k^\prime}^{\mathcal{N}}$ so as the colour assigned to a point $x \in \mathcal{M}$ (or $\mathcal{N}$) completely captures the set of quantifier-free formulae with parameters $a_1^{r_1}, \ldots, a_k^{r_k}$ (respectively $b_1^{r_1}, \ldots, b_k^{r_k}$) that are satisfied by $x$. \[Th:ColouringsCaptureOneTypes\] Let $n \in \mathbb{N}$ and let $\theta(x_1^{r_1}, \ldots, x_k^{r_k}, x^n)$ be a quantifier-free $\mathcal{L}_{\mathrm{TST}}$-formula. If $x \in M_n$ and $y \in N_n$ are such that $c_{n, k^\prime}^{\mathcal{M}}(x)= c_{n, k^\prime}^{\mathcal{M}}(y)$ then $$\mathcal{M} \models \theta[a_1^{r_1}, \ldots, a_k^{r_k}, x] \textrm{ if and only if } \mathcal{N} \models \theta[b_1^{r_1}, \ldots, b_k^{r_k}, y]$$ This follows immediately from the definition of the colourings $c_{n, k^\prime}^{\mathcal{M}}$ and $c_{n, k^\prime}^{\mathcal{N}}$. Our construction also ensures that if $x \in M_{n+1}$ (or $N_{n+1}$) then the colour assigned to $x$ by $c_{n+1, k^{\prime}}^{\mathcal{M}}$ (respectively $c_{n+1, k^\prime}^{\mathcal{N}}$) tells us, for all $\alpha \in \mathcal{C}_{n, k^\prime}$, whether there exists a point $y \in M_n$ (respectively $N_n$) such that $c_{n, k^\prime}^{\mathcal{M}}(y)= \alpha$ (respectively $c_{n, k^\prime}^{\mathcal{N}}(y)= \alpha$) and $y$ is in the relationship $\in_n$ or $\notin_n$ to $x$ in $\mathcal{M}$ (respectively $\mathcal{N}$). \[Th:KeepingTrackOfColousLemma\] Let $x \in M_{n+1}$ and $y \in N_{n+1}$, and let $\alpha \in \mathcal{C}_{n, k^\prime}$. If $c_{n+1, k^\prime}^{\mathcal{M}}(x)= c_{n+1, k^\prime}^{\mathcal{N}}(y)$ then $$(\exists z \in M_n)(c_{n, k^\prime}^{\mathcal{M}}(z)= \alpha \land \mathcal{M} \models z \in_n x) \textrm{ if and only if } (\exists z \in N_n)(c_{n, k^\prime}^{\mathcal{N}}(z)= \alpha \land \mathcal{N} \models z \in_n y),$$ $$\textrm{and }(\exists z \in M_n)(c_{n, k^\prime}^{\mathcal{M}}(z)= \alpha \land \mathcal{M} \models z \notin_n x) \textrm{ if and only if } (\exists z \in N_n)(c_{n, k^\prime}^{\mathcal{N}}(z)= \alpha \land \mathcal{N} \models z \notin_n y).$$ This follows immediately from the definition of the colourings $c_{n+1, k^\prime}^{\mathcal{M}}$ and $c_{n+1, k^\prime}^{\mathcal{N}}$. This allows us to show that an $\mathcal{L}_{\mathrm{TST}}$-sentence in the form (A) or (B) which is true $\mathcal{N}$ is also true in $\mathcal{M}$. \[Th:SentencesOfTypeBHoldInM\] Let $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s} \cdots \exists y_l^{s} \theta$ be an $\mathcal{L}_{\mathrm{TST}}$-formula with $\theta$ is quantifier-free. If $\mathcal{N} \models \phi$ then $\mathcal{M} \models \phi$. Suppose that $\mathcal{N} \models \phi$. Let $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{M}$. Using $a_1^{r_1}, \ldots, a_k^{r_k}$ and the construction we presented above we can define the colour classes $\mathcal{C}_{n, k^\prime}$ and colourings $c_{n, k^\prime}^\mathcal{M}$ and $c_{n, k^\prime}^\mathcal{N}$ for all $n \in \mathbb{N}$, and elements $b_1^{r_1}, \ldots, b_k^{r_k} \in \mathcal{N}$. The colourings $c_{n, k^\prime}^\mathcal{M}$ and $c_{n, k^\prime}^\mathcal{N}$ are $2^K$-similar and satisfy Lemma \[Th:ColouringsCaptureOneTypes\]. Let $e_1, \ldots, e_l \in N_s$ be such that $$\mathcal{N} \models \theta[b_1^{r_1}, \ldots, b_k^{r_k}, e_1, \ldots, e_l].$$ For all $1 \leq i \leq l$, let $d_i \in M_s$ such that $c_{s, k^\prime}^\mathcal{M}(d_i)= c_{s, k^\prime}^{\mathcal{N}}(e_i)$ and for all $1 \leq j < i$, $d_j \neq d_i$ if and only if $e_i \neq e_j$. The fact that $l < 2^K$ and $c_{s, k^\prime}^\mathcal{M}$ and $c_{s, k^\prime}^\mathcal{N}$ are $2^K$-similar ensures we can find $d_1, \ldots, d_l \in M_s$ satisfying these conditions. Now, since the variables $y_1^s, \ldots y_l^s$ all have the same type in $\theta$, the only atomic or negatomic subformulae of $\theta$ are in the form $y_i^s = y_j^s$, $y_i^s \in_s x_j^{r_j}$ if $r_j= s+1$, $x_i^{r_i} \in_{r_i} y_j^s$ if $s= r_i +1$ or $x_i^{r_i} \in_{r_i} x_j^{r_j}$ if $r_j= r_i +1$ or one of negations of these. Therefore, by Lemma \[Th:ColouringsCaptureOneTypes\], $$\mathcal{M} \models \theta[a_1^{r_1}, \ldots, a_k^{r_k}, d_1, \ldots, d_l].$$ Since the $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{M}$ were arbitrary this shows that $\mathcal{M} \models \phi$. \[Th:SentencesOfTypeAHoldInM\] Let $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s_1} \cdots \exists y_l^{s_l} \theta$ be an $\mathcal{L}_{\mathrm{TST}}$-sentence with $s_1 > \ldots > s_l$ and $\theta$ quantifier-free. If $\mathcal{N} \models \phi$ then $\mathcal{M} \models \phi$. Suppose that $\mathcal{N} \models \phi$. Let $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{M}$. Using $a_1^{r_1}, \ldots, a_k^{r_k}$ and the construction we presented above we can define the colour classes $\mathcal{C}_{n, k^\prime}$ and colourings $c_{n, k^\prime}^\mathcal{M}$ and $c_{n, k^\prime}^\mathcal{N}$ for all $n \in \mathbb{N}$, and elements $b_1^{r_1}, \ldots, b_k^{r_k} \in \mathcal{N}$. The colourings $c_{n, k^\prime}^\mathcal{M}$ and $c_{n, k^\prime}^\mathcal{N}$ are $2^K$-similar and satisfy Lemma \[Th:ColouringsCaptureOneTypes\]. Let $e_1^{s_1}, \ldots, e_l^{s_l} \in \mathcal{N}$ be such that $$\mathcal{N} \models \theta[b_1^{r_1}, \ldots, b_k^{r_k}, e_1^{s_1}, \ldots, e_l^{s_l}].$$ We inductively choose $d_1^{s_1}, \ldots, d_l^{s_l} \in \mathcal{M}$. Let $d_1^{s_1} \in \mathcal{M}$ be such that $c_{s_1, k^\prime}^\mathcal{M}(d_1^{s_1})= c_{s_1, k^\prime}^{\mathcal{N}}(e_1^{s_1})$. Suppose that $1 \leq i < l$ and we have chosen $d_i^{s_i} \in \mathcal{M}$ with $c_{s_i, k^\prime}^\mathcal{M}(d_i^{s_i})= c_{s_i, k^\prime}^{\mathcal{N}}(e_i^{s_i})$. If $s_i \neq s_{i+1} + 1$ then let $d_{i+1}^{s_{i+1}} \in \mathcal{M}$ be such that $c_{s_{i+1}, k^\prime}^\mathcal{M}(d_{i+1}^{s_{i+1}})= c_{s_{i+1}, k^\prime}^{\mathcal{N}}(e_{i+1}^{s_{i+1}})$. If $s_i= s_{i+1} + 1$ and $\mathcal{N} \models e_{i+1}^{s_{i+1}} \in_{s_{i+1}} e_i^{s_i}$ then let $d_{i+1}^{s_{i+1}} \in \mathcal{M}$ be such that $c_{s_{i+1}, k^\prime}^\mathcal{M}(d_{i+1}^{s_{i+1}})= c_{s_{i+1}, k^\prime}^{\mathcal{N}}(e_{i+1}^{s_{i+1}})$ and $\mathcal{M} \models d_{i+1}^{s_{i+1}} \in_{s_{i+1}} d_i^{s_i}$. If $s_i= s_{i+1} + 1$ and $\mathcal{N} \models e_{i+1}^{s_{i+1}} \notin_{s_{i+1}} e_i^{s_i}$ then let $d_{i+1}^{s_{i+1}} \in \mathcal{M}$ be such that $c_{s_{i+1}, k^\prime}^\mathcal{M}(d_{i+1}^{s_{i+1}})= c_{s_{i+1}, k^\prime}^{\mathcal{N}}(e_{i+1}^{s_{i+1}})$ and $\mathcal{M} \models d_{i+1}^{s_{i+1}} \notin_{s_{i+1}} d_i^{s_i}$. Lemma \[Th:KeepingTrackOfColousLemma\] and the fact that $1 < 2^K$, and $c_{s_{i+1}, k^\prime}^\mathcal{M}$ and $c_{s_{i+1}, k^\prime}^\mathcal{N}$ are $2^K$-similar ensure that we can find $d_{i+1}^{s_{i+1}} \in \mathcal{M}$ satisfying these conditions. Now, since the variables $y_1^{s_1}, \ldots y_l^{s_l}$ all have distinct types in $\theta$, the only atomic or negatomic subformulae of $\theta$ are in the form $y_{i+1}^{s_{i+1}} \in_{s_{i+1}} y_i^{s_i}$ if $s_i= s_{i+1}+1$, $y_i^{s_i} \in_{s_i} x_j^{r_j}$ if $r_j= s_i+1$, $x_i^{r_i} \in_{r_i} y_j^{s_j}$ if $s_j= r_i+1$, or $x_i^{r_i} \in_{r_i} x_j^{r_j}$ if $r_j= r_i+1$, or one of the negations of these. Therefore, by Lemma \[Th:ColouringsCaptureOneTypes\], $$\mathcal{M} \models \theta[a_1^{r_k}, \ldots, a_k^{r_k}, d_1^{s_1}, \ldots, d_l^{s_l}].$$ Since the $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{M}$ were arbitrary this shows that $\mathcal{M} \models \phi$. Since $\mathcal{N}$ is an arbitrary model of $\mathrm{TSTI}$ and $\mathcal{M}$ is an arbitrary sufficiently large finitely generated model of $\mathrm{TST}$, Theorems \[Th:SentencesOfTypeBHoldInM\] and \[Th:SentencesOfTypeAHoldInM\] show that any $\mathcal{L}_{\mathrm{TST}}$-sentence in the form (A) or (B) has the finitely generated model property. Combining this with Theorem \[Th:ExistentialUniversalSentencesHaveFinitelyGeneratedModelProperty\] shows that $\mathrm{TSTI}$ decides any sentence in the form (A) or (B). If $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s_1} \cdots \exists y_l^{s_l} \theta$ is an $\mathcal{L}_{\mathrm{TST}}$-sentence with $s_1 > \ldots > s_l$ and $\theta$ quantifier free then $\mathrm{TST}$ decides $\phi$. If $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s} \cdots \exists y_l^{s} \theta$ is an $\mathcal{L}_{\mathrm{TST}}$-sentence with $\theta$ quantifier-free then $\mathrm{TST}$ decides $\phi$. Combining these results with Theorem \[Th:DecidableFragmentsOfNF\] shows that sentences in the form (A’) or (B’) are decided by $\mathrm{NF}$. If $\phi= \forall x_1 \cdots \forall x_k \exists y_1 \cdots \exists y_l \theta$ is an $\mathcal{L}$-formula with $\theta$ quantifier-free and $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$ is a stratification of $\phi$ that assigns the same value to all of the variables $y_1, \ldots, y_l$ then $\mathrm{NF}$ decides $\phi$. If $\phi= \forall x_1 \cdots \forall x_k \exists y_1 \cdots \exists y_l \theta$ is an $\mathcal{L}$-formula with $\theta$ quantifier-free and $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$ is a stratification of $\phi$ that assigns distinct values to all of the variable $y_1, \ldots, y_l$ then $\mathrm{NF}$ decides $\phi$. It is interesting to note that the only use of the Axiom of Infinity in the above arguments was to ensure that the bottom type is externally infinite. Thus our arguments show that all models of $\mathrm{TST}$ with infinite bottom type agree on all sentences in the form (A) and all sentences in the form (B). [^1]: The research of the first and the third author was supported in part by EPSRC grant EP/H026835.

--- abstract: 'This paper proposes a novel scheme which can efficiently reduce the energy consumption of Optical Line Terminals (OLTs) in Time Division Multiplexing (TDM) Passive Optical Networks (PONs) such as EPON and GPON. Currently, OLTs consume a significant amount of energy in PON, which is one of the major FTTx technologies. To be environmentally friendly, it is desirable to reduce energy consumption of OLT as much as possible; such requirement becomes even more urgent as OLT keeps increasing its provisioning data rate, and higher data rate provisioning usually implies higher energy consumption. In this paper, we propose a novel energy-efficient OLT structure which guarantees services of end users with the smallest number of power-on OLT line cards. More specifically, we adapt the number of power-on OLT line cards to the real-time incoming traffic. Also, in order to avoid service disruption resulted by powering off OLT line cards, proper optical switches are equipped in OLT to dynamically configure the communications between OLT line cards and ONUs.' --- --------------------------- --  --------------------------- -- <span style="font-variant:small-caps;">Design and Analysis of Green Optical Line Terminal for TDM Passive Optical Networks</span>\ [<span style="font-variant:small-caps;">mina taheri</span>]{}\ [<span style="font-variant:small-caps;">nirwan ansari</span>]{}\ [<span style="font-variant:small-caps;">TR-ANL-2015-004</span>\ ]{}\ [<span style="font-variant:small-caps;">Advanced Networking Laboratory</span>]{}\ [<span style="font-variant:small-caps;">Department of Electrical and Computer Engineering</span>]{}\ [<span style="font-variant:small-caps;">New Jersy Institute of Technology</span>]{}\ Introduction ============ As energy consumption is becoming an environmental and therefore social and economic issue, green Information and Communication Technology (ICT) has attracted significant research attention recently. It was reported that Internet consumes as much as $\sim 1\%-2.5\% $ of the total electricity in broadband enabled countries[@BalEne09; @pickavet2009worldwide; @fettweisict], and currently and in the medium term future, the majority of the energy of Internet is consumed by access networks owing to the large quantity of access nodes [@BalEne08]. Energy consumptions of access networks depend on the access technologies. Among various access technologies including WiMAX, FTTN, and point to point optical access networks, passive optical networks (PONs) consume the smallest energy per transmission bit attributed to the proximity of optical fibers to the end users and the passive nature of the remote node [@LanOnt08]. However, as PON is deployed worldwide, it still consumes a significant amount of energy. It is desirable to reduce the energy consumption of PONs since every single watt saving will end up with overall terawatt and even larger power saving. Reducing energy consumption of PONs becomes even more important as the current PON systems evolve into next-generation PONs with increased data rate provisioning [@ZhaNex09; @ansari2013media]. In PONs, energy is consumed by optical line terminal (OLT) and optical network units (ONUs). Owing the large quantity, ONUs consume a large portion of the overall PON energy [@ZhaTow11]. Although OLT consumes a less amount of power than the total aggregated ONUs, one OLT line card does consume a much larger amount of power than one ONU. Reducing energy consumption of OLT is as important as reducing energy consumption of ONUs especially from the operators’ and home users’ perspectives. For the network operators, decreasing the energy consumption of OLT can significantly reduce the energy consumption of the central office, while decreasing the energy consumption of ONUs has small and likely negligible impacts on that of home users who have many other electrical appliances with much higher energy consumption. Formerly, sleep mode and adaptive line rate have been proposed to efficiently reduce the power consumption of ONUs by taking advantages of the bursty nature of the traffic at the user side [@zhang2013standards; @KubStu10; @WonSle09; @ChoEne10; @taheri2014multi]. It is, however, challenging to introduce “sleep” mode into OLT to reduce its energy consumption for the following reasons. In PONs, OLT serves as the central access node which controls the network resource access of ONUs. Putting OLT into sleep can easily result in service disruption of ONUs in communicating with the OLT. Thus, a proper scheme is needed to reduce the energy consumption of OLT without degrading services of end users. In this paper, we propose a novel energy-efficient OLT structure which can adapt its power-on OLT line cards according to the real-time arrival traffic. To avoid service degradation during the process of powering on/off OLT line cards, proper devices are added into the legacy OLT chassis to facilitate all ONUs communicate with power-on line cards. To the best of our knowledge, this is the first work focusing on reducing the energy consumption of OLT [^1]. Framework of the energy-efficient OLT design {#sec:I} ============================================  In the central office, one OLT chassis typically comprises of multiple OLT line cards that transmit downstream signals and receive upstream signals at different wavelengths. Each line card communicates with a number of ONUs. Two wavelengths for the uplink and the downlink are assigned to each ONU. In the currently deployed EPON and GPON systems, one OLT line card usually communicates with either $16$ or $32$ ONUs and such an arrangement is referred to as a PON segment. To avoid service disruptions of ONUs connected to the central office, all these OLT line cards in the OLT chassis are usually power-on all the time. To reduce the energy consumption of OLT, our main idea is to adapt the number of power-on OLT line cards in the OLT chassis to the real-time incoming traffic. There are two types of subscribers that each network serves; Business subscribers and residential subscribers. Business and residential areas are usually disjunct. It is more likely that each PON segment serves either business customers or residential customers. These two types of customers have different traffic profiles. Business users demand high bandwidth during the day and low bandwidth at night while residential customers request high bandwidth in the evening and low bandwidth during the day. During the day time, residential segments are lightly loaded. Therefore, one OLT line card can serve several residential segments. In the similar way, the traffic from the business segments can combined to traverse a smaller number of line cards in the evening. Business and residential segments usually have low bandwidth demands during the midnight. In these situations, the whole network is lightly loaded. In order to save energy, the number of line cards can be reduced based on the traffic volume. Parameters of the proposed model are notated below: $C_u$: Data rate of one OLT line card in the upstream direction. $C_d$: Data rate of one OLT line card in the downstream direction. $L$: Total number of line cards (PON segments). $N_j$: Number of ONUs connected to PON segment $j$. $T$: Fixed traffic cycle in TDM PON. $u_{i,j}(t)$: Arrival upstream traffic rate from ONU $i$ of PON segment $j$ at time $t$. $d_{i,j}(t)$: Arrival downstream traffic rate to ONU $i$ of PON segment $j$ at time $t$. $l(t)$: smallest number of required OLT line cards at time $t$. By powering on all the OLT line cards, the overall upstream data rate and downstream data rate accommodated by the OLT chassis equal to $C_u \cdot L$ and $C_d \cdot L$, respectively. $C_u \cdot L$ (or $C_d \cdot L$) may be greater than the real-time upstream (or downstream) traffic. The traffic rate of each segment cannot be more than the provisioned capacity of the dedicated fiber. Therefore, the following constraints have to be satisfied for any segment $j$: $$\sum_{i=1}^{N_j}{u_{i,j}(t)} \leq C_u$$ $$\sum_{i=1}^{N_j}{d_{i,j}(t)} \leq C_d$$ The real time incoming upstream and downstream traffics are defined as $\mathop{\sum_{j=1}^L\sum_{i=1}^{N_j}}{u_{i,j}(t)}$ and $\mathop{\sum_{j=1}^L\sum_{i=1}^{N_j}}{d_{i,j}(t)}$, respectively. Then, $$l(t)= max ({\lceil\mathop{\sum_{j=1}^L\sum_{i=1}^{N_j}}{u_{i,j}(t)}/{C_u}\rceil , \lceil\mathop{\sum_{j=1}^L\sum_{i=1}^{N_j}}{d_{i,j}(t)}/C_d\rceil })$$ Our ultimate objective is to **power on only $l(t)$ OLT line cards to serve all $N$ ONUs** at a given time $t$ instead of powering on all $L$ line cards. However, powering off OLT line cards may result in service disruptions of ONUs communicating with these OLT line cards. To avoid service disruption, power-on OLT line cards should be able to provision bandwidth to all ONUs connected to the OLT chassis. To address this issue, we propose several modifications over the legacy OLT chassis to realize the dynamic configuration of OLT as will be presented next. OLT with optical switch ======================= To dynamically configure the communications between OLT line cards and ONUs, one scheme we propose is to place an optical switch in front of all OLT line cards as shown in Fig.\[fig:engolt\] (a). The function of the optical switch is to dynamically configure the connections between OLT line cards and ONUs. When the network is heavily loaded, the switches can be configured such that each PON system communicates with one OLT line card. As discussed in Section \[sec:I\], when the network is lightly loaded, the switches can be configured such that multiple PON systems communicate with one line card. Then, some OLT line cards can be powered off, thus reducing energy consumption. Assume the energy consumption of the optical switch is negligible. As compared to the scheme of always powering on all $L$ line cards, the scheme of powering on only $l(t)$ line cards at time $t$ can achieve relative energy saving as large as$$1-\frac{l(t)}{L}$$ Then, the average energy saving over time span $T$ equals to $$\frac{1}{T}{\int_{t=0}^{T}\frac{1-l(t)}{L}dt}$$ We define the traffic load as the maximum of upstream and downstream traffic loads to determine the required number of line cards: $$load = max (\mathop{\sum_{j=1}^L\sum_{i=1}^{N_j}}{u_{i,j}(t)}/{C_u} L \; , \; \mathop{\sum_{j=1}^L\sum_{i=1}^{N_j}}{d_{i,j}(t)}/{C_d L} )$$The process of changing the switch configuration is time consuming, frequent change of each line card’s status may degrade the ONU performance. Since the traffic of the ONUs is also bursty and changes dynamically, it is more efficient to monitor the traffic for an observation period ($T_O$) before changing the switch configuration. If the traffic load of the network remains below a threshold for $T_O$, the number of line cards will be adjusted accordingly. By dynamically configuring switches, the number of power-on OLT line cards is reduced from $L$ to $x$ when the traffic load falls between $(x-1)/L$ and $x/L$ for a period of $T_O$. Thus, a significant amount of power can be potentially saved. Fig. \[fig:engolt\] (b)-(e) illustrates the configuration of switches for the case that one OLT chassis contains four OLT line cards. The number of power-on OLT line cards is reduced from four to three, two, and one when the traffic load falls within the range $[50\%,75\%]$, $[25\%,50\%]$, and $[0,25\%]$, respectively. By equipping the OLT chassis with proper optical switches, the communications between OLT line cards and ONUs can be dynamically configured. The new OLT structure appears to be promising and cost-effective as compared to the WDM based solutions. Optical switch specifications ============================= Configuration time of the optical switches and power consumption of each switch are two main specifications that should be considered in the analysis. Switch configuration time ------------------------- Switches take time to change configurations. The switch reconfiguration time may affect the ONU performances when powering on/off OLT line cards. We investigate the impacts of the switch reconfiguration time on EPON and GPON, respectively. - EPON: We argue that services of EPON ONUs are not affected when the switch configuration time is as large as $50$ ms. In EPON, the upstream bandwidth allocation is controlled by OLT. ONUs transmit the upstream traffic using the allocated time durations stated in the GATE message. IEEE 802.3ah [@8023av] specifies that ONUs need to send GATE messages every $50$ ms to maintain registration even if they do not have traffic to transmit. Thus, the disrupted servicing time can be transparent to EPON ONUs when the switch reconfiguration time is as large as $50$ ms, which can be satisfied by most optical switches. - GPON: We argue that services of GPON ONUs are not affected when the switching time is no greater than $125 \mu s$. For GPON, ITU-T G. 984.3 [@984] specifies a fixed frame length of 125 $\mu s$. An ONU receives downstream control messages and sends its upstream data or control traffic every GPON frame, i.e., 125 $\mu s$. Therefore, services of GPON ONUs are not affected when the switch configuration time is no greater than $125 \mu s$. Power consumption of optical switches ------------------------------------- So far, we have not considered the impact of the power consumption of optical switches on the saved energy of OLT chassis. Optical switches consume some power, and the nonzero power consumption of optical switches may reduce the saved energy of the OLT chassis. Denote $p(s)$ and $p(l)$ as the power consumption of optical switch and one OLT line card, respectively. By incorporating considering the energy consumption of the optical switch, the power consumption of the proposed OLT chassis at any given time $t$ equals to $$p(l)\cdot l(t) + p(s)$$ Hence, the energy saving is reduced to $$1-\frac{p(l)/T\int_{t=0}^T{l(t)}dt+p(s)}{p(l) \cdot L}$$ The proposed OLT chassis can save some energy as compared to the legacy OLT chassis when the power consumption $p(s)$ of the optical switch satisfy the following condition $$p(s)<(L-\frac{1}{T}\int_{t=0}^T{l(t)}dt)\cdot p(l)$$ The power consumption of optical switch differs from the switch size as well as the manufacturing technology. It is reported that one $2\times 2$ optomechanic switch consumes around $0.2$ w [@opto], while an OLT line card consumes around $5$ w [@Vereecken2011]. Then, for the $L=2$ case, energy can be saved as long as the average number of power-on OLT line cards $\frac{1}{T}\int_{t=0}^T{l(t)}dt$ is less than $1.96$. This condition can be easily satisfied. Therefore, even considering the power consumption of optical switches, the proposed OLT structure can still achieve a significant amount of power saving as compared to the legacy OLT. System model ============  We employ semi-Markov chains to analyze the system. Different observation periods have been considered for decreasing and increasing the number of line cards. $T_D$ is the observation time period for decreasing the number of active line cards. $T_I$ is defined as an observation period for increasing the number of line cards. In order to ease the analysis, from now on, the formulations are based on the downstream traffic only. Poisson processes with the average rate of $\lambda_a$ packets per second is assumed for downlink packet arrivals. A fixed traffic scheduling cycle is assumed in the performance analysis. Consider $P_a(\alpha,T)$ as the probability that $\alpha$ downstream packets arrive at the OLT chassis during $T$ traffic scheduling cycle: $$P_a{(\alpha,T)}=e^{-\lambda_a T} \cdot \frac{(\lambda_a T)^ \alpha}{\alpha!}$$ The same process can be considered for uplink packet arrival in the case of having downstream and upstream traffic simultaneously. OLT chassis state ----------------- In the defined Markov chain model, the total number of active line cards vary from $1$ to $L$. Therefore, there are $L$ possible states of active line cards. $A_i$ represents $i$ active line cards. Transition from each active state to another active state should be done through the listening states. There is no direct transition between active states. There are two different types of listening states; $D$ (listening states for decreasing the number of line cards) and $I$ (listening states for increasing the number of line cards). Figure \[fig:mrkv\] shows the state transitions. $D_{i,i-1}(j)$ refers to the state that the total traffic load of the OLT chassis remains below $i/L$ for $j$ cycles. The number of active line cards during $M=T_D/T$ time cycles remains as $i$ line cards. If the traffic load goes higher than $i/L$ amount during a traffic scheduling cycle, the transition to $A_i$ occurs. Otherwise, the transition from $D_{i,i-1}(j)$ to $D_{i,i-1}(j+1)$ happens. Whenever the number of listening cycles reaches to $M$, the OLT chassis switches to the next lower active state ($A_{i-1}$). $I_{i, i+1}(k)$ refers to the state that the total traffic load of the OLT chassis remains above $i/L$ for $k$ time cycles. The number of active line cards during $N=T_I/T$ remains as $i$ line cards. The excess amount of the traffic will be buffered during $T_I$. If the traffic load becomes lower than $i/L$ amount during a traffic scheduling cycle, the transition to $A_i$ occurs. Otherwise, the transition from $I_{i,i+1}(k)$ to $I_{i, i+1}(k+1)$ happens. If the total traffic load stays beyond the defined threshold ($i/L$) for $N$ traffic scheduling cycles, the OLT chassis switches to the next upper active state ($A_{i+1}$). State transitions ----------------- - State transitions from $A_i$ to $D_{(i,i-1)}(1)$, $D_{(i,i-1)}(j)$ to $D_{(i,i-1)}(j+1)$, or from $D_{(i,i-1)}(M)$ to $A_{i-1}$ $for \medspace i=2,..,L$ $and$ $j=1,...,M-1$ happen when the traffic load in a traffic scheduling time $T$ is less than $(i-1)/L$. Therefore, the probability that the number of arrival packets is smaller than $max=\lfloor (i-1)C_d/Packet \medspace Size \rfloor$ equals to: $\sum_{\alpha=0}^{max}P_a(\alpha,T)$. - State transitions from $I_{i,i+1}(k)$ to $A_i$ $for$ $i=2,..,L$ $and$ $k=1,...,N$ happen when the number of arrival packets in a traffic scheduling time $T$ is smaller than $max=\lfloor iC_d/Packet \medspace Size \rfloor$, which is equals to: $\sum_{\alpha=0}^{max}P_a(\alpha,T)$ - State transitions from $A_i$ to $I_{(i,i+1)}(1)$, from $I_{(i,i+1)}(k)$ to $I_{(i,i+1)}(k+1)$, or from $I_{(i,i+1)}(N)$ to $A_{i+1}$ $for \medspace i=2,..,L$ $and$ $k=1,...,N-1$ happen when the number of arrival packets in a traffic scheduling time $T$ is greater than $min=\lceil iC_d/Packet \medspace Size \rceil$. The probability equals to $\sum_{\alpha=min}^{\infty}P_a(\alpha,T)$. - State transitions from $D_{i,i-1}(j)$ to $A_i$ $for$ $i=2,..,L$ $and$ $j=1,...,M$ happen when the number of arrival packets in a traffic scheduling time $T$ is greater than $min=\lceil (i-1)C_d/Packet \medspace Size \rceil$, which is equals to: $\sum_{\alpha=min}^{\infty}P_a(\alpha,T)$ - State transition from $A_i$ to itself happens when the number of arrival packets in a traffic scheduling time $T$ remains between $min=\lfloor (i-1)C_d/Packet \medspace Size \rfloor$ and $max=\lceil iC_d/Packet \medspace Size \rceil$. The probability is equal to $\sum_{\alpha=min}^{max}P_a(\alpha,T)$. Steady state probabilities -------------------------- Denote $P(A_i)$, $P(D_{i,i-1}(j))$ $(i=2,...,L$ $j=1,...,M)$, and $P(I_{i,i+1}(k))$ $(i=1,...,L-1$; $k=1,...,N)$ as the probability of the OLT state when the network is at its steady state. Therefore, the following constraints are satisfied. Steady state probabilities of all the active states except $A(1)$ and $A(L)$ are as follows: $$\begin{aligned} \label{eq:first} &P(A_i) [pr\{A_i \negmedspace \rightarrow \negmedspace D_{i,i-1}(1)\}+pr\{A_i \negmedspace \rightarrow \negmedspace I_{i,i+1}(1)\}] \nonumber\\ &=[\sum_{k=1}^{\infty}P(I_{i,i+1}(k))pr\{I_{i,i+1}(k) \rightarrow A_i\}]\nonumber\\ &+P(I_{i-1,i}(N))pr\{I_{i-1,i}(N) \rightarrow A_i\}\nonumber\\ &+\sum_{j=1}^{M}P(D_{i,i-1}(j))pr\{D_{i,i-1}(j) \rightarrow A_i\} \nonumber\\ &+P(D-{i+1,i}(M))pr\{D_{i+1,i}(M) \rightarrow A_i\} \nonumber\\ &(i=2,...,L-1)\end{aligned}$$ Steady state probability of $A_1$ is calculated as follows: $$\begin{aligned} &P(A_1)pr\{A_1 \rightarrow I_{1,2}(1)\} \nonumber\\ &=\sum_{k=1}^{N}P(I_{1,2}(K)pr\{I_{1,2}(k) \negmedspace \rightarrow \negmedspace A_1\} \nonumber\\ &+P(D_{2,1}(M))pr\{D_{2,1}(M) \negmedspace \rightarrow \negmedspace A_1\}\end{aligned}$$ Steady state probability of $A_L$ is acheived as follows: $$\begin{aligned} &P(A_L)pr{A_L \rightarrow D_{L,L-1}(1)} \nonumber\\ &=\sum_{j=1}^{M}P(D_{L,L-1}(j))pr\{D_{L,L-1}(j) \rightarrow A_L\} \nonumber\\ &+P(I_{L-1,L}(N))pr\{P(I_{L-1,L}(N)) \rightarrow A_L\}\end{aligned}$$ Steady state probability of listening states $(D)$ except the first and last states are as follows: $$\begin{aligned} &P(D_{i,i-1}(j))[pr\{D_{i,i-1}(j) \rightarrow D_{i,i-1}(j+1)\} \nonumber\\ &+ pr\{D_{i,i-1}(j) \rightarrow A_i\}]\nonumber\\ &= \negmedspace P(D_{i,i-1}(j-1))pr\{ \negmedspace D_{i,i-1}(j-1) \negmedspace \rightarrow \negmedspace D_{i,i-1}(j) \negmedspace \} \nonumber\\ &(i=1,...,L) \& (j=2,...,M-1)\end{aligned}$$ Steady state probability of $D_{i,i-1}(1)$ equals to: $$\begin{aligned} &P(D_{i,i-1}(1))[pr\{D_{i,i-1}(1) \rightarrow D_{i,i-1}(2)\} \nonumber\\ &+pr\{D_{i,i-1}(1) \rightarrow A_i\}] \nonumber\\ &=P(A_i)pr{A_i \rightarrow D_{i,i-1}(1)} (i=1,...,L)\end{aligned}$$ Steady state probability of $D_{i,i-1}(M)$ is as follows: $$\begin{aligned} &D_{i,i-1}(M)[pr\{D_{i,i-1}(M) \rightarrow A_i\} \nonumber\\ &+pr\{D_{i,i-1}(M) \rightarrow A_i-1\}] \nonumber\\ &= \negmedspace P(D_{i,i-1}(M \negmedspace - \negmedspace 1))pr\{ \negmedspace D_{i,i-1}(M \negmedspace - \negmedspace 1) \negmedspace \negmedspace \rightarrow \negmedspace D_{i,i-1}(M)\negmedspace \}\end{aligned}$$ The following is the steady state probability of the listening states $I$ except the first and last states: $$\begin{aligned} &p(I_{i,i+1}(k))[pr\{I_{i,i+1}(k) \rightarrow A_i\}+ \nonumber\\ &pr\{I_{i,i+1}(k) \rightarrow I_{i,i+1}(k+1)\}] \nonumber\\ &=P(I_{i,i+1}(k-1))pr\{I_{i,i+1}(k-1) \rightarrow I_{i,i+1}(k)\} \nonumber\\ &(i=1,...,L) \& (k=2,...N-1)\end{aligned}$$ Steady state probability of $I_{i,i+1}(1)$ is as follows: $$\begin{aligned} &P(I_{i,i+1}(1))[pr\{I_{i,i+1}(1) \rightarrow A_i\} \nonumber\\ &+pr\{I_{i,i+1}(1) \rightarrow I_{i,i+1}(2)\}] \nonumber\\ &=P(A_i)pr\{A_i \rightarrow I{i,i+1}(1)\} (i=1,...,L)\end{aligned}$$ Steady state probability of $I_{i,i+1}(N)$ is obtained as follows: $$\begin{aligned} &P(I_{i,i+1}(L))[pr\{I_{i,i+1}(L) \rightarrow A_i\} \nonumber\\ &+pr\{I_{i,i+1}(L) \rightarrow A_{i+1}\}] \nonumber\\ &=P(I_{i,i+1}(L-1))pr\{I_{i,i+1}(L-1) \rightarrow I{i,i+1}(L)\}\end{aligned}$$ Moreover, the sum of the probabilities of all the states is equal to $1$, i.e.: $$\begin{aligned} \label{eq:last} \sum_{i=1}^{L}P(A_i) + \sum_{i=1}^{L} \sum_{j=1}^{M} P(D_{i,i-1}(j)) + \sum_{i=1}^{L} \sum_{k=1}^{N} P(I_{i,i+1}(k))=1\end{aligned}$$ Therefore, the steady state probabilities of all the state can be calculated by solving the above equations \[eq:first\]-\[eq:last\]. Performance analysis -------------------- Denote $p(l)$ as the power consumption of one OLT line card and $P(A_i)$, $P(D_{i,i-1}(j))$ $(i=2,...,L$ $j=1,...,M)$, and $P(I_{i,i+1}(k))$ $(i=1,...,L-1$; $k=1,...,N)$ as the probability of the OLT state when the network is at its steady state. Therefore, the average power consumption equals to: $$p(l)[\sum_{i=1}^{L}ip(A_i)+\sum_{i=2}^{L}\sum_{j=1}^{M}ip(D_{i,i-1}(j))+\sum_{i=1}^{L-1}\sum_{k=1}^{N}ip(I_{i,i+1}(k))]$$  Cost reduction ============== OLT with cascaded $2 \times 2$ switches --------------------------------------- Another problem with optical switches is their high prices. The prices of optical switches vary from their manufacturing techniques [@MaxOpt03ofc]. At present, there are generally four kinds of optical switches: opto-mechanical switches, micro-electro-mechanical system (MEMS), electro-optic switches, and semiconductor optical amplifier switches. Currently, the opto-mechanic switches are less expensive than the other three kinds. Simply because of their low prices, opto-mechanic switches are generally the adopted choices in designing energy-efficient OLT. For opto-mechanic switches, an important constraint is their limited port counts. Popular sizes of opto-mechanic switches are $1\times 2$ and $2\times 2$. Considering the port count constraints, we further propose the cascaded $2 \times 2$ switches structure to achieve the dynamic configuration of OLT. More specifically, to replace an $N \times N$ switch, the cascaded $2 \times 2$ switch contains $\log_2^N$ stages and $(N-1)$ $2 \times 2$ switches. Fig. \[fig:casc\] (a) illustrates the proposed cascaded switches. In the switch, the $k$th stage contains $2^{(k-1)}$ switches. Fig. \[fig:casc\] (b) shows a two-stage cascaded $2 \times 2$ switches to replace a $4 \times 4$ switch. As illustrated in Fig. \[fig:casc\](c)-(e), when the traffic load is greater than $50\%$, one PON system is connected with one OLT line card; when the traffic load is between $25\%$ and $50\%$, two PON systems are connected with one OLT line card; when the traffic load is less than $25\%$, all PON sytems are connected with a single OLT line card. Here, we analyze the saved energy of the proposed OLT equipped with cascaded switches. Assume the traffic is uniform among all ONUs. Then, when the traffic load is between $50\%$ and $100\%$, all OLT line cards need be power-on; when the traffic load is between $25\%$ and $50\%$, half of the OLT line cards are powered on. Generally, when the traffic load is between $1/2^k$ and $1/2^{k+1}$, $1/2^k$ of the OLT line cards are power-on. Therefore, $$1/2^{k+1} \leq load \leq 1/2^k$$ $$k={\lfloor \log_2{(1/load)} \rfloor}$$ $$l(t)=1/2^{\lfloor \log_2{(1/load)}\rfloor}$$ The saved energy equals to $$1-1/2^{\lfloor \log_2{(1/load)}\rfloor}$$ As compared to the OLT with an $N\times N$ switch, the OLT with cascaded $2\times 2$ switches saves a less amount of energy. Note that the typical switching speed of the opto-mechanical switch is around $5$ ms. As discussed before, it does not affect the performances of users in EPON, but may have impacts on the performances of users in GPON. OLT with electrical switches ============================ Another scheme of avoiding the significant cost increase is to use electrical switches instead. However, before aggregating traffic using electrical switches, the optical transceivers are required to convert the optical signals into electrical signals. Thus, only the energy consumption of the electrical part in an OLT line card can be saved. The energy saving is limited as compared to the scheme of using optical switches. Let $p(e)$ be the energy consumption of the electrical part of an OLT line card. Then, the average energy saving of the OLT equipped with an electrical switch equals to: $$1-\frac{l(t)\cdot p(e)}{L\cdot p(l)}$$ The efficiency of energy saving of this scheme depends on the ratio $p(e)/p(l)$. Performance evaluation ====================== In this section, we study the performance of the sleep control scheme of the OLT line cards for Poisson and non-Poison traffic. In the Markov chain model, we consider Poisson traffic arrival for the ease of analysis. The total number of line cards is assumed to be $4$ with the capacity of $10Gb/s$ for each line card. The time duration of each traffic scheduling cycle is considered to be $2ms$. Figure \[fig:numerical\] depicts the numerical results of the Markov model. In Figure \[fig:loadnum\], the energy saving performance under different traffic arrival rates is illustrated. We assume that the maximum number of “D” listen cycles ($M$) as well as the maximum number of “I” listen cycles ($N$) equal to $2$. The arrival traffic is increased up to $40Gb/s$. With the increase of traffic arrival, the energy saving decreases as the OLT chassis need to have more active line cards to support the arrival traffic. When the arrival traffic is less than $10Gb/s$, one line card is sufficient to satisfy the traffic and the other $3$ line cards are shut down. Therefore, the maximum energy saving is achieved. Before decreasing the number of OLT line cards, the OLT chassis needs to monitor the traffic for the maximum of $M$ listen cycles. Figure \[fig:Dlistennum\] shows the effect of changing the number of M on energy saving. With the increase of M, the OLT chassis needs to keep a larger number of line cards active for the time duration of $M \cdot T$. Thus, the energy saving decreases by increaing $M$. $N$ is the maximum number of traffic cycles during which the OLT chassis needs to monitor the traffic before increasing the number of line cards. With the increase of $N$, the energy saving increases as the number of active line cards is smaller for the total duration of $N \cdot T$. We also study the performance of the proposed scheme for non-Poisson traffic. Since the actual network traffic with bursts is self-similar, we conduct our simulation for self-similar traffic with the Hurst parameter of $0.8$. The packet length is uniformly distributed between $64$ bytes and $1518$ bytes. As it can be seen in Figure \[fig:loadsim\], the trend of energy saving performance by increasing the traffic load for self-similar traffic is similar to that of the Markov model for Poisson traffic. Figure \[fig:Dlistensim\] shows the impact of different $M$ listen cycles on the system performance. Similar to the theoretical analysis of Poisson traffic, energy saving decreases as the number of listen cycles increases. However, increasing the $N$ cycles helps saving more energy. As mentioned earlier, during “I” listen cycles, the OLT chassis receives traffic load more than the available line cards. Before increasing the number of line cards, the OLT chassis needs to monitor the traffic for $N$ cycles. During theses cycles, the number of line cards stay the same, and the OLT buffers the extra traffic. The buffered packets encounter some delay as depicted in Figure \[fig:Ilistensim\]. Therefore, proper setting of $N$ needs to be considered to save energy without impairing the QoS of the users. Conclusion ========== We have proposed a novel energy-efficient OLT structure which adapts its power-on line cards to the real-time arrival traffic. Specifically, we have added a switch into the legacy OLT chassis to dynamically configure the connection between OLT line cards and ONUs. We first describe an OLT equipped with an $N\times N$ switch, and investigate the impacts of the power consumption of the optical switch and switch configuration time on the saved energy of the whole OLT chassis. Then, to avoid a dramatic cost increase, we advocate the use of opto-mechanical switches among all currently commercially-available switches, and further propose to use a cascaded $2\times 2$ switch structure. Our analysis demonstrates that the proposed OLT achieves significant power savings as compared to the legacy OLT. [15]{} J. Baliga, R. Ayre, K. Hinton, W. Sorin, and R. Tucker, “[Energy consumption in optical IP networks]{},” *IEEE/OSA Journal of Lightwave Technology*, vol. 27, no. 13, pp. 2391–2403, 2009. M. Pickavet, *et al.*, “[Worldwide energy needs for ICT: the rise of power-aware networking]{},” in *2nd International Symposium on Advanced Networks and Telecommunication Systems*.1em plus 0.5em minus 0.4emIEEE, 2008, pp. 1–3. G. Fettweis and E. Zimmermann, “[ICT energy consumption-trends and challenges]{},” in *The 11th International Symposium on Wireless Personal Multimedia Communications*.vol. 2, no. 4, 2008. J. Baliga, R. Ayre, W. Sorin, K. Hinton, and R. Tucker, “[Energy consumption in access networks]{},” in *Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference*, 2008. C. Lange, M. Braune, and N. Gieschen, “[On the energy consumption of FTTB and FTTH access networks]{},” in *National Fiber Optic Engineers Conference*, 2008. J. Zhang, N. Ansari, Y. Luo, F. Effenberger, and F. Ye, “Next-generation [PON]{}s: a performance investigation of candidate architectures for next-generation access stage 1,” *IEEE Communications Magazine*, vol. 47, no. 8, pp. 49–57, August 2009. N. Ansari and J. Zhang, *Media Access Control and Resource Allocation for Next Generation Passive Optical Networks*, Springer, ISBN: ISBN: 978-1461439387, 2013. J. Zhang and N. Ansari, “Towards energy-efficient [1G-EPON]{} and [10G-EPON]{} with sleep-aware [MAC]{} control and scheduling,” *IEEE Communications Magazine*, vol. 49, no. 2, pp. S33-S38, February 2011. J. Zhang, M. Taheri, and N. Ansari, “Standards-compliant EPON sleep control for energy efficiency: Design and analysis,” *Journal of Optical Communications and Networking*, vol. 5, no. 7, pp. 677–685, 2013. R. Kubo, J. Kani, H. Ujikawa, T. Sakamoto, Y. Fujimoto, N. Yoshimoto, and H. Hadama, “[Study and demonstration of sleep and adaptive lLink rate control mechanisms for energy efficient 10G-EPON]{},” *IEEE/OSA Journal of Optical Communications and Networking*, vol. 2, no. 9, pp. 716–729, 2010. S. Wong, L. Valcarenghi, S. Yen, D. Campelo, S. Yamashita, and L. Kazovsky, “[Sleep Mode for Energy Saving PONs: Advantages and Drawbacks]{},” in *2009 IEEE GLOBECOM Workshops*, 2009, pp. 1–6. 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--- author: - 'A.S. Umar[^1], V.E. Oberacker,' - 'J.A. Maruhn' date: 'Received: / Revised version: date' title: | Neutron Transfer Dynamics and Doorway to Fusion\ in Time-Dependent Hartree-Fock Theory --- =10000 =10000 =10000 Introduction ============ Heavy-ion fusion reactions are a sensitive probe of the size, shape, and structure of atomic nuclei as well as the collision dynamics. With the increasing availability of radioactive ion-beams the study of fusion reactions of neutron-rich nuclei are now possible [@Li03; @Li05; @Ji04]. Other experimental frontiers are the synthesis of superheavy nuclei in cold and hot fusion reactions [@Ho02; @Og04; @Gi03; @Mo04; @II05], and weakly bound light systems [@YZ06; @PF04; @KG98; @TS00]. Microscopic descriptions of nuclear fusion may provide us with a better understanding of the interplay between the strong, Coulomb, and the weak interactions as well as the enhanced static and dynamic correlations present in these many-body systems. Recently, two aspects of the collision dynamics leading to fusion that involve pre-compound neutrons have been of interest. Over the last decade a number of fusion studies have reported that the average number of neutrons evaporated by the compound nucleus is considerably less than what is predicted by statistical fusion evaporation calculations [@WB06]. This phenomenon is quite possibly linked to the excitation of the pre-compound collective dipole mode, which is likely to occur when ions have significantly different $N/Z$ ratio, and is a reflection of dynamical charge equilibration. This was studied in the context of TDHF in Refs. [@SC01; @SC07; @US85] and we have recently observed this phenomenon in the $^{64}$Ni+$^{132}$Sn system [@UO07a]. Similarly, considerable attention has been given to the influence of neutron transfer on fusion cross-sections. Studies suggest that the transfer of neutrons with positive $Q$ value strongly enhances the fusion cross-section in comparison to systems having negative $Q$ value [@Li07; @ZSW07; @DLW83]. This may explain the fact that lowering of the potential barrier for neutron-rich systems does not always lead to higher fusion cross-sections. In Ref. [@ZSW07] near-barrier fusion of neutron-rich nuclei was studied within a channel coupling model for intermediate neutron rearrangement using a semi-empirical time-dependent three-body Schrödinger equation. Studies showed that for the $^{40}$Ca+$^{96}$Zr system neutrons were transferred in the early stages of the collision from the $2d_{5/2}$ state of $^{96}$Zr to the unoccupied levels of the $^{40}$Ca nucleus. It is generally acknowledged that the TDHF method provides a useful foundation for a fully microscopic many-body theory of low-energy heavy-ion reactions [@Ne82]. Historically, fusion in TDHF has been viewed as a final product of two colliding heavy-ions, and the dynamical details influencing the formation of the compound system have not been carefully dissected in terms of the pre-compound properties. Due to the availability of much richer fusion data and considerable advances in TDHF codes that make no symmetry assumptions and use better effective interactions, it may now be possible to examine these effects more carefully. In TDHF complete fusion proceeds by converting the entire relative kinetic energy in the entrance channel into internal excitations of a single well-defined compound nucleus. The dissipation of the relative kinetic energy into internal excitations is due to the collisions of the nucleons with the “walls” of the self-consistent mean-field potential. TDHF studies demonstrate that the randomization of the single-particle motion occurs through repeated exchange of nucleons from one nucleus into the other. Consequently, the equilibration of excitations is very slow, and it is sensitive to the details of the evolution of the shape of the composite system. This is in contrast to most classical pictures of nuclear fusion, which generally assume near instantaneous, isotropic equilibration. The relaxation of the final compound system is a long-time process occurring on a time scale on the order of a few thousand [*fm/c*]{}. In contrast, the pre-compound stage corresponds to a time scale of a few hundred [*fm/c*]{}. In this manuscript we focus on the analysis of transfer during the early stages of the collision. In particular, we confirm the findings of Ref. [@ZSW07]. We also show that in TDHF different single-particle states seem to see different potential barriers in comparison to the generic ion-ion barrier. This influences the overall dynamics leading to fusion and consequently the effective potential barrier. Transfer in TDHF ================ The TDHF calculations have been carried out using our new three-dimensional unrestricted TDHF code [@UO06]. For the effective interaction we have used the Skyrme SLy4 force [@CB98], including all of the time-odd terms. Static Hartree-Fock calculations for all the nuclei studied here produce spherically symmetric systems. The chosen mesh spacing was $1$ fm in all three directions, which yield a binding energy accuracy of about $50$ keV in comparison to a spherical Hartree-Fock code. For these calculations we have in addition required that the fluctuations in energy be as low as $10^{-4}$-$10^{-5}$, the corresponding accuracy in binding energy is about $10^{-12}$. This ensures that the tails of the wavefunctions are well converged in the numerical box. The box size used was $60$ fm in the direction of the collision axis and $30$ fm in the other two directions. The initial nuclear separations were $25$ fm. ![\[fig:vrOO\] Potential barrier, $V(R)$, for the $^{16}$O+$^{24}$O system obtained from density constrained TDHF calculations (black curve). Also shown is the point Coulomb potential (red curve).](fig1.eps) $^{16}$O+$^{24}$O system ------------------------ As an example of a collision involving one neutron rich nucleus we studied the $^{16}$O+$^{24}$O system. In order to determine the potential barrier for the system we have used the DC-TDHF method as described in Ref. [@UO06b]. In this approach the TDHF time-evolution takes place with no restrictions. At certain times during the evolution the instantaneous density is used to perform a static Hartree-Fock minimization while holding the neutron and proton densities constrained to be the corresponding instantaneous TDHF densities. Some of the effects naturally included in the DC-TDHF calculations are: neck formation, particle exchange, internal excitations, and deformation effects to all order. The heavy-ion potential was obtained by initializing the system at $E_{\mathrm{c.m.}}=9.5$ MeV, which is slightly above the barrier shown in Fig. \[fig:vrOO\]. The peak of the barrier is about $8.4$ MeV at a nuclear separation of $9.9$ fm. This is lower than the barrier of the $^{16}$O+$^{16}$O system, which has a height of about $10$ MeV. Here and in the following, the heavy-ion interaction potential has been calculated with a constant mass parameter corresponding to the reduced mass of the ions. This is a good approximation as long as one is only interested in the value of the potential barrier height (as is the case here). For the calculation of sub-barrier fusion cross sections, however, it is essential that coordinate-dependent mass parameters be utilized [@UO07a] because the cross sections depend sensitively on the shape of the potential in the interior region. ![\[fig:rhoz\] Partially integrated neutron densities calculated from Eq.(\[eq:rhoz\]) for the $^{24}$O nucleus plotted on a logarithmic scale versus the collision axis coordinate $z$ for the $^{16}$O+$^{24}$O system at three energies, $E_{\mathrm{c.m.}}=7,8$, and $9$ MeV. The black-solid curves correspond to the initial partial density, the red-dashed curves are the same quantity at the distance of closest approach, and the blue-solid curves are partial densities long after the recoil. Filled spheres near the bottom axis approximately show the initial and final location of the two nuclei.](fig2.eps) In order to examine the center-of-mass energy dependence of mass exchange below and above the barrier we have initiated TDHF collisions at energies $E_{\mathrm{c.m.}}=6$, 7, 8, 9, and 9.5 MeV. Interestingly, the head-on (zero impact parameter) TDHF collisions for the lowest four energies behave as a typical sub-barrier collision: the two ions approach a minimum distance with no visible overlap, then recoil and move away from each other. This is also true for $E_{\mathrm{c.m.}}=9$ MeV despite the fact that this energy lies above the ion-ion barrier. This suggests that while we can talk about an [*effective*]{} ion-ion barrier the individual single-particle states may see a barrier somewhat different than the effective one. This is in agreement with the findings of Ref. [@ZSW07], and we shall come back to this point again later in the manuscript. Even though we are dealing with sub-barrier energies, we observe mass exchange (mainly neutron) from $^{24}$O to $^{16}$O for all these energies. Similarly, we observe a small dissipation of the relative kinetic energy, ranging from $0.05$ MeV for $E_{\mathrm{c.m.}}=6$ MeV to $0.40$ MeV for $E_{\mathrm{c.m.}}=9$ MeV. For comparison, the difference in total energy of the system before and after the collision is on the order of $0.02$ MeV, i.e., the numerical error in total energy conservation is less than the dissipated energy, even at very low energies. At $E_{\mathrm{c.m.}}=9.5$ MeV the system fuses. ![\[fig:pa\] Mass transfer probability defined in Eq. (\[eq:pa\]) for the transfer in the $^{16}$O+$^{24}$O system.](fig3.eps) To gain further insight into the mass exchange we have looked at the change in nuclear density along the collision axis ($z$-axis). We define $$\rho(z,t)=\int dx\int dy \; \rho(x,y,z,t)\;, \label{eq:rhoz}$$ where $\rho(x,y,z,t)$ is the instantaneous TDHF density. In Fig. \[fig:rhoz\] we plot $\rho(z,t)$ for three energies, $E_{\mathrm{c.m.}}=7$, 8, and $9$ MeV, on a logarithmic scale versus the collision axis coordinate $z$. We emphasize that the plotted density contains only states that correspond to the $^{24}$O nucleus. The black-solid curves correspond to the initial partial density, the red-dashed curves are the same quantity at the distant of closest approach, and the blue-solid curves are partial densities long after the recoil. Filled spheres near the bottom axis denote the approximate initial and final location of the two nuclei. The distance of closest approach for energies $E_{\mathrm{c.m.}}=$7, 8, and 9 MeV are 13.2 fm, 11.6 fm, and 10.1 fm, respectively. Two features are worth noting: first, we see a buildup of the density on the $^{16}$O side of the reaction plane, and secondly, we note that the buildup gets substantially larger with increasing energy. We also observe that the recoiled density profile remains very close to the initial profile except in the tail region. The corresponding mass transfer $\Delta A$ at sub-barrier energies can be found by integrating $\rho(z,t)$ over the $z$ values on the left side of the minimum. At energies $E_{\mathrm{c.m.}}=7$, 8, and 9 MeV we find mass transfer values of 0.024, 0.1464, and 0.6768, respectively. The mass transfer at $E_{\mathrm{c.m.}}=6$ MeV is 0.0022. We emphasize that unitarity (or total mass number) was conserved with an accuracy of about one part in $10^5$. In Fig. \[fig:pa\] we plot the energy dependence of the transfer probability defined as $$P(E_{\mathrm{c.m.}})=\frac{\Delta A(E_{\mathrm{c.m.}})}{A_{in}}\;, \label{eq:pa}$$ where $\Delta A$ is the mass exchange value mentioned above and $A_{in}$ is the mass of the incoming nucleus, in this case $24$. We observe that for sub-barrier energies the curve is essentially linear and jumps to one when fusion occurs. ![\[fig:palpha\] Neutron single-particle probability densities given by Eq. (\[eq:palpha\]) for the $^{24}$O nucleus at $E_{\mathrm{c.m.}}=7$ MeV at $t=T_{final}$. Filled spheres near the bottom axis approximately show the final location of the two nuclei.](fig4.eps) In order to identify which of the single-particle states are predominantly responsible for transfer at sub-barrier energies we define the quantity $$P_{\alpha}(z,t)=\int dx \int dy \; |\psi_{\alpha}(x,y,z,t)|^2\;, \label{eq:palpha}$$ where the $\psi_{\alpha}$’s are single-particle states. In Fig. \[fig:palpha\] we plot this quantity on a logarithmic scale for the neutron states of the $^{24}$O nucleus at $E_{\mathrm{c.m.}}=7$ MeV, long after the recoil, at which time the ion-ion separation is about $R=25$ fm. Although our single-particle wave functions are calculated on a 3-D Cartesian grid and thus do not carry the same good quantum numbers as the spherical representation, for a well converged spherical nucleus it is possible to calculate the expectation values of orbital angular momentum, spin, and parity to identify these states. As expected, the contribution to the sub-barrier mass transfer is primarily coming from the $2s_{1/2}$ neutron state of the $^{24}$O nucleus. The contribution from the $d_{5/2}$ and $d_{3/2}$ states is an order of magnitude lower than that of the $2s_{1/2}$ state. This again suggests that the barrier seen by the $2s_{1/2}$ state is different than the effective barrier for the entire nucleus. ![\[fig:vrCaZr\] Potential barrier, $V(R)$, for the $^{40}$Ca+$^{96}$Zr system obtained from density constrained TDHF calculations (black curve). Also shown is the point Coulomb potential (red curve).](fig5.eps) $^{40}$C+$^{96}$Z system ------------------------ In Ref. [@ZSW07] fusion for neutron-rich systems was studied in the vicinity of the Coulomb barrier using a semi-empirical time-dependent three-body Schrödinger equation. Their studies showed that for the $^{40}$Ca+$^{96}$Zr system at $E_{\mathrm{c.m.}}=97$ MeV, which is in the vicinity of the Coulomb barrier, neutrons were transferred in the early stages of the collision ($R=$11-14 fm) from the $2d_{5/2}$ state of $^{96}$Zr to the unoccupied levels of the $^{40}$Ca nucleus. Here, we shall study the same system using the TDHF theory. We have used the DC-TDHF method to find the effective potential barrier for this system as shown in Fig. \[fig:vrCaZr\]. The barrier was calculated by initializing the TDHF run at $E_{\mathrm{c.m.}}=97$ MeV. The barrier peak is about $95$ MeV and is located at about $R=11.5$ fm. For this energy the TDHF collision results in fusion. In order to establish the early mass exchange we have plotted the nuclear density and the corresponding neutron density of the $^{96}$Zr nucleus (which is on the right half of the collision plane) in Fig. \[fig:dens\]. The neutron density was plotted on a logarithmic scale to emphasize the low-density contours. The neutron tail seen in the lower frame of Fig. \[fig:dens\] is very similar to the neutron density contours shown in Fig. 10 of Ref. [@ZSW07]. ![\[fig:dens\] Contours of the total density for the dinuclear system $^{40}$Ca+$^{96}$Zr at ion-ion separation of $R=12.3$ fm (top frame), and contours of the $^{96}$Zr neutron states, plotted on a logarithmic scale at the same separation (bottom frame).](fig6a.eps "fig:")![\[fig:dens\] Contours of the total density for the dinuclear system $^{40}$Ca+$^{96}$Zr at ion-ion separation of $R=12.3$ fm (top frame), and contours of the $^{96}$Zr neutron states, plotted on a logarithmic scale at the same separation (bottom frame).](fig6b.eps "fig:") For the same energy we have also calculated the partially integrated neutron density as a function of the collision axis coordinate $z$ using Eq.(\[eq:rhoz\]) for the $^{96}$Zr nucleus, shown in Fig. \[fig:rhoz2\]. In this case we do not have a recoiled final state since the system actually fuses. ![\[fig:rhoz2\] Partially integrated neutron density calculated from Eq.(\[eq:rhoz\]) for the $^{96}$Zr nucleus plotted on a logarithmic scale versus the collision axis coordinate $z$ for the $^{40}$Ca+$^{96}$Zr system at $E_{\mathrm{c.m.}}=97$ MeV. The solid black curve corresponds to the initial partial density and the dashed red curve to the same quantity at $R=12.3$ fm.](fig7.eps) The integrated neutron transfer at the ion-ion separation distance of about $R=12$ fm is approximately $0.5$. To identify which states actually contribute to the probability of mass exchange from $^{96}$Zr to $^{40}$Ca we have plotted the individual neutron single-particle probabilities given by Eq. (\[eq:palpha\]) for the $^{96}$Zr nucleus as a function of collision coordinate axis $z$ at the same ion-ion separation $R=12.3$ fm, as shown in Fig. \[fig:palpha2\]. ![\[fig:palpha2\] Neutron single-particle probability densities given by Eq. (\[eq:palpha\]) for the $^{96}$Zr nucleus at $E_{\mathrm{c.m.}}=97$ MeV at $R=12.3$ fm.](fig8.eps) Again, the static Hartree-Fock calculation for the $^{96}$Zr nucleus helps us identify these states. The states up to $N=50$ can be enumerated (by using parity, degeneracy of eigenvalues, etc.) and exactly match the shell-model states with the spin-orbit term. The last six states (we do not impose time-reversal invariance and thus have one state per nucleon) are almost degenerate in energy and have positive parity values. This is consistent with Ref. [@ZSW07], which finds the neutron to be in the $2d_{5/2}$ state. The corresponding three states (time-reversed pairs) are the red, cyan, and magenta colored curves drawn thicker than others and pointed out by arrows. We observe that while two of the states are the largest contributors to the transmission probability, one of the states practically makes no contribution. This is true despite the fact that all these states are the highest energy states and are degenerate in energy. After the second highest $2d_{5/2}$ state at -7.05 MeV (cyan) the next three states that have the largest contribution to the transmission probability are; one of the two $2p_{3/2}$ states at -18.06 MeV, the $2p_{1/2}$ state at -16.17 MeV, and only one of the five $1g_{9/2}$ states at -12.26 MeV. The single-particle quadrupole moment is a good indicator of which of the states, specially sub-states belonging to the same quantum number, make the largest contribution to the transmission probability, since the most stretched states have the largest positive quadrupole moments. For example the three substates of the $2d_{5/2}$ state have single-particle quadrupole moments of $11.8$, $3.7$, and $-15.6$ fm$^2$ corresponding to the three states shown in Fig. \[fig:palpha2\] in descending order. Similarly, the $2p_{3/2}$ state with a quadrupole moment of $8.0$ fm$^2$ is the next largest contributor whereas the other sub-state with a quadrupole moment of $-8.0$ fm$^2$ has a transmission probability which is orders of magnitude smaller. ![\[fig:palpha3\] Neutron single-particle probability densities given by Eq. (\[eq:palpha\]) for the $^{96}$Zr nucleus at $E_{\mathrm{c.m.}}=91$ MeV and after the recoiling nuclei are at a separation of $R=19$ fm.](fig9.eps) It is interesting to note that some of the sub-states of the same quantum number give a much larger contribution, and states like $1f_{5/2}$, which is higher in energy than $2p_{3/2}$, do not contribute appreciably. This seems to indicate that the transmission probability depends not only on the energies of the single-particle states, but that it has an additional strong dependence on other properties of the states. We have also repeated the same study for the $^{40}$Ca+$^{96}$Zr system at $E_{\mathrm{c.m.}}=91$ MeV, which is below the effective barrier. In this case the nuclei recoil with a closest approach distance of $12.6$ fm. Again, in order to identify which states actually contribute to the probability of mass exchange from $^{96}$Zr to $^{40}$Ca we have plotted the individual neutron single-particle probabilities given by Eq. (\[eq:palpha\]) for the $^{96}$Zr nucleus as a function of collision coordinate axis $z$ when the recoiled ions are about $R=19$ fm apart, as shown in Fig. \[fig:palpha3\]. We observe that the single-particle transmission probabilities after the recoiled nuclei are well separated are similar to the findings for the $E_{\mathrm{c.m.}}=97$ MeV case, except with reduced probabilities. In addition, the observed sub-barrier behavior for the $^{40}$Ca+$^{96}$Zr system is analogous to the $^{16}$O+$^{24}$O system. Conclusions =========== We have performed a detailed analysis of mass exchange in the vicinity of the Coulomb barrier for systems involving a neutron-rich nucleus using the TDHF theory. Our work was motivated by Ref. [@ZSW07] where the same phenomenon was studied using a quantum mechanical three-body model. For the $^{40}$Ca+$^{96}$Zr system at $E_{\mathrm{c.m.}}=97$ MeV, which is slightly above the barrier, we can essentially confirm the results of Ref. [@ZSW07]. We confirm that at relatively large ion-ion distances the neutron transfer probability begins to build up at the location of the receiving nucleus. We have analyzed this aspect of neutron transmission in TDHF using the lighter $^{16}$O+$^{24}$O system. We find that in the vicinity of the Coulomb barrier and below the barrier different single-particle states see barriers that may differ from the effective ion-ion barrier with a fixed center-of-mass. 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--- abstract: 'When a highly charged globular macromolecule, such as a dendritic polyelectrolyte or charged nanogel, is immersed into a physiological electrolyte solution, monovalent and divalent counterions from the solution bind to the macromolecule in a certain ratio and thereby almost completely electroneutralize it. For charged macromolecules in biological media, the number ratio of bound mono- versus divalent ions is decisive for the desired function. A theoretical prediction of such a sorption ratio is challenging because of the competition of electrostatic (valency), ion-specific, and binding saturation effects. Here, we devise and discuss a few approximate models to predict such an equilibrium sorption ratio by extending and combining established electrostatic binding theories such as Donnan, Langmuir, Manning as well as Poisson–Boltzmann approaches, to systematically study the competitive uptake of mono- and divalent counterions by the macromolecule. We compare and fit our models to coarse-grained (implicit-solvent) computer simulation data of the globular polyelectrolyte dendritic polyglycerol sulfate (dPGS) in salt solutions of mixed valencies. The dPGS has high potential to serve in macromolecular carrier applications in biological systems and at the same time constitutes a good model system for a highly charged macromolecule. We finally use the simulation-informed models to extrapolate and predict electrostatic features such as the effective charge as a function of the divalent ion concentration for a wide range of dPGS generations (sizes).' author: - Rohit Nikam - Xiao Xu - Matej Kanduč - Joachim Dzubiella title: 'Competitive sorption of mono- versus divalent ions by highly charged globular macromolecules' --- \[intro\] Introduction ====================== Polyelectrolytes in polar solvents such as water are important and ubiquitous in biological as well as in synthetic matter. [@Muthukumar2017; @Katchalsky1964; @Rubinstein2012; @Boroudjerdi2005; @Forster1995; @Dobrynin2005; @Liu2003] In these systems, electrostatic interactions, regulated by free ions and water, play a dominant role in shaping the structural and electrostatic characteristics of the polyelectrolyte, and the subsequent function of the system. [@Muthukumar2017; @Rubinstein2012; @Boroudjerdi2005] The electrostatic attraction between the isolated polyelectrolyte molecule and the oppositely charged counterions in the solution leads to strong counterion condensation on the molecule. This significantly modifies its interaction with other charged molecules (*e.g.*, proteins, DNA, etc.) and its electric properties such as the electrophoretic mobility in an external electric field. [@Boroudjerdi2005; @Forster1995; @Liu2003] Therefore, understanding counterion condensation is of utmost importance in order to understand the properties of polyelectrolytes and their implications in the biological and synthetic environments. [@Chremos2016; @Boroudjerdi2005] Condensation effectively leads to neutralizing an equivalent amount of the structural charge ${Z_\mathrm{d}}$ of the macromolecule. [@alexander1984charge; @belloni1998ionic] Hence, the charged substrate plus its confined counterions may be considered as a single entity with an effective (or renormalized) charge ${Z_\mathrm{eff}}$, which is significantly lower than the bare structural charge ${Z_\mathrm{d}}$. One can then identify the difference ${Z_\mathrm{d}}-{Z_\mathrm{eff}}$ as the amount of counterions condensed in the surface region. [@Bocquet2002] The phenomenon of counterion condensation and the effect of ionic strength on the configurational properties of different types of polyelectrolyte molecules such as chains, [@Forster1992; @dobrynin1995; @Wenner2002; @Raspaud1998; @Dobrynin2005; @Liu2002; @Liu2003; @Muthukumar2004; @Chremos2016] brushes, [@ruhe2004polyelectrolyte; @pincus1991colloid; @borisov1991collapse; @Zhulina1995; @Zhulina2000] or polyelectrolyte nanogels [@nanogel1; @nanogel2; @nanogel3; @arturo2017; @arturo2018] have been studied extensively in the past. Through the knowledge of the distribution of the salt ions around the polyelectrolyte, *e.g.*, measured in terms of the radial distribution function in simulations and experiments, it is possible to derive important properties such as charge–charge correlation, osmotic compressibility and shear viscosity of the system. [@forster1995polyelectrolytes] Muthukumar, in his extensive and comprehensive review of the experimental, theoretical and simulation based research done on polyelectrolyte chains, described the effect of salt concentration, valency of counterions, chain length and polyelectrolyte concentration on counterion condensation. [@Muthukumar2017; @manning2012poisson] Besides the properties of a single isolated polyelectrolyte molecule, the ionic strength of the solution also influences the interaction of polyelectrolytes with other entities, such as adsorption on substrates, [@VandeSteeg1992; @Dahlgren1993; @Netz1999; @Hariharan1998; @Gittins2001; @caruso2000hollow] formation of ultra-thin polyelectrolyte multilayer membranes, [@Decher1992; @Decher1997; @Ladam2000; @McAloney2001; @Dubas2001] the structure and solubility of polyelectrolyte complexes [@hugerth1997effect; @rusu2003formation; @winkler2002complex; @Kudlay2004a; @Mende2002] or coacervates. [@spruijt2010binodal; @Gucht2011; @biesheuvel2004electrostatic; @Perry2014] As an emerging class of functional polyelectrolytes, polyelectrolyte nanogels [@nanogel1; @nanogel2; @nanogel3; @arturo2017; @arturo2018] and dendritic or hyperbranched polyelectrolytes [@JensDernedde2010; @Khandare2012; @Groeger2013; @Maysinger2015; @Reimann2015] have attracted considerable interest in the scientific community in the last years due to their multifaceted bioapplications, such as biological imaging, drug delivery and tissue engineering. [@Leereview; @Ballauff2004; @Tian2013] In particular, the hyperbranched or dendritic polyglycerol sulfate molecules (hPGS or dPGS, respectively) are found to possess strong anti-inflammatory properties,[@Maysinger2015; @Reimann2015] act as a transport vehicle for drugs towards tumor cells,[@Sousa-Herves2015; @Groeger2013; @Vonnemann2014] and can be used as imaging agents for the diagnosis of rheumatoid arthritis. [@Vonnemann2014] This wide variety of applications, thus, have proven them to be high potential candidates for the use in medical treatments. [@Khandare2012] Hence, the understanding of dPGS interaction with the *in vivo* environment becomes important. The highly symmetric dendritic topology, terminated with monovalent negatively charged sulfate groups, makes dPGS also an excellent representative model in the class of highly charged globular polyelectrolytes. [@xu2017charged; @nikam2018charge] Because of the charged terminal groups, dPGS mainly interacts through electrostatics, rendering counterion condensation and subsequent charge renormalization effects to become substantial for function. There have been past efforts to investigate the counterion condensation and to define the effective charge as a result of the charge renormalization on charged hard-sphere colloids. [@Ohshima1982; @Zimm1983; @alexander1984charge; @Belloni1984; @belloni1998ionic; @Ramanath1988; @Manning2007; @Bocquet2002; @Gillespie2014] However, the characterization of open-structure nanogel particles or dendrites such dPGS, which in part are penetrable to ions and a surface is not well defined, remains challenging. [@Ohshima2008] Recently, Xu *et al.* implemented a simple but accurate scheme to define and determine the effective surface potential and its location for dPGS, by mapping potentials obtained from simulations to the Debye–Hückel potential in the far-field regime. [@xu2017charged] This scheme is widely known as the Alexander prescription. [@alexander1984charge; @Trizac2002; @bocquet2002effective; @Levin2004] Based on this criterion, a systematic electrostatic characterization of dPGS has been performed via coarse-grained [@xu2017charged] and all-atom [@nikam2018charge] simulations by defining the number of condensed (bound) ions. It was then established that the strong binding of dPGS to lysozyme – an abundant protein in the human biological environment – and a sequential formation of a protein corona around dPGS in the presence of NaCl salt solution, is dominantly governed by the entropic gain due to the release of a few Na$^+$ counterions during binding. [@xu:biomacro] Proteins typically bind strongly to the macromolecular surface, thereby forming a protein ‘corona’, a dense shell of proteins that can entirely coat the macromolecule. [@Owens2006; @Cedervall2007; @Lindman2007; @Monopoli2012; @wang2013biomolecular; @LoGiudice2016; @Boselli2017] Considering the medicinal applications of dPGS, it is important to study its interactions with [*divalent metal cations*]{}, *viz.* magnesium(II) and calcium(II) ions, which are key constituents of the human blood serum. Mg$^{2+}$ is essential for the stabilization of proteins, polysaccharides, lipids and DNA/RNA molecules, while Ca$^{2+}$ is critical for bone formation and plays a key role in signal transduction. [@Friesen2019a; @da2001biological] Human serum blood contains approximately $0.75-0.95$mM Mg$^{2+}$ ions, $1-4$mM Ca$^{2+}$ ions and around $150$mM NaCl salt in a dissociated form. [@meyers2004encyclopedia; @kretsinger2013encyclopedia] Thus, upon the administration of dPGS into the human biological environment, it is imperative for the competitive adsorption between the divalent (Mg$^{2+}$/Ca$^{2+}$) and monovalent (Na$^+$) ions to establish on the dPGS molecule, which can change the effective charge, and subsequently the interaction properties of dPGS with other charged entities such as proteins. This microscopic mechanism has a potential to significantly alter the attributes of protein corona around dPGS, [@XiaoCPS] thus, the biological immune response to the dPGS–protein corona complex, its metabolic fate, and the function of such a complex in biomedical and biotechnological applications. The competitive ion binding can be observed also in a wide variety of the biological and industrial ion-exchange processes such as the alkaline-earth/alkali-metal ion-exchange onto polyelectrolytes, [@Pochard1999] desalination of saline water to produce potable water, [@Birnhack2019] demineralization of whey, acid and alkali recovery from waste acid [@kobuchi1986application] and alkali solutions [@sata1993new] by diffusion dialysis, [@Sata2002] etc. Interactions of multivalent ions with polyelectrolyte solutions have been theoretically studied in the past, in terms of their thermodynamic properties, [@Kuhn1999] ionic and potential distributions, [@Gavryushov1997] accurate calculation of the effective charge, [@DosSantos2010] and the effect on the interaction between polyelectrolyte macromolecules. [@Arenzon1999; @Naji2004; @Kanduc2010; @Rudi2016] In this paper, the focus is to theoretically analyze the competitive sorption of [*mono- versus divalent counterions*]{} by highly charged spherical dPGS-like polyelectrolytes with the help of mean-field continuum and discrete binding site models, informed by coarse-grained computer simulations of dPGS of various generations. The theoretical models are generally formulated for globular charged macromolecules and include ion-specific effects in a parametric way and can thus be straightforwardly modified or adapted to other charged globules, where mono-/divalent ion-exchange plays a role. In particular, we begin with the simple Donnan model, modified for ion-specific uptake, assuming that the electrostatic potential and the ionic concentrations are constant within the macromolecule phase and the bulk phase. [@Basser1993; @arturo2016; @Ahualli2014] Despite being simple, still, for the mixed case of monovalent and divalent ions the resultant composition is a non-trivial outcome. We continue with the mean-field Poisson–Boltzmann (PB) model, widely used in colloidal science and electrochemistry, [@Rubinstein2012; @israelachvili2011intermolecular; @adamson1967physical; @verwey1947theory; @Borukhov2000] and with the limitations well known and discussed, in particular the neglect of electrostatic and steric correlations, [@eigen1954; @kralj1996; @Cuvillier1997; @Borukhov1997; @DosSantos2010] or ion-specific sorption effects. [@Kalcher2010; @Kalcher2010a; @Chudoba2018; @arturo2014; @Yaakov2009a; @Lima2008; @Koelsch2007; @okur2017beyond; @LoNostro2012; @Schwierz2013] The PB model has also been implemented to address the problem of competitive counterion binding in a mixed salt for the cases of linear polyelectrolytes such as DNA [@Burak2004; @Chen2002; @Rouzina1997; @Misra1994; @Paulsen1987] and planar geometries. [@Rouzina1994] We also devise a two-state approximation model for an ion condensation around a charged globule. {width="14cm" height="14cm"} The two-state approach was firstly used in the Oosawa–Manning model [@Manning1969; @Oosawa] for the counterion condensation around polyelectrolyte chains, according to which, counterions in a solution can be classified into two categories: ‘free’ counterions, which are able to explore the whole solution volume $V$ and the ‘condensed’ (or ‘bound’) counterions, which are localized within a small volume surrounding the polyelectrolyte macromolecule. An equivalent model for an impenetrable sphere with a surface charge was developed by Manning, where the number of condensed counterions on the macromolecule per bare unit surface charge is obtained by a free energy minimization, pointing to the competition between the electrostatic binding of counterions to the macromolecule and their dissociation entropy. [@Manning2007] We extend this model by introducing a discrete binding site model by considering the finite configurational volume of the ion in the condensed state and that the macromolecule has a finite number of charged binding sites by adopting the mixing entropy from the works of McGhee and von Hippel. [@Mcghee1974] Ion-binding models in the same spirit have been developed in the past to describe the ionization equilibrium of [*linear*]{} polyelectrolytes in monovalent salt [@flory1953molecular; @Raphael1990; @Muthukumar2004], multivalent salt, [@Friedman1984] and in mixtures of mono- and divalent salts. [@Kundagrami2008] All our models are compared to molecular simulations and used to study systematically the key electrostatic features of a highly charged globule, such as the effect of competitive adsorption on the variation of the number of condensed monovalent and divalent counterions, effective charge, and its variation with divalent ion concentration. Coarse-grained computer simulations {#cg_sim} =================================== Simulation methods, force fields, and systems {#sec:cg_ff} --------------------------------------------- The coarse-grained (CG) monomer-resolved models of the dPGS macromolecule have been developed previously [@xu2017charged] and maintain the essential dPGS structural and electrostatic features with affordable computing expense. In brief, the dPGS branching units (–) and inner core (–) (both of which are a part of the glycerol chemical group, respectively), and the terminal sulfate groups (–) are individually represented by the CG segments of specific type. The gross number of the CG segments is equal to the dendrimer polymerization $N_g = 3 \times 2^{n+1} - 2$ of generation index $n$. Only the terminal segments are charged with $-1e$ (where $e$ is the elementary charge), leading to the dPGS bare valency $|Z_n| = 3 \times 2^{n+1}$. The CG segments are connected by bonded and angular potentials both in harmonic form. In the previous work [@xu2017charged] we only studied monovalent ions. Here we extend it to study the competitive uptake of mono- and divalent ions for generations 2 and 4. The bare charge valencies of the G$_2$-dPGS are thus ${Z_\mathrm{d}}= Z_{n=2} = -24$ and ${Z_\mathrm{d}}= Z_{n=4} = -96$. Snapshots are shown in Fig. \[snap\]. The non-bonded interactions between CG beads are described by the Lennard-Jones (LJ) potential together with the Lorentz–Berthelot mixing rules. In particular, the energy parameter $\epsilon_{\rm LJ} = 0.1\,{k_\mathrm{B}T}$ and the diameter $\sigma_{\rm LJ} = 0.4$ nm are set identical for all ions (mono- and divalent) and thus any ion-specific effects are not explicitly included. In our simulations we place the dPGS in the center of a periodically repeated cubic box with a volume of $V$ (side-length of $L = 30$ nm). The solvent is implicitly assumed as a dielectric continuum with a dielectric constant $\epsilon_{\rm w} = 78$. The CG simulations employ the stochastic dynamics (SD) integrator in Gromacs $4.5.5$ as in our previous work. [@xu2017charged] {width="12.5cm" height="12.5cm"} {width="13cm" height="13cm"} All simulations are performed in the canonical ensemble. The divalent cations (DCs), monovalent cations (MCs) and monovalent anions in the system are referred to with subscripts $++,\; +$ and $-$, respectively. The dPGS is accompanied by the corresponding number of monovalent counterions ${N_\mathrm{s}}$ ($24$ for G$_2$-dPGS and $96$ for G$_4$-dPGS) electrically neutralizing the macromolecule and having the same chemical identity as the MCs of the salt. The number of salt ions $i$ ($i = +\!+,+,-$) is denoted as $n_i$, while the corresponding total salt concentrations are denoted as ${c^0_{i}}=n_i/V$. Bulk concentrations are defined as ${c^{\mathrm{b}}_{i}}=(n_i - {N^\mathrm{b}_{i}})/(V - {v_\mathrm{eff}})$ (for $i=+\!+,-$) and ${c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}=(n_{{\scalebox{.8}{$\scriptscriptstyle +$}}} +{N_\mathrm{s}}- {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}})/(V - {v_\mathrm{eff}})$, where ${v_\mathrm{eff}}= 4\pi {r_\mathrm{eff}}^3/3$ is the volume enclosed by the effective radius ${r_\mathrm{eff}}$ of dPGS and ${N^\mathrm{b}_{i}}$ is the number of ions $i$ condensed (bound) on the dPGS. The definitions of both ${r_\mathrm{eff}}$ and ${N^\mathrm{b}_{i}}$ are discussed in Section \[sim\_analysis\]. The simulations are performed at the total DC concentrations ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ of 0.98, 2.95, 3.75, 9.96 and 14.94mM. G$_2$-dPGS simulation snapshots for different ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ values are shown in Fig. \[snap\](a)-(c), while the whole simulation box is displayed in Fig. \[snap\](d). The MC concentration ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ is fixed to $150.37$mM and the monovalent anion concentration is adjusted in a way to ensure electroneutrality in the simulation box. The bulk ionic strength $I = \frac{1}{2}\sum_i z^2_i {c^{\mathrm{b}}_{i}}$ ($i = +,+\!+,-$ with the charge valency $z_i$) ranges from $150.5$mM to $195$mM. The corresponding Debye screening length $\kappa^{-1} = \left(8\pi {l_\mathrm{B}}I \right)^{-1/2}$ (where ${l_\mathrm{B}}$ is the Bjerrum length) ranges from $0.8$nm (${c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}=0$ and ${c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}=150.5$mM) to $0.7$nm (${c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}=14.94$ and ${c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}=150.5$mM). As a reference, we also perform CG simulations in the limit of only monovalent salt, with total concentrations ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ of 10.02, 25.06 and 150.37mM. Simulation results: radial density distributions {#sim_results} ------------------------------------------------ The dPGS structure and its response to the addition of the DCs, is examined by the density distribution of the terminal sulfate beads ${c_\mathrm{s}}(r)$ as a function of the distance $r$ from the center-of-mass (COM) of the dPGS, for different DC concentrations ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$, as shown in Fig. \[rho\_sulfate\]. Interestingly, the presence of DCs does not lead to a notable change in the dPGS structure. Instead, the ${c_\mathrm{s}}(r)$ profiles in the operated range of ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ and for both G$_2$-dPGS and G$_4$-dPGS are reasonably coincident. Fig. \[rho\_sulfate\](a) shows that for G$_2$-dPGS, a single-peak distribution is found, indicating that most of the sulfate beads reside on the molecular surface. However, in Fig. \[rho\_sulfate\](b), a bimodal distribution is seen for G$_4$-dPGS with a small peak at $r \simeq 0.6$nm. This backfolding phenomenon, contributing to a dense-core arrangement due to the dense macromolecular shell, [@Ballauff2004] is also found in our previous works [@xu2017charged; @nikam2018charge] and has been detected for other terminally charged CG dendrimer models. [@Huismann2010; @Huismann2010B; @Klos2010; @Klos2011] After the major peak, ${c_\mathrm{s}}(r)$ gradually subsides to zero. The location where the charge density ${c_\mathrm{s}}(r)$ falls to $150$mM, which we set as the physiological NaCl concentration, is defined as the bare (intrinsic) radius of dPGS ${r_\mathrm{d}}$, [^1] shown as vertical dashed blue lines in Fig. \[rho\_sulfate\]. The ${r_\mathrm{d}}$ values for G$_2$-dPGS and G$_4$-dPGS are obtained as $1.40$nm and $2.11$nm, respectively. Fig. \[rho\_sulfate\](b) also shows that a slight shift in the location of the major peak and an enrichment of the lower peak appears as ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ increases, indicating a slow shrinking of the dPGS molecule due to the condensation of DCs (see Fig. \[density\_ions\]). Figs. \[density\_ions\](a) and (b) show the cation density distributions $c_i(r)$ ($i =+,+\!+$) for G$_2$-dPGS and G$_4$-dPGS, respectively. Let us focus first on G$_2$ in Fig. \[density\_ions\](a). The MC distribution $c_{{\scalebox{.8}{$\scriptscriptstyle +$}}}(r)$ shows a high accumulation of counterions close to the sulfate groups, with a global maximum at distances $r \sim 1.2$nm, slightly larger than the sulfate peak (peaking roughly at $\sim 1$nm). This means that the most strongly bound ‘condensed’ MCs are thus distributed more on the surface layers of the dPGS. At larger distances, $r \sim 2$nm, a Debye–Hückel like decay is observed. Adding more DCs, the MC distribution gradually diminishes, as expected from the exchange of MCs with DCs within the dPGS. However, interestingly, the DC distribution peaks at distances distinctively smaller than the location of the sulfate peak, roughly $0.5-0.6$nm shifted towards the dPGS center away from the peak of the MC distribution. This more interior binding might be attributed to different binding mechanisms between DCs and sulfate, *e.g.*, bridging of two sulfate groups by one DC, which might be sterically favored closer to the dPGS core. These subtle structural effects may have important consequences in the context of the counterion-release mechanism driving the dPGS–protein binding, [@xu:biomacro] which should be interesting for future studies. The ion profiles for G$_4$ shown in Fig. \[density\_ions\](b) show qualitatively the same behavior but are broader and double-peaked because of the significant sulfate backfolding as previously presented in Fig. \[rho\_sulfate\](b). It is worth noting that simulations of DCs in general are more challenging than for MCs only. DC are more heavily hydrated than MCs (*e.g.*, Mg$^{2+}$ and Na$^+$ ions), [@Stokes1948; @Marcus2006] therefore future studies should scrutinize the ionic size used in the implicit solvent. Furthermore, quantum mechanical charge transfer effects as a result of the ion-induced powerful electronic polarization of the surrounding media, [@Yao2015] which are much more prevalent in the case of DCs [@Pavlov1998; @Kohagen2014] than MCs, may also be subsumed in ionic sizes in the implicit water. These model details may subtly change the density profiles shown in Figs. \[density\_ions\](a) and (b). However, the effects on total competitive uptake should be relatively minor as they are dominantly driven by valency and electrostatic correlations, and size effects are typically of second order importance. Using the density distributions of the charged entities shown above, the electrostatic properties of dPGS can be studied in the presence of the mixture of DCs and MCs. The analysis methods described in Section \[sim\_analysis\] are used to define the effective radius ${r_\mathrm{eff}}$, charge valency ${Z_\mathrm{eff}}$ and potential ${\phi_\mathrm{eff}}$ of dPGS. Electrostatic properties of dPGS {#sim_analysis} -------------------------------- **dPGS effective radius** The first step to study the ion condensation behavior is to adopt a characteristic distance ${r_\mathrm{eff}}$ to distinguish a bound ion from an unbound one. A practical method in that respect has been summarized in our previous work. [@xu2017charged] In short, we first consider the dPGS radial electrostatic potential profile $\phi$ (scaled by ${k_\mathrm{B}T}/e$), through the framework of the Poisson’s equation $$\nabla^2 \phi = -4\pi {l_\mathrm{B}}\sum_{i} z_i c_i(r) \qquad i= s, ++, +, - \label{possion}$$ Here, $c_i(r)$ refers to the radial number density profiles with respect to the distance to the dPGS-COM $r$ for all charged species in the CG simulation, namely, sulfates ($s$), DCs, MCs, and monovalent anions. For all ionic species, $c_i(r)$ reaches the bulk number density ${c^{\mathrm{b}}_{i}}$ in the far-field. The simulation results for the profiles are shown in Figs. \[rho\_sulfate\] and \[density\_ions\]. The Poisson’s equation is numerically integrated twice to obtain $\phi(r)$, which is then compared with the dimensionless Debye–Hückel potential $\phi^{}_\mathrm{DH}$, given by [@alexander1984charge; @xu2017charged; @nikam2018charge] $$\phi^{}_\mathrm{DH}(r) = {Z_\mathrm{eff}}{l_\mathrm{B}}\frac{\mathrm{e}^{\kappa {r_\mathrm{eff}}}}{1+\kappa {r_\mathrm{eff}}} \frac{\mathrm{e}^{-\kappa r}}{r}. \label{DH_eq}$$ $\phi^{}_{\rm DH}$ is applicable to a charged sphere with radius ${r_\mathrm{eff}}$ and valency ${Z_\mathrm{eff}}$. It approaches to $\phi$ only after the distance $r^*$ where non-linear effects, including the correlation and condensation of ions, subside. Thus, $r^* = {r_\mathrm{eff}}$ is eligible to serve as the dPGS effective radius to define the bound ions. The effective surface potential of dPGS obtained from simulations is then defined as ${\phi_\mathrm{eff}}= \phi({r_\mathrm{eff}})$, which is shown in Table \[cg\_table\]. Comparing Eq.  to the radial electrostatic potentials from the simulations, [@xu2017charged] the value of ${r_\mathrm{eff}}$ for dPGS in the simulations for G$_2$ and G$_4$ was found to be $1.65$nm and $2.40$nm, respectively, under the operated concentration range in the mixture of DCs with MCs as well as in the monovalent limit, as shown in Table \[cg\_table\]. These values are different than the ones obtained in our previous work, [@xu2017charged] which operates at ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= 10$mM, unlike the current work where ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= 150.37$mM. The newly obtained ${r_\mathrm{eff}}$ values in this work are then used as an input for the MMvH model, as discussed in Section \[mvh\], to describe the competitive sorption. It is thus implicitly assumed that ${r_\mathrm{eff}}$ does not depend on the sorption of DCs, within the operated range of DC concentrations. The same prescription will be used to define ${r_\mathrm{eff}}$ (denoted as ${r_\mathrm{eff}}^\mathrm{PB}$) from the solutions of the PPB model, as discussed in Section \[sec:ppbl\]. The results for ${r_\mathrm{eff}}^\mathrm{PB}$ are also shown in Table \[cg\_table\]. ---------------------------------------------------- ------------------ -------------------- -------------------- ----------------------- ------------------ -------------------- -------------------- ----------------------- -------------------------------- -------------------------------- ----------------------------------- -------------------------------- -------------------------------- ----------------------------------- ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ ${r_\mathrm{d}}$ ${r_\mathrm{eff}}$ ${Z_\mathrm{eff}}$ ${\phi_\mathrm{eff}}$ ${r_\mathrm{d}}$ ${r_\mathrm{eff}}$ ${Z_\mathrm{eff}}$ ${\phi_\mathrm{eff}}$ ${r_\mathrm{eff}}^\mathrm{PB}$ ${Z_\mathrm{eff}}^\mathrm{PB}$ ${\phi_\mathrm{eff}}^\mathrm{PB}$ ${r_\mathrm{eff}}^\mathrm{PB}$ ${Z_\mathrm{eff}}^\mathrm{PB}$ ${\phi_\mathrm{eff}}^\mathrm{PB}$ ([mM]{}) ([nm]{}) ([nm]{}) ([nm]{}) ([nm]{}) ([nm]{}) ([nm]{}) $0.00$ $-10.09$ $-1.26$ $-20.04$ $-1.27$ $-11.72$ $ -1.32$ $-23.60$ $-1.56$ $0.98$ $-8.85$ $-1.15$ $-17.75$ $-1.14$ $-9.79$ $-1.12$ $-20.03$ $-1.38$ $2.95$ $-7.40$ $-0.98$ $-14.21$ $-0.93$ $-8.89$ $-0.88$ $-15.54$ $-1.05$ $3.75$ $-6.84$ $-0.85$ $-12.25$ $-0.77$ $-8.29$ $-0.83$ $-14.34$ $-0.97$ $9.96$ $-6.33$ $-0.75$ $-10.11$ $-0.62$ $-7.03$ $-0.57$ $-8.86$ $-0.60$ $14.94$ $-5.86$ $-0.68$ $-9.65$ $-0.55$ $-6.36$ $-0.46$ $-6.13$ $-0.44$ ---------------------------------------------------- ------------------ -------------------- -------------------- ----------------------- ------------------ -------------------- -------------------- ----------------------- -------------------------------- -------------------------------- ----------------------------------- -------------------------------- -------------------------------- ----------------------------------- **Number of bound ions and effective charge** The cumulative number of ions of species $i$ as a function of the distance $r$ from the COM of dPGS is calculated as $$N_{\mathrm{acc},i}(r) = \int^{r}_{0} c_i(r') 4\pi {r'}^2 \mathrm{d}r' \quad \quad i = ++,+,-. \label{nbi}$$ Summing up the contribution of all charged species, the cumulative charge valency of the system as a function of the distance $r$ reads $$Z_\mathrm{acc}(r) = {Z_\mathrm{d}}(r) + 2N_{\mathrm{acc},{\scalebox{.8}{$\scriptscriptstyle ++$}}}(r) + N_{\mathrm{acc},{\scalebox{.8}{$\scriptscriptstyle +$}}}(r) - N_{\mathrm{acc},{\scalebox{.8}{$\scriptscriptstyle -$}}}(r), \label{neff}$$ where ${Z_\mathrm{d}}(r)$ denotes the spatial distribution of bare charge valency of the dPGS, obtained from the simulation. With that, the number of the bound ions and the effective charge valency of the dPGS follow as ${N^\mathrm{b}_{i}}= N_{\mathrm{acc},i}({r_\mathrm{eff}})$ and ${Z_\mathrm{eff}}= Z_{\mathrm{acc}}({r_\mathrm{eff}})$, respectively. The values are shown in Table \[cg\_table\]. ${Z_\mathrm{eff}}$ and ${\phi_\mathrm{eff}}$ exhibit a strong decrease in the magnitude with higher ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$, indicating an enhanced dPGS charge renormalization. Theoretical models ================== Basic model {#basic} ----------- In our theoretical models, the macromolecule is represented as a perfect sphere with the bare radius ${r_\mathrm{d}}$, the bare charge valency ${Z_\mathrm{d}}$, the effective radius ${r_\mathrm{eff}}$ and the effective charge valency ${Z_\mathrm{eff}}$, enclosed in a spherical domain of radius $R$ and volume $V$, as shown in Fig. \[fig:theoretical-model\]. The total number of charged monomers in the macromolecule is ${N_\mathrm{s}}$, each of which is negatively charged with a charge valency ${z_\mathrm{s}}$. ![\[fig:theoretical-model\] Schematic of a theoretical model representing the system shown in Fig. \[snap\](d). The computational cell domain (blue) is assumed to be spherical with the same volume as that of the simulation box, $V$, and with a uniform dielectric constant of water $\epsilon_\mathrm{w}=78$. dPGS is assumed to be a perfect sphere (orange) at the center of the domain. The dPGS bare and effective charge valencies are ${Z_\mathrm{d}}$ and ${Z_\mathrm{eff}}$, respectively. ${r_\mathrm{d}}$ is the bare radius of dPGS, while ${r_\mathrm{eff}}$, the effective radius, representing the distance separating the electric double layer regime ($r > {r_\mathrm{eff}}$) from the non-linear counterion ’condensation’ regime ($r < {r_\mathrm{eff}}$).](images/fig4.pdf){width="5.5cm" height="5.5cm"} All ionic species and the macromolecule are assumed to be in an aqueous bath with an implicitly modeled solvent, having a uniform dielectric constant $\epsilon_\mathrm{w}=78$ at a temperature $T=298$K. \[dm\] The Donnan model (DM) ---------------------------- The arguably simplest model for competitive uptake is the Donnan model. The Donnan equilibrium assumes two strictly electroneutral and mutually exclusive regions, *i.e.*, the macromolecule region with the Donnan radius set to be the bare radius ${r_\mathrm{d}}$ taken from simulations (*i.e.*, with a bare macromolecular volume ${v_\mathrm{d}}=4\pi {r_\mathrm{d}}^3 /3$) and total homogeneously distributed bare charge of valency ${Z_\mathrm{d}}= {z_\mathrm{s}}{N_\mathrm{s}}$ with a concentration ${c_\mathrm{s}}= {N_\mathrm{s}}/{v_\mathrm{d}}$ of charged groups of the macromolecule, and the bulk region outside the molecule with a bulk ion concentration ${c^{\mathrm{b}}_{i}}$ ($i=+,+\!+,-$). Charge neutralization of the macromolecule by the counterions leads to the Donnan potential, which is a potential having a constant non-zero value in the macromolecule region. The potential in the bulk region is set to zero. The equilibrium distribution (partitioning) of ions among the regions results in the concentrations of ionic species $i$ as ${c^\mathrm{m}_{i}}$ and ${c^{\mathrm{b}}_{i}}$ in the macromolecule and bulk regions, respectively. These concentrations are related via the partition coefficient ${\mathcal{K}_{i}}$, given by $${\mathcal{K}_{i}}= \frac{{c^\mathrm{m}_{i}}}{{c^{\mathrm{b}}_{i}}} \quad \quad i=+\!+,+,- \label{ki-don1}$$ Neglecting ion–ion correlations, an approximate expression for ${\mathcal{K}_{i}}$ can be obtained using the condition that the equilibrium electrochemical potential of ion $i$ is equal in both the macromolecule and bulk regions, implying that $${\mathrm{ln}}\,{c^{\mathrm{b}}_{i}}= z_i{\phi^{}_\mathrm{D}}+ {\mathrm{ln}}\,{c^\mathrm{m}_{i}}+ \beta {\Delta \mu_{\mathrm{int},\,i}}\label{phase_eqm}$$ where ${\phi^{}_\mathrm{D}}$ is the dimensionless Donnan potential (scaled by ${k_\mathrm{B}T}/e$) in the macromolecule region and $\beta^{-1} ={k_\mathrm{B}T}$ is the thermal energy. With ${\Delta \mu_{\mathrm{int},\,i}}$ we account for additional non-electrostatic effects that can drive adsorption, *e.g.*, dispersion and hydrophobic forces in the net ion–macromolecule interaction, and is termed the ion-specific binding chemical potential of the condensed ion. The inclusion of ${\Delta \mu_{\mathrm{int},\,i}}$ has been considered in previous work, for example, as a term reflecting the steric ion–ion packing effects in a Donnan model for ion binding by polyelectrolytes or charged hydrogels. [@Chudoba2018; @arturo2014; @Ahualli2014] Eq.  with the help of Eq.  then leads to $${\mathcal{K}_{i}}= \frac{{c^\mathrm{m}_{i}}}{{c^{\mathrm{b}}_{i}}} = \mathrm{e}^{-\beta {\Delta \mu_{\mathrm{bind},\,i}}}= \mathrm{e}^{-\beta {\Delta \mu_{\mathrm{int},\,i}}} \mathrm{e}^{-z_i {\phi^{}_\mathrm{D}}} \label{ki_don2}$$ where ${\Delta \mu_{\mathrm{bind},\,i}}$ is the total transfer chemical potential for ion $i$ from the bulk to the macromolecule region. This allows us to define the intrinsic partition ratio for ionic species $i$ as $${\mathcal{K}_{\mathrm{int},\,{i}}}= \mathrm{e}^{-\beta {\Delta \mu_{\mathrm{int},\,i}}} \quad \quad i=+,+\!+ \label{kiz}$$ and the Donnan partition ratio as a contribution from pure electrostatic interaction between the ion and the macromolecule environment as $${\mathcal{K}_{\mathrm{el},\,{i}}}= \mathrm{e}^{-z_i {\phi^{}_\mathrm{D}}} \quad \quad i=+,+\!+ \label{kid}$$ The electrostatic component of total binding chemical potential of a counterion $i$ is then defined as $\beta {\Delta \mu_{\mathrm{el},\,i}}= -{\mathrm{ln}}\,{\mathcal{K}_{\mathrm{el},\,{i}}}= z_i{\phi^{}_\mathrm{D}}$. Eq.  can then be conveniently shortened as $${\mathcal{K}_{i}}= {\mathcal{K}_{\mathrm{el},\,{i}}}\, {\mathcal{K}_{\mathrm{int},\,{i}}}\label{ki-don3}$$ where ${\mathcal{K}_{i}}$ is shown as a composition of intrinsic and electrostatic effects. The signature assumption behind the Donnan model is the electroneutrality in the macromolecule region expressed as $${z_\mathrm{s}}{c_\mathrm{s}}+ \sum_i z_i {c^{\mathrm{b}}_{i}}\,{\mathcal{K}_{i}}= 0 \label{neutral}$$ Solving Eq.  for ${\phi^{}_\mathrm{D}}$ enables us to evaluate the net partition coefficient ${\mathcal{K}_{i}}$. Eq.  has no closed solution for multivalent ions, but it exists for the case of only monovalent ions in the system, ($i=\pm$) and is given as [@cemil2] $${\phi^{}_\mathrm{D}}= -{\mathrm{ln}}\left(-\frac{ \sqrt{1 + \chi_{{\scalebox{.8}{$\scriptscriptstyle +$}}} \chi_{{\scalebox{.8}{$\scriptscriptstyle -$}}}} + 1 }{\chi_{{\scalebox{.8}{$\scriptscriptstyle +$}}}} \right) \label{donnan-mono-expr}$$ where $\chi^{}_i = 2{\mathcal{K}_{\mathrm{int},\,{i}}}\, {c^{\mathrm{b}}_{i}}/{z_\mathrm{s}}{c_\mathrm{s}}$. Note that $\chi^{}_i<0$, since the valency of charged groups ${z_\mathrm{s}}$ is negative. Using Eqs. , and , for the monovalent-only case, the number of ions of species $i(=\pm)$ partitioned into the macromolecule region is then given as $$N^\mathrm{b}_{\pm} = c^0_{\pm} {v_\mathrm{d}}\, \mathcal{K}_{\mathrm{int}, \pm} \left( -\frac{ \sqrt{1 + \chi_{{\scalebox{.8}{$\scriptscriptstyle +$}}} \chi_{{\scalebox{.8}{$\scriptscriptstyle -$}}}} + 1 }{\chi_{{\scalebox{.8}{$\scriptscriptstyle +$}}}} \,\right)^{\pm 1} \label{Nb-don}$$ To evaluate the competition between MCs and DCs in the Donnan model we evaluate Eqs.  and numerically, *cf.* section \[numerical\]. Because of the electroneutrality assumption, the Donnan prediction for the amount of counterion sorption by the macromolecule in the monovalent-only case is given by ${N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= |Z| + N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle -$}}}$. For highly charged macromolecules, *i.e.*, $\chi_i \rightarrow 0$, Eq.  this trivially yields ${N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\simeq |Z|$. For the competitive sorption case, however, the result is non-trivial and can give a useful orientation with little effort. The Donnan model should become more quantitative for large dPGS generations, i.e., large size and/or high salt concentrations, so that $\kappa {r_\mathrm{d}}\gg 1$, for which the electroneutrality assumption is then very well justified. \[sec:ppbl\] Ion-specific Penetrable Poisson–Boltzmann (PPB) model ------------------------------------------------------------------ We now put forward a penetrable PB (PPB) model in which the charge profiles can be resolved in $r$, the radial distance from the macromolecular center. Since our charged macromolecules we have in mind (dPGS above and similar) are polymer-based with open structures and typically internally smeared out charge distributions, we opted (as in the Donnan model) for a penetrable model instead of a PB model for surface adsorption as typically used in studies of colloidal charge renormalization. [@Wall1957; @Ohshima1982; @Ohshima2008] Based on the parametrization described in the basic model (Section \[basic\]), we assume the macromolecule as a perfect penetrable sphere with a charge valency ${Z_\mathrm{d}}= {z_\mathrm{s}}{N_\mathrm{s}}$ and radius ${r_\mathrm{d}}$, as shown in Fig. \[fig:theoretical-model\]. ${r_\mathrm{d}}$ is taken from the dPGS internal charge distribution obtained from simulations, *cf.* Section \[sim\_results\] and Fig. \[rho\_sulfate\]. The charged monomers of the macromolecule, thus, have a uniform number distribution ${c_\mathrm{s}}= {N_\mathrm{s}}/{v_\mathrm{d}}$ (where ${v_\mathrm{d}}= 4\pi {r_\mathrm{d}}^3/3$) within the volume ${v_\mathrm{d}}$. ${c_\mathrm{s}}$ is applicable only within the macromolecule domain, *i.e.*, ${c_\mathrm{s}}(r) = {c_\mathrm{s}}\left(1-H(r-{r_\mathrm{d}})\right)$, where $H(r)$ is the Heaviside-step function. As an improvement to the standard PB model, here we also consider a contribution of the intrinsic non-electrostatic ion-specific interaction ${\Delta \mu_{\mathrm{int},\,i}}$ between the ion and the macromolecule, [@Kalcher2010; @Kalcher2010a] analogous to Eq.  in the Donnan model above. Assuming the electrostatic potential far away from the macromolecule, $\phi\left(r \rightarrow R \right)=0$, we first balance the chemical potential for each ion, between the bulk regime far from the macromolecule and the regime at the finite distance $r$ from the center of the macromolecule $${\mathrm{ln}}\,{c^{\mathrm{b}}_{i}}= z_i\phi(r) + {\mathrm{ln}}\,c_{i}(r) + \beta {\Delta \mu_{\mathrm{int},\,i}}(r), \label{pb_chem_pot_bal}$$ which is similar to Eq. , but in a distance-resolved manner. ${\Delta \mu_{\mathrm{int},\,i}}$ is considered on a local level, *i.e.*, ${\Delta \mu_{\mathrm{int},\,i}}(r) = {\Delta \mu_{\mathrm{int},\,i}}\left(1-H(r-{r_\mathrm{d}})\right)$. The Boltzmann ansatz then becomes $$c_i(r) = {c^{\mathrm{b}}_{i}}\,\mathrm{e}^{-z_i\phi(r) - \beta {\Delta \mu_{\mathrm{int},\,i}}(r) } \label{pb_ansatz0}$$ The distance-resolved electrostatic potential can be calculated from Eq.  together with the Poisson’s equation as $$\nabla^2 \phi(r) = -4\pi {l_\mathrm{B}}\left(\sum_{i} z_i c_i(r) + {z_\mathrm{s}}{c_\mathrm{s}}(r) \right) \quad \quad i = +\!+,+,- \label{pb-pot}$$ which establishes the PPB model including ion-specific binding effects. The boundary conditions used are $({{\mathop{}\!\mathrm{d}}}\phi/{{\mathop{}\!\mathrm{d}}}r)\left(r \rightarrow 0 \right) = 0$ and $({{\mathop{}\!\mathrm{d}}}\phi/{{\mathop{}\!\mathrm{d}}}r)\left(r \rightarrow R \right)= 0$. An effective radius for dPGS is calculated independently for this model (labeled ${r_\mathrm{eff}}^\mathrm{PB}$) using the Alexander prescription [@alexander1984charge; @Trizac2002; @bocquet2002effective; @Levin2004] on the obtained potential $\phi$, the same recipe used to calculate ${r_\mathrm{eff}}$ from simulations, *cf.* Section \[sim\_analysis\]. The values of ${r_\mathrm{eff}}^\mathrm{PB}$ for G$_2$-dPGS and G$_4$-dPGS are obtained as $1.42$nm and $2.36$nm, respectively, under the operated range of ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ and at ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}=150.37$mM. The ${r_\mathrm{eff}}^\mathrm{PB}$ values are thus found to be close to those obtained from the simulations, as shown in Table \[cg\_table\]. The effective surface potential of the macromolecule is then given by ${\phi_\mathrm{eff}}^\mathrm{PB} = \phi({r_\mathrm{eff}}^\mathrm{PB})$. The number of bound ions of species $i$ within ${r_\mathrm{eff}}$, is then given by $${N^\mathrm{b}_{i}}= \int_0^{{r_\mathrm{eff}}} c_i(r)\, 4 \pi r^2 {\mathop{}\!\mathrm{d}}r \quad \quad i=+,+\!+ \label{Nib}$$ The corresponding effective charge valency ${Z_\mathrm{eff}}^\mathrm{PB}$ is calculated using Eq. . The PPB equations are solved numerically, *cf.* Section \[numerical\]. \[mvh\] Manning–McGhee–von Hippel binding model (MMvH) ------------------------------------------------------ In this section, we introduce a model based on a discrete two-state (condensed or free) perspective for the counterions, built to capture the essential physics of polyelectrolyte–ion binding in an accurate but minimalistic fashion. The model is an extension of ideas by Manning, [@Manning2007] in which ion-condensation on charged spherical surfaces was described on a mean-field free energy level as a competition between the charging (Born) self-energy of the macromolecule in salt solution and the entropy cost of binding for one-component counterions. Here, we extend this model to the case of mixtures of MCs and DCs, including binding saturation for a fixed number of binding sites like in Langmuir isotherms. The extension of the latter to binary binding of one or two binding sites by mono- or divalent solutes, respectively, was put forward buy McGhee and von Hippel. [@Mcghee1974] Therefore, we name the model Manning–McGhee–von Hippel binding model (MMvH). Following Manning, [@Manning2007] we treat the macromolecule as an impenetrable sphere of radius ${r_\mathrm{eff}}$ and charge valency ${Z_\mathrm{d}}= z_s{N_\mathrm{s}}$ taken from simulations, and extend the Manning’s model into a discrete binding site model, where the ${N_\mathrm{s}}$ charged monomers act as a finite collection of discrete binding sites for both the MCs and DCs. For the case of the DCs, two adjacent charged monomers can collectively act as a single binding site for a DC. The resulting combinatorial ways to arrange the bound MCs and DCs lead to mixing entropies worked out by McGhee and von Hippel. [@Mcghee1974] Pertaining to the canonical ensemble, we fix the total number of salt ions $n_i$, the corresponding concentrations ${c^0_{i}}$ ($i= +\!+,+,-$), the number of monovalent counterions ${N_\mathrm{s}}$ to the macromolecule, the total number of binding sites on the macromolecule and the total domain volume $V$. The coions in this model simply serve the function of maintaining electroneutrality in the total domain and their explicit adsorption is neglected. A counterion $i$ ($=+,+\!+$) is assumed to bind to the macromolecule and to occupy $f_i$ consecutive (spatially adjacent) charged terminal groups of the macromolecule. We designate $f_{{\scalebox{.8}{$\scriptscriptstyle +$}}}=1$ and $f_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}=2$ for MCs and DCs, respectively, implying that, in a bound state, one MC occupies only one charged terminal group, while one DC occupies two consecutive charged terminal groups, owing to the fact that each terminal group has a charge valency ${z_\mathrm{s}}= -1$. Consider at a given state, ${N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ MCs and ${N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ DCs are bound to the macromolecule. The *binding density*, *i.e.*, the number of bound counterions per charged terminal group is then ${N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}/{N_\mathrm{s}}$ and ${N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}/{N_\mathrm{s}}$ for MCs and DCs, respectively. By multiplying with $f_i$, we then define the fraction of the binding sites occupied by the counterions, *i.e.*, coverages ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= f_{{\scalebox{.8}{$\scriptscriptstyle +$}}} {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}/ {N_\mathrm{s}}= {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}/ {N_\mathrm{s}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}= f_{{\scalebox{.8}{$\scriptscriptstyle ++$}}} {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}/{N_\mathrm{s}}= 2{N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}/{N_\mathrm{s}}$. Thus, the total number of binding sites on the macromolecule available for MCs, is $N_{{\scalebox{.8}{$\scriptscriptstyle +$}}} = {N_\mathrm{s}}/f_{{\scalebox{.8}{$\scriptscriptstyle +$}}} = {N_\mathrm{s}}$, and those available for DCs, is $N_{{\scalebox{.8}{$\scriptscriptstyle ++$}}} = {N_\mathrm{s}}/f_{{\scalebox{.8}{$\scriptscriptstyle ++$}}} = {N_\mathrm{s}}/2$. The effective charge valency of the macromolecule is then ${Z_\mathrm{eff}}= -{N_\mathrm{s}}+ {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}+ 2{N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}= {-{N_\mathrm{s}}(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})}$. The total Helmholtz free energy ${\mathcal{F}_\mathrm{tot}}$ depends on the coverages ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ and the ionic concentrations ${c^0_{i}}$. The coverages can then be obtained by minimizing ${\mathcal{F}_\mathrm{tot}}$ simultaneously with respect to ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$. The total Helmholtz free energy ${\mathcal{F}_\mathrm{tot}}$ is given by the expression $${\mathcal{F}_\mathrm{tot}}= {\mathcal{F}_\mathrm{el}}+ {\mathcal{F}_\mathrm{tr}}+ {\mathcal{F}_\mathrm{mix}}+ {\mathcal{F}_\mathrm{int}}\label{eq:mvh_ftot}$$ where the four additive contributions, ${\mathcal{F}_\mathrm{el}}$, ${\mathcal{F}_\mathrm{tr}}$, ${\mathcal{F}_\mathrm{mix}}$ and ${\mathcal{F}_\mathrm{int}}$ are defined respectively as (i) electrostatic (Born) self-energy of charge renormalized macromolecule, (ii) ideal gas entropy of free ions in the bulk regime, (iii) mixing entropy of the condensed counterions in the macromolecule, and (iv) the non-electrostatic ion-specific binding free energy between the condensed counterion and the corresponding binding site on the macromolecule. The Born charging self-energy of the macromolecule immersed in an electrolyte solution associated with the Debye screening length $\kappa^{-1}$, refers to the work required to charge the macromolecule from its electroneutral to a certain charged state. Following Manning, such a charged state is associated with the effective charge ${Z_\mathrm{eff}}e$, corresponding to the sum of the intrinsic bare charge of the macromolecule ${Z_\mathrm{d}}$ and its captive, neutralizing counterions. [@Manning2007] Thus, the expression for the Born charging free energy of the macromolecule (or the self energy of the charge renormalized macromolecule) per monovalent binding site is thus expressed as $$\beta {\mathcal{F}_\mathrm{el}}= \frac{{Z_\mathrm{eff}}^2 {l_\mathrm{B}}}{2 {N_\mathrm{s}}{r_\mathrm{eff}}(1+\kappa {r_\mathrm{eff}})} = \frac{\zeta}{2} (1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})^2 \label{fel1}$$ where $\zeta/2$ is the Born free energy per monovalent binding site in the absence of counterion condensation, and $\zeta$ is given for surface charging by[@mcquarrie2000statistical] $$\zeta = \frac{{N_\mathrm{s}}{l_\mathrm{B}}}{{r_\mathrm{eff}}(1 + \kappa{r_\mathrm{eff}})} \label{zeta}$$ Considering the effective volume of dPGS ${v_\mathrm{eff}}$ to be very small compared to the total volume $V$ (${v_\mathrm{eff}}\ll V$), the bulk concentrations of MCs and DCs are given by $$\begin{aligned} \begin{split} &{c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= {c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}+ \frac{{N_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}})}{V} \\ &{c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}= {c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}- \frac{{N_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{V} \end{split} \label{eq:bulk_conc_mvh}\end{aligned}$$ owing to the depletion of the ions in the bulk due to partitioning. ${c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ above is calculated considering the monovalent counterions remaining in the solution, in the salt-free limit. We assume that no anions are bound to the macromolecule binding sites, hence their bulk concentration is assumed to be the same as their salt concentration, *i.e.*, ${c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle -$}}}}= {c^0_{{\scalebox{.8}{$\scriptscriptstyle -$}}}}$. The ideal gas free energy of free cations in the bulk, normalized by the number of monovalent binding sites ${N_\mathrm{s}}$, is given as\ $$\begin{aligned} \begin{split} &\beta {\mathcal{F}_\mathrm{tr}}= -\frac{S_\mathrm{id}}{{N_\mathrm{s}}{k_\mathrm{B}}} = \sum_{i=+,++} \left(\frac{n_i - {N^\mathrm{b}_{i}}}{{N_\mathrm{s}}}\right)\left({\mathrm{ln}}\,{c^{\mathrm{b}}_{i}}\Lambda^3_i -1\right) \\ &= \sum_{i=+,++} \left(\frac{n_i - N_i {\Theta_{i}}}{{N_\mathrm{s}}}\right)\left[{\mathrm{ln}}\left({c^0_{i}}\Lambda^3_i - \frac{N_i {\Theta_{i}}\Lambda_i^3}{V}\right) -1\right] \end{split}\end{aligned}$$ where $\Lambda_i$ and $n_i$ are the thermal (de Broglie) wavelength and the number of salt ions $i$. The bound DCs and MCs can occupy the binding sites on the macromolecule in different proportions, and can distribute among the occupied sites in multiple ways at a certain bound coverages ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$. We exert constraints to such possibilities of binding compositions and configurations, such that, ($i$) one bound DC can only bind to two adjacent monovalent binding sites, ($ii$) all non-overlapping configurations between the bound ions are possible, ($iii$) there are no designated binding sites for DCs, and ($iv$) the position of the bound DC can be shifted by a single adjacent monovalent binding site. The number of possible combinatorial binding arrangements under these constraints, adopted from the work by McGhee and von Hippel, [@Mcghee1974] is given by $$W = \frac{ \gamma^{{N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}} \gamma^{{N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}} ({N_\mathrm{s}}- {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})!}{{N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}!{N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}!({N_\mathrm{s}}- 2{N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}- {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}})!} \label{w}$$ where we define $\gamma_i={v^0_{i}}/\Lambda^3_i$ in terms of the effective configurational volume ${v^0_{i}}$ in the bound state. [@XiaoCPS] ${v^0_{i}}$ takes into account the rotational and vibrational degrees of freedom of a bound counterion $i$. We now define the free energy associated with the partition function $W$, normalized by the number of monovalent binding sites ${N_\mathrm{s}}$, as the free energy of mixing of the bound ions per binding site, $$\begin{aligned} \begin{split} &\beta {\mathcal{F}_\mathrm{mix}}= -\frac{S_\mathrm{mix}}{{N_\mathrm{s}}k_{\rm B}}= -\frac{1}{{N_\mathrm{s}}}{\mathrm{ln}}\, W\\ & \simeq {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}{\mathrm{ln}}{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}+ \frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{2}{\mathrm{ln}}\frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{2} -\left(1-\frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{2}\right){\mathrm{ln}}\left(1-\frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{2}\right) \\ &+ (1 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}){\mathrm{ln}}(1 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}) \\ &- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}{\mathrm{ln}}\frac{{v^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}{\Lambda_+^3} - \frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{2} {\mathrm{ln}}\frac{{v^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{\Lambda_{++}^3} \end{split} \label{fmx}\end{aligned}$$ where the Stirling approximation has been used for the logarithm of the factorials. This description of condensed counterion entropy is different than the ion-binding models proposed in previous works for linear polyelectrolytes [@flory1953molecular; @Raphael1990; @Muthukumar2004] in terms of the localization of counterions within volume ${v^0_{i}}$. We express this intrinsic interaction ${\mathcal{F}_\mathrm{int}}$ by the intrinsic binding chemical potential ${\Delta \mu_{\mathrm{int},\,i}}$ of each bound ion $i$. The sum of such interactions for all bound ions, normalized by the total number of monovalent binding sites gives $$\begin{aligned} \begin{split} \beta {\mathcal{F}_\mathrm{int}}&= \frac{1}{{N_\mathrm{s}}}\left({N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\beta {\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}+ {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}\beta {\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}\right) \\ &= {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\beta {\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}+ \frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{2}\beta {\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}\end{split}\end{aligned}$$ The equilibrium coverages $\Theta_i$ are then obtained by the minimization condition $$\frac{\partial}{\partial {\Theta_{i}}} {\mathcal{F}_\mathrm{tot}}\overset{!}{=} 0 \qquad \quad i=+,+\!+$$ This leads to the relation $${\Delta \mu_{\mathrm{tr},\,i}}+ {\Delta \mu_{\mathrm{el},\,i}}+ {\Delta \mu_{\mathrm{mix},\,i}}+ {\Delta \mu_{\mathrm{int},\,i}}= 0 \quad \quad i=+,+\!+ \label{k_and_g_all}$$ where ${\Delta \mu_{\mathrm{tr},\,i}}$ denotes the translational entropy change associated with one ion $i$ when it transfers from the bulk environment to the bound state in the macromolecule. ${\Delta \mu_{\mathrm{el},\,i}}$ is the *electrostatic binding chemical potential* and ${\Delta \mu_{\mathrm{mix},\,i}}$ is the *mixing chemical potential*. Eq. , similar to the PPB (Eq. ) and DM (Eq. ) models, indicates the counterion chemical potential components contributing to its condensation on the macromolecule. The expressions for the constituent chemical potential contributions in Eq.  are given by $$\begin{aligned} \begin{split} &\beta {\Delta \mu_{\mathrm{tr},\,i}}=-{\mathrm{ln}}\,{c^{\mathrm{b}}_{i}}{v^0_{i}}\qquad \qquad \qquad \quad i=+,+\!+\\ &\beta {\Delta \mu_{\mathrm{el},\,i}}=-z_i\zeta(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}) \quad \quad i=+,+\!+\\ &\beta {\Delta \mu_{\mathrm{mix},\,i}}= \begin{cases} \begin{aligned} {\mathrm{ln}}\frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}\left(2-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}\right)}{4(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})^2} \quad \quad i=++\\[1.1ex] {\mathrm{ln}}\frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}{(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})} \quad \quad i=+ \end{aligned} \end{cases} \end{split} \label{all_free_energies}\end{aligned}$$ Using Eqs.  and leads to the final form of the MMvH model, given by $$K_{{\scalebox{.8}{$\scriptscriptstyle ++$}}} = v^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}} {\mathcal{K}_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}\mathrm{e}^{2\zeta \left( 1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}\right)} = \frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}(2 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})}{4 {c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}{(1 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})}^2} \label{Kb}$$ $$K_{{\scalebox{.8}{$\scriptscriptstyle +$}}} = v^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}} {\mathcal{K}_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}\mathrm{e}^{\zeta \left(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}\right)} = \frac{ {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}{{c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}(1 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})} \label{Ka}$$ where $K_i$ are the *equilibrium binding constant* associated with the binding of ion $i$ to its corresponding binding site on the macromolecule. The relationship between $K_i$, the total binding chemical potential ${\Delta \mu_{\mathrm{bind},\,i}}$ and the total partition ratio ${\mathcal{K}_{i}}$ is given as $$\beta {\Delta \mu_{\mathrm{bind},\,i}}= -{\mathrm{ln}}\,\frac{K_i}{{v^0_{i}}} = -{\mathrm{ln}}\,{\mathcal{K}_{i}}\quad \quad i=+,+\!+ \label{kbind}$$ Or in other words, referring back to Eq. , $${\mathcal{K}_{i}}= {\mathcal{K}_{\mathrm{int},\,{i}}}\,{\mathcal{K}_{\mathrm{el},\,{i}}}= {\mathcal{K}_{\mathrm{int},\,{i}}}\, \mathrm{e}^{z_i \zeta \left(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}\right)}$$ where the electrostatic contribution of the total partition ratio is defined as $${\mathcal{K}_{\mathrm{el},\,{i}}}= \mathrm{e}^{-\beta {\Delta \mu_{\mathrm{el},\,i}}} = \mathrm{e}^{z_i\zeta(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})} \quad \quad i=+,+\!+ \label{kiel-mvh}$$ From Eq. , for a given magnitude of $K_i$, the absolute magnitude of ${\Delta \mu_{\mathrm{bind},\,i}}$ depends on ${v^0_{i}}$, which we calculate from our simulations and predict respective values of ${\Delta \mu_{\mathrm{bind},\,i}}$. Finally, we consider the limit of the MMvH model for vanishing DCs (MCs only). Without DCs, we have $$\begin{aligned} \begin{split} &\beta {\Delta \mu_\mathrm{tr}}=-{\mathrm{ln}}\,{c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}v^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}} \\ &\beta {\Delta \mu_\mathrm{el}}=-\zeta(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}) \\ &\beta {\Delta \mu_\mathrm{mix}}= {\mathrm{ln}}\,\frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}{(1 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}})} \label{g_langmuir} \end{split}\end{aligned}$$ Combining Eqs.  and leads to the “Manning–Langmuir" (ML) model $$K_{{\scalebox{.8}{$\scriptscriptstyle +$}}} = v^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}} {\mathcal{K}_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}\mathrm{e}^{\zeta (1 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}})} = \frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}{{c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}(1 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}})} \label{manning-langmuir}$$ The McGhee–von Hippel combinatorics here reduces to the standard one-component Langmuir picture, *i.e.*, the right-hand-side of Eq.  reflects the Langmuir isotherm. The standard Langmuir model is thus extended to include charging free energies by ion condensation (charge renormalization) and ion-specific binding. From another perspective, it extends the Manning model for the counterion condensation on spheres [@Manning2007; @Gillespie2014] to include ion-specific effects as well as the saturation of binding sites in terms of the translation entropy of the condensed ions. Numerical evaluation {#numerical} -------------------- The PPB model, with the assumption of the uniform intrinsic macromolecular volume charge distribution ${c_\mathrm{s}}(r)e$ and with the knowledge of the bare radius ${r_\mathrm{d}}$ of the macromolecule inherited from simulations, generates the distance-resolved number density profiles of charged species, similar to Fig. \[density\_ions\]. Hence, it performs the same analysis as that for simulations (*cf.* Section \[sim\_analysis\]), to calculate the effective radius ${r_\mathrm{eff}}$ and other electrostatic properties of the macromolecule, such as ${Z_\mathrm{eff}}$, ${\phi_\mathrm{eff}}$, etc. The DM model also assumes uniform ${c_\mathrm{s}}(r)e$ and requires the knowledge of the electroneutrality radius, which is taken as ${r_\mathrm{d}}$ from simulations as an input parameter, similar to the PPB model. The MMvH (ML) model, on the other hand, assumes the macromolecule as a hard sphere with a uniform surface charge distribution. The effective radius of the hard sphere ${r_\mathrm{eff}}$ is taken from simulations as an input parameter. The results from the DM, PPB and MMvH (ML) models and simulations are compared in terms of the coverages ${\Theta_{i}}$ ($i=+\!+, +$), which are defined as ${\Theta_{i}}={N^\mathrm{b}_{i}}/{N_{i}}$, where ${N^\mathrm{b}_{i}}$ is the number of condensed counterions $i$ and ${N_{i}}$ is the corresponding number of binding sites available on dPGS, defined in Section \[mvh\]. Since the PPB model deals with a volume sorption, while the DM model deals with the ion partitioning between two electroneutral phases, “coverage" ${\Theta_{i}}$ in these cases are interpreted as a load or an extent of neutralization of dPGS. For the DM, PPB and MMvH (ML) models, the intrinsic partition coefficients ${\mathcal{K}_{\mathrm{int},\,{i}}}$ for both ions ($i=+\!+,+$) are unknowns and taken as fitting parameters in order to match the coverages from the simulations, which are described in Section \[sec:cg\_ff\]. Regarding the PPB and DM models, we make a further assumption that intrinsic non-electrostatic ion–binding site interaction for the MCs is identical to that for the monovalent anions, *i.e.*, ${\mathcal{K}_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}= {\mathcal{K}_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle -$}}}}}$. Mathematically, the PPB model represents a boundary-value problem having a second order differential equation (Eq. ) non-linear in the electrostatic potential paired with the boundary conditions, while the MMvH model (Eqs.  and ) represents two non-linear simultaneous equations in coverages ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$. Both PPB and MMvH models are evaluated self consistently for the potential and coverages, respectively. To solve Eq. , we employ the `solve_bvp` function in the SciPy library (version 1.3.1) from Python (version 3.7.4), which solves a boundary-value problem for a system of ordinary differential equations using the fourth order collocation algorithm. [@kierzenka2001bvp] The bulk concentration ${c^{\mathrm{b}}_{i}}$ is obtained using the law of conservation of mass, in an iterative manner. Eqs.  and are solved using `fsolve` function from the SciPy library, which is also used to evaluate the DM model (Eq. ) representing the single non-linear equation in the Donnan potential ${\phi^{}_\mathrm{D}}$. The effective configurational volume ${v^0_{i}}$ of bound counterions, used in the MMvH model is assumed to be equal for both counterions, *i.e.*, $v_{{\scalebox{.8}{$\scriptscriptstyle ++$}}} = v_{{\scalebox{.8}{$\scriptscriptstyle +$}}} = v_0$. It is worth considering that the volume $v_0$ depends on the precise nature of the bound state and it is infeasible to have its knowledge in experiments due to unknown microscopic details, although it can be computed using simulations. [@xu:biomacro; @Yu2015] According to the convention in experiments, the standard volume is defined as $v_0 = 1\,\mathrm{M}^{-1} \simeq 1.6$nm$^{3}$, corresponding to the standard concentration $c^\mathrm{std} = 1$M. [@atkins; @Gilson2007; @General2010a] In this case, the total binding chemical potential ${\Delta \mu_{\mathrm{bind},\,i}}$ can be referred to as the standard binding energy $\Delta G^0$. [@Gilson2007; @General2010a] Results and Discussion ====================== Monovalent limit: theoretical comparison and best fit to simulations -------------------------------------------------------------------- {width="12cm" height="12cm"} Considering the monovalent limit as reference case, we now start with the application of aforementioned theoretical binding models. Fig. \[monocg\](a) shows the predictions of the PPB and ML (monovalent-only limit of MMvH) models for the variation of the binding coverage of MCs, ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$, as a function of the MC concentration, ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$. It can be observed that ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ increases sharply for a small increase in ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ from $0$ to $\sim 10$mM, while it increases gradually for larger ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$. This is attributed to the combined contribution of the electrostatics and an entropy of a bound counterion, facilitating condensation. In the low ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ regime, the bare charge of G$_2$-dPGS is weakly renormalized, and some of the dPGS binding sites are unoccupied. This leaves a high propensity of condensation for new incoming counterions. This can be conveniently explained via the ML model. Referring to Eq. , the increase in the condensation of MCs at the limit of low ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$, $\lim_{{c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\to 0} {{\mathop{}\!\mathrm{d}}}{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}/{{\mathop{}\!\mathrm{d}}}{c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ is directly proportional to the total binding constant $K_{{\scalebox{.8}{$\scriptscriptstyle +$}}}$, while at high ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$, $\lim_{{c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\to \infty} {{\mathop{}\!\mathrm{d}}}{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}/{{\mathop{}\!\mathrm{d}}}{c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= 0$. This implies that at low ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$, the resultant low coverage ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ leads to a high electrostatic driving force for condensation as well as entropy of a bound counterion, thus a high amount of condensation. On the other hand, at high ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$, the macromolecule charge is almost entirely renormalized and most of the binding sites are occupied, resulting in hardly any increase in condensation. Comparing the coverage profiles from PPB and ML models that neglect ion-specific effects, *i.e.*, with ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}=0$ (dotted curves), we find that the PPB coverage values are close to the ML values in the low ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ regime, however, attain higher values than the ML counterpart at high ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$. This is attributed to the effects of discrete binding sites incorporated in the ML model, in the form of the configurational volume $v_0$ (here, we used $v_0=1.04$ M$^{-1}$ obtained from our previous simulations [@xu:biomacro]). The PPB model, on the other hand, assumes the condensed ions as point charges, leaving no entropic penalty for new incoming counterions as they condense on the binding sites. Another reason is that the PPB model also incorporates, to some extent, the non-linear effects in the electrostatic interactions, which are not considered in the DH-level Born energy used in the ML model. Both models, however, underestimate the simulations if we do not include corrections via ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$. The reason is likely the approximative treatments of the electrostatic energy in both models, PPB and ML, which are mean-field and do not include the discrete nature of the charged binding sites and the complex spatial charge correlations inside the macromolecule. The DM model, in addition to these assumptions, takes the macroscopic view of macromolecule and bulk phases in a segregated form. The model then predicts the ion partitioning while imposing electroneutralities of phases. In that respect, for highly charged macromolecules like dPGS, the DM model predicts ${N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\simeq {N_\mathrm{s}}$, implying ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\simeq 1$. This plot is not shown, since it does not provide a useful insight for us in the context of counterion condensation. The case of salt concentration ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= 0$ is referred to as the counterion-only case, and gives ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\sim 0.28$ for the PPB model. Note that ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ in this limit is system specific, since the size of the simulation box/computational domain determines the counterion concentration and subsequently the coverage. The coverage ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ in the ML model in this limit is undefined, since the electrostatic binding energy of MCs depends on the screening length $\kappa^{-1}$, which is undefined in this model in the absence of the salt. In the next step, ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ values for PPB and ML models are fitted (bold curves in Fig. \[monocg\](a)) to the simulation results for G$_2$-dPGS in the monovalent limit by allowing ion-specific effects in the counterion–macromolecule binding, *i.e.*, ${\Delta \mu_{\mathrm{int},\,{+}}}$ as a fitting parameter. The values of ${\Delta \mu_{\mathrm{int},\,{+}}}$ are found to be $-0.45\,{k_\mathrm{B}T}$ and $-1.81\,{k_\mathrm{B}T}$ for PPB and ML models, respectively. Recall that the simulations have not really included ion-specific effects in terms of specific hydration phenomena, etc., still, they include excluded-volume, dispersion attraction, and importantly, all electrostatic charge–charge correlations, not captured in the mean-field theories. Hence, the ion-specific fitting parameters can be viewed in general as correction factors, including all ionic contributions that are beyond the mean-field treatment of the PPB and ML models. The larger fitting parameter for ML than PPB (in the absolute value) may indicate the higher level of approximations in the ML model. Having the models now informed using the benchmark data from simulations, they can be utilized to predict the binding at other ion concentrations. Fig. \[monocg\](b) shows the numerical fitting of ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ values (dashed curves) to those obtained from G$_4$-dPGS simulations. The values of ${\Delta \mu_{\mathrm{int},\,{+}}}$ as a fitting parameter are $-0.56\,{k_\mathrm{B}T}$ and $-1.85\,{k_\mathrm{B}T}$ for PPB and ML models, respectively, which are close to those obtained for G$_2$-dPGS, within the error difference of $\sim 0.1\,{k_\mathrm{B}T}$. The ML model fits better to both G$_2$-dPGS and G$_4$-dPGS CG results than the PPB model at large ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$, which may indicate that the dPGS charge in the simulations acts more as finite binding sites, as assumed in the ML model. Divalent case: theoretical comparison and best fit to simulations ----------------------------------------------------------------- {width="14cm" height="14cm"} We now aspire to use the obtained ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$ to inform the MMvH and PPB models with the help of the reference data obtained from simulations, in order to capture the competitive ion binding in a mixture of MCs and DCs. The models fitted to the benchmark data can then be used to predict the binding coverages ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ for different dPGS generations and salt concentrations. In practice, we perform the numerical fitting of ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ obtained from the MMvH and PPB models to those from simulations, by fixing ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$ for MCs obtained from the monovalent-only case, and then subsequently fitting ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}$ for DCs. The values of ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$ for MCs obtained from the monovalent limit are, for a given binding model (ML or PPB), found to be approximately independent of the dPGS generation (with $\sim 0.1\,{k_\mathrm{B}T}$ as margin of error). Therefore, ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$ is averaged over generations (G$_2$ and G$_4$), as shown in Table \[mvh\_ppb\_kint\_table\]. Fig. \[divalent\] depicts the behavior of MMvH, PPB and the DM model in terms of the binding coverages ${\Theta_{i}}$, in a mixture of DCs and MCs. The MMvH model uses the effective configurational volumes $v_0=1.04$ M$^{-1}$ and $0.57$ M$^{-1}$ for G$_2$-dPGS and G$_4$-dPGS, respectively, as obtained from our previous simulations. [@xu:biomacro] At low DC concentration, *i.e.* in the monovalent limit, MCs act as the only counterions to the macromolecule, resulting in the highest MC coverage ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$. In this limit at ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= 150.37$mM, both MMvH and PPB models show ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\simeq 0.57$ for G$_2$-dPGS, and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\simeq 0.8$ for G$_4$-dPGS. As ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ increases, more DCs bind to the macromolecule and more of the previously bound MCs get released into the bulk. Table \[cg\_table\] shows the resultant effective charge valency ${Z_\mathrm{eff}}^\mathrm{PB}$ and potential ${\phi_\mathrm{eff}}^\mathrm{PB}$ of G$_2$-dPGS and G$_4$-dPGS evaluated by the PPB model. Quantitatively consistent with the ${Z_\mathrm{eff}}$ and ${\phi_\mathrm{eff}}$ obtained from simulations, ${Z_\mathrm{eff}}^\mathrm{PB}$ and ${\phi_\mathrm{eff}}^\mathrm{PB}$ show a strong decrease in magnitude with a higher ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$, depicting higher dPGS charge renormalization. [cdd]{} &\ & ++ & +\ & 5.13 & 3.37\ & -1.87 & -0.50\ ($v_0$ [CG]{}) & -2.85 & -1.83\ ($v_0$ [Std.]{}) & -2.86 & -1.44\ {width="11.8cm" height="11.8cm"} Corresponding to the fitting of binding coverages ${\Theta_{i}}$ on G$_2$-dPGS and G$_4$-dPGS binding sites, as shown in Fig. \[divalent\], the resulting ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}$ values are calculated as $-2.73\,{k_\mathrm{B}T}$ (G$_2$) and $-2.98\,{k_\mathrm{B}T}$ (G$_4$) for the MMvH model, whereas $-1.77\,{k_\mathrm{B}T}$ (G$_2$) and $-1.98\,{k_\mathrm{B}T}$ (G$_4$) for the PPB model. Table \[mvh\_ppb\_kint\_table\] shows the values of ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}$ averaged over G$_2$-dPGS and G$_4$-dPGS cases. It can be observed that both ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}$ and ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$ values from the MMvH model exceed (in magnitude) those from the PPB model across the whole ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}\sim 0-25$mM range. This can again be attributed to higher approximations in the electrostatic partition coefficient designed in the MMvH model, based on the Debye–Hückel charging free energy, as compared to that from the PPB model, incorporating non-linear effects in the electrostatic potential near the macromolecule vicinity. The standard intrinsic chemical potentials ${\Delta \mu_{\mathrm{int},\,i}}^0$ after fitting the MMvH model ${\Theta_{i}}$ with those from simulations are also given in Table \[mvh\_ppb\_kint\_table\]. Unlike the other models, we simultaneously fit both ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$ and ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}$ to perform numerical fitting of ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ obtained from the DM model with the simulation data. As shown in Table \[mvh\_ppb\_kint\_table\], the values of ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$ and ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}$ for the model turn out large and positive compared with those from other models, since the DM model tries to neutralize the entire dPGS charge via the electroneutrality condition in the dPGS phase. The DM fits for ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ differ to an extent with those from simulations, while those for ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ are found to be reasonably good. The DM provides much better fits for ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ in the case of G$_4$-dPGS as compared to G$_2$-dPGS. This is attributed to the bigger size of G$_4$-dPGS, which better satisfies the criterion $\kappa {r_\mathrm{d}}\gg 1$, under which the DM electroneutrality condition holds comparatively well. Having established the model frameworks by informing ${\Delta \mu_{\mathrm{int},\,i}}$ by fitting the coverages ${\Theta_{i}}$ to those from simulations and averaging the values of obtained ${\Delta \mu_{\mathrm{int},\,i}}$ over generations (See Table \[mvh\_ppb\_kint\_table\]), we finally utilize their predictive ability to explore the electrostatic characterization of dPGS for different generations and salt concentrations. As an example, Fig. \[prediction\](d) shows the MMvH model predictions for the binding coverages ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ for the case of a competitive ion binding on G$_6$-dPGS, similar to Fig. \[divalent\] on G$_2$-dPGS and G$_4$-dPGS. We also study the effective charge valency ${Z_\mathrm{eff}}$ of dPGS along with the composition of condensed ions on the molecule. Figs. \[prediction\](a) and \[prediction\](b) show the variation of the effective charge valency ${Z_\mathrm{eff}}$ of G$_2$-dPGS and its normalized form ${Z_\mathrm{eff}}/{Z_\mathrm{d}}$, respectively, as a function of the DC concentration ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$, as predicted by the MMvH model. It can be clearly seen from Fig. \[prediction\](a) that the introduction of DCs leads to a net charge renormalization of dPGS, which further decreases its ${Z_\mathrm{eff}}$. The inset shows that, with reference to the monovalent limit, the dPGS effective charge is $30-35\%$ further renormalized upon introducing DCs in the range of $1-4$mM, which is close to the physiological concentration range for calcium(II) ions. Fig. \[prediction\](b) shows that the fraction of the bare dPGS charge that gets renormalized increases with the dPGS generation. The inset shows the variation for ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ varying from $0$mM to $10$mM. The rate of dPGS charge renormalization with respect to ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ is the highest at the low ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ regime and subsides as ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ increases, since the charge renormalized dPGS results in lower electrostatic binding chemical potential ${\Delta \mu_{\mathrm{el},\,i}}$. The reduced amount of renormalization is not attributed to the ion packing, which is evident from Fig. \[prediction\](c) showing the total number of condensed ions (including both DCs and MCs) per dPGS sulfate group. As ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ increases, the total number of condensed ions decreases, indicating that the ion packing effects diminish as ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ increases. The decrease in the amount of renormalization thus predominantly has electrostatic origin. Fig. \[prediction\](a) shows that $80-90\%$ of the dPGS bare charge is renormalized as ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ increases from $0-100$mM, however, the total number of condensed counterions effectively decreases, according to Fig. \[prediction\](c). This in effect would significantly hamper the binding affinity of protein with dPGS. It has been well established through our previous works that the dPGS–protein complexation is dominantly influenced by the release of a few MCs that were highly confined due to strong charge renormalization. [@xu:biomacro] The introduction of DCs, however, decreases the confinement of these condensed counterions, thus less counterions to be released during dPGS-protein binding. In addition, the strongly charge renormalized dPGS leads to lower electrostatic contribution to its overall binding affinity with the protein or any other multivalent ligand. Conclusion ========== In this paper, we have addressed the biologically and industrially relevant problem of the competitive sorption of mono- and divalent counterions into a highly charged globular polyelectrolyte, with direct comparison to CG simulations of the dendritic macromolecule dPGS. Beyond simple Donnan and ion-specific penetrable PB models, we introduced a two-state discrete binding site model (MMvH) applicable for heterogeneous ligand systems (counterions with mixed valencies/stoichiometries). The broad classification of surrounding counterions as “bound" and “free" gives the MMvH model a computationally unique advantage over the PPB model, which involves the calculation of the distance-resolved counterion density profiles. The fitting results with simulations highlight the key differences in the MMvH and PPB models. Although being on a mean-field level, the PPB model incorporates non-linear electrostatic effects, which become more prominent near the surface of dPGS, delivering a relatively accurate picture of the dPGS–counterion electrostatic binding affinity, compared to the MMvH model, which approximates dPGS–counterion electrostatic interaction on a linearized PB (DH) level by absorbing these non-linear electrostatic effects into the effective charge valency ${Z_\mathrm{eff}}$ of dPGS. On the contrary, the MMvH model provides more accurate values of the extent of counterion adsorption $\Theta$ at high concentrations (*i.e.*, in the binding site saturation regime) than the PPB model. The reason is that the MMvH model assumes discrete binding sites, whereas the PPB model treats dPGS charge as continuum and allows an unlimited uptake of counterions, which is not realistic. Future extensions of the MMvH model could include an extra level of competition between adsorbed ions explicitly, namely through a non-linear term in Eq.  (of the type used in the regular solution theory or the Flory–Huggins approximation in polymer theories) that describes the interaction between two adsorbed ions in proximal positions (sites). The effects of this generalization in a different context can be found in a study on ion induced lamellar-lamellar phase transition in charged surfactant systems. [@Harries2006] In general, this type of competition results in non-continuous adsorption equilibria and could be interesting in the present context. The simplest presented model, the Donnan model (DM) extended for ion-specific effects, is also useful for a quick, qualitative prediction of the adsorption ratio. Per construction it should become more accurate for large globules and/or large salt concentrations (for which the globule size becomes larger than the DH screening length), where the electroneutrality condition is better justified. The models presented in this work can be used to accurately extrapolate and predict the competitive ionic sorption in experiments for a wide range of salt concentrations and salt compositions. They can be also easily generalized to more ionic components and valencies. The electroneutrality radius required for the DM model and the intrinsic macromolecular charge distribution required for the PPB model as an input parameter (in the form of the bare radius ${r_\mathrm{d}}$), are taken from simulations. However, they can also be derived by measuring the form factors from, *e.g.*, neutron scattering. [@Boris1996; @Berndt2006] The MMvH (ML) model requires the effective radius ${r_\mathrm{eff}}$ of the macromolecule as an input parameter, which besides simulations, can also be derived from independent experiments such as electrophoresis and fitting structure factors (of non-dilute colloidal suspensions) by DLVO interactions. [@hunter; @israelachvili2011intermolecular] As we showed, ${r_\mathrm{eff}}$ can also be obtained using PB models and related theories provided the intrinsic macromolecular charge distribution is available. The authors are indebted to Matthias Ballauff for insightful discussions. R.N. thanks Jacek Walkowiak for fruitful discussion. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 646659). X. 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**Quantum Kaluza-Klein Cosmologies (V)** Zhong Chao Wu Dept. of Physics Beijing Normal University Beijing 100875, P.R. China **Abstract** In the No-boundary Universe with $d=11$ supergravity, under the $S_n \times S_{11-n}$ Kaluza-Klein ansatz, the only seed instanton for the universe creation is a $S_7 \times S_4$ space. It is proven that for the Freund-Rubin, Englert and Awada-Duff-Pope models the macroscopic universe in which we are living must be 4- instead of 7-dimensional without appealing to the anthropic principle. PACS number(s): 98.80.Hw, 11.30.Pb, 04.60.+n, 04.70.Dy Key words: quantum cosmology, Kaluza-Klein theory, supergravity, gravitational instanton In a series of papers \[1\] the origin of the dimension of the universe was investigated for the first time in quantum cosmology. As far as I am aware, in the No-Boundary Universe \[2\], the only way to tackle the dimensionality of the universe is through Kaluza-Klein cosmologies. In the Kaluza-Klein model with $d=11$ supergravity, under the $S_n \times S_{11-n}$ ansatz, it has been shown that the macroscopic universe must be 4- or 7-dimensional. The motivation of this paper is to prove that the universe must be 4-dimensional. In $d=11$ simple supergravity, in addition to fermion fields, a 3-index antisymmetric tensor $A_{MNP}$ is introduced into the theory by supersymmetry \[3\]. In the classical background of the $WKB$ approximation, one sets the fermion fields to vanish. Then the action of the bosonic fields can be written $$\bar{I}= \int \sqrt{-g_{11}}\left ( \frac{1}{2} R - \frac{1}{48} F_{MNPQ}F^{MNPQ} + \frac{\sqrt{2}}{6\cdot (4!)^2} \eta^{M_1M_2\cdots M_{11}}F_{M_1M_2M_3M_4}F_{M_5M_6M_7M_8}A_{M_9M_{10}M_{11}} \right )d^{11}x,$$ where $$F_{MNPQ} \equiv 4! \partial_{[M}A_{NPQ]},$$ $$\eta^{A\cdots N} = \frac{1}{\sqrt{-g_{11}}} \epsilon^{A\cdots N}$$ and $R$ is the scalar curvature of the spacetime with metric signature $(-, +, +, \cdots +)$. The theory is invariant under the Abelian gauge transformation $$\delta A_{MNP} = \partial_{[M}\zeta_{NP]}.$$ It is also noticed that the action is invariant under the combined symmetry of time reversal with $A_{MNP} \rightarrow -A_{MNP}$. The field equations are $$R_{MN} - \frac{1}{2}Rg_{MN} = \frac{1}{48} (8F_{MPQR}F_N^{\;\;\;PQR} -g_{MN}F_{SPQR}F^{SPQR}),$$ and $$F^{MNPQ}_{\;\;\;\;\;\;\;\;\;;M}= \left [\frac{-\sqrt{2}}{2\cdot(4!)^2 }\right ]\cdot \eta^{M_1 \cdots M_8NPQ}F_{M_1\cdots M_4}F_{M_5\cdots M_8}.$$ At the $WKB$ level, it is believed that the Lorentzian evolution of the universe originates from a compact instanton solution, i.e. a stationary action solution of the Euclidean Einstein and other field equations. In order to investigate the origin of the dimension of the universe, we are trying to find the following minisuperspace instantons: the $d=11$ spacetime takes a product form $S_n\times S_{11-n}$ with an arbitrary metric signature and all components of the $F$ field with mixed indices in the two factor spaces to be zero. In the factor space $S_n \;(n =1,2,3)$ the $F$ components must be vanish due to the antisymmetry of the indices. Then $F$ must be a harmonic in $S_{11-n}$ since the right hand side of the field equation (6) vanishes. It is known in de Rham cohomology that $H^4(S_4) =1$ and $H^4(S_m) =0 \;\;(m\neq 4)$. So there is no nontrivial instanton for $n = 1,2,3$. For $n=5,6$, both $F$ components in $S_5$ and $S_6$ must be harmonics and so vanish. By the dimensional duality, there does not exit nontrivial instanton either for $n= 10, 9, 8$. The case $S_4 \times S_7$ is the only possibility for the existence of a nontrivial instanton, the $F$ components must be a harmonic in $S_4$, but do not have to in $S_7$. The no-boundary proposal and the ansatz are very strong, otherwise the nonzero $F$ components could live in open or closed $n$-dimensional factor spaces $( 4\leq n\leq 10)$ \[1\]. Four compact instantons are known, their Lorentzian versions are the Freund-Rubin, Englert, Awada-Duff-Pope and Englert-Rooman-Spindel spaces \[4\]\[5\]\[6\]\[7\]. They are products of a 4-dimensional anti-de Sitter space and a round or squashed 7-sphere. These spaces are distinguished by their symmetries from other infinitely many solutions with the same $F$ field. From now on, Greek letters run from 0 to 3 for the indices in $S_4$ and small Latin letters from 4 to 10 for the indices in $S_7$. One can analytically continue the $S_7$ or $S_4$ space at the equator to form a 7- or 4-dimensional de Sitter or anti-de Sitter space, which is identified as our macroscopic spacetime, and the $S_4$ or $S_7$ space as the internal space. One may naively think, since in either case the seed instanton is the same, that the creation of a macroscopic 7- or 4-dimensional universe should be equally likely. However, a closer investigation shows that this is not the case, it turns out that the macroscopic universe must be 4-dimensional, regardless whether the universe is habitable. The Freund-Rubin is of the $N=8$ supersymmetry \[4\]. Here the only nonzero $F$ components are in the $S_4$ factor space of the instanton $$F_{\mu \nu \sigma \delta} = i\kappa \sqrt{g_4}\epsilon_{\mu \nu \sigma \delta },$$ where $g_4$ is the determinant of the $S_4$ metric, the $F$ components are set imaginary in $S^4$ such that their values become real in the anti-de Sitter space, which is an analytic continuation of the $S_4$ space, as shown below. The $F$ field plays the role of an anisotropic effective cosmological constant, which is $\Lambda_7 = \kappa^2/3$ for $S_7$ and $\Lambda_4 = - 2\kappa^2/3$ for $S_4$, in the sense that $R_{mn} = \Lambda_7 \; g_{mn}$ and $R_{\mu \nu} = \Lambda_4 \; g_{\mu \nu}$, respectively. The $S_4$ space must have radius $r_4 = (3/\Lambda_4)^{1/2}$ and metric signature $(-,-,-,-)$, while the $S_7$ space is of radius $r_7 =(6/\Lambda_7)^{1/2}$ and metric signature $(+,+, \cdots +)$. Since the metric signature of the factor space $S_4$ is not appropriate, one has to analytically continue the $S_4$ manifold into an anti-de Sitter space with the right metric signature $(-,+,+,+)$. The $S_4$ metric can be written $$ds_4^2= -dt^2 - \frac{3}{\Lambda_4} \sin^2\left (\sqrt{\frac{\Lambda_4}{3}}t \right )(d\chi^2 + \sin^2 \chi (d\theta^2 + \sin^2 \theta d\phi^2)).$$ One can obtain the 4-dimensional anti-de Sitter space by setting $ \rho = i\chi$. However, if one looks closely in the quantum creation scenario, this continuation takes two steps. First, one has to continue on a three surface where the metric is stationary. One can choose $\chi = \frac{\pi}{2}$ as the surface, set $\omega = i(\chi - \frac{\pi}{2})$ and obtain the metric with signature $(-,-,-, +)$ $$ds_4^2= -dt^2 - \frac{3}{\Lambda_4} \sin^2\left (\sqrt{\frac{\Lambda_4}{3}}t \right )(-d\omega^2 + \mbox{cosh}^2 \omega (d\theta^2 + \mbox{cos}^2 \theta d\phi^2)).$$ Then one can analytically continue the metric through the null surface at $t = 0$ by redefining $ \rho = \omega +\frac{i\pi}{2}$ and get the anti-de Sitter metric $$ds_4^2= -dt^2 + \frac{3}{\Lambda_4} \sin^2\left (\sqrt{\frac{\Lambda_4}{3}}t \right )(d\rho^2 + \mbox{sinh}^2 \rho (d\theta^2 + \sin^2 \theta d\phi^2)).$$ In the No-Boundary Universe, the relative probability of the creation, at the $WKB$ level, is the exponential to the negative of the Euclidean action of the instanton $S_7 \times S_4$ $$P =\Psi^* \cdot \Psi \approx \exp -I ,$$ where $\Psi$ is the wave function of the configuration at the quantum transition. The configuration is the metric and the matter field at the equator. $I$ is the Euclidean action. If we are living in the section of the 7-dimensional de Sitter universe with the $S_4$ space of metric (8) or the Euclidean version of (10) as the internal space, then the Euclidean action $I$ should take the form $$I=- \int \sqrt{g_{11}}\left ( \frac{1}{2} R - \frac{1}{48} F_{MNPQ}F^{MNPQ} + \frac{\sqrt{2}i}{6\cdot (4!)^2} \eta^{M_1M_2\cdots M_{11}}F_{M_1M_2M_3M_4}F_{M_5M_6M_7M_8}A_{M_9M_{10}M_{11}} \right )d^{11}x.$$ This is obtained through analytical continuation as in the usual 4-dimensional Euclidean quantum gravity. However, if we are living in the section of the 4-dimensional anti-de Sitter universe, due to the metric signature, the Euclidean action will gain an extra negative sign in the continuation. This is also supported by cosmological implications. The $R$ term in the actions can be decomposed into $R_7 - R_4$, where $R_7$ and $R_4$ are the scalar curvatures for the two factor spaces with the positive-definite metric signatures. The negative sign in front of $R_4$ is required so that the perturbation modes of the gravitational field in the $S_4$ background would take the minimum excitation state allowed by the Heisenberg uncertainty principle \[8\]. The perturbation modes are the origin for the structure of the Lorentzian universe in both the closed and open models. By the same argument, if we consider 7-dimensional factor space as our macroscopic spacetime, then one has to turn the sign around, as the analytic continuation has taken care of automatically. The Euclidean action $I$ of the $AdS_4 \times S_7$ space can be calculated $$I =\frac{1}{3}\kappa^2 V_7V_4 ,$$ where the volume $V_7$ $\;(V_4)$ of $S_7$ $\;(S_4)$ is $\pi^4r_7^7/3$ $\;(8\pi^2r^4_4/3)$. The field equation (6) is derived from the action (1) for the condition that the tensor $A_{MNP}$ is given at the boundary. Therefore, if one uses the action (1) in the evaluation of the wave function and the probability, then the induced metric and tensor $A$ on it must be the configuration of the wave function. The wave function is expressed by a path integral over all histories with the configuration as the only boundary. In deriving Eq. (11), one adjoins the histories in the summation of the wave function to their time reversals at the equator to form a manifold without boundary and discontinuity. If the configuration is given, then one obtains a constrained instanton for the stationary action solution. If one lifts the restriction at the equator, the stationary action solution is a regular instanton. The induced metric and scalar field (if there is any) at the equator will remain intact under the reversal operation. However, for other fields, one has to be cautious. This occurs to our $A_{MNP}$ field. For convenience, we choose the following gauge potential $$A = i\kappa \left (\frac{3}{\Lambda_4} \right )^2 \left(\sin \left (\sqrt{\frac{\Lambda_4}{3}} \tau \right ) - \frac{1}{3}\sin^3 \left (\sqrt{\frac{\Lambda_4}{3}} \tau \right )+ \frac{2}{3} \right ) \sin^2 \chi \sin \theta d\chi \wedge d \theta \wedge d \phi,$$ where $\tau = i(t- \frac{\pi}{2})$, the gauge is chosen such that $A$ is regular at the south pole $(\tau = - \pi/2)$ of the hemisphere $(0\geq \tau \geq - \pi/2)$. The gauge potential for the north hemisphere will take the same form with a negative sign in front of the constant term $\frac{2}{3}$. The sign change of the potential is consistent with the time reversal, as we mentioned earlier. One can see that $A_{MNP}$ is subjected to a discontinuity at the equator. Therefore, $A_{MNP}$ is not allowed to be the argument for the instanton probability calculation in (11). In order for the instanton approach to be valid, one has to use the canonical conjugate representation. One can make a Fourier transform of the wave function $\Psi (h_{ij}, A_{123})$ to get the wave function $\Psi (h_{ij}, P^{123})$, $$\Psi ((h_{ij}, P^{123}) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{iA_{123}P^{123}} \Psi(h_{ij}, A_{123}).$$ where $P^{123}$ is the canonical momentum conjugate to $A_{123}$, the only degree of freedom of the matter content under the minisuperspace ansatz $$P^{123} = \int_{\Sigma}\sqrt {- g_{11}} \left (- F^{0123} + \frac{\sqrt{2}}{3(4!)} \eta^{0123m_5\cdots m_{11}} F_{m_5m_6m_7m_8} A_{m_9m_{10} m_{11}}\right ) d^{10}x,$$ where $\Sigma$ denotes the 10-dimensional surface $ t = const$. The quantum transition should occur at the equator $\chi = \pi/2$. However, the calculation at $\tau =0$ or $t = \pi/2$ is simpler. Apparently, the result does not depend on the choice of the equator (this has been confirmed), since all equators are congruent for the round $S_4$ sphere. Strictly speaking, one cannot use $A_{123}$ as the argument of the wave function without gauge condition. The only meaning of this argument is its flux at the surface. We shall not use the wave function $\Psi (h_{ij}, A_{123})$ anyway. The discontinuity occurred at the equator instanton is thus avoided using the momentum representation, although it is due to the two distinct patches covering the whole sphere and can be glued through a gauge transformation. At the $WKB$ level, the Fourier transform of the wave function is equivalent to the Legendre transform of the action. The Legendre transform has introduced an extra contribution $-2 A_{123}P^{123}$ to the Euclidean action, where all quantities are in the Euclidean version, and the factor 2 is due to the two sides of the equator in the adjoining. Then the effective action becomes $$I_{effect} = - \frac{2}{3} \kappa^2 V_7V_4.$$ If we consider the quantum transition to occur at the equator of $S_7$ instead, using the same argument, then it turns out that the corresponding canonical momentum using the time coordinate in $S_7$ vanishes, and the effective action should be the negative of (13), taking account of the sign of the factor $\sqrt{g_{11}}$ in the action (12). Since the creation probability is the exponential to the negative of the Euclidean action, the probability of creating a 7-dimensional macroscopic universe is exponentially suppressed relative to that of the 4-dimensional case. In the classical framework, the $S_7$ factor space in the Freund-Rubin model can be replaced by $S_2\times S_5$, $S_2 \times S_2 \times S_3$, $S_4 \times S_3$ or other Einstein spaces. However, all these product spaces have volumes smaller than that of $S_7$. It would lead to an exponential suppression of the creation probability. Therefore, the internal space must be the round $S_7$ space. Now we consider the Englert model \[5\]. Then, in addition to the components of the space $S_4$ in (7), the $F_{mnpq}$ components of the $S_7$ space can be non-vanishing and satisfying $$F^{mnpq}_{\;\;\;\;\;\;\;;m} = \left [ \frac{\sqrt{2}}{(4!)\sqrt{g_7}} \right ] \kappa \epsilon^{npqrstu} F_{rstu}.$$ Two nontrivial solutions are $$F_{mnpq} = \frac{4}{\kappa} \partial_{[m} S^{\pm}_{npq]},$$ where $S^{\pm}_{ mnp}= S^{\pm}_{[mnp]}$ are the two torsion tensors which can flatten the $S_7$ space in the Cartan-Schouten sense \[9\] $$R^m_{\;\;npq}\{\Gamma^r_{st} + S^r_{st}\} = 0,$$ where $+\; (-)$ is for the case $\kappa > 0\; (\kappa < 0)$. It is noted that $S_7$ is the only compact manifold to allow this, apart from group manifolds. The potential can be chosen as $$A_{mnp} = \frac{1}{6\kappa} S^\pm_{mnp}.$$ The anisotropic cosmological constants are $\Lambda_7 = 3\kappa^2/4$ and $\Lambda_4=-5\kappa^2/4$. The tensor $A_{mnp}$ satisfies the gauge condition $A^{mnp}_{\;\;\;\; ;p} = 0$. The following properties of the torsion tensor will be used in later calculations $$S^{tr}_{\;\;\;\; m}S_{trn} = \frac{3}{4} \kappa^2 g_{mn},$$ $$S^{\pm \;mnp} = \mp \frac{2\sqrt{2}}{4!|\kappa | \sqrt{g_7}} \epsilon^{mnpqrst}S^\pm_{ [rst,q]}.$$ As in the Freund-Rubin model, before we take account of the Legendre term, the Euclidean action of the Englert $AdS_4 \times S_7$ space is $$I =-\frac{1}{4} \kappa^2V_7V_4.$$ After including the Legendre term the effective action becomes $$I_{effect} =-\frac{2}{3} \kappa^2V_7V_4.$$ It is surprising that after the long calculation, the effective action remains the same as that in the Freund-Rubin case. If the quantum transition occurred at an equator of the $S_7$ space, one has to include the Legendre terms correspondingly. In contrast to the Freund-Rubin model, the canonical momenta do not vanish. Fortunately, due to the symmetries of the torsion tensor, the sum of the $C^3_6 = 20$ Legendre terms cancel exactly. The action is the negative of that in (24) $$I_{effect} =\frac{1}{4} \kappa^2V_7V_4.$$ Again, comparing the results of (25) and (26), one can conclude that the universe we are living is most likely 4-dimensional. In the Freund-Rubin model, the $S_7$ factor space can be replaced by a general Einstein space with the same cosmological constant $\Lambda_7$. The Awada-Duff-Pope model \[6\] is most interesting. The round 7-sphere is replaced by a squashed one, so that the $N=8$ supersymmetry breaks down to $N = 1$. As far as the scenario of the quantum creation is concerned , the argument for the Freund-Rubin model remains intact, the only alternations are that the quantum transition should occur at one of its stationary equators and $V_7$ should be the volume of the squashed 7-sphere. There is no supersymmetry in the Englert model \[5\]. Englert, Rooman and Spindel also discussed the model with a squashed $S_7$ factor space \[7\]. Here the $A$ components in the $S_7$ space are proportional to the torsion which renders the squashed sphere Ricci-flat, instead. It is believed that our conclusion should remain the same. The right configuration for the wave function has also been chosen in the problem of quantum creation of magnetic and electric black holes \[10\]. If one considers the quantum creation of a general charged and rotating black holes, this point is even more critical. It is become so acute that unless the right configuration is used, one even cannot find a seed constrained instanton \[11\]. Many previous studies on dimensionality have essentially been restricted to the classical framework. For $d=11$ supergravity, there is no way to discriminate the $d=4$ and $d= 7$ macroscopic universes in the classical framework, as in other similar but more artificial models. This discrimination can be realized only through quantum cosmology. **References:** 1\. Z.C. Wu, *Phys. Lett. **B, 307 (1984). X.M. Hu and Z.C. Wu, *Phys. Lett. **B, 87 (1984); *Phys. Lett. **B, 237 (1985); *Phys. Lett. **B, 305 (1986).************ 2\. J.B. Hartle and S.W. Hawking, *Phys. Rev. **D, 2960 (1983).*** 3\. E. Cremmer, B. Julia and J. Scherk, *Phys. Lett. **B, 409 (1978).*** 4\. G.O. Freund and M.A. Rubin, *Phys. Lett. **B, 233 (1980).*** 5\. F. Englert, *Phys. Lett. **B, 339 (1982).*** 6\. M.A. Awada, M.J. Duff and C.N. Pope, *Phys. Rev. Lett. , 294 (1983).* 7\. F. Englert, M. Rooman and P. Spindel, *Phys. Lett. **B, 47 (1983).*** 8\. J.J. Halliwell and S.W. Hawking, *Phys. Rev. **D, 346 (1985)*** 9\. E. Cartan and J.A. Schouten, *Proc. K. Acad. Wet. Amsterdam , 933 (1926).* 10\. S.W. Hawking and S.F. Ross, *Phys. Rev. **D, 5865 (1995). R.B. Mann and S.F. Ross, *Phys. Rev. **D, 2254 (1995).****** 11\. Z.C. Wu, *Int. J. Mod. Phys. **D, 199 (1997), gr-qc/9801020; *Phys. Lett. **B, 174 (1998), gr-qc/9810012. .******

--- address: | Laboratoire d’Astrophysique, UMR 5572, Observatoire Midi-Pyrénées\ 14 avenue E.-Belin, F-31400 Toulouse, France author: - 'G. GOLSE, J.-P. KNEIB and G. SOUCAIL' title: CONSTRAINING THE COSMOLOGICAL PARAMETERS FROM GRAVITATIONAL LENSES WITH SEVERAL FAMILIES OF IMAGES --- Introduction ============ Recent works on constraining the cosmological parameters using the CMB and the high redshift supernovae seem to converge to a new “standard cosmological model” favouring a flat universe with $\Omega_m\sim 0.3$ and $\Omega_\lambda\sim 0.7$: White [@White] and references therein. However these results are still uncertain and depend on some physical assumptions, so the flat $\Omega_m=1$ model is still possible (Le Dour [*et al.*]{} [@LeDour]). It is therefore important to explore other independent techniques to constrain these cosmological parameters. In cluster gravitational lensing, the existence of multiple images – with known redshifts – given by the same source allows to calibrate in an absolute way the total cluster mass deduced from the lens model. The great improvement in the mass modeling of cluster-lenses that includes the cluster galaxies halos (Kneib [*et al.*]{} [@Kneib96], Natarajan & Kneib [@Natarajan]) leads to the hope that clusters can also be used to constrain the geometry of the Universe, through the ratio of angular size distances, which only depends on the redshifts of the lens and the sources, and on the cosmological parameters. The observations of cluster-lenses containing large number of multiple images lead Link & Pierce [@Link] (hereafter LP98) to investigate this expectation. They considered a simple cluster potential and on-axis sources, so that images appear as Einstein rings. The ratio of such rings is then independent of the cluster potential and depends only on $\Omega_m$ and $\Omega_\lambda$, assuming known redshifts for the sources. According to them, this would allow marginal discrimination between extreme cosmological cases. But real gravitational lens systems are more complex concerning not only the potential but also off-axis positions of sources. They conclude that this method is ill-suited for application to real systems. We have re-analyzed this problem building up on the modeling technique developed by us. As demonstrated below, we reach a rather different conclusion showing that it is possible to constrain $\Omega_m$ and $\Omega_\lambda$ using the positions of multiple images at different redshifts and some physically motivated lens models. Troughout this paper we have assumed $H_0=65$ km s$^{-1}$ Mpc$^{-1}$, however the proposed method is independant of the value of $H_0$. Influence of $\Omega_m$ and $\Omega_\lambda$ on the images formation ==================================================================== Angular size distances ratio term --------------------------------- In the lens equation: $\mathbf{\theta_{S}}= \mathbf{\theta_{I}} - \displaystyle{\frac{2}{c^2}\frac{D_{OL}D_{LS}}{D_{OS}}} \mathbf\nabla \phi_\theta(\mathbf{\theta_{I}}) $, the dependence on $\Omega_m$ and $\Omega_\lambda$ is solely contained in the term $F=\displaystyle{{D_{OL}}{D_{LS}}/{D_{OS}}}$. For a given lens plane, $F(z_s)$ increases rapidly up to a certain redshift and then stalls, with significant differences for various values of the cosmological parameters (see Fig. \[F\_zs\]). Thus in order to constrain the actual shape of $F(z_s)$ several families of multiple images are needed, ideally with their redshifts regularly distributed in $F(z_s)$ to maximize the range in the $F$ variation. If we consider fixed redshifts for both the lens and the sources, at least 2 multiple images are needed to derive cosmological constraints. In that case $F$ has only an influence on the modulus of $\mathbf{\theta_{I}}-\mathbf{\theta_{S}}$. So taking the ratio of two different $F$ terms provides the intrinsic dependence on cosmological scenarios, independently of $H_0$. A typical configuration leads to the Fig. \[F\_zs\] plot. The discrepancy between the different cosmological parameters is not very large, less than 3% between an EdS model and a flat low matter density one. The figure also illustrates the expected degeneracy of the method, also confirmed by weak lensing analyzes, with a continuous distribution of background sources ([*e.g.*]{} Lombardi & Bertin [@Lombardi] ). Relative influence of the different parameters ---------------------------------------------- We now look at the relative influence of the different parameters, including the lens parameters, to derive expected error bars on $\Omega_m$ and $\Omega_\lambda$. To model the potential we choose the mass density distribution proposed by Hjorth & Kneib [@Hjorth], characterized by a core radius, $a$, and a cut-off radius $s\gg a$. We can then get the expression of the deviation angle modulus $D_{\theta_{I}}=\parallel\mathbf{\theta_{I}}-\mathbf{\theta_{S}}\parallel$. For 2 families of multiple images, the relevant quantity becomes the ratio of 2 deviation angles for 2 images $\theta_{I1}$ and $\theta_{I2}$ belonging to 2 different families at redshifts $z_{s1}$ and $z_{s2}$. Let’s define $R_{\theta_{I1},\theta_{I2}}=\displaystyle{\frac{D_{\theta_{I1}}}{D_{\theta_{I2}}}}$. With several families, the problem is highly constrained because a single potential must reproduce the whole set of images. In practice we calculate $\displaystyle{\frac{dR_{\theta_{I1},\theta_{I2}}}{R_{\theta_{I1},\theta_{I2}}}}$ versus the different parameters it depends on. We chose a typical configuration to get a numerical evaluation of the errors on the cosmological parameters: $z_l=0.3$, $z_{s1}=0.7$, $z_{s2}=2$, $\displaystyle{\frac{\theta_{I2}}{\theta_{I1}}}=2$, $\displaystyle{\frac{\theta_{s}}{\theta_{a}}}=10$ ($\theta_a=a/D_{OL}$,$\theta_s=s/D_{OL}$) and we assume $\Omega_m=0.3$ and $\Omega_\lambda=0.7$. We then obtain the following orders of magnitudes for the different contributions : = 0.57 + 0.74 + 0.17 + 0.4( - ) - 0.1 - 0.06 - 0.015 + 0.02 As expected, even with 2 families of multiple images the influence of the cosmological parameters is of the second order. The precise value of the redshifts is quite fundamental, therefore a spectroscopic determination ($dz=0.001$) is essential. The position of the (flux-weighted) centers of the images are also important. With HST observations we assume $d\theta_I=0.1$”. So even if the problem is less dependent on the core and cut-off radii (in other word the mass profile), they will represent the main sources of error. Taking $d\theta_a/\theta_a= d\theta_s/\theta_s= 20$ %, we then derive the errors $d\Omega_m$ and $d\Omega_{\lambda}$ from the above relation in the flat low matter density we chose. We did this computation for different sets of cosmological models. Indeed the errors we will obtain with this method change significantly with respect to $\Omega_m$ and $\Omega_\lambda$. All other things being equal apart from the cosmological parameters, we plot $d\Omega_m$ and $d\Omega_\lambda$ for a continuous set of universe models (Fig. \[erreurs\]). For instance in the 2 popular cosmological scenarios, we have : $\Omega_m=0.3\pm0.24 $ $\Omega_\lambda=0.7\pm0.5$ or $\Omega_m=1\pm0.33$ $\Omega_\lambda=0\pm1.2$ As this can be easily understood from the Fig. \[F\_zs\] degeneracy plot, the method is in general far more sensitive to the matter density than to the cosmological constant, for which the error bars are larger. However the results we could obtain this way are as precise as the ones given by other constraints. But these errors are just typical; provided spectroscopic and HST observations, they depend mostly on the particular cluster and the potential model chosen to describe it. They could be quite tightened with a precise model, and by increasing the number of clusters with multiple images. \[simul\]Constraint on $(\Omega_m,\Omega_\lambda)$ from strong lensing ====================================================================== Method and algorithm for numerical simulations ----------------------------------------------- We consider basically the potential introduced in section 2.2. After considering the lens equation, fixing arbitrary values $(\Omega_m^0$,$\Omega_\lambda^0)$ and a cluster lens redshift $z_l$, our code can determine the images of a source galaxy at a redshift $z_s$. Then taking as single observables these sets of images as well as the different redshifts, we can recover some parameters (the more important ones being $\sigma_0$, $\theta_a$ or $\theta_s$) of the potential we left free for each point of a grid $(\Omega_m$,$\Omega_\lambda)$. The likelihood of the result is obtained via a $\chi^2$-minimization, where the $\chi^2$ is computed in the source plane as follows : \^2= The subscript $i$ refers to the families and the subscript $j$ to the images of a family. There is a total of $\sum_{i=1}^n n^i=N$ images. $\theta_{Sj}^i$ is the source found for the image $\theta_{Ij}^i$ in the inversion. $\theta_{SG}^i$ is the barycenter of the $\theta_{Sj}^i$ (belonging to a same family). Finally if $\sigma_{Ii}^{j}$ is the error on the position of the center of $\theta_{Ij}^i$ and $A_j^i$ the amplification for this image, then $\sigma_{Si}^{j}=\sigma_{Ii}^{j}/\sqrt{A_j^i}$. Numerical simulations in a typical configuration ------------------------------------------------ To recover the parameters of the potential ([*i.e.*]{} $\sigma_0$, $\theta_a$, $\theta_s$ and adjusted lens parameters), we generated 3 families of images with regularly distributed source redshifts. For starting values $(\Omega_m^0,\Omega_\lambda^0)=(0.3,0.7)$ we obtained confidence levels shown in Fig. \[3fam\]. The method puts forward a good constraint, better on $\Omega_m$ than on $\Omega_\lambda$, and the degeneracy is the expected one (see Fig. \[F\_zs\]). Concerning the free parameters, we also recovered in a rather good way the potential, the variations being $\Delta\sigma_0\sim150$ km/s, $\Delta\theta_a\sim3$”and $\Delta\theta_s\sim20$”. This is an “ideal” case, of course, because we tried to recover the same type of potential we used to generate the images, the morphology of the cluster being quite regular and the redshift range of the sources being wide enough to check each part of the $F$ curve. Such a simple approach can be applied to regular clusters like MS2137-23, which shows at least 3 families of multiple images including a radial one, see Fig. \[MS2137\]. But the spectroscopic redshifts are still currently missing for this cluster. Conclusions & prospects ======================= Following the work of LP98, we discussed a method to obtain informations on the cosmological parameters $\Omega_m$ and $\Omega_\lambda$ while reconstructing the lens gravitational potential of clusters with multiple image systems at different redshifts. This technique gives degenerate constraints, $\Omega_m$ and $\Omega_\lambda$ being negatively correlated, with a better constraint of the matter density. With a single cluster in a typical lensing configuration we can expect the following error bars : $\Omega_m=0.3{\pm 0.24}$, $\Omega_\lambda=0.7{\pm 0.5}$. To perform that, several general conditions must be fulfilled: $\ast$ a cluster with a rather regular morphology, so that a few parameters are needed to describe the gravitational potential ; this is not so restrictive because we saw that a bimodal cluster can also provide a constraint, $\ast$ “numerous” systems of multiple images, probing each part of the cluster, $\ast$ a good spatial resolution image (HST observations) of the cluster and arcs to compute relatively precise – 0.1” – (flux weighted) images positions, $\ast$ spectroscopic precision for the different redshifts that should be also regularly distributed from $z_l$ to high values – this requires deep spectroscopy on 8-10m class telescopes due to the faintness of the multiple images . It is important to notice that one cluster could provide one constraint on the geometry of the whole universe. And it is possible to combine data from different clusters to tighten the error bars. Combining the study of about 10 clusters would lead to meaningful constraints. The dashed lines confidence levels in the Fig. \[3fam\] are the result of a numerical simulation made with 10 [*identical*]{} clusters. Actually the degeneracy depends only on the different redshifts involved that we will have various sets of when applying the method to real configurations. This should lead to a more reduced area of allowed cosmological parameters. We are encouraged by more and more known observations including systems with multiple sources and we plan to apply this technique to clusters like MS2137-23, MS0440+02, A370, AC114 and A1689. References {#references .unnumbered} ========== [99]{} J. Hjorth and J.-P. Kneib, J.-P. Kneib, R. Ellis, I. Smail, W. Couch & R. Sharples, M. Le Dour, M. Douspis, J. Bartlett & A. Blanchard, R. Link and M. Pierce, M. Lombardi and G. Bertin, P. Nararajan & J.-P. Kneib, M. White,

--- abstract: | The main difficulty in solving the Helmholtz equation within polygons is due to non-analytic vertices. By using a method nearly identical to that used by Fox, Henrici, and Moler in their 1967 paper; it is demonstrated that such eigenvalue calculations can be extended to unprecedented precision, very often to well over a hundred digits, and sometimes to over a thousand digits. A curious observation is that as one increases the number of terms in the eigenfunction expansion, the approximate eigenvalue may be made to alternate above and below the exact eigenvalue. This alternation provides a new method to bound eigenvalues, by inspection. Symmetry must be exploited to simplify the geometry, reduce the number of non-analytic vertices and disentangle degeneracies. The symmetry-reduced polygons considered here have at most one non-analytic vertex from which all edges can be seen. Dirichlet, Neumann, and periodic-type edge conditions, are independently imposed on each polygon edge. The full shapes include the regular polygons and some with re-entrant angles (cut-square, L-shape, 5-point star). Thousand-digit results are obtained for the lowest Dirichlet eigenvalue of the L-shape, and regular pentagon and hexagon. author: - Stephen bibliography: - 'references.bib' title: 'Computing ultra-precise eigenvalues of the Laplacian within polygons' --- \[sec:intro\]Introduction {#secintrointroduction .unnumbered} ========================= The task is to calculate very precise eigenvalues of the Laplacian within the shapes shown in Fig. \[fig:allshapes\], on which may be imposed either Neumann or Dirichlet boundary conditions. ![Four geometries (with abbreviated names) in which a variety of Dirichlet and Neumann eigenvalues are calculated to within at least 100 digits. Of the regular polygons, only the regular pentagon is shown.[]{data-label="fig:allshapes"}](allshapes) The technique is substantially identical to the method used by Fox, Henrici, and Moler [@fhm1967] (hereafter referred to as “FHM”) who used a method of particular solutions (MPS) called the “point-matching” or “collocation” method to calculate Dirichlet eigenvalues within the now-famous L-shape. To see what is possible using this method, an assorted set of eigenvalues, all truncated to 100 digits of precision, for the chosen shapes (Fig. \[fig:allshapes\]) is presented in Table \[tab:ultraprecise\]. In addition, three “thousand-digit” results are submitted to the “On-line Encyclopedia of Integer Sequences” [@oeisorg] (OEIS.org). These results far exceed all previous published results.[^1] It is generally a good idea to plot some eigenfunctions if only to inspect the contours and nodal patterns to ensure that one is actually calculating eigenvalues. Several eigenfunction contour plots are shown in Figs. \[fig:stareigenfunctions\], \[fig:pentagonplots\], and \[fig:cutsquareeigenfunctions\]. Of course, published concerns regarding the numerically ill-conditioned nature of this method must be addressed. Some such concerns were actually identified by FHM, but more recently clarified by Betcke and Trefethen [@bt2004] in 2004, when they demonstrated the numerical advantages of the so-called “generalized singular-value decomposition” (GSVD) method, a relative to the point-matching method. Fortunately, the answer to make the point-matching method work is short and simple: One must both (a) select adequate matching points (both number and distribution) and (b) keep enough precision in the intermediate calculations. My empirical observations are that Chebyshev nodes chosen as matching points (as suggested by Betcke and Trefethen [@bt2004]) often work well, even where equally-spaced points do not, and the precision of the intermediate calculations must be significantly higher, often several times, than the precision of the eigenvalue. The ability to calculate many digits depends on the convergence rate, and a good way to improve it is to exploit symmetries. Fortunately, this exploitation has the important side-effects of (a) reducing the complexity of the region in which one must work, (b) classifying eigenmodes, and (c) disentangling geometrically degenerate eigenfunctions. Do not underestimate the importance of exploiting symmetry. At virtually every step in this project, only free software[^2] running on modern commodity computer hardware has been used, and with great success. The software of choice is the [GP/PARI]{} calculator [@PARI2] using the GNU Multiple Precision Arithmetic Library [gmp]{} [@gmp6] running on a laptop computer with a [GNU/linux]{} operating system and its countless ancillary programs. (Some use was also made of [maxima]{} [@maxima] for the occasional symbolic calculation.) This programming environment permits efficient numerical computations while “natively” retaining up to several thousand digits of precision. Several personal computers were used, but the best was a modern laptop computer with and a processor with eight threads.[^3] CPU times are reported using that laptop, and are unavoidably approximate due to multitasking. Before beginning, it should be made clear that this is a classical and very well-known problem, worked on by many people over the last two-hundred years—with many applications and results. As such, I shall limit the discussion to only those facts that are required to reproduce and possibly extend the present calculations. A thirty-year-old, but still popular and relevant survey of the problem was given by Kuttler and Sigillito [@ks1984]. More practically, active investigators, Barnett and Betcke have created [ MPSpack]{} [@bb2010; @bh2014] that helps bring sophisticated eigenvalue calculations closer to the rest of us. Despite the long history, except for the recent work of P. Amore, et al. [@abfr2015] , who incidentally make this same observation, I am unaware of any published, non-closed-form eigenvalues accurate to just beyond a dozen or so digits for [*any*]{} shape not related to the closed-form solutions within the equilateral triangle, rectangle, or circle (ellipse). All eigenvalue results in this report are likely unprecedented. \[sec:eigen\]The eigenvalue problem {#seceigenthe-eigenvalue-problem .unnumbered} =================================== Let $\mathbf{r}$ be a point in the plane described by either Cartesian coordinates or polar coordinates , where and , and where notation ambiguity is removed by context. The two-dimensional Helmholtz equation is $$\Delta\Psi(k;\mathbf{r})+\lambda\Psi(k;\mathbf{r}) = 0 \label{eq:helmholtz1}$$ where $\Delta$ is the Laplacian and $k=\sqrt{\lambda}$ is the usual “wavenumber”. Without a boundary, $\lambda$ (or $k$) is treated as a continuous eigen-parameter, and $\Psi(k;\mathbf{r})$ may describe a free wave with wavelength $\Lambda=2\pi/k$. The “interior” Helmholtz eigenvalue problem is obtained by restricting $\mathbf{r}$ to the interior of a region (one of Fig. \[fig:allshapes\]), and imposing relevant boundary conditions (Neumann or Dirichlet), which constrains $\lambda$ to a non-accumulating set of discrete eigenvalues, some of which may be degenerate. (The trivial, “closed-form”, Neumann solution, i.e., $\Psi=\mbox{constant}$ with $\lambda=0$, is completely ignored in this project.) The point-matching method works if the eigenvalues are non-degenerate. Thus it is very important to deal with degeneracies either by (a) dismissing them or (b) disentangling them using symmetry. First, closed-form solutions are not only known[^4], but also arbitrarily high in “accidental” degeneracy as one climbs the eigenvalue towers[^5]; so after identifying them, simply exclude them from the calculations. Second, if there is a geometric or reflection symmetry, all eigenfunctions can be sorted into separate symmetry classes—some of which may be closed-form. Doing so effectively splits the problem up into a set of sub-problems, each one with a corresponding symmetry-reduced polygon and edge conditions, and a resulting, non-accumulating, infinite tower of distinct eigenvalues $$0 < \lambda_1 < \lambda_2 < \lambda_3 < \cdots < \lambda_\alpha < \cdots,$$ where $\alpha$ labels the eigenvalue within that tower. Considering only the non-closed-form symmetry classes, it is indeed assumed that there are no degeneracies within such a tower.[^6] Thus, to each non-closed-form symmetry class, we associate a sub-problem, effectively consisting of non-degenerate eigenmodes. A tower of such eigen-pairs for a given symmetry class shall be written as the set $$\left\{ \lambda_\alpha, \Psi_\alpha(\mathbf{r})\,\middle|\, \alpha=1,2,3,... \right\} \label{eq:exacteigenpair}$$ where $\Psi(k_\alpha;\mathbf{r})\equiv\Psi_\alpha(\mathbf{r})$. Degeneracies, of course, may exist between separate symmetry classes, so it may be possible to choose a subset of non-closed-form classes and calculate only those eigenvalues. It is these symmetry-reduced, sub-problems that we focus on. In the end, one can piece together all of the eigenmodes of each symmetry class to construct the complete spectrum for the chosen shapes in Fig. \[fig:allshapes\], which restores the rich assortment of symmetries and degeneracies. See, for example, Fig. \[fig:starSpectrum\] depicting the lowest seventy-nine eigenvalues of the star. As is conventional, an edge or a line of symmetry within the shape is said to be “even” or “odd” according to how a corresponding eigenfunction transforms under Riemann-Schwarz reflection. Also, for the purposes of this report, the analyticity of a vertex is defined by whether or not an eigenfunction can be (locally) Riemann-Schwarz reflected around that vertex and remain single valued.[^7] In the current project, not all boundary edges will be either even or odd. Indeed, types of symmetry arise where (a) the eigenfunction satisfies a periodic-like condition such that values along one edge are proportional to values along another edge (star and regular polygon), or (b) a reflection symmetry may be exploited, effectively excluding terms from the eigenfunction expansion (L-shape and cut-square). To demonstrate the technique, it is necessary to introduce a polygon, $\Omega$, as illustrated in the example of Fig. \[fig:starA\]. The specific form of $\Omega$ depends on the symmetry group of the problem, and for this project, define it to be an $s$-sided polygon with at most one non-analytic vertex from which all edges can be seen. Label the vertices $V_a$, where $a=1,2,...,s$; beginning with the non-analytic vertex and proceeding clockwise. (In the example, $s=4$.) The edges are labeled similarly, specifically, $\partial\Omega_a$ connects $V_a$ to $V_{a+1}$ (, as required). The “adjacent” edges are $\partial\Omega_1$ and $\partial\Omega_s$, while the “point-matched” edges are $\partial\Omega_a$, where $a=2,3,...,s-1$. In this project, the shapes are such that only the adjacent edges lie along lines passing through the non-analytic vertex. Define the [*canonical position*]{} of $\Omega$ as follows. First locate the non-analytic vertex, $V_1$, at the origin, and orient as needed so that $\partial\Omega_1$ and $\partial\Omega_2$ are independently even or odd. Then, rotate around $V_1$ until $\partial\Omega_2$ is parallel to the at a positive . In this canonical position, each edge $\partial\Omega_a$ is a segment of the line $$\mathcal{L}_a : \, A_a x+ B_a y = C_a \label{eq:linea1}$$ where the constants ($A_a$, $B_a$, $C_a$) are uniquely chosen by requiring that $$\widehat{\mathbf{r}}_{n,a} = A_a \widehat{\mathbf{x}} + B_a \widehat{\mathbf{y}} \label{eq:linea2}$$ be an outward pointing unit normal. ![(LEFT) Star shape showing how (RIGHT) an arrowhead polygon, $\Omega$, in its canonical position, is constructed for the $\mathcal{B}_e$ and $\mathcal{C}_e$ eigenfunctions of the star. The vertical bisector of the star for these eigenfunctions is an even (dashed) line, a portion of which becomes $\partial\Omega_2$. Edges $\partial\Omega_1$ and $\partial\Omega_4$ have same boundary conditions as the star, while edge $\partial\Omega_3$ satisfies a periodic-type boundary condition (values on $\partial\Omega_3$ are proportional to corresponding values on $\partial\Omega_2$). Also shown are $N/2=10$ Chebyshev distributed matching points on $\partial\Omega_2$. The other $N/2$ points would be symmetrically located on $\Omega_3$. $V_1$ is non-analytic for all symmetry classes of the star.[]{data-label="fig:starA"}](starA) The two adjacent edges, $\partial\Omega_1$ and $\partial\Omega_s$, at polar angles $\phi_1$ and $\phi_s$, respectively, form the non-analytic vertex, which has internal angle $$\Delta\phi=\phi_s-\phi_1.$$ It is also convenient to introduce the abbreviation, $$\tilde\theta \equiv \theta - \phi_1 \label{eq:thetaabbreviation}$$ Having stated the problem, with some specific restrictions and conventions, the next task is to describe the method of solution, the so-called “point-matching” method. The point-matching method {#the-point-matching-method .unnumbered} ========================= Precisely following FHM, expand the eigenfunction in a truncated Fourier-Bessel series $$\Psi^{[N]}(k;\mathbf{r}) = \sum_{\nu=1}^{N} c_\nu^{[N]}\, \psi_\nu(k;\mathbf{r}) \vspace{1ex} \label{eq:fourierbessel}$$ where the so-called “basis functions” are $$\psi_\nu(k;\mathbf{r}) = J_{m_\nu}(kr)\,\times\, \left\{ \begin{array}{ll} \sin(m_\nu\tilde{\theta}), & \mbox{\ $\partial\Omega_1$(odd)} \vspace{1ex} \\ \cos(m_\nu\tilde{\theta}), & \mbox{\ $\partial\Omega_1$(even)} \end{array} \right. \label{eq:basisfunctions}$$ where, as indicated, the choice is based solely on if the adjacent edge $\partial\Omega_1$ is odd or even; and $J_m(x)$ is the Bessel function of the first kind of order $m$.[^8] The index $\nu$ runs from $1$ to $N$, here labeling the term in the expansion; and the $m_\mu$ are not yet specified. When the symmetry requirements and boundary conditions are enforced along [*both*]{} adjacent edges, the are restricted. It often happens that the adjacent edges are independently even or odd, in which case, the $m$-values are given by $$m_\nu = \frac{\pi}{\Delta\phi} \times \left\{ \begin{array}{ll} (\nu-1) & \mbox{ both even} \\ (\nu-\frac{1}{2}) & \mbox{ one odd, other even} \\ \hphantom{(}\nu & \mbox{ both odd} \\ \end{array} \right. \label{eq:mvalues1}$$ Among the chosen shapes, the L-shape and cut-square have other interesting symmetry-related restrictions on the $m$-values due to the fact that these shapes can be obtained by reflecting a 45-90-45 triangle, respectively, six and seven times. Once the required $m$-values are selected, it is important to realize that $\Psi^{[N]}(k;\mathbf{r})$ is an [*exact*]{} solution within the infinite sector . This is true for any positive integer $N$ and real $\lambda=k^2>0$ (excluding $\lambda=0$). The function is specifically crafted to handle the non-analyticity of  and the boundary or symmetry conditions on the adjacent edges  and , exactly. The above formalism generally applies to the MPS. The next step, unique to the “point-matching” method, is to select $N$ points, $$\mathcal{P}_N: \left\{\mathbf{r_\mu} \,\middle|\,\mu=1,2,3,\cdots,N\right\}$$ on the non-adjacent edges where the edge conditions may be nontrivially enforced on those edges.[^9] This is the same $N$ that appears in the eigenfunction expansion, Eq. (\[eq:fourierbessel\]). Selecting adequate sets of matching points is a nontrivial task that is key to making the point-matching method work. For the immediate following, assume this has been done. Enforcing the boundary conditions or symmetry-related edge conditions at the $N$ matching points yields $N$ linear equations in the $N$ expansion coefficients (the $c_\nu^{[N]}$ in Eq. (\[eq:fourierbessel\])), and a resulting “point-matching matrix”, $\mathcal{M}^{[N]}(\lambda)$. Linear algebra suggests that the determinant of this matrix must be zero for nontrivial solutions to exist, i.e., $$\left|\mathcal{M}^{[N]}(\lambda)\right| = 0 \label{eq:pointmatchingdeterminant}$$ This is the so-called “point-matching determinant”, and finding roots of this equation for increasing values of $N$ is the name of the game. For a given $N$, there is an infinite number of roots of the point-matching determinant, but only the first $\kappa_N$ of them, $$0 < \lambda_1^{[N]} < \lambda_2^{[N]} < \cdots < \lambda_{\kappa_N}^{[N]}$$ are guaranteed to correspond to actual eigenvalues. The number $\kappa_N$ is such that if $N$ is increased to , this set of roots may become better estimates of the eigenvalues, but new roots may be introduced only for values of $\lambda$ a little bigger than . Thus, for a given eigenvalue, $\lambda_\alpha$, there is a minimum $N$-value, say $N_{\alpha,1}$, above which Eq. (\[eq:pointmatchingdeterminant\]) may yield increasingly better estimates as $N$ is incremented. I conjecture that a more detailed picture shall require $N_{\alpha,1}$ be big enough so that a gently “wiggling” contour passes continuously through all the matching points—a contour along which the edge conditions are satisfied exactly.[^10] The smallest, problem-dependent value of $\Delta N$ defines a “properly incremented” set of $N$-values, say $\{N_{\alpha}\}$, where its members are $$N_{\alpha,i} = N_{\alpha,1}+(i-1)\Delta N \qquad i=1,2,3,\cdots; \label{eq:properincrement}$$ that yield a corresponding set of approximate eigenvalues $$\left\{ \lambda^{[N_{\alpha,1}]}_\alpha,\lambda^{[N_{\alpha,2}]}_\alpha,\lambda^{[N_{\alpha,3}]}_\alpha,\cdots\right\}. \label{eq:approximateeigenvaluesequence}$$ Provided everything “works”, we then tacitly assume that the approximate eigen-pair converges to the corresponding exact eigen-pair (see Eq. (\[eq:exacteigenpair\])), specifically, $$\lim_{N\to\infty} \left\{\lambda^{[N]}_{\alpha},\Psi^{[N]}(k_\alpha^{[N]};\mathbf{r})\right\} = \left\{ \lambda_\alpha, \Psi_\alpha(\mathbf{r}) \right\} \label{eq:eigenpairlimit}$$ where $N\in\{N_\alpha\}$. The procedure depends on several factors, most notably, the nature of the vertices and the distribution of the matching points. Eigenvalue bounds {#eigenvalue-bounds .unnumbered} ================= If the rate of convergence is “exponential”, a very interesting thing happens that might be used to obtain eigenvalue bounds, by inspection. Specifically, the problem may be set up so that that sequence of approximate eigenvalues, Eq. (\[eq:approximateeigenvaluesequence\]), as $N$ increases, can be made to alternate above and below an asymptote—assumed to correspond to an eigenvalue—in such a way that the peaks of that alternation effectively provide an increasingly narrow bound for that eigenvalue.[^11] This means we can write $$\lambda_\alpha^{[N_\downarrow]} < \lambda_\alpha < \lambda_\alpha^{[N_\uparrow]} \label{eq:eigenvaluebound}$$ where $N_\downarrow$ and $N_\uparrow$ correspond to of a given minimum and the previous or next maximum, respectively. Table \[tab:Lshapeloworder\] clearly illustrates that alternation for the L-shape calculation—not only for my modern calculation, but (interestingly) also for the original FHM data. To calculate many digits, one must estimate the roots of the point-matching determinant, Eq. (\[eq:pointmatchingdeterminant\]), to a precision somewhat higher than the observed bound, Eq. (\[eq:eigenvaluebound\]), at a given . By construction, the eigenvalues are not degenerate, meaning the point-matching determinant only has simple roots, so it is actually quite straightforward to calculate those roots to very high precision.[^12] This key ingredient—the ability to precisely locate those roots—is a significant advantage of the point-matching method over some other methods, like the GSVD method (which seeks minima of a function). Another important feature of the point-matching method is that the number of basis functions is maximized for a given number of matching points. With more terms, the approximate eigenfunction better matches the exact eigenfunction. With a given eigenvalue bound in hand, there are several related and useful numbers one can calculate. They are relative gap, approximate number of correct digits, and global convergence rate; respectively, $$\mbox{(a)}\quad\epsilon_\alpha^{[\widehat{N}]} = \frac{\lambda^{[N_\uparrow]}_\alpha-\lambda^{[N_\downarrow]}_\alpha} {\frac12\left[{\lambda^{[ N_\uparrow ]}_\alpha+\lambda^{[N_\downarrow]}_\alpha}\right]} \qquad\qquad\mbox{(b)}\quad D_\alpha^{[\widehat{N}]} = -\log_{10}\left(\epsilon_\alpha^{[\widehat{N}]}\right) \qquad\qquad\mbox{(c)}\quad \rho^{[\widehat{N}]}_\alpha = \frac{D_\alpha^{[\widehat{N}]}}{\widehat{N}} \label{eq:relativegap}$$ where the $\widehat{N}=\max(N_\uparrow,N_\downarrow)$ form a new set of $N$-values, , a subset of $\{N_\alpha\}$. Note that Eqs.  provide a practical, numerical definition of what it means to be “exponentially convergent”. Such a practical definition is useful since the exact value of the eigenvalue is never really known. Specifically, if $\rho_\alpha^{[\widehat{N}]}=\rho_\alpha$ is constant wrt $\widehat{N}$, then the eigenvalue [*relative gap*]{} decays exponentially, i.e., $$\epsilon_\alpha^{[\widehat{N}]} \sim 10^{- \rho_\alpha \widehat{N}}$$ as $\widehat{N}\to\infty$. My observation is that the global convergence rate $\rho$ for non-closed-form solutions is typically a deceasing function of $\widehat{N}$. Convergence rates that permit relatively easy results range down to about $1/6$ or so, and the best convergence rates for the chosen shapes are typically a little better than $1/2$. Note that this reciprocal notation is useful because it reveals how many matching points—or terms in the expansion—must be added to achieve each additional eigenvalue digit: Thus, if $\rho=1/4$, then four additional terms in the expansion and a corresponding four additional matching points will add an additional eigenvalue digit. To report a bound, suffix the matching digits with a rounded-up superscript and a rounded-down subscript. This conservative approach helps ensure that the reported bound will include the true eigenvalue. To illustrate, consider the relatively difficult-to-calculate, lowest Dirichlet eigenvalue within a 256-sided regular polygon (area=$\pi$). It is first precisely calculated to $$\lambda=\left\{ \begin{array}{ll} 5.7831876203689428757 2892661 ... & \mbox{ for $N_\downarrow= 930$} \\ 5.7831876203689428757 8671658 ... & \mbox{ for $N_\uparrow= 929$} \\ \end{array}\right.$$ This calculation used $N=282, 283,...,929,930$; while judiciously skipping $N$-values, and stopping when $\epsilon<10^{-20}$. Very reliably, even and odd $N$-values provided lower and upper bounds, respectively. From that result, one may write the bound $$\lambda= 5.7831876203689428757\,_{289}^{868} \label{eq:polygon256}$$ for $\widehat{N}=930$. For this example, , , and . To get that twenty-digit result took almost two CPU days. Despite achieving [*only*]{} twenty digits, this example does exhibit exponential convergence. Indeed, more detailed inspection reveals the relationship $$\epsilon^{[\widehat{N}]}\approx 10^{-\left(5.73+ \widehat{N}/65.1\right)}$$ based on $280\le\widehat{N}\le 930$, which specifically indicates exponential convergence.[^13] Incidentally, for comparison, the best “published” value may be calculated with Eq. (\[eq:regpolasymp\]), yielding , of which the first fifteen (underlined) digits appear to be correct. I am unaware of any efforts to calculate this particular eigenvalue. Note that above a few dozen eigenvalue digits, either the “correct digits” or the “correctly rounded digits” may seem more impressive than the actual bound. But the bound is nevertheless useful because the exact value is not known, and one may not know otherwise where to stop counting correct digits. It is the alternation property that provides the bounds, but what causes that alternation? By considering the necessary continuous contour through the matching points along which the boundary or symmetry conditions are satisfied exactly, I offer a simple heuristic explanation.[^14] Such a contour defines an area $\mathrm{A}_N$ in which $\{\lambda^{[N]}_\alpha,\Psi^{[N]}(k_\alpha,\mathbf{r})\}$ is an [*exact*]{} solution. As matching points are added, that area not only approaches the polygon area, $\mathrm{A}$, but I conjecture that the contour “flips”, analogous to “$\sin\to-\sin$”; and that such a flipping causes the area to alternate above and below the polygon area. Since the area difference is very small[^15] , using $\lambda^{[N]}_\alpha\mathrm{A}_N\approx \lambda_\alpha\mathrm{A}$, it becomes plausible that such a flipping causes the alternation. My empirical observation is that if there is exponential convergence, then it becomes more likely that flipping occurs with each proper increment the closer the matching-point distribution is to being equally-spaced. More optimal (or necessary) Chebyshev distributions give better convergence rates, but quite often require several proper increments to flip. In that case, the number of proper increments is often irregular, but may become more regular for higher . Any automatic program should make sure that what appear to be upper and lower limits are actually so. Note that this eigenvalue bounding technique is related to, but slightly different from the conventional approach of using the approximate eigenfunction along the point-matched edges, as in the Moler-Payne method [@mp1969]. A practical difference is that one need not calculate the coefficients (to evaluate the eigenfunction along the point-matched edges) to calculate the upper and lower limits to the eigenvalue. Instead, the eigenvalue bound here is identified by “simply” watching roots of the point-matching determinant as $N$ is increased. However, ease of calculation may not be the only advantage: It may indeed provide [*better*]{} bounds, as demonstrated with the L-shape calculation below. To make all of this work well, one must identify symmetries, select polygons and adequate matching point distributions, and calculate precise values of the point-matching determinant for a properly incremented set of $N$-values. For a given problem, this may be an iterative procedure as one discovers new things. The matching points {#the-matching-points .unnumbered} =================== Next consider the nontrivial task of choosing matching points. Each problem will have a specific choice, but there are some common strategies. Suppose each point-matched edge $\partial\Omega_a$ has $n_a$ matching points at which the edge conditions may be nontrivially satisfied. For the point-matching method to work well, it is somewhat important that—as $N$ is incremented—the same proportion of matching points be used on each edge, and the spacing and distribution should also be approximately similar on each edge. These requirements help define a “proper increment”, Eq. (\[eq:properincrement\]). It is also recommended that the maximum gap between matching points is always smaller than some empirically determined fraction of the free wavelength, . This will help ensure that the contour connecting matching points does indeed pass continuously through all the matching points. Equally-spaced matching-points seem quite popular and easy to program, but these do not always work and are certainly not the best. It seems that crowding points near vertices is good practice since that is usually where the contour tends to deviate from the polygon edge the most: Crowding the points constrains the contour better. Chebyshev nodes usually yield excellent results and happen to be quite easy to program. Indeed, let  to $\ell$ measure the position along a point-matched edge $\partial\Omega_a$, then $$s_\mu = \frac{\ell}{2}\left[1 - \cos\left(\frac{\mu-\frac{1}{2}}{n_a}\,\pi\right)\right] \label{eq:canonicalchebyshevnodes}$$ where $\mu=1,2,3,...,n_a$ is a canonical set of Chebyshev nodes. This distribution has points very close to the end-points, and a relative crowding near those end-points. Variations may be used, for example, to ensure more crowding near acute, distant vertices. The point-matching matrix {#the-point-matching-matrix .unnumbered} ========================= Armed with an adequate selection of matching points, the next task is to describe the matrix elements. Each point-matched edge, $\partial\Omega_a$ with $n_a$ matching points, can be odd, even, or satisfy a periodic-type requirement; and each will yield $n_a$ rows of the point-matching matrix. Odd point-matched edge {#odd-point-matched-edge .unnumbered} ---------------------- If the point-matched edge $\partial\Omega_a$ is odd, the $n_a$ rows are obtained by equating at the $n_a$ matching points to zero. Specifically, $$\Psi^{[N]}(k;\mathbf{r_\mu}) = 0 \qquad \mbox{$\mathbf{r_\mu}\in\partial\Omega_a$(odd)},$$ which yields $n_a$ rows, $$\mathcal{M}^{[N]}_{\mu\nu}(\lambda) = J_{m_\nu}(kr_\mu)\,\times\, \left\{ \begin{array}{l} \sin(m_\nu\tilde{\theta}_\mu),\mbox{ $\partial\Omega_1$(odd)} \vspace{1ex} \\ \cos(m_\nu\tilde{\theta}_\mu),\mbox{ $\partial\Omega_1$(even)} \end{array} \right. \label{eq:matrixelementsodd}$$ The vast majority of published accounts of all variants of the MPS use Dirichlet boundary conditions, so this is perhaps a well-known result. It is also relatively easy to program. Even point-matched edge {#even-point-matched-edge .unnumbered} ----------------------- If the point-matched edge $\partial\Omega_a$ is even, the formulas are also elementary and easy to program, but a little long. These results appear to be less well-known, so they may be of some utility. First recall equations (\[eq:linea1\]) and (\[eq:linea2\]), which define a segment of which forms an “outward-pointing” unit normal, $\widehat{\mathbf{r}}_{n,a}$. Equating the normal derivative of at the $n_a$ matching points to zero shall yield the corresponding $n_a$ rows of the matrix. Specifically, $$\left. \frac{\partial\Psi^{[N]}(k;\mathbf{r})}{\partial r_{n,a}}\right|_{\mathbf{r}_\mu} = 0 \qquad \mbox{$\mathbf{r_\mu}\in\partial\Omega_a$(even)}$$ where $$\displaystyle \frac{\partial\Psi}{\partial r_{n,a}} = A_a\,\frac{\partial\Psi}{\partial x} + B_a\, \frac{\partial\Psi}{\partial y}.$$ To make the formulas less cumbersome, split $\mathcal{M}$ into two $N\times N$ matrices, $\mathcal{X}$ and $\mathcal{Y}$, such that $$\mathcal{M}^{[N]}_{\mu\nu}=A_a\mathcal{X}^{[N]}_{\mu\nu}+B_a \mathcal{Y}^{[N]}_{\mu\nu}.$$ Differentiating the basis functions and evaluating at the matching points yields $$\begin{aligned} \displaystyle r_\mu^2\mathcal{X}^{[N]}_{\mu\nu}(\lambda) &=& \left\{ \begin{array}{ll}\displaystyle m_\nu \left[ x_\mu \sin(m_\nu\tilde{\theta}_\mu) - y_\mu \cos(m_\nu\tilde{\theta}_\mu) \right] \, J_{m_\nu}(kr_\mu) -kr_\mu x_\mu \sin(m_\nu\tilde{\theta}_\mu) \, J_{1+m_\nu}(kr_\mu) , & \mbox{$\partial\Omega_1$(odd)}\vspace{1ex} \\ \displaystyle m_\nu \left[ x_\mu \cos(m_\nu\tilde{\theta}_\mu) + y_\mu \sin(m_\nu\tilde{\theta}_\mu) \right] \, J_{m_\nu}(kr_\mu) -kr_\mu x_\mu\cos(m_\nu\tilde{\theta}_\mu) \, J_{1+m_\nu}(kr_\mu) , &\mbox{$\partial\Omega_1$(even)} \end{array}\right. \\ r_\mu^2\mathcal{Y}^{[N]}_{\mu\nu}(\lambda) &=& \left\{ \begin{array}{ll}\displaystyle m_\nu \left[ x_\mu \cos(m_\nu\tilde{\theta}_\mu) + y_\mu \sin(m_\nu\tilde{\theta}_\mu) \right] \, J_{m_\nu}(kr_\mu) -kr_\mu y_\mu \sin(m_\nu\tilde{\theta}_\mu) \, J_{1+m_\nu}(kr_\mu), & \mbox{$\partial\Omega_1$(odd)}\vspace{1ex} \\ \displaystyle m_\nu \left[ -x_\mu \sin(m_\nu\tilde{\theta}_\mu) + y_\mu \cos(m_\nu\tilde{\theta}_\mu) \right] \, J_{m_\nu}(kr_\mu) -kr_\mu y_\mu\cos(m_\nu\tilde{\theta}_\mu) \, J_{1+m_\nu}(kr_\mu),&\mbox{$\partial\Omega_1$(even)} \end{array}\right.\qquad \label{eq:matrixelementseven}\end{aligned}$$ To calculate eigenvalues, one may absorb the factor $r_\mu^{-2}$ into the expansion coefficients since it is common to each term in the row. In most of my examples, $A_2=1$ and $B_2=0$, but Neumann eigenvalues of the cut square shall also require $A_3=0$ and $B_3=1$. Periodic-type edge (dihedral symmetry) {#periodic-type-edge-dihedral-symmetry .unnumbered} -------------------------------------- Next consider the periodic-type boundary conditions that arise if the geometry has dihedral symmetry. Although this analysis is rather elementary, it too seems to be rarely discussed in the context of this problem. Since the regular polygon[^16] has dihedral symmetry, it will be used to most efficiently develop the matrix elements; but it also applies to another chosen shape, the 5-point star. ![(LEFT) A regular polygon with an even apothem (dashed) on the positive $x$-axis, (CENTER) the triangular fundamental region, and (RIGHT) the kite-shaped region for the doubly degenerate eigenmodes. Values of an even , doubly-degenerate eigenfunction are proportional to each other at points $P$ and $P'$.[]{data-label="fig:regularpolygons"}](regularpolygon/regpoly) To that end, consider a $\sigma$-sided regular polygon (and other diagrams) shown in Fig. \[fig:regularpolygons\], with vertex angle $\beta=(\sigma-2)\pi/\sigma$. The symmetry group of this shape is the dihedral group, $D_\sigma$, of degree $\sigma$ or order $2\sigma$. To work out the symmetry properties, first center the polygon at the origin with an apothem on the positive , as shown in . The dihedral group is generated by $R$, a counter-clockwise rotation by , and $Q$, a reflection through the , i.e., . The dihedral group has only and irreducible representations (irreps), so let $\eta_1$ and $\eta_2$ count those irreps. If $\sigma$ is odd or even, then $\eta_1=2$ or $\eta_1=4$, respectively. Knowing $\eta_1$, we have . These 1-dim and 2-dim irreps lead to, respectively, non-degenerate and doubly-degenerate towers of eigenvalues. Since $Q^2=1$, define a such that any non-degenerate eigenfunction is either even or odd according to that parity. For the doubly degenerate pairs of eigenfunctions, one can always be made odd and the other even, and assume that this is done. Use a subscript to denote the , as in $$\begin{aligned} \varphi_e(x,-y)=+\varphi_e(x,y) \qquad \varphi_o(x,-y)=-\varphi_o(x,y)\qquad &&\mbox{Cartesian}\vspace{1ex}\\ \varphi_e(r,-\theta)=+\varphi_e(r,\theta) \qquad \varphi_o(r,-\theta)=-\varphi_o(r,\theta)\qquad &&\mbox{Polar}\end{aligned}$$ where “$\varphi(\mathbf{r})$” denotes a function in the regular polygon centered at the origin. An obvious but relevant fact is that all odd $Q$-parity functions are zero on the , i.e., . For the regular polygon, the fundamental region may be chosen to be the shaded triangle of Fig. \[fig:regularpolygons\](LEFT) because the entire polygon may be obtained from that triangle via group operations. Relevant to this project, it is a right triangle with two other angles $$\frac{\alpha}{2}=\frac{\pi}{\sigma}\qquad\mbox{and}\qquad \frac{\beta}{2} = \frac{(\sigma-2)\pi}{2\,\sigma},$$ exactly one of which, $\beta/2$, is non-analytic (except for $\sigma=3$, $4$, and sometimes $6$). For the non-degenerate eigenfunctions, the triangular fundamental region becomes $\Omega$. If $\sigma$ is even, there are eight possible sets of edge conditions on this triangle since all its edges can be independently even or odd. But, if $\sigma$ is odd, there are only four possible sets because the regular polygon’s apothem must have the same symmetry as its circumradius (line connecting its center to a vertex).[^17] When $\Omega$ is put in its canonical position, , it should become clear how to calculate the matrix elements for the non-degenerate eigenvalues. For the doubly-degenerate eigenfunctions, we have an interesting choice. Since the group transformations form linear combinations of degenerate eigenfunctions, we may either (a) solve for [*both*]{} of the degenerate eigenfunctions within the fundamental region or (b) solve for [*one*]{} of the degenerate eigenfunctions in an area twice as large. In both cases, we can reconstruct both functions using the $2\sigma$ group transformations. Since choice (b) requires only one function, it shall be the better choice. By reflecting the fundamental region about its hypotenuse to form the kite-shaped quadrilateral, we obtain the only polygon that is both twice as large and has (at most) one non-analytic vertex from which all edges can be seen. This kite quadrilateral thus becomes $\Omega$ for the doubly-degenerate eigenmodes.[^18] Cut this kite-quadrilateral out of the polygon as shown in , but before re-orienting it, observe that if we “rotate” the [*even*]{} $Q$-parity eigenfunction, we get the simple but important result that $$\varphi_e(r,\alpha)=\cos(\gamma\alpha)\,\varphi_e(r,0) \label{eq:periodicphi}$$ where $\gamma=1,2,...,\eta_2$. The effective purpose of $\gamma$ is to identify to which one of the $\eta_2$ doubly-degenerate eigenvalue towers this eigenfunction belongs.[^19] (Note that this is where we used ). That periodic-type symmetry relationship, Eq. (\[eq:periodicphi\]), is useful because it relates values of an (even) eigenfunction on the positive $x$-axis (an apothem) to values of the same (even) eigenfunction on a neighboring apothem, i.e., at points $P$ and $P'$. When that kite quadrilateral is re-oriented to its canonical position, , the apothems become point-matched edges and the regular-polygon boundary segments become adjacent edges; so that Eq. (\[eq:periodicphi\]) gets transformed into $$\Psi_e(k;r,\beta-\theta)=\cos(\gamma\alpha)\,\Psi_e(k;r,\theta), \label{eq:periodicpsi}$$ where the even apothem becomes $\partial\Omega_2$. Note how the points $P$ and $P'$ are mapped. The subscript “$e$” is retained to remind us that we only need to consider this member of the degenerate pair to determine the eigenvalue tower. Since the regular polygon is assumed to have either Dirichlet or Neumann boundary conditions, the adjacent edges ($\partial\Omega_1$ and $\partial\Omega_4$) are either both odd or both even. In this case, referring to Eq. (\[eq:mvalues1\]), we can use $$\begin{aligned} && \sin(\nu\,(\beta-\theta)) = -(-1)^\nu \sin(\nu\,\theta) \\ && \cos((\nu-1)(\beta-\theta)) = -(-1)^\nu \cos((\nu-1)\,\theta)\end{aligned}$$ to combine Eqs. (\[eq:fourierbessel\]) and (\[eq:periodicpsi\]), yielding, $$0 = \sum_{\nu=1}^N c_\nu^{[N]}\, \left\{(-1)^\nu + \cos(\gamma\alpha)\right\} \, J_{m_\nu}(kr) \times\left\{\begin{array}{ll} \sin(m_\nu\theta),&\mbox{$\partial\Omega_1$(odd) and $\partial\Omega_4$(odd)} \vspace{0.5ex}\\ \cos(m_\nu\theta),&\mbox{$\partial\Omega_1$(even) and $\partial\Omega_4$(even)} \end{array}\right. \label{eq:evensymmetrycondition}$$ Evaluating this equation at each of the matching points on $\partial\Omega_2$ establishes $N/2$ equations for the $N$ expansion coefficients; while the other $N/2$ equations are obtained by using the fact that $\partial\Omega_2$ is even. Requiring nontrivial solutions (for the expansion coefficients) leads to these $N/2$ rows of the point-matching matrix $$\mathcal{M}^{[N]}_{\mu\nu}(\lambda) = \left\{(-1)^\nu + \cos(\gamma\alpha) \right\} \, J_{m_\nu}(kr_\mu) \times\left\{\begin{array}{ll} \sin(m_\nu\theta_\mu),&\mbox{$\partial\Omega_1$(odd) and $\partial\Omega_4$(odd)} \vspace{0.5ex}\\ \cos(m_\nu\theta_\mu),&\mbox{$\partial\Omega_1$(even) and $\partial\Omega_4$(even)} \end{array}\right. \label{eq:matrixelementsmixed}$$ while the other $N/2$ rows are obtained using the fact that $\partial\Omega_2$ is even. There are really two point-matched edges, but that periodic-type boundary condition, Eq. (\[eq:periodicpsi\]), enables us to use one point-matched edge, twice. This requires that $N$ be even, with $N/2$ points on edge $\partial\Omega_2$, and $N/2$ corresponding “phantom” points on $\partial\Omega_3$ which aren’t actually numerically needed. As it happens, the symmetry group of the star is $D_5$, i.e., the same as the regular pentagon. The fundamental region is a triangle with one non-analytic vertex (see Fig. \[fig:starB\]), and the region $\Omega$ for the doubly degenerate eigenfunctions is an arrowhead quadrilateral (see Fig. \[fig:starA\]), also with one non-analytic vertex—this time, a re-entrant vertex. To accommodate the star (and other polygons with dihedral symmetry), Eq. (\[eq:matrixelementsmixed\]) is changed by replacing $\theta_\mu$ with , see Eq. (\[eq:thetaabbreviation\]). As is common practice, the eigenfunctions can be identified by their nodal and antinodal patterns. With dihedral symmetry, the non-degenerate eigenfunctions have a maximal set of criss-crossing even and odd lines of symmetry. These are obtained by Riemann-Schwarz reflecting the eigenfunction within the triangular fundamental region to flesh out the full eigenfunction. Every problem with such dihedral symmetry[^20] will have at least two towers of non-degenerate eigenfunctions, here named the “symmetric” $\mathcal{S}$ and the “antisymmetric” $\mathcal{A}$, which have all even and odd lines of symmetry, respectively. If $\sigma$, as in “$D_\sigma$”, is even, then there are two more non-degenerate symmetry classes that have lines of symmetry that alternate, even and odd, as one proceeds around the polygon. These might be named $\mathcal{S}'$ and $\mathcal{A}'$ according to even and odd $Q$-parity, respectively. The doubly degenerate eigenfunctions can be similarly identified by the nodal and antinodal patterns since these functions have \[at least\] $\gamma$ nodal [*curves*]{} crossing at the origin. Since the complexity of the nodal pattern increases with $\gamma=1,2,...,\eta_2$; and we already used the symbol $\mathcal{A}$, label these $\eta_2$ symmetry classes using $\mathcal{B}$, $\mathcal{C}$, $\mathcal{D}$, ...; and when identifying a particular eigenfunction, suffix with an $e$ or $o$ to indicate $Q$-parity, as in $\mathcal{B}_e$ or $\mathcal{B}_o$.[^21] Some numerical considerations {#some-numerical-considerations .unnumbered} ============================= When calculating the point-matching determinant, it is important that all calculations be carried out to relatively high precision. To get the process started, first calculate a low precision eigenvalue bound in order to estimate the convergence rate, $\rho$. Then use that to estimate the targeting a high precision bound, accurate to, say, , i.e., . To set the precision for this new calculation, use the rule-of-thumb requiring at least $1.2N$ digits in the intermediate calculations. If the process works, the precision was adequate. If not, try increasing it. The roots of the determinant must certainly be found to a precision better than the desired $D$ digits of the eigenvalue, perhaps to within $1.2D$ digits. I observed that for $D$ beyond a few dozen digits, the point-matching determinant becomes quite linear in $\lambda$, so a simple secant-based root-finding algorithm works very well. If one pursues hundreds of digits, the computer language must be able to handle arbitrary-precision calculations efficiently. Whatever programming environment you use, be sure that the numerics are good, especially the fractional-order Bessel functions.[^22] Examples {#examples .unnumbered} ======== My main project is the regular pentagon, for which I have so-far calculated the lowest 8139 Dirichlet eigenvalues to at least 60 correctly rounded digits. See Figs. \[fig:stareigenfunctions\] and \[fig:pentagonplots\] for several plots, including the highest one in that set; and—as presented in Table \[tab:ultraprecise\]—some hundred-digit eigenvalue results. Except for those examples, I find it more interesting to use other shapes to demonstrate the eigenvalue calculation and bounding methods. L-shape {#l-shape .unnumbered} ------- This project would not be complete without considering the famous L-shape formed by joining two unit-edged squares to adjacent edges of a third, as shown in Fig. \[fig:Lshape\]. For this, I shall limit the scope to calculating the lowest Dirichlet eigenvalue. The first one-hundred digits of this eigenvalue appear in Table \[tab:ultraprecise\], and an abbreviated bound is $$\lambda_1 = 9.6397238440219410527\cdots 27494610057016061858\,_{4831}^{5300}$$ where the leading and trailing digits of the 1001-digit result are shown. The first 1001 decimal digits of this number are listed as <http://www.oeis.org/A262701> [@oeisorg]. The 2004 influential “Reviving the Method of Particular Solutions” by Betcke and Trefethen [@bt2004], which, among other things, examines the L-shape as an example, and apparently provides the best modern reference for this shape. Despite that, I regress back to 1967 and use FHM as a starting point. ![L-shape showing a line of symmetry (even line $V_1A$) and region $\Omega$ (shaded square , in its canonical position) with ten evenly-spaced matching points.[]{data-label="fig:Lshape"}](lshape) Following FHM, fully exploit symmetry to yield a square region, $\Omega$, in its canonical position as shown by the shaded region of Fig. \[fig:Lshape\]. This means we can use Eq. (\[eq:fourierbessel\]) with $$\qquad m_\nu=\frac{2}{3}\left( 2 \left\lfloor\frac{3\nu}{2}\right\rfloor-1\right)$$ where $\lfloor x\rfloor$ is $\mathop{\mathrm {floor}}(x)$. The expression in parentheses gives the required sequence $\{1, 5, 7, 11, 13, ...\}$; i.e., (positive) integers on either side of the integers in $\{0,6,12,...\}$, which are appropriate for the lowest Dirichlet eigenmode. Now, consider a near repeat of the FHM calculation for an “apples-to-apples” comparison—this time using arbitrary precision and much higher . To do this exercise, (a) choose equally-spaced matching points on $\partial\Omega_2$ and $\partial\Omega_3$; (b) impose at $V_2$ and $V_3$; and (c) use the proper increment of [$\Delta N=2$]{}. Then, to numerically verify the alternating/converging nature, calculate every approximate eigenvalue for even $N$ from 4 to just over 260.[^23] The results for are shown in Table \[tab:Lshapeloworder\], which illustrates the alternation. FHM reported the eight-digit $9.6397238\,_{05}^{84}$ (FHM Table 3) eigenvalue bound, and they indicate calculations up to $N=26$ (FHM Table 2). FHM round-off error appears to become significant for $N>14$, so use my very low-order and $14$ numbers for that “apples-to-apples” comparison. By inspection, the bound is $9.6397238\,_{43}^{55}$. This happens to be slightly better, but the emphasis should be placed on the ease with which this bound is obtained. Interestingly, FHM data is essentially the same as mine out to ninth decimal place, so that original FHM ( and $14$) data might be used to bound the eigenvalue to around eight digits without using Moler-Payne, however, without more precision and higher N-values, it is not obvious that the alternation may be used to bound the eigenvalue. Note that the $N$-value used in the FHM Moler-Payne calculation was not indicated. [rll]{} & $\lambda$ & FHM $\lambda^*$\ 4 & 9.58161723 & 9.658161723\ 6 & 9.63624491 & 9.639624491\ 8 & 9.639766319 & 9.639726632\ 10 & 9.6397270221 & 3703\ 12 & 9.63972354826 & 3855\ 14 & 9.639723830369 & 3844\ 16 & 9.6397238412442 & 3844\ 18 & 9.63972384410281 & 3845\ 20 & 9.639723844033611 & 3845\ 22 & 9.6397238440275875 & 3845\ 24 & 9.63972384402165466 & 3844\ 26 & 9.63972384402137668 & 3846\ 28 &\ 30 &\ 32 &\ $\vdots$ &&\ 254 &\ 256 &\ 258 &\ 260 &\ \[tab:Lshapeloworder\] Except for the P. Amore et al. [@abfr2015] recent effort, I believe the best published result is the decade-old, 13-digit correctly rounded result of Betcke and Trefethen [@bt2004], . To make a reasonable comparison, they used the GSVD method, no symmetry reduction, equal-spaced boundary matching points (excluding all vertices) on four point-matched segments, 50 randomly-selected interior points, only 15 basis-functions, and machine-precision. That required minimizing the smallest generalized singular value of a $(50+240)\times 240$ matrix and (to obtain the MP-type bounds) estimating the function values on the point-matched edge. Using the current method, which has a better distribution of points (by including vertices and actually being evenly spaced) and taking full advantage of symmetry, a similar looking bound is achieved by comparing the and $22$ approximate eigenvalues. This result required a few seconds of CPU time. Although not an “apples-to-apples” comparison, it does indicate significantly less numerical effort is required to obtain a similar result. Next, I extend the calculations from $N=814$ up to $826$, still using equally-spaced matching points[^24], to obtain a result with an overall convergence rate of $1/2.75$. This took about a day of CPU time. Then, I switched to Chebyshev-distributed points (very similar to those shown in Fig. \[fig:cutsquare\] for the cut-square), which improved the convergence rate to very close to $1/2$. This allowed calculation of a result at a similar $N=816$, after another CPU day. By extending up to about $2100$, “thousand-digit” results may be obtained after several weeks of CPU effort. This illustrates how exploiting symmetry and judiciously choosing matching points permits one to extend results to very high precision. Cut-square {#cut-square .unnumbered} ---------- The sole and limited purpose of this example is to illustrate the bounding method for the apparently difficult shape shown in Fig. \[fig:cutsquare\], which I call the “cut square” since it is formed by cutting a triangle out of a unit-edged square. It has six edges and a $7\pi/4$ re-entrant vertex. It was inspired entirely by an example in Reference [@yh2009]. Those authors, Yuan and He, were apparently unaware of the then-recent relevant work by Trefethen and Betcke [@bt2004; @tb2006], but nevertheless provide an interesting discussion on the L-shape and this cut-square shape—as they attempt to bound eigenvalues. Of note is that those authors point out that this shape has no symmetry, but indeed it does.[^25] ![Example contour plots. Blue and yellow areas are opposite sign, and light-dark steps indicate level contours. These are (LEFT) the lowest three Neumann and (RIGHT) the lowest three Dirichlet eigenfunctions. Their eigenvalues are listed in Table \[tab:ultraprecise\], and they are not degenerate.[]{data-label="fig:cutsquareeigenfunctions"}](cutsquare/N1a "fig:")  ![Example contour plots. Blue and yellow areas are opposite sign, and light-dark steps indicate level contours. These are (LEFT) the lowest three Neumann and (RIGHT) the lowest three Dirichlet eigenfunctions. Their eigenvalues are listed in Table \[tab:ultraprecise\], and they are not degenerate.[]{data-label="fig:cutsquareeigenfunctions"}](cutsquare/N2a "fig:")  ![Example contour plots. Blue and yellow areas are opposite sign, and light-dark steps indicate level contours. These are (LEFT) the lowest three Neumann and (RIGHT) the lowest three Dirichlet eigenfunctions. Their eigenvalues are listed in Table \[tab:ultraprecise\], and they are not degenerate.[]{data-label="fig:cutsquareeigenfunctions"}](cutsquare/N3a "fig:")     ![Example contour plots. Blue and yellow areas are opposite sign, and light-dark steps indicate level contours. These are (LEFT) the lowest three Neumann and (RIGHT) the lowest three Dirichlet eigenfunctions. Their eigenvalues are listed in Table \[tab:ultraprecise\], and they are not degenerate.[]{data-label="fig:cutsquareeigenfunctions"}](cutsquare/D1a "fig:")  ![Example contour plots. Blue and yellow areas are opposite sign, and light-dark steps indicate level contours. These are (LEFT) the lowest three Neumann and (RIGHT) the lowest three Dirichlet eigenfunctions. Their eigenvalues are listed in Table \[tab:ultraprecise\], and they are not degenerate.[]{data-label="fig:cutsquareeigenfunctions"}](cutsquare/D2a "fig:")  ![Example contour plots. Blue and yellow areas are opposite sign, and light-dark steps indicate level contours. These are (LEFT) the lowest three Neumann and (RIGHT) the lowest three Dirichlet eigenfunctions. Their eigenvalues are listed in Table \[tab:ultraprecise\], and they are not degenerate.[]{data-label="fig:cutsquareeigenfunctions"}](cutsquare/D3a "fig:") For this polygon, calculations of both Dirichlet and Neumann eigenvalues are demonstrated. Because of the very high convergence rate of , with modest effort, the lowest three eigenvalues for each type are calculated to 200 digits, with 100-digit truncated values presented in Table \[tab:ultraprecise\]. The first challenge is to identify the symmetries and classify the eigenmodes. To do this, first consider the Dirichlet modes and expand the eigenfunction (Eqs. \[eq:fourierbessel\] and \[eq:basisfunctions\], choosing “$\sin$”) about the re-entrant vertex with $$m_\nu=\frac{4\nu}{7} \qquad\mbox{where $\nu=1,2,3,...,N$}.$$ By calculating the eigenfunction coefficients (the $c_\nu$ in Eq. (\[eq:fourierbessel\])), a pattern is revealed where, for a given eigenfunction, many coefficients are zero. That lead to a simple discovery[^26] of a symmetry quite analogous to that of the L-shape. By inspection of the calculated coefficients, it became obvious that (a) closed-form modes are selected by choosing integral $m$, while (b) any non-closed-form eigenmode belongs to one of three classes (here labeled A, B, and C) obtained by choosing positive on either side of multiples of seven. To best reveal the pattern, line up the possible , $$m = \textstyle [0,\!]\, \overset{A}{\frac{4}{7}}, \overset{B}{\frac{8}{7}}, \overset{C}{\frac{12}{7}}, \overset{C}{\frac{16}{7}}, \overset{B}{\frac{20}{7}}, \overset{A}{\frac{24}{7}}, 4, \overset{A}{\frac{32}{7}}, \overset{B}{\frac{36}{7}}, \overset{C}{\frac{40}{7}}, \overset{C}{\frac{44}{7}}, \overset{B}{\frac{48}{7}}, \overset{A}{\frac{52}{7}}, 8, \overset{A}{\frac{60}{7}},\cdots \label{eq:cutsquaremvalues}$$ where the zero is shown but used only for the closed-form Neumann solutions. The fundamental region for the closed-form modes appears to be the 45-90-45 triangle. However, the smallest region in which I can solve the problem for the non-closed-form classes appears to be twice as large, $2/7$ of the total area, which can be chosen to be the square region in the first quadrant. Thus, for the non-closed-form modes, there are two point-matched edges meeting at right angles—as with the L-shape. I found that a Chebyshev-like distribution of matching points, equally divided between the two matching edges on the square in the first quadrant, and crowded near the 90-degree corners ($V_2$ and $V_3$), yields excellent convergence rates. Because of these choices, $N$ must be even, and very specifically, $$\mathcal{P}_N:\, \left(x_\mu,y_\mu\right) = \left\{ \begin{array}{cl} \displaystyle \left(\frac12, \frac{1}{4}\left[ 1 - \cos \left(\frac{[\mu-\frac12]}{N/2}\,\pi \right) \right] \right) &\qquad\mbox{for}\quad \mu=1,2,\cdots,\frac{N}{2}\\ \displaystyle \left(\frac{1}{2}\, \sin \left(\frac{[\mu-\frac12 ]}{N}\,\pi \right) , \frac12 \right) & \qquad\mbox{for}\quad \mu=\frac{N}{2}+1,\frac{N}{2}+2,\cdots,N \end{array}\right.$$ Incrementing $\mu$, these matching points start near $V_2$, and end near the midpoint of $V_3$ and $V_4$, as shown in Fig. \[fig:cutsquare\]. ![Cut-square hexagon and example $N=8$ matching points ($N$ even and $N/2$ on each edge, as shown). There are two matching points very close to vertex $V_3$ (one on each edge), and one very close to vertex $V_2$. The shaded region is the most convenient and smallest $\Omega$ region needed to solve for the non-closed-form eigenvalues.[]{data-label="fig:cutsquare"}](cutsquare/cutsquare) A sweep of the interval from to $100$ reveals the first three Dirichlet eigenvalues, which belong to class A, B, and C; respectively. Initial bounds (within $\pm 1$) of these three are , , and ; which are then calculated to 200 digits, with 100-digit truncated values presented in Table \[tab:ultraprecise\]. The lowest closed-form Dirichlet eigenvalue[^27] is relatively high at $\lambda\approx 197.4$, which appears to be the tenth Dirichlet eigenvalue. The discussion for the Neumann modes is nearly identical, except that one must choose “$\cos$” (in Eq. \[eq:basisfunctions\]) and, for the closed-form modes, one must include the $m=0$ term. Also, this example is unique among the chosen examples because there are two even point-matched edges. Like the Dirichlet results, the lowest three Neumann eigenvalues belong to symmetry classes A, B, and C; respectively. They are initially bounded (within $\pm 1$) with , , and ; which are then calculated to 200 digits, with 100-digit truncated values presented in Table \[tab:ultraprecise\]. The lowest closed-form Neumann eigenvalue[^28] is $\lambda\approx 39.5$, which is apparently the next one up. For these lowest Neumann and Dirichlet eigenvalues, the convergence rate is quite rapid at about $1/2$, which means about 400 matching points are needed to yield the results. Prior to this work and that of Ref. [@abfr2015], the only published eigenvalue for this shape was the lowest Dirichlet eigenvalue, and the best bound is the six-digit result, $\lambda_1=35.6315\,_{15}^{22}$. To get that, Yuan and He [@yh2009] used a similar MPS starting point, fifty matching points in total on all but the two adjacent edges forming the “cut”, and a far more complicated bound calculation. (Their distribution of points was unspecified.) To obtain an “apples-to-apples” comparison with that publish result, I temporarily abandoned the symmetry considerations and repeated the calculation for up to the relatively low matching points (using $N$—a multiple of seven—Chebyshev matching points, distributed on all four non-adjacent edges), and was able to write down a nine-digit estimate , by inspection. That “low-precision” result took a few seconds of CPU time. It is interesting to note that by simply restoring the symmetry reduction, the convergence rate becomes about three times faster: At $50$ expansion terms (i.e., the same numerical effort, requiring a few CPU seconds), I can write down thirty digits, , again, by inspection. In hindsight, this problem was not as difficult (to solve) as one might have been led. But there are interesting features that do suggest further research, such as the nature of the symmetry. Star (regular concave decagon) {#star-regular-concave-decagon .unnumbered} ------------------------------ The five-pointed star provides another relatively difficult shape with which to demonstrate the method. It is also known as a regular concave decagon or the outline of the pentagram star. The size is such that points of the star coincide with the vertices of a unit-edged, regular pentagon. This shape shares the same symmetry group as the regular pentagon. Since I observed some regular-pentagon Dirichlet eigenfunctions develop nodal lines approximating a regular [*pentagram*]{}—two examples are illustrated in Figs. \[fig:starandpentagon\] and \[fig:DS065a\]; I was inspired to examine this shape, at least for a few low eigenvalues. A subset of eigenmodes of the regular pentagon do have an approximate symmetry leading to that observation, however, I reserve that for a future discussion. This star can become a research project in and of itself: It is an interesting and familiar shape for which I have been unable to find any published eigenvalues for comparison. ![(LEFT) Star shape with all even lines connecting opposite vertices and (RIGHT) the fundamental region forming the polygon, $\Omega$, in its canonical position and with $N=10$ Chebyshev-like distributed matching points.[]{data-label="fig:starB"}](starB) ![The spectrum of the concave decagon (star) for $\lambda < 1002$. This diagram includes the lowest 79 eigenvalues, which is composed of 31 Dirichlet and 48 Neumann eigenvalues. They are sorted according to symmetry. Except that every B and C eigenvalue is doubly degenerate, there are actually no other degeneracies in this set—even though some come close.[]{data-label="fig:starSpectrum"}](starspectrum) Like the regular pentagon, all star eigenvalues can sorted into one of two one-dimensional or two two-dimensional classes. The non-degenerate $\mathcal{S}$ and $\mathcal{A}$ eigenvalues require the triangular fundamental region shown in Fig. \[fig:starB\]. The doubly-degenerate $\mathcal{B}$ and $\mathcal{C}$ “mixed” modes require the arrow-head shape shown in Fig. \[fig:starA\], which is formed by reflecting the fundamental triangle through it shortest edge. With this problem, unlike some other examples, I could not get the point-matching method to converge with evenly-spaced matching points, however, it does work with points crowded near the vertices. A distribution that yields good convergence rates is the Chebyshev-like distribution $$y_\nu = \frac{1}{2}\left\{\left[ y_{C}+y_{B} \right] +\left[ y_{C}-y_{B}\right]\cos\left(\frac{\nu\pi}{n_2+1}\right)\right\}$$ for $\nu=1,2,...,n_2$; where $n_2=N$ ($\mathcal{A}$, $\mathcal{S}$) or $N/2$ ($\mathcal{B}$, $\mathcal{C}$), and where $y_B$ and $y_C$ are the $y$-values of the matching-edge endpoints. With this distribution, there is not a simple alternation of approximate eigenvalues with incremented values of $N$. Instead, it usually takes several (about three) steps to alternate. The convergence rate is nevertheless high enough that the peaks can be interpreted as upper and lower bounds. The lowest seventy-nine eigenvalues (including both Dirichlet and Neumann modes) were calculated for $\epsilon<10^{-20}$, and since so little is known about this problem, the rich spectrum is simply and concisely presented in Fig. \[fig:starSpectrum\]. Several representative eigenfunctions are presented in Fig. \[fig:stareigenfunctions\], and the lowest four Dirichlet and lowest four Neumann eigenvalues are calculated to at least 100 digits and presented in Table \[tab:ultraprecise\]. Regular polygons {#regular-polygons .unnumbered} ---------------- Of the regular polygons, it is well known that the equilateral triangle and the square are fully solved in closed form, and the regular hexagon has a subset of closed-form solutions (by piecing together equilateral triangle solutions). Otherwise, regular polygons have no closed-form solutions. Quite surprisingly, (non-closed-form) regular polygon results are rather fragmentary and spread far and wide in the published literature. This represents a gaping hole in our collective understanding. To illustrate, consider the regular hexagon. In 1978, Bauer and Rice [@br1978] calculated the lowest 21 Dirichlet eigenvalues to at least five or so digits. In 1993, I [@phdthesis] extended that by a factor of three, to about six digits; and in 1998, Cureton and Kuttler [@ck1998] almost doubled that tally, and to about eight digits. The regular pentagon is no better. In 2010, Lanz [@lanz2010] published what appears to be the most comprehensive list (that I am aware of) consisting of the ten lowest Dirichlet eigenvalues, accurate to half-a-dozen digits. Jumping to the more extreme regular polygon: In 2004, Strang and Grinfeld [@sg2004], and in 2008, Guidotti and Lambers [@gl2008], both calculated the ten lowest simple Dirichlet eigenvalues, accurate to about half-a-dozen digits, the lowest of which ranges from $5.78319$ to $5.78320$ (adjusting for the area). In addition, those authors, and—more specifically—Oikonomou [@o2010], offer an asymptotic expansion in “$1/\sigma$” where $\sigma$ is the number of sides on the regular polygon. Recently, Mark Broady[^29] [@phdthesisBroady] added two more terms to that expansion, which, for the lowest Dirichlet eigenvalue, is $$\frac{\lambda_1}{j_{0,1}^2} = \left\{ 1 + \frac{4\,\zeta(3)}{\sigma^3} + \frac{\left[ 12 - 2\, j_{0,1}^2 \right]\,\zeta(5)}{\sigma^5} + \frac{\left[ 8 + 4\, j_{0,1}^2 \right]\,\zeta^2(3)}{\sigma^6} + \mathcal{O}\left(\frac{1}{\sigma^{7}}\right) \right\}. \label{eq:regpolasymp}$$ where the regular polygon has area equal to $\pi$, is the first root of the Bessel function $J_0(x)$, and $\zeta(x)$ is the Riemann-zeta function.[^30] Incidentally, and as far as I know, that has been the only effort to express any non-closed-form eigenvalues analytically in terms of other “known” constants. To illustrate the method, I find it interesting to calculate the lowest Dirichlet eigenvalues of $\sigma$-sided regular polygons with up to $\sigma=130$ at a precision of $\epsilon<10^{-30}$. The tail end of that calculation is shown in Table \[tab:thirtydigitregularpolygon\], with truncated 100-digits results for the pentagon to the decagon presented in Table \[tab:ultraprecise\]. See also Eq. (\[eq:polygon256\]) for a result for the 256-sided regular polygon, which was used to illustrate the notation. Of note is that above the dodecagon, i.e., $\sigma>12$, the canonical Chebyshev distribution given by Eq. (\[eq:canonicalchebyshevnodes\]) doesn’t work. Instead, using half of it does seem to work for all regular polygons, $$y_\nu = y_{\mathrm max} \sin(\nu\pi/(N+1))$$ where $\nu=1,2,...,N$, and where $y_{\mathrm max}$ is the $y$-value of the highest point of the fundamental triangle in Fig. \[fig:regularpolygons\](CENTER); here $y_{\mathrm max}=\cos(\pi/\sigma)$. This distribution crowds matching points near the “distant” acute angle ($\alpha/2=\pi/\sigma$), and spreads them out near the right angle. It also seems to very reliably yield alternating approximations at each increment, $\Delta N=1$. This exercise illustrates specifically that even though exponential convergence is present, as the number of sides increases, the convergence rate becomes quite poor. Beginning at the pentagon, $\rho\approx 1/1.14$ when $\epsilon\approx 10^{-1000}$. When one reaches the 130-sided regular polygon, that convergence rate has dropped to about $\rho\approx 1/28.1$ at the 30-digit result. A rough estimate for the convergence rate for this problem is $\rho\approx 5/\sigma$, at least for the 30-digit results. Also of note is that when the convergence rate is good, more precision in the calculations is needed: The regular pentagon required $1.7N$ digits. Above the decagon, $1.4N$ seemed adequate. For situations where $\sigma$ (number of polygon sides) is low (5 or 6), with a little practice, it becomes relatively straightforward to calculate hundred-digit results, for both Neumann and Dirichlet boundary conditions, for up to perhaps the lowest ten thousand eigenvalues. For example, the 8139^th^ Dirichlet eigenvalue (same symmetry class as the lowest) of the (unit-edged) regular pentagon to 300 correct digits is bound with $$\lambda_{8139}=100001.28198274831240\cdots 32644630143129793681\,_{3271}^{5381} \label{eq:8139}$$ which, as indicated, displays the leading and trailing digits. Turning attention to how far one can extend the precision of these numbers using a laptop, two “1000+” digit eigenvalues are calculated. The lowest Dirichlet eigenvalue of the unit-edged regular pentagon, to 1502 digits, $$\lambda_1 = 10.996427084559806648 \cdots 82166474404544652968\,^{9936}_{8251}$$ was submitted to <http://www.oeis.org/A262823> [@oeisorg]. This hexagon number required just under one month of computation time. Similarly, the lowest Dirichlet eigenvalue of the unit-edged regular hexagon, to 1001 correct digits, $$\lambda_1 = 7.1553391339260551282 \cdots 77979918378681828158\,^{1503}_{0889}$$ was submitted to <http://www.oeis.org/A263202> [@oeisorg]. This required about five days of computation time. [clcc]{} $\sigma$ & $\lambda_1$ ($\epsilon<10^{-30}$) & $\widehat{N}$ & $\epsilon$\ 126 & $5.7831998639169811697955997275\,_{4738}^{5287}$ & 818 & $9.5\times 10^{-31}$\ 127 & $5.7831995381236804121745520138\,_{6591}^{7147}$ & 824 & $9.6\times 10^{-31}$\ 128 & $5.78319922243209895698523832013\,_{030}^{555}$ & 831 & $9.7\times 10^{-31}$\ 129 & $5.78319891645372682901545245421\,_{112}^{643}$ & 837 & $9.2\times 10^{-31}$\ 130 & $5.78319861981784749432269771828\,_{013}^{587}$ & 842 & $9.9\times 10^{-31}$\ $\infty$ & $5.783185962946784521175995758456$ & &\ Acknowledgements {#acknowledgements .unnumbered} ================ I wish to thank Alex Barnett for making the specific suggestion to expand about the non-analytic vertex using fractional-order Bessel functions, i.e., non-integral $m$-values (private communication, December 2014). Indeed, after sharing some regular pentagon results with him, he suggested that instead of expanding about the center of the regular pentagon, I should expand about one of its vertices. That one simple, and—in hind-sight—obvious suggestion, immediately turned my eight-digit calculations into multi-hundred-digit calculations because of the exponential convergence. I also wish to thank James Kuttler and Nick Trefethen for suggestions and encouragement. I am also encouraged by the recent independent efforts of Mark Broady and Paolo Amore, et al., and wish to thank them for interesting dialogs. Of course, this project was made possible by free software, most specifically [GMP]{} [@gmp6] and the [GP/PARI]{} [@PARI2] calculator. Conclusion {#conclusion .unnumbered} ========== By using a method substantially identical to that of FHM, but using modern hardware and \[free\] software, and a little patience, I have demonstrated that it becomes relatively easy to exploit the well-known “exponential convergence” to calculate eigenvalues of the Laplacian to very high precision, very typically hundreds of digits, and about a thousand digits for the lowest Dirichlet eigenvalue of the L-shape (1001 digits), regular pentagon (1502 digits), and regular hexagon (1001 digits). I believe my unique contribution to this problem includes (a) the observation that the sequence of approximate eigenvalues alternates as one adds matching points—thus yielding an easy and excellent method to bound eigenvalues, (b) revealing a very simple way to overcome the numerical ill-conditioning of the point-matching method and exploit its advantages to permit easy calculation of those eigenvalues, and (c) computing the first hundred-digit and thousand-digit non-closed-form eigenvalue results for a variety of shapes. As I explore this interesting and classic problem, I see theoretical and pedagogical gaps that need to be filled, and a need to compile results. My hope is to inspire others pick a shape and start calculating. Appendix {#appendix .unnumbered} ======== [l]{}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[tab:ultraprecise\] [^1]: Very recently, P. Amore, et al. [@abfr2015] have independently calculated several highly precise eigenvalues (up to dozens of digits) for various shapes, including the L-shape and the cut-square. They also used the FHM method for some problems, but their focus was on a high-order Richardson extrapolation method with finite elements, which, among other things, can handle more varied shapes. Fortunately, their calculations provide (partial) independent validation of my results. [^2]: Typically per the GPL <http://www.gnu.org/philosophy/free-sw.html>. [^3]: At times, using up to swap space. [^4]: Among the chosen shapes, the L-shape, cut-square, and regular hexagon have subsets of closed-form solutions. For polygons, a closed-form solution can be written as a finite sum of plane waves, and its eigenvalue is related to $\pi$, presently known to $13.3\times 10^{12}$ digits. [^5]: This is related to the increasing number of integer solutions to Diophantine equations $a^2=b^2+c^2$ (rectangle) and $a^2=b^2+bc+c^2$ (equilateral triangle). [^6]: Such non-closed-form degeneracies are possible but presumably quite rare. This assumption should be independently verified. [^7]: All non-closed-form solutions in this project have an $\Omega$ with exactly one non-analytic vertex. For all eigenfunctions within a given symmetry class, it seems that a vertex will be either analytic or not. [^8]: To calculate only eigenvalues, we need not normalize these basis functions. [^9]: For example, if $\partial\Omega_1$ and $\partial\Omega_2$ are both odd, then $V_2$ cannot be used as a matching point. [^10]: For example, if the point-matched edge is odd, then that contour would be a nodal curve passing through the matching points. If $N$ is too small for a given eigenvalue, then, for example, two adjacent matching points might not be directly joined by nodal curve, although nodal curves will pass through those matching points. [^11]: I first observed this alternation property in 1993 [@phdthesis] with exponentially-convergent closed-form solutions. [^12]: The point-matching determinant changes sign as one passes a root, i.e., provided that $\delta\lambda$ is sufficiently small, then if , then the root is in the interval $(\lambda,\lambda+\delta\lambda)$. Without knowing the smallest gap between eigenvalues, it is important to independently locate each eigenvalue. The GSVD method is perhaps better suited to sweep an interval to locate eigenvalues. [^13]: Going from $\widehat{N}=280\to 930$, the global convergence rate decreases from $1/27.9\to 1/46.5$. Asymptotically, $\rho^{[\widehat{N}]}\sim 1/65.1$. [^14]: This is certainly not a mathematical proof, but is quite plausible. [^15]: The area difference for a 100-digit eigenvalue is $\Delta A/A=10^{-100}$: Whimsically compare the area of a drum (vibrating membrane) with a diameter matching that of the universe, i.e., $R=47\times 10^9 \,\ell \mathrm{y}$, $A=6.1\times 10^{53}\,\mathrm{m}^2$, to the area change equal to the cross-sectional area of a proton, i.e., $R=0.88\,\mathrm{fm}$, $\Delta A = 2.4\times 10^{-30}\,\mathrm{m}^2$. This $\Delta A/A=4.0\times 10^{-84}$ is some fifteen orders of magnitude larger. [^16]: Although most of this analysis applies to all regular polygons, I shall exclude the closed-form solutions, as explained above. [^17]: This counting and the value of $\eta_1$ are related. [^18]: The other obvious choice, the triangle, has [*two*]{} non-analytic vertices. [^19]: To unify the expressions, one may include $\gamma=0$ (all $\sigma$) and $\gamma=\sigma/2$ (even $\sigma$) for the even-parity, non-degenerate towers, but this is not done here because $P'$ is not on the boundary of triangular fundamental region. It is easier to keep the solutions corresponding to and irreps separated. [^20]: Not just the regular polygons or the star. It is an exercise in geometry to figure out all of the specific shapes with dihedral-symmetry that also yield exponentially-convergent eigenfunction expansions (per the current procedure). [^21]: Of course the division of eigenfunctions according to dihedral symmetry is hardly new. For example, Cureton and Kuttler [@ck1998] identify the symmetry classes for the regular hexagon, which can be lined up using , , , , , , , and ; where the are class names in that reference. [^22]: For example, and rather unfortunately, the [maxima]{} function [ bessel\_j]{} does not seem to respect [fpprec]{}. [^23]: That highest $N$ value yields 100 correct digits in the eigenvalue, and altogether, takes about one hour of CPU time. Also note that requirement (b) was imposed by FHM for numerical reasons, I impose it here to accurately reproduce their results. [^24]: ... but abandoning the $\partial\Psi/\partial\theta=0$ requirement at $V_2$ and $V_3$ [^25]: Very recently, P. Amore et al. [@abfr2015] also examined this shape and discovered other interesting symmetry properties, and were also able to calculate very precise Dirichlet eigenvalues. They reported 40-digit MPS results, which do agree with my results. [^26]: This generally identifies a clever technique that can be used to empirically discover symmetries if they are not obvious or one is not inspired to figure it out from first principles. FHM has indications of this technique. [^27]: $\lambda=(1^2+2^2)(\pi/L)^2=20\pi^2\approx 197.4$, where $\Psi(x,y)\propto \sin(\pi x/L)\sin(2\pi y/L)-\sin(2\pi x/L)\sin(\pi y/L)$ and $L=1/2$. Not degenerate. [^28]: $\lambda=(1^2+0^2)(\pi/L)^2=20\pi^2\approx 39.5$, where $\Psi(x,y)\propto \cos(\pi x/L)+\cos(\pi y/L)$ and $L=1/2$. Not degenerate. [^29]: Private communication. [^30]: Area-$\pi$ regular polygons are preferred since it factors out the well-known area dependence on the eigenvalue; and, as $\sigma\to\infty$, $\lambda_1\to j_{0,1}^2$.

--- author: - 'Thomas Reeves[^1]' - 'Anil Damle[^2]' - 'Austin R. Benson' title: 'Network Interpolation[^3]' --- [^1]: Center for Applied Mathematics, Cornell University, Ithaca, NY 14853 (). [^2]: Department of Computer Science, Cornell University, Ithaca, NY 14853 (, ). [^3]: Submitted to the editors on June 28, 2019.

--- abstract: 'We present a dimensional analysis of two characteristic time scales in the boundary layer where the disk adjusts to the rotating neutron star (NS). The boundary layer is treated as a transition region between the NS surface and the first Keplerian orbit. The radial transport of the angular momentum in this layer is controlled by a viscous force defined by the Reynolds number, which in turn is related to the mass accretion rate. We show that the observed low- Lorentzian frequency is associated with radial oscillations in the boundary layer, where the observed break frequency is determined by the characteristic diffusion time of the inward motion of the matter in the accretion flow. Predictions of our model regarding relations between those two frequencies and frequencies of kHz QPO’s compare favorably with recent observations for the source 4U 1728-34. This Letter contains a theoretical classification of kHz QPO’s in NS binaries and the related low frequency features. Thus, results concerning the relationship of the low-Lorentzian frequency of viscous oscillations and the break frequency are presented in the framework of our model of kHz QPO’s viewed as Keplerian oscillations in a rotating frame of reference.' author: - Lev Titarchuk - Vladimir Osherovich title: Correlations between kHz QPO and Low Frequency Features Attributed to Radial Oscillations and Diffusive Propagation in the Viscous Boundary Layer Around a Neutron Star --- \#1[0= -0.015em0-0 0.03em0-0 -0.015em0.0433em0 ]{} 0.5 truecm Introduction ============ The discovery of kilohertz quasiperiodic oscillations (QPO’s) in the low mass X-ray neutron star (NS) binaries (Strohmayer 1996; Van der Klis 1996 and Zhang 1996) has stimulated both theoretical and observational studies of these sources. In the upper part of the spectrum (400- 1200 Hz) for most of these sources, two frequencies $\nu_k$ and $\nu_h$ have been seen. Initially, the fact that for some sources, the peak separation frequency $\Delta \nu=\nu_h-\nu_k$ does not change much led to the beat frequency interpretation (Strohmayer 1996; Van der Klis 1998) which was presented as a concept for the first time in the paper by Alpar & Shaham (1985). Beat-frequency models, where the peak separation is identified with the NS spin rate have been challenged by observations: for Sco X-1, $\Delta\nu$ varies by 40% (van der Klis 1997 hereafter VK97) and for source 4U 1608-52, $\Delta\nu$ varies by 26% (Mendez 1998). Mounting observational evidence that $\Delta\nu$ is not constant demands a new theoretical approach. For Sco X-1, in the lower part of the spectrum, VK97 identified two branches (presumably the first and second harmonics) with frequencies 45 and 90 Hz which slowly increase in frequency when $\nu_k$ and $\nu_h$ increase. Furthermore, in the spectra observed by Rossi X-ray Timing Explorer (RXTE) for 4U 1728-34, Ford and van der Klis (1998, herein FV98) found low frequency Lorentzian (LFL) oscillations with frequencies between 10 and 50 Hz. These frequencies as well as break frequency, $\nu_{break}$ of the power spectrum density (PSD) for the same source were shown to be correlated with $\nu_k$ and $\nu_h$. It is clear that the low and high parts of the PSD of the kHz QPO sources should be related within the framework of the same theory. Difficulties which the beat frequency model faces are amplified by the requirement of relating the observed low frequency features, described above, with $\nu_k$ and $\nu_h$. Recently, a different approach to this problem has been suggested: kHz QPO’s in the NS binaries have been modeled by Osherovich & Titarchuk (1999) as Keplerian oscillations in a rotating frame of reference. In this new model the fundamental frequency is the Keplerian frequency $\nu_k$ (the lower frequency of two kHz QPO’s) $$\nu_k={{1}\over{2\pi}}\left({{GM}\over{R^3}}\right)^{1/2},$$ where G is the gravitational constant, M is the NS mass, and R is the radius of the corresponding Keplerian orbit. The high QPO frequency $\nu_h$ is interpreted as the upper hybrid frequency of the Keplerian oscillator under the influence of the Coriolis force $$\nu_h=[\nu_k^2+(\Omega/\pi)^2]^{1/2},$$ where $\Omega$ is the angular rotational frequency of the NS magnetosphere. For three sources (Sco X-1, 4U 1608-52 and 4U 1702-429), we demonstrated that the solid body rotation ($\Omega=\Omega_0=const$) is a good first order approximation. Slow variation of $\Omega$ as a function of $\nu_k$ within the second order approximation is related to the differential rotation of the magnetosphere controlled by a frozen-in magnetic structure. This model allows us to address the relation between the high and low frequency features in the PSD of the neutron systems. We interpreted the $\sim 45$ and $90$ Hz oscillations as 1st and 2nd harmonics of the lower branch of the Keplerian oscillations in the rotating frame of reference: $$\nu_L=(\Omega/\pi)(\nu_k/\nu_h)\sin\delta,$$ where $\delta$ is the angle between ${\bf \Omega}$ and the vector normal to the plane of the Keplerian oscillations. For Sco X-1, we found that the angle $\delta=5.5^o$ fits the observations. In this Letter we include the LFL oscillations and related break frequency phenomenon in our classification. We attribute LFL oscillations to radial oscillations in the viscous boundary layer surrounding a neutron star. According to the model of Shakura & Sunyaev (1973, hereafter SS73), the innermost part of the Keplerian disk adjusts itself to the rotating central object (i.e. neutron star). The recent modelling by Titarchuk, Lapidus & Muslimov (1998, hereafter TLM) led to the determination of the characteristic thickness of the viscous boundary layer $L$. In the following section, we present the extension of this work to relate the frequency of the viscous oscillations $\nu_v$ and $\nu_{break}$ with $\nu_k$. Comparison with the observations is carried out for 4U 1728-34. The last section of this Letter contains our theoretical classification of kHz QPO’s and related low frequency phenomena. Radial Oscillations and Diffusion in the Viscous Boundary Layer =============================================================== We define the boundary layer as a transition region confined between the NS surface and the first Keplerian orbit. The radial motion in the disk is controlled by the friction and the angular momentum exchange between adjacent layers resulting in the loss of the initial angular momentum by an accreting matter. The corresponding radial transport of the angular momentum in a disk is described by the equation (e.g. SS73): $$\dot M {d\over {dR}}(\omega R^2) = 2\pi {d\over {dR}} (W_{r\varphi}R^2),$$ where $\dot{M}$ is the accretion rate, and $ W_{r\varphi}$ is the component of a viscous stress tensor which is related to the gradient of the rotational frequency $\omega$, namely $$W_{r\varphi}=-2\eta HR{{d\omega}\over{dR}},$$ where $H$ is a half-thickness of a disk, and $\eta$ is the turbulent viscosity. The nondimensional parameter which is essential for equation (4) is the Reynolds number for the accretion flow $$\gamma={{\dot M}\over{4\pi\eta H}}={{3R v_r}\over {{\it v}_t{\it l}_t}},$$ which is the inverse $\alpha-$parameter in the SS73-model; $v_r$ is a characteristic velocity, $v_t$ and $l_t$ are a turbulent velocity and related turbulent scale respectively. Equations $\rm \omega=\omega_0~{\rm~at}~R=R_0$ (NS radius) and ${\rm \omega=\omega_K~at~R=R_{out}}$ (radius where the boundary layer adjusts to the Keplerian motion), and $\rm {{d\omega}\over{dr}}={{d\omega_k}\over{dr}}~ at~\rm R=R_{out}$ were assumed by TLM as boundary conditions. Thus the profile $\omega(R)$ and the outer radius of the viscous boundary layer $R_{out}$ are uniquely determined by these boundary conditions. Presenting $\omega(R)$ in terms of dimensionless variables: namely angular velocity $\theta=\omega/\omega_0$, radius $ r=R/R_0$ ($ R_0=x_0R_s$, $ R_s=2GM/c^2$ is the Schwarzschild radius), and mass $ m=M/M_{\odot}$, we express Keplerian angular velocity as $$\theta_K={{6}/(a_K r^{3/2}}),$$ where $a_K=m(x_0/3)^{3/2}(\nu_0/363~{\rm Hz})$ and the NS rotational frequency $\nu_0$ has a particular value for each star. The particular coefficient, 6, presented in formula (7) is obtained for the frequency of nearly coherent (burst) oscillations for 4U 1728-34, i.e. for $\nu_0=363$ Hz. The solution of equations (4-5 ) satisfying the above boundary conditions is $$\theta(r)=D_1 r^{-\gamma} + (1-D_1) r^{-2},$$ where $D_1=(\theta_{out}-r_{out}^{-2})/(r_{out}^{-\gamma} -r_{out}^{-2})$ and $\theta_{out}=\theta_K(r_{out})$. Equation $\theta^{\prime}(r_{out})=\theta_K^{\prime}(r_{out})$ determines $r_{out}$: $${3\over2} \theta_{out}=D_1 \gamma r_{out}^{-\gamma}+2(1-D_1)r_{out}^{-2}.$$ The solution of equations (4-5) subject to the inner sub-Keplerian boundary condition has a regime corresponding to the super-Keplerian rotation (TLM). For such a regime matter piles up in the vertical direction thus disturbing the hydrostatic equilibrium. The vertical component of the gravitational force prevents this matter from further accumulation in a vertical direction and drives relaxation oscillations. The radiation drag force, which is proportional to the vertical velocity, determines the characteristic decay time of the vertical oscillations (TLM). The characteristic time $t_r$, over which the matter moves inward through this region, bounded between the innermost disk and relaxation oscillations zone is $$t_r\sim {L\over v_r},$$ where $L=R_{out}-R_0$ is the characteristic thickness of this region. Even though the specific mechanism providing the modulation of the observed X-ray flux over this timescale needs to be understood, this timescale apparently “controls” the supply of accreting matter into the innermost region of the accretion disk. Any local perturbation in the transition region would propagate diffusively outward over a timescale $$t_{diff}\sim \left({L\over {l_{fp}}}\right)^2 {{l_{fp}}\over v_r},$$ where $l_{fp}$ is the mean free path of a particle. Note, that the $\gamma-$parameter is proportional to the accretion rate (see Eq. 6), and therefore $v_r\propto \gamma$. Using this relationship, we can exclude $v_r$ from the above equations and get the relations for the corresponding inverse timescales (frequencies). For the frequency of viscous oscillations $$\nu_{v}\propto {\gamma\over {r_{out}-r_0}},$$ and for the break frequency, related to the diffusion $$\nu_{break}\propto{\gamma\over {(r_{out}-r_0)^2}}.$$ In the following section, we compare the predictions of this model with the observations and also establish the theoretical relation between $\nu_v$ and $\nu_{break}$. Comparisons with Observations ============================= The results of FV98 for the low frequency Lorentzian in the X-ray binary 4U 1728-34 are presented in Figure 1 and for the break frequency $\nu_{break}$ in Figure 2. In Figure 1, crosses represent the frequencies (with the appropriate error bars) observed during four days. Data collected on February 16 (open circles) are situated apart from the rest of observations and they are not included in the empirical power law fit which is suggested by FV98. In the work discussed above, the authors plotted the observed low frequencies versus high-frequency QPO which for all days, except February 16 was $\nu_k$ and apparently for February 16 it was $\nu_h$. Our theoretical curve for $\nu_v$ versus $\nu_k$ is based on equation (12). The $\chi^2$ dependence on this parameter is rather strong: the parabola $\chi^2=38024-73076\cdot a_k+35732\cdot a_k^2$ has a minimum at $a_k=1.03$, which determines the best fit. Using $\Omega/2\pi=340$ Hz in the upper hybrid relation (2), we calculate $\nu_k$ for the points observed on February 16 and show that they belong to the set of frequencies modeled by our theoretical curve for the viscous radial oscillations (closed circles). Identification of the observed $\nu_{break}$ with the inverse diffusion time (formulas 11 and 13) is illustrated by theoretical curves in Figure 2. It is worth noting that these two correlations with kHz frequencies are fit by two theoretical curves using [*only one parameter $a_k$*]{}. The $\chi^2-$ dependence on $a_k$ is obtained with inclusion of all data points for the break and low frequency correlations (75 data points). The theoretical dependences of $\nu_k$ and $r_{out}$ on $\gamma-$parameter are calculated numerically using equations (7) and (9) and employed here for calculations of the theoretical curves in Figures 2 and 3 using equations (12) and (13). We were unable to interpret data for $\nu_{break}$ collected in February 16 (open circles). Neither $\nu_v$ nor $\nu_{break}$ in our theory have a power law relation with $\nu_k$. However, the theoretical relation between $\nu_{break}$ and $\nu_v$, shown in Figure 3 by a solid curve, is close to the straight line (in log-log diagram), suggesting an approximate power law $$\nu_{break}=0.041\nu_v^{1.61}.$$ This relation is derived from the theoretical dependence for the best fit parameter $a_k=1.03$. Observations of FV98 (except February 16) are also presented in the Figure 3. **4. Discussion and Conclusions** We present a model for the radial oscillations and diffusion in the viscous boundary layer surrounding the neutron star. Our dimensional analysis has identified the corresponding frequencies $\nu_v$ and $\nu_{break}$ which are consistent with the low Lorentzian and break frequencies for 4U 1728-34. and predicted values for $\nu_{break}$ related to the diffusion in the boundary layer are consistent with the break frequency observed for the same source. Both oscillations (Keplerian and radial) and diffusion in the viscous boundary layer are controlled by the same parameter - Reynolds number $\gamma$ which in turn is related to the accretion rate. It is shown in TLM that $\nu_k$ is a monotonic function of $\gamma$. Therefore, the observed range of $\nu_k$, (350-900 Hz) corresponds to the range $1<\gamma<5$ (or $0.2<\alpha<1$). The results in this Letter extend the classification of kHz QPO’s and the related low frequency phenomena suggested by Osherovich & Titarchuk 1999. Figure 4 summarizes the new classification. Solid lines represent our theoretical curves and open circles observations for Sco X-1 (from VK97). As one can see, formulas (2) and (3), for the Keplerian oscillator under the influence of the Coriolis force, reproduce the observations well. Indeed, $\Delta\nu=\nu_h-\nu_k$ is not constant, as observed (see OT99 for details of comparisons of the data with the theory). Effectively, the main viscous frequency $\nu_v$ and the diffusive $\nu_{break}$ introduce the second oscillator with two new branches in the lower part of the spectra. The unifying characteristic of spectra for both oscillators is the strong dependence on $\nu_k$. This common dependence on $\nu_k$ can be viewed as a result of the interaction between Keplerian oscillator and the viscous oscillator which share the common boundary at the outer edge of the viscous transition layer. Our parametric study indicates that the power law index 1.6 in Eq. (14) should be the same for different neutron stars. We expect a similar relation for black holes but with a distinctly different index. The found value of $a_k$ leads ultimately to independent constraints in the determination of mass and radius for the neutron star (Haberl & Titarchuk 1995). LT thanks NASA for support under grants NAS-5-32484 and RXTE Guest Observing Program. The authors acknowledge discussions with Alex Muslimov, Jean Swank, Lorella Angelini, Will Zhang, Joe Fainberg and fruitful suggestions by the referee. Particularly, we are grateful to Eric Ford, and Michiel van der Klis, for the data which enable us to make comparisons with the data in detail. Alpar, M., & Shaham, J. 1985, Nature, 316, 239 Ford, E., & van der Klis 1998, , 506, L39 (FV98) Haberl, F., & Titarchuk, L. 1995, A&A , 299, 414 Mendez, M., van der Klis, M., Ford, E.C., Wijnand, R., Ford, E.C., van Paradijs, J., Vaughan, B.A. 1998, , 505, L23 Osherovich, V.A. & Titarchuk, L.G. 1999, submitted to ApJLetters (OT99) Shakura, N. I. & Sunyaev, R. A. 1973, A&A, 24, 337 (SS73) Strohmayer, T. E., Zhang, W., Swank, J. H., Smale, A., Titarchuk, L., Day, C., & Lee, U. 1996, , 469, L9 Titarchuk, L., Lapidus, I.I.,& Muslimov, A. 1998, , 499, 315 (TLM) van der Klis, M. in AIP Conf Proc 431, 361 (VK98) van der Klis, M., Wijnands, A.D., Horne, K. & Chen, W. 1997, , 481, L97 (VK97) van der Klis, M. 1996, IAUC 6319, , 469, L1 Zhang, W., Lapidus, I.I., White, N. E., & Titarchuk, L. G. 1996, , 469, L17

--- abstract: 'We consider the $\pi^+\pi^-\pi_0\gamma$ final state in electron-positron annihilation at cms energies not far from the threshold. Both initial and final state radiations of the hard photon are considered, but without interference between them. The amplitude for the final state radiation is obtained by using the effective Wess-Zumino-Witten Lagrangian for pion-photon interactions valid for low energies. In real experiments energies are never so small that $\rho$ and $\omega$ mesons would have a negligible effect. So a phenomenological Breit-Wigner factor is introduced in the final state radiation amplitude to account for the vector mesons influence. Using radiative 3$\pi$ production amplitudes, a Monte Carlo event generator is developed which could be useful in experimental studies.' author: - | A.Ahmedov$^a$, G.V.Fedotovich$^b$, E.A.Kuraev$^c$, Z.K.Silagadze$^b$\ $^a$[Laboratory of Particle Physics, JINR, 141980, Dubna, 141980 Russia ]{}\ $^b$[Budker Institute of Nuclear Physics, 630 090, Novosibirsk, Russia ]{}\ $^c$[Laboratory of Theoretical Physics, JINR, 141980, Dubna, Russia ]{} title: 'Near threshold radiative 3$\pi$ production in $e^+e^-$ annihilation' --- Introduction ============ The new Brookhaven experimental result for the anomalous magnetic moment of the muon [@1] aroused considerable interest in the physics community, because it was interpreted as indicating a new physics beyond the Standard Model [@2]. However, such claims, too premature in our opinion, assume that the theoretical prediction for the muon anomaly is well understood at the level of necessary precision. Hadronic uncertainties become of main concern [@3]. Fortunately the leading hadronic contribution is related to the hadronic corrections to the photon vacuum polarization function, which can be accurately calculated provided that the precise experimental data on the low-energy hadronic cross sections in the $e^+e^-$ annihilation are at our disposal. In the last few years high-statistics experimental data were collected in the $\rho$-$\omega$ region in Novosibirsk experiments at the VEPP-2M collider [@4]. In this region the hadronic cross sections are dominated by the $e^+e^-\to 2\pi$ and $e^+e^-\to 3\pi$ channels. The former is of uppermost importance for reduction of errors in the evaluation of the hadronic vacuum polarization contribution to the muon g-2. Considerable progress was reported for this channel by the CMD-2 collaboration [@5]. The $e^+e^-\to 3\pi$ channel, which gives a less important but still significant contribution to the hadronic error, was also investigated in the same experiment in the $\omega$-meson region [@6]. Such high precision experiments require accurate knowledge of various backgrounds. Among them the $e^+e^-\to 3\pi\gamma$ channel provides an important background needed to be well understood. This experimental necessity motivated our investigation of the three pion radiative production presented here. Besides, being of interest as an important background source, this process could be of interest by itself, because a detailed experimental study of the final state radiation will allow one to get important information about pion-photon dynamics at low energies. However, such an experimental investigation will require much more statistics than available in VEPP-2M experiments and maybe would be feasible only at $\phi$-factories where the low energy region can be reached by radiative return technique as was recently been demonstrated in the KLOE experiment [@7]. Initial state radiation ======================= Let $J_\mu$ be the matrix element of the electromagnetic current between the vacuum and the $\pi^+\pi^-\pi^0$ final state. Then the initial state radiation (ISR) contribution to the $e^+e^-\to\pi^+\pi^-\pi^0\gamma$ process cross section is given at $O(\alpha^3)$ by the standard expression [@8] $$\begin{aligned} & & d\sigma_{ISR}(e^+e^-\to 3\pi\gamma)=\frac{e^6}{4(2\pi)^8 (Q^2)^2}\left \{ \frac{Q^2}{4E^2}~J\cdot J^* \left (\frac{p_+}{k\cdot p_+}- \frac{p_-}{k\cdot p_-} \right )^2- \right . \nonumber \\ & & \left . -\frac{Q^2}{2E^2}~\frac{p_+\cdot J~ p_+\cdot J^* + p_-\cdot J~ p_-\cdot J^*}{k\cdot p_+ k\cdot p_-}- \frac{J\cdot J^*}{2E^2}\left ( \frac{k\cdot p_+}{k\cdot p_-}+\frac{k\cdot p_-}{k\cdot p_+}\right ) + \right . \label{eq1} \\ & & \left . +\frac{m_e^2}{E^2} \left ( \frac{p_+\cdot J} {k\cdot p_-}-\frac{p_-\cdot J}{k\cdot p_+}\right ) \left (\frac{p_+\cdot J^*}{k\cdot p_-}- \frac{p_-\cdot J^*}{k\cdot p_+}\right ) \right \}d\Phi \equiv \frac{e^6}{4(2\pi)^8}~|A_{ISR}|^2d\Phi, \nonumber\end{aligned}$$ where $d\Phi$ stands for the Lorentz invariant phase space $$d\Phi=\frac{d\vec{k}}{2\omega}~\frac{d\vec{q}_+}{2E_+}~ \frac{d\vec{q}_-}{2E_-}~ \frac{d\vec{q}_0}{2E_0}~\delta(p_++p_--k-q_+-q_--q_0)$$ and $Q^2=(q_++q_-+q_0)^2=4E(E-\omega)$ is the photon virtuality, $E$ being the beam energy and $\omega$ – the energy of the $\gamma$ quantum. Particle 4-momenta assignment can be read from the corresponding diagrams presented in Fig.\[Fig1\]. The current matrix element $J_\mu$ has a general form $$J_\mu=\epsilon_{\mu\nu\sigma\tau}q_+^\nu q_-^\sigma q_0^\tau ~F_{3\pi}(q_+,q_-,q_0). \label{eq2}$$ For the $F_{3\pi}$ form-factor, which depends only on invariants constructed from the pions 4-momenta, we will take the expression from [@9] $$F_{3\pi}=\frac{\sqrt{3}}{(2\pi)^2f_\pi^3}\left [\sin{\theta}\cos{\eta} ~R_\omega (Q^2)-\cos{\theta}\sin{\eta}~R_\phi (Q^2)\right ] \left ( 1-3\alpha_K-\alpha_K H \right ) . \label{eq3}$$ Here $\alpha_K\approx 0.5$, $f_\pi\approx 93~\mathrm{MeV}$ is the pion decay constant, $\eta=\theta-\arcsin{\frac{1}{\sqrt{3}}}\approx 3.4^\circ$ characterizes the departure of the $\omega$-$\phi$ mixing from the ideal one, and $$H=R_\rho(Q_0^2)+R_\rho(Q_+^2)+R_\rho(Q_-^2),$$ where $$Q_0^2=(q_++q_-)^2,\;\; Q_+^2=(q_0+q_+)^2,\;\;Q_-^2=(q_0+q_-)^2.$$ The dimensionless Breit-Wigner factors have the form $$R_V(Q^2)=\left [ \frac{Q^2}{M_V^2}-1+i\frac{\Gamma_V}{M_V}\right ]^{-1}, \;\; R_\rho(Q^2)=\left [ \frac{Q^2}{M_\rho^2}-1+ i\frac{\sqrt{Q^2}\Gamma_\rho(Q^2)}{M_\rho^2}\right ]^{-1},$$ where $V=\omega, \phi$ and for the $\rho$ meson the energy dependent width is used $$\Gamma_\rho(Q^2)=\Gamma_\rho \frac{M_\rho^2}{Q^2}\left (\frac{Q^2-4m\pi^2} {M_\rho^2-4m\pi^2}\right )^{3/2}.$$ The last term in (\[eq1\]) is completely irrelevant for VEPP-2M energies if the hard photon is emitted at a large angle. So we will neglect it in the following. Final state radiation ===================== To describe final state radiation (FSR), we use the effective low-energy Wess-Zumino-Witten Lagrangian [@10]. The relevant piece of this Lagrangian is reproduced below $$\begin{aligned} & & \hspace*{30mm}\left . \left . {\cal{L}}= \frac{f_\pi^2}{4}Sp \right [D_\mu U(D_\mu U)^+ +\chi U^++U\chi^+ \right ]- \nonumber \\ & & \hspace*{-9.2mm} \left . \left . \left . \left . -\frac{e}{16\pi^2}\epsilon^{\mu\nu\alpha\beta} A_\mu Sp \right [Q \right \{ (\partial_\nu U)(\partial_\alpha U^+)(\partial_\beta U)U^+- (\partial_\nu U^+)(\partial_\alpha U)(\partial_\beta U^+)U \right \} \right ] - \nonumber \\ & & \hspace*{-9mm} - \frac{ie^2}{8\pi^2}\epsilon^{\mu\nu\alpha\beta}(\partial_\mu A_\nu) A_\alpha Sp\left [Q^2(\partial_\beta U)U^++Q^2U^+(\partial_\beta U)+ \frac{1}{2} QUQU^+(\partial_\beta U)U^+ - \right .\nonumber \\ & & \hspace*{30mm} \left . -\frac{1}{2}QU^+QU(\partial_\beta U^+)U \right ]. \label{eq4} \end{aligned}$$ Here $U=\exp{\left (i\frac{\sqrt{2}P}{f_\pi}\right )}$, $D_\mu U=\partial_\mu U+ieA_\mu [Q,U]$, $Q=\mathrm{diag}\left ( \frac{2}{3},-\frac{1}{3}, -\frac{1}{3}\right )$ is the quark charge matrix, and the terms with $\chi=B\, \mathrm{diag}\left (m_u,m_d,m_s \right )$ introduce explicit chiral symmetry breaking due to nonzero quark masses. The constant $B$ has dimension of mass and is determined through the equation $Bm_q=m_\pi^2,\; m_q=m_u\approx m_d.$ The pseudoscalar meson matrix $P$ has its standard form $$P = \left ( \begin{array}{ccc} \frac{1}{\sqrt{2}}\pi^0+\frac{1} {\sqrt{6}}\eta & \pi^+ & K^+ \\ \pi^- & -\frac{1}{\sqrt{2}}\pi^0+\frac{1}{\sqrt{6}}\eta & K^0 \\ K^- & \bar K^0 & -\frac{2}{\sqrt{6}}\eta \end{array} \right ).$$ It is straightforward to get from (\[eq4\]) the relevant interaction vertices shown in Fig.\[Fig2\]. Using these Feynman rules, one can calculate the $\gamma^*\to \pi^+\pi^-\pi^0\gamma$ amplitude originated from the diagrams shown in Fig.\[Fig3\]. The result is $$A_{\mu\nu}(\gamma_\mu^*\to 3\pi\gamma_\nu)=\frac{ie^2}{4\pi^2f_\pi^3} T_{\mu\nu}, \label{eq5}$$ where ($Q=q_++q_-+q_0+k$ is the virtual photon 4-momentum) $$\begin{aligned} & & T_{\mu\nu}=\epsilon_{\mu\nu\alpha\beta}Q^\alpha k^\beta \left ( 1- \frac{(q_++q_-)^2-m_\pi^2}{(Q-k)^2-m_\pi^2}\right )+ \epsilon_{\mu\nu\alpha\beta}(Q+k)^\alpha q_0^\beta + \nonumber \\ & & +\epsilon_{\mu\lambda\alpha\beta}Q^\alpha q_0^\beta \left ( \frac{(2q_-+k)_\nu q_+^\lambda}{2q_-\cdot k}+ \frac{(2q_++k)_\nu q_-^\lambda}{2q_+\cdot k}\right )- \label{eq6} \\ & & -\epsilon_{\nu\lambda\alpha\beta} k^\alpha q_0^\beta \left (\frac{(2q_--Q)_\mu q_+^\lambda}{Q^2-2q_-\cdot Q}+ \frac{(2q_+-Q)_\mu q_-^\lambda}{Q^2-2q_+\cdot Q} \right ). \nonumber\end{aligned}$$ The tensor $T_{\mu\nu}$ is gauge invariant $$Q^\mu T_{\mu\nu}=k^\nu T_{\mu\nu}=0.$$ Note that our result for $A_{\mu\nu}(\gamma_\mu^*\to 3\pi \gamma_\nu)$ is in agreement with the known result [@11; @12] for the $\gamma^*\gamma^* \to 3\pi$ amplitude (these two amplitudes are connected by crossing symmetry, of course). If $J_\mu^{(\gamma)}$ is the amplitude of the transition $\gamma^*_\mu\to\pi^+\pi^-\pi^0\gamma$, then the final state radiation (FSR) contribution to the $e^+e^-\to\gamma^*\to \pi^+\pi^-\pi^0\gamma$ process cross section is given by [@8] $$d\sigma_{FSR}=\frac{e^2}{(2\pi)^8~64 E^4}\sum \limits_\epsilon \left \{\frac{Re(p_+\cdot J^{(\gamma)}~p_-\cdot J^{(\gamma)*})}{E^2}-J^{(\gamma)}\cdot J^{(\gamma)*} \right \}~ d\Phi\approx$$ $$\approx \frac{e^2}{(2\pi)^8~ 64 E^4}\sum\limits_\epsilon \left [ |J_1^{(\gamma)}|^2+|J_2^{(\gamma)}|^2 \right ]~ d\Phi,$$ where the sum is over the photon polarization $\epsilon$ and the$z$ axis was assumed to be along $\vec{p}_-$, but $J_\mu^{(\gamma)}=\epsilon^\nu A_{\mu\nu}(\gamma^*\to 3\pi\gamma)$. So we can perform the polarization sum by using $\sum\limits_\epsilon \epsilon_\mu \epsilon_\nu^*= -g_{\mu\nu}$. By introducing gauge invariant real 4-vectors $t_1$ and $t_2$ via $t_{1\mu}=T_{1\mu},\; t_{2\mu}=T_{2\mu}$, the result can be casted in the form (note that the norm of the gauge invariant 4-vector is always negative) $$d\sigma_{FSR}(e^+e^-\to 3\pi\gamma)=\frac{e^6}{(2\pi)^8~ 64 E^4}~\frac{1} {(2\pi)^4f_\pi^6} \left [ -t_1\cdot t_1 -t_2\cdot t_2\right ]~d\Phi. \label{eq7}$$ However, for the photon virtualities of real experimental interest vector meson effects can no longer be neglected. So we replace (\[eq7\]) by $$d\sigma_{FSR}(e^+e^-\to 3\pi\gamma)=\frac{e^6}{(2\pi)^8~ 64 E^4}~\frac{1} {(2\pi)^4f_\pi^6} \left [ -t_1\cdot t_1 -t_2\cdot t_2\right ]~K_{BW} ~d\Phi\equiv$$ $$\equiv \frac{e^6}{4(2\pi)^8~}~|A_{FSR}|^2 d\Phi, \label{eq8}$$ where we have introduced the phenomenological Breit-Wigner factor $$K_{BW}=3\left |\sin{\theta}\cos{\eta}R_\omega(4E^2)- \cos{\theta}\sin{\eta}R_\phi(4E^2)\right |^2.$$ This factor is similar to one presented in ISR ( see (\[eq3\]) ) and tends to unity as $E\to 0$. It gives about an order of magnitude increase in $\sigma_{FSR}$ for energies $2E=0.65\div 0.7~\mathrm{GeV}$. Monte-Carlo event generator =========================== Although what follows can be considered as a textbook material [@13], we will nevertheless give a somewhat detailed description of the Monte Carlo algorithm for the sake of convenience. The important first step is the following transformation of the Lorentz invariant phase space. Let $R_n(p^2;m_1^2,\ldots,m_n^2)$ be $n$-particle phase space $$R_n(p^2;m_1^2,\ldots,m_n^2)=\int\prod\limits_{i=1}^n \frac{d\vec{q}_i} {2E_i}~\delta (p-\sum\limits_{i=1}^n q_i).$$ Inserting the identity $$1=\int d k_1~d\mu_1^2~\delta (p-q_1-k_1)\delta(k_1^2-\mu_1^2)$$ we get $$R_4(p^2;m_1^2,m_2^2,m_3^2,m_4^2)=\int \frac{d\vec{q}_1}{2E_1} R_3((p-q_1)^2;m_2^2,m_3^2,m_4^2)=$$ $$= \int \frac{d\vec{q}_1}{2E_1} dk_1~d\mu_1^2 R_3(k_1^2;m_2^2,m_3^2,m_4^2)\delta (p-q_1-k_1) \delta(k_1^2-\mu_1^2).$$ However, (note that $(p-q_1)_0=E_2+E_3+E_4>0$) $$\int \frac{d\vec{q}_1}{2E_1}dk_1\delta(k_1^2-\mu_1^2)\delta (p-q_1-k_1)= \int \frac{d\vec{q}_1}{2E_1} \frac{d\vec{k}_1}{2k_{10}} \delta (p-q_1-k_1)=R_2(p^2;m_1^2,\mu_1^2).$$ Therefore, $$R_4(p^2;m_1^2,m_2^2,m_3^2,m_4^2)=\int d\mu_1^2 R_3(\mu_1^2;m_2^2,m_3^2,m_4^2)R_2(p^2;m_1^2,\mu_1^2). \label{eq9}$$ But [@13] $$R_2(p^2;m_1^2,\mu_1^2)=\int\frac{\lambda^{1/2}(p^2;m_1^2,\mu_1^2)}{8p^2} d\Omega_1^*$$ where $\lambda$ stands for the triangle function and $\Omega_1^*$ describes the orientation of the $\vec{q}_1$ vector in the $p$-particle rest frame. It is more convenient to integrate over $q$-particle energy $E^*$ instead of mass $\mu$, the two being interconnected by the relation $\mu^2=p^2+q^2- 2\sqrt{p^2}E^*$ in the $p$-particle rest frame. Using the relation [@13] $$\frac{\lambda^{1/2}(p^2;m^2,\mu^2)}{2\sqrt{p^2}}=\mu \sqrt{{\bar \gamma}^2-1},$$ where $\bar \gamma$ is the $\gamma$-factor of the “particle” (subsystem) with the invariant mass $\mu$, after repeatedly using (\[eq9\]) we get $$R_4=\int\frac{1}{2}\sqrt{{\bar \gamma}_1^2-1}~dE_1^*d\Omega_1^* \frac{1}{2}\mu_1\sqrt{{\bar \gamma}_2^2-1}dE_2^*d\Omega_2^*\frac{1}{2}|\vec{ p}\,_3^*|d\Omega_3^*,$$ where $\vec{p}\,_3^*$ momentum is in the rest frame of the (3,4) subsystem, and $E_2^*,$ $\Omega_2^*,$ $\;{\bar \gamma}_2$ are in the rest frame of the (2,3,4) subsystem. Now it is straightforward to rewrite the differential cross-section in the following form: $$d\sigma(e^+e^-\to 3\pi\gamma)=\frac{\alpha^3}{2\pi^2}|A|^2f d\Phi^*, \label{eq10}$$ where $|A|^2= |A_{ISR}|^2+|A_{FSR}|^2$ (we do not take into account the interference between the initial and final state radiations. This interference integrates to zero if we do not distinguish between negative and positive $\pi$-mesons), $$f=\mu_1 (\omega_{max}-\omega_{min}) (E_{0\, max}^*-E_{0\, min}^*) \sqrt{(E_-^{*2}-m_\pi^2)({\bar \gamma}_1^2-1)({\bar \gamma}_2^2-1)}, \label{eq11}$$ and $$d\Phi^*=\frac{d\omega}{(\omega_{max}-\omega_{min})}~\frac{d\varphi}{2\pi} \frac{ d\cos{\theta}}{2}~\frac{dE_0^*}{(E_{0\, max}^*-E_{0\, min}^*)} \frac{d\varphi_0^*}{2\pi}\frac{d\cos{\theta_0^*}}{2} \frac{d\varphi_-^*}{2\pi}\frac{d\cos{\theta_-^*}}{2}. \label{eq12}$$ The upper and lower limits for energies are $$\omega_{max}=\frac{s-9m_\pi^2}{2\sqrt{s}},\;\; E_{0\, max}^*=\frac{\mu_1^2-3m_\pi^2}{2\mu_1},\;\; E_{0\, min}^*=m_\pi.$$ The minimal photon energy $\omega_{min}$ is an external experimental cut. At last, $|A_{ISR}|^2$ and $|A_{FSR}|^2$ can be read from the corresponding expressions (\[eq1\]) and (\[eq8\]), respectively. According to (\[eq10\]), we can generate $e^+e^-\to\pi^+\pi^-\pi^0\gamma$ events in the cms frame by the following algorithm. $\bullet$ generate the photon energy $\omega$ as a random number uniformly distributed from $\omega_{min}$ to $\omega_{max}$. Calculate for the $S_1=(\pi^+\pi^-\pi^0)$ subsystem the energy ${\bar E}_1=2E-\omega$, invariant mass ${\bar \mu}_1=\sqrt{4E(E-\omega)}$ and the Lorentz factor ${\bar \gamma}_1={\bar E}_1/{\bar \mu}_1$. $\bullet$ generate a random number ${\bar\varphi}_1$ uniformly distributed in the interval $[0,2\pi]$ and take it as an azimuthal angle of the $S_1$ subsystem velocity vector in the cms frame. Generate another uniform random number in the interval $[-\cos{\theta_{min}},\cos{\theta_{min}}]$ and take it as a $\cos{{\bar \theta}_1},\;{\bar \theta}_1$ being the polar angle of the $S_1$ subsystem velocity vector in the cms frame. This defines the unit vector $\vec{n}_1=(\sin{{\bar \theta}_1}\cos{{\bar \varphi}_1}, \sin{{\bar \theta}_1}\sin{{\bar \varphi}_1}, \cos{{\bar \theta}_1})$ along the $S_1$ subsystem velocity; $\theta_{min}$ is the minimal photon radiation angle – an external experimental cut. $\bullet$ construct the photon momentum in the cms frame $\vec{k}= -\omega \vec{n}_1$. $\bullet$ generate the $\pi^0$-meson energy $E_0^*$ in the $S_1$ rest frame as a random number uniformly distributed from $E_{0\, min}^*$ to $E_{0\, max}^*$. Calculate for the $S_2=(\pi^+,\pi^-)$ subsystem the energy ${\bar E}_2={\bar \mu}_1-E_0^*$, invariant mass ${\bar \mu}_2=\sqrt{{\bar \mu}_1^2+m_\pi^2-2{\bar \mu}_1E_0^*}$ and the Lorentz factor ${\bar \gamma}_2={\bar E}_2/{\bar \mu}_2$. $\bullet$ generate a random number ${\bar\varphi}_2$ uniformly distributed in the interval $[0,2\pi]$ and take it as an azimuthal angle of the $S_2$ subsystem velocity vector in the $S_1$ rest frame. Generate another uniform random number in the interval $[-1,1]$ and take it as a $\cos{{\bar \theta}_2},\;{\bar \theta}_2$ being the polar angle of the $S_2$ subsystem velocity vector in the $S_1$ rest frame. This defines the unit vector along the $S_2$ subsystem velocity in the $S_1$ rest frame $\vec{n}_2=(\sin{{\bar \theta}_2}\cos{{\bar \varphi}_2}, \sin{{\bar \theta}_2}\sin{{\bar \varphi}_2}, \cos{{\bar \theta}_2})$. $\bullet$ construct $\vec{q}_0^*=-\sqrt{E_0^{*2}-m_\pi^2}~\vec{n}_2$ – the $\pi^0$-meson momentum in the $S_1$ rest frame. $\bullet$ generate $\varphi_-^*$ and $\cos{\theta_-^*}$ in the manner analogous to what was described above for ${\bar\varphi}_2$ and $\cos{{\bar \theta}_2}$ and construct the unit vector along the $\pi^-$ meson velocity in the $S_2$ rest frame $\vec{n}_3=(\sin{\theta_-^*}\cos{\varphi_-^*}, \sin{\theta_-^*} \sin{\varphi_-^*},\cos{\theta_-^*})$. $\bullet$ construct the $\pi^-$-meson 4-momentum in the $S_2$ rest frame $E_-^*={\bar \mu}_2/2$, $\vec{q}_-^*=\sqrt{E_-^{*2}-m_\pi^2}~\vec{n}_3$. $\bullet$ construct the $\pi^+$-meson 4-momentum in the $S_2$ rest frame $E_+^*={\bar \mu}_2/2$, $\vec{q}_+^*=-\vec{q}_-^*$. $\bullet$ transform $\pi^0$-meson 4-momentum from the $S_1$ rest frame back to the cms frame. $\bullet$ transform $\pi^-$ and $\pi^+$ mesons 4-momenta firstly from the $S_2$ rest frame to the $S_1$ rest frame and then back to the cms frame. $\bullet$ for the generated 4-momenta of the final state particles, calculate $z=|A|^2f$. $\bullet$ generate a random number $z_R$ uniformly distributed in the interval from 0 to $z_{max}$ where $z_{max}$ is some number majoring $|A|^2f$ for all final state 4-momenta allowed by 4-momentum conservation. $\bullet$ if $z\ge z_R$, accept the event that is the generated 4-momenta of the $\pi^+,\;\pi^-$ and $\pi^0$ mesons and the photon. Otherwise repeat the whole procedure. Soft and collinear photon corrections ===================================== We assume that the photon in the $e^+e^-\to 3\pi\gamma$ reaction is hard enough $\omega>\omega_{min}$ and radiated at large angle $\theta>\theta_{min}$ so that it could be detected by experimental equipment (a detector). In any process with accelerated charged particles soft photons are emitted without being detected because a detector has finite energy resolution. Even moderately hard photons can escape detection in some circumstances. How important are such effects? Naively every photon emitted brings extra factor $e$ in the amplitude and so a small correction is expected. But this argument (as well as the perturbation theory) breaks down for soft photons. When an electron (positron) emits a soft enough photon, it nearly remains on the mass shell, thus bringing a very large propagator in the amplitude. Formal application of the perturbation theory gives an infinite answer to the correction due to soft photon emission because of this pole singularity. It is well known [@14] how to deal with this infrared divergence. In real experiments very low energy photons never have enough time and space to be formed, because of a finite size of the laboratory. So we have a natural low energy cut-off. A remarkable fact, however, is that the observable cross sections do not depend on the actual form of the cut-off because singularities due to real and virtual soft photons cancel each other [@15]. The net effect is that the soft photon corrections, summed to all orders of perturbation theory, factor out as some calculable, so called Yennie-Frautschi-Suura exponent [@14]. Collinear radiation of (not necessarily soft) photons by highly relativistic initial electrons (positrons) is another source of big corrections which should also be treated non-perturbatively. Unlike soft photons, however, the matrix element for a radiation of an arbitrary number of collinear photons is unknown. Nevertheless, there is a nice method (the so called Structure Functions method) [@16] which enables one to sum leading collinear (and soft) logarithms. The corrected cross-section, when radiation of unnoticed photons with total energy less than $\Delta E\ll E$ is allowed, looks like [@16] $$\tilde \sigma (s)=\int\limits_0^{\Delta E}\frac{d\omega}{\omega}~ \sigma(4E(E-\omega))~\beta \left (\frac{\omega}{E}\right )^\beta \left [ 1+ \frac{3}{4}\beta+\frac{\alpha}{\pi}\left (\frac{\pi^2}{3}-\frac{1}{2}\right ) \right ], \label{eq13}$$ where $\beta=\frac{2\alpha}{\pi}\left (\ln{\frac{s}{m_e^2}}-1\right )$ and we have omitted some higher order terms. In our case the hard photon is well separated (because of $\omega> \omega_{min},\;\theta>\theta_{min}$ cuts) from the soft and collinear regions of the phase space. So equation (\[eq13\]) is applicable and it indicates that the soft and collinear corrections to the cross-section of the process $e^+e^-\to 3\pi\gamma$ do not exceed 20% when $\Delta E\sim\omega_{min}=30~{\mathrm{MeV}},\;\theta_{min}= 20^\circ$ and $E=0.7~ {\mathrm{GeV}}$. Such corrections are irrelevant for the present VEPP-2M statistics but may become important in future high statistics experiments. Numerical results and conclusions ================================= In Fig.\[Fig4\], numerical results are shown for $\sigma(e^+e^-\to 3\pi \gamma)$ with $\omega_{min}=30~\mathrm{MeV},\; \theta_{min}=20^\circ$. As expected, the cross section is small, only few picobarns, for energies $0.65\div 0.7~\mathrm{GeV}$. This figure shows also that FSR contributes significantly at such low energies. So if future $\phi$-factory experiments produce high enough statistics in this energy region, the study of FSR will become realistic. FSR and ISR give different angular and energy distributions for the photon as illustrated by Fig.\[Fig5\] and Fig.\[Fig6\]. This fact can be used for the FSR separation from a somewhat more intensive ISR. Let us note, however, that the model considered here is not valid in the $\phi$-meson region – very far from the threshold. At that the status of uncertainties in the ISR and FSR contributions are different. We believe that the ISR amplitude remains accurate enough even in the $\phi$-meson region. This belief stems from the fact that all relevant vector meson effects are already included in the ISR amplitude (\[eq3\]). The situation with the FSR amplitude is different. Our phenomenological Breit-Wigner factor mimics only some part of the vector meson effects. To estimate the corresponding uncertainty in $\sigma_{FSR}$, let us try some other choices for $K_{BW}$ which also have the correct low energy limit. If in the expression for the $K_{BW}$ factor we make the change $$R_\omega(4E^2)\to \frac{1}{2} \left [R_\omega(4E^2)+ R_\rho(4E^2) \right ],$$ $\sigma_{FSR}$ will be lowered by $\sim 5\%$ for $2E=0.65~\mathrm{GeV}$, and by $\sim 25\%$ for $2E=0.7~\mathrm{GeV}$. In the FSR amplitude the $\rho$-meson contributes via a number of various diagrams. For example, the $\gamma^*\to\rho\to\rho^+\rho^-\to\pi^+\pi^0\pi^-\gamma$ intermediate state, which has no counterpart in the $\omega$ meson contribution, gives the following piece of the $T_{\mu\nu}$ tensor $$T_{\mu\nu}^{(3\rho)}=\left . \left. -\frac{\alpha_K}{2} R_\rho(4E^2) \right \{A_1-2A_2-2A_3\right \},$$ where $$A_1= \epsilon_{\nu\alpha\beta\lambda}~(Q-2q_0)^\alpha k^\beta \left [ (q_++q_0-q_- -k)_\mu~ q_-^\lambda \,Y_- + \right .$$ $$\left .+ (q_-+q_0-q_+ -k)_\mu~ q_+^\lambda \,Y_+\right ],$$ $$A_2=\epsilon_{\nu\alpha\beta\lambda}~Q^\alpha k^\beta \left [ (q_+-q_0)_\mu~ q_-^\lambda \,Y_- + (q_--q_0)_\mu~ q_+^\lambda \,Y_+\right ],$$ $$A_3=\epsilon_{\mu\nu\beta\lambda}~k^\beta \left [ Q\cdot(q_+-q_0)~ q_-^\lambda \,Y_- +Q\cdot(q_--q_0)~q_+^\lambda \, Y_+\right ],$$ and $$Y_\mp=\frac{M_\rho^2}{\left [(q_\mp+k)^2-M_\rho^2\right ] \left [(q_\pm+q_0)^2-M_\rho^2\right ]}.$$ If we include this contribution, and besides ensure that the remaining part of the FSR amplitude (\[eq6\]) also takes $$K_{BW}= |R_\rho(4E^2)|^2$$ in the role of the phenomenological Breit-Wigner factor, the FSR cross section will be lowered by $\sim 10\%$ for $2E=0.65~ \mathrm{GeV}$, and by $\sim 35\%$ for $2E=0.7~\mathrm{GeV}$. This uncertainty in the FSR magnitude is irrelevant for the present VEPP-2M statistics. For future high precision experiments a systematic inclusion of all relevant vector meson effects in the FSR amplitude is desired. Acknowledgments {#acknowledgments .unnumbered} =============== One of us (EAK) is grateful to Heisenberg-Landau Fund 2000-02 and to RFFI Grant 99-02-17730. We are grateful to G. Sandukovskaya for the help. [99]{} H. N. Brown [*et al.*]{} \[Muon g-2 Collaboration\], Phys. Rev. Lett.  [**86**]{}, 2227 (2001). For review see, for example A. Czarnecki and W. J. Marciano, Phys. Rev. D [**64**]{}, 013014 (2001). K. Melnikov, Int. J. Mod. Phys. A [**16**]{}, 4591 (2001); J. F. De Troconiz and F. J. Yndurain, hep-ph/0106025; M. Knecht and A. Nyffeler, hep-ph/0111058. M. N. Achasov [*et al.*]{}, hep-ex/0010077. R. R. Akhmetshin [*et al.*]{} \[CMD-2 Collaboration\], Nucl. Phys. A [**675**]{}, 424C (2000). R. R. Akhmetshin [*et al.*]{} \[CMD-2 Collaboration\], hep-ex/9904027. R. R. Akhmetshin [*et al.*]{} \[CMD-2 Collaboration\], Phys. Lett. B [**476**]{}, 33 (2000). A. Aloisio [*et al.*]{} \[KLOE Collaboration\], hep-ex/0107023. G. Bonneau and F. Martin, Nucl. Phys. B [**27**]{}, 381 (1971). E. A. Kuraev and Z. K. Silagadze, Phys. Atom. Nucl.  [**58**]{}, 1589 (1995) E. Witten, Nucl. Phys. B [**223**]{}, 422 (1983). J. Wess and B. Zumino, Phys. Lett. B [**37**]{}, 95 (1971). J. W. Bos, Y. C. Lin and H. H. Shih, Phys. Lett. B [**337**]{}, 152 (1994). P. Talavera, L. Ametller, J. Bijnens, A. Bramon and F. Cornet, Phys. Lett. B [**376**]{}, 186 (1996); L. Ametller, J. Kambor, M. Knecht and P. Talavera, Phys. Rev. D [**60**]{}, 094003 (1999). E. Byckling and K. Kajantie, [*Particle Kinematics*]{}. Wiley, 1973. D. R. Yennie, S. C. Frautschi and H. Suura, Annals Phys.  [**13**]{}, 379 (1961). F. Bloch and A. Nordsieck, Phys. Rev.  [**52**]{}, 54 (1937). D. R. Yennie, [*in*]{} Lectures on strong and electromagnetic interactions, p. 166 (Brandies Summer Institute in Theoretical Physics, Waltman, Mass., 1963) E. A. Kuraev and V. S. Fadin, Sov. J. Nucl. Phys.  [**41**]{}, 466 (1985). For review see M. Skrzypek, Acta Phys. Polon. B [**23**]{}, 135 (1992).

KOBE–FHD–95–06\ October        1995 [**[$\Delta s$]{} density in a proton and unpolarized\ lepton–polarized proton scatterings**]{}\ T. Morii\ \ [*Sciences for Natural Environment*]{}\ [*and*]{}\ [*Graduate School of Science and Technology,*]{}\ [*Kobe University, Nada, Kobe 657, Japan*]{}\ Alexander I. Titov\ \ [*Joint Institute for Nuclear Research,*]{}\ [*141980, Dubna, Moscow region, Russia*]{}\ and\ T. Yamanishi\ \ [*Osaka University, Ibaraki, Osaka 567, Japan*]{}\ It is shown that the parity–violating deep–inelastic scatterings of unpolarized charged leptons on polarized protons, $\ell^{\mp} + \vec P\to \stackrel{\scriptscriptstyle(-)}{\nu_{\ell}} + X$, could provide a sensitive test for the behavior and magnitude of the polarized strange–quark density in a proton. Below charm threshold these processes are also helpful to uniquely determine the magnitude of individual polarized parton distributions. There have been interesting problems on strange–quark (s–quark) contents in contemporary hadron physics. Deductions of the $\sigma$–term from pion-nucleon scatterings imply an existence of significant s–quark contents in a nucleon[@sigmaterm]. New analysis suggests that about one third of the rest mass of the proton comes from $s\bar s$ pairs. So far, interesting experimental proposals[@newexp] have been presented to measure the neutral weak form factors of the nucleon which might be sensitive to the s–quarks inside the nucleon. A different idea is also proposed to directly probe the s–quark content of the proton by using the lepto– and photo–production of $\phi$–meson that is essentially 100% $s\bar s$[@phi]. Another surprizing results on the s–quark contents in the nucleon have been drawn from the data of polarized deep inelastic scatterings[@PDIS]. To our surprise, the experimental data have suggested that, contrary to the prediction of the naive quark model, there is a large and negative contribution of s–quarks to the proton spin, $i.e.$ $\Delta s=-0.12$, and furthermore very little of the proton spin is carried by quarks. For low-energy properties of baryons, conventional phenomenological quark models treat nucleons as consisting of only u– and d–quarks and thus it naturally comes as a big surprise when some recent measurements and theoretical analyses have indicated a possible existence of a sizable s–quark. In order to get deep understanding of hadron dynamics, it is very important to investigate the behavior of s–quarks in a nucleon. In this paper, we concentrate on the behavior of the polarized s–quark and study the processes sensitive to its polarized distributions in the nucleon. So far, several people have suggested various processes, which are sensitive to polarized s–quark distributions, such as Drell–Yan processes[@Leader], inclusive $W^{\pm}$– and $Z^0$–productions[@Soffer94] in polarized proton–polarized proton collisions, and also inclusive $\pi^{\pm}$– and $K^{\pm}$–productions in polarized lepton–polarized proton scatterings[@Close]. However, since the differential cross sections for Drell–Yan processes and inclusive $W^{\pm}$– /$Z^0$–hadroproductions are described by the product of two parton distributions participating in such processes, one cannot extract the $x$–dependence of polarized s–quark distributions without ambiguities from such cross sections. In addition, those of inclusive $\pi^{\pm}$– and $K^{\pm}$–leptoproductions include the fragmentation functions of $\pi^{\pm}$– and $K^{\pm}$–decays which possess some theoretical ambiguities, and hence it is also difficult to derive the exact behavior of polarized s–quark distributions from these processes. Recently, it has been pointed out that parity–violating polarized electron elastic scatterings on unpolarized protons can give informations on the matrix elements, $\langle p|\bar s\Gamma_{\mu}s|p\rangle$ with $\Gamma_{\mu}$ $=\gamma_{\mu}$ and $\gamma_{\mu}\gamma_5$[@Fayyazuddin]. However, since its differential cross section includes not only the spin–dependent but also spin–independent proton form factors, one cannot extract the polarized s–quark content without ambiguities even from such processes. Here we consider a different process for examining the polarized s–quark density, which is the parity–violating polarized deep inelastic scattering at high energy. It must be advantageous to study such a process because its differential cross section includes only the spin–dependent structure function of the proton and is explicitly described as a function of $x$. In parity–violating deep inelastic scatterings of unpolarized charged lepton on longitudinally polarized proton, an interesting parameter is the single–spin asymmetry $A_L^{W^{\mp}}$ defined as $$\begin{aligned} A_L^{W^{\mp}}&=&\frac{(d\sigma_{++}^{W^{\mp}}+d\sigma_{-+}^{W^{\mp}})- (d\sigma_{+-}^{W^{\mp}}+d\sigma_{--}^{W^{\mp}})} {(d\sigma_{++}^{W^{\mp}}+d\sigma_{-+}^{W^{\mp}})+ (d\sigma_{+-}^{W^{\mp}}+d\sigma_{--}^{W^{\mp}})}\nonumber\\ &=&\frac{d\sigma_{0+}^{W^{\mp}}-d\sigma_{0-}^{W^{\mp}}} {d\sigma_{0+}^{W^{\mp}}+d\sigma_{0-}^{W^{\mp}}} =\frac{d\Delta_L\sigma^{W^{\mp}}/dx}{d\sigma^{W^{\mp}}/dx}~, \label{eqn:A_L}\end{aligned}$$ where $d\sigma_{0-}^{W^{\mp}}$, for instance, denotes that the lepton is unpolarized and the helicity of the proton is negative. Note that since a fast incoming negatively (positively) charged lepton, $\ell^-$ ($\ell^+$), couples to a $W$–boson only when it has a negative (positive) helicity, part of spin–dependent cross sections in eq.(\[eqn:A\_L\]) should be zero. For parity–violating weak–interacting reactions with $W^{\mp}$ exchanges, $\ell^{\mp} + \vec P\to \stackrel{\scriptscriptstyle(-)}{\nu_{\ell}} + X$, the spin–dependent and spin–independent differential cross sections as a function of momentun fraction $x$ are given by[@Anselmino] $$\begin{aligned} &&\frac{d\Delta_L\sigma^{W^{\mp}}}{dx} =16\pi M_NE\frac{\alpha^2}{Q^4}\eta\left\{\pm (\frac{2}{3}+\frac{xM_N}{6E})x~ g_1^{W^{\mp}}(x, Q^2)+(\frac{2}{3}-\frac{xM_N}{12E})~g_3^{W^{\mp}}(x, Q^2) \right\}~,\nonumber\\ &&\label{eqn:dDs}\\ &&\frac{d\sigma^{W^{\mp}}}{dx} =16\pi M_NE\frac{\alpha^2}{Q^4}\eta\left\{(\frac{2}{3}-\frac{xM_N}{4E})~ F_2^{W^{\mp}}(x, Q^2)\pm\frac{1}{3}x~F_3^{W^{\mp}}(x, Q^2)\right\}~, \label{eqn:ds}\end{aligned}$$ where $E$ is the energy of the charged lepton beam and $M_N$ the mass of the proton. $\eta$ is written in terms of the $W$–boson mass $M_W$ as $$\eta=\frac{1}{2}\left(\frac{G_FM^2_W}{4\pi\alpha}\frac{Q^2}{Q^2+M_W^2}\right)~. \label{eqn:eta}$$ $g_1^{W^{\mp}}$, $g_3^{W^{\mp}}$ in eq.(\[eqn:dDs\]) and $F_2^{W^{\mp}}$, $F_3^{W^{\mp}}$ in eq.(\[eqn:ds\]) represent spin–dependent and spin–independent proton structure functions, respectively. Below charm threshold, the region of which could be investigated by SMC and/or E143 Collabolations, we can describe these structure functions for the W$^-$ exchange as $$\begin{aligned} &&F_2^{W^-}(x, Q^2)=2x\left[c_1\{u_v(x, Q^2)+u_s(x, Q^2)\}+ c_2~\bar d_s(x, Q^2)+c_3~\bar s_s(x, Q^2)\right]~, \nonumber\\ &&F_3^{W^-}(x, Q^2)=2\left[c_1\{u_v(x, Q^2)+u_s(x, Q^2)\}- c_2~\bar d_s(x, Q^2)-c_3~\bar s_s(x, Q^2)\right]~, \label{eqn:str1}\\ &&g_1^{W^-}(x, Q^2)=\left[c_1\{\delta u_v(x, Q^2)+\delta u_s(x, Q^2)\}+ c_2~\delta\bar d_s(x, Q^2)+c_3~\delta\bar s_s(x, Q^2)\right]~,\nonumber\\ &&g_3^{W^-}(x, Q^2)=2x\left[c_1\{\delta u_v(x, Q^2)+\delta u_s(x, Q^2)\}- c_2~\delta\bar d_s(x, Q^2)-c_3~\delta\bar s_s(x, Q^2)\right]~,\nonumber\end{aligned}$$ and similarly for the W$^+$ exchange $$\begin{aligned} &&F_2^{W^+}(x, Q^2)=2x\left[c_1~\bar u_s(x, Q^2)+c_2\{d_v(x, Q^2)+ d_s(x, Q^2)\}+c_3~s_s(x, Q^2)\right]~,\nonumber\\ &&F_3^{W^+}(x, Q^2)=2\left[-c_1~\bar u_s(x, Q^2)+c_2\{d_v(x, Q^2)+ d_s(x, Q^2)\}+c_3~s_s(x, Q^2)\right]~, \label{eqn:str2}\\ &&g_1^{W^+}(x, Q^2)=\left[c_1~\delta\bar u_s(x, Q^2)+ c_2\{\delta d_v(x, Q^2)+\delta d_s(x, Q^2)\}+ c_3~\delta s_s(x, Q^2)\right]~,\nonumber\\ &&g_3^{W^+}(x, Q^2)=2x\left[-c_1~\delta\bar u_s(x, Q^2)+ c_2\{\delta d_v(x, Q^2)+\delta d_s(x, Q^2)\}+ c_3~\delta s_s(x, Q^2)\right]\nonumber\end{aligned}$$ with CKM matrix elements $$c_1~=~|U_{ud}|^2+|U_{us}|^2~,~~~c_2~=~|U_{ud}|^2~,~~~c_3~=~|U_{us}|^2~. \label{eqn:kmm}$$ Here $\delta q(x, Q^2)=q_+(x, Q^2)-q_-(x, Q^2)$ ($q(x, Q^2)=q_+(x, Q^2)+q_-(x, Q^2)$) stands for the polarized (unpolarized) quark distribution in a proton, and $q_+(x, Q^2)$ ($q_-(x, Q^2)$) the quark density having a momentum fraction $x$ with the helicity parallel (anti–parallel) to the proton helicity. It should be noticed that since $g_1^{W^-}$ and $g_1^{W^+}$ have no relation to the flavor singlet axial–vector current, it is not affected by the axial anomaly which, in deep inelastic reactions with one–photon exchanges, leads to rather large contributions of the polarized gluons to polarized quark distribution functions[@anomaly]. In order to examine how the observed parameter is affected by the behavior of polarized s–quark distributions, we calculate $A_L^{W^{\mp}}$ by substituting eqs.(\[eqn:dDs\]) and (\[eqn:ds\]) into eq.(\[eqn:A\_L\]) as follows, $$A_L^{W^{\mp}}~=~\frac{\pm (\frac{2}{3}+\frac{xM_N}{6E})x~ g_1^{W^{\mp}}(x, Q^2)+(\frac{2}{3}-\frac{xM_N}{12E})~g_3^{W^{\mp}}(x, Q^2)} {(\frac{2}{3}-\frac{xM_N}{4E})~F_2^{W^{\mp}}(x, Q^2)\pm\frac{1}{3}x~ F_3^{W^{\mp}}(x, Q^2)}~. \label{eqn:dDs/ds}$$ As typical examples of the polarized s–quark distribution functions, we take the following three different types; (i) negative and large $\Delta s$ with $\Delta s(Q^2=4$GeV$^2)=-0.11$ (BBS model)[@BBS] , (ii) zero $\Delta s$ with $\Delta s(Q^2=10$GeV$^2$)=0[@Cheng90] , (iii) positive and small $\Delta s$ with $\Delta s(Q^2=10$GeV$^2)=0.02$ (KMTY model)[@Kobayakawa] , where $\Delta s(Q^2)$ is the first moment of $\delta s(x, Q^2)$ and its value means the amount of the proton spin carried by the s–quark. All of the models, (i), (ii) and (iii), can reproduce the EMC and SMC data on $xg_1^p(x)$ equally well. The $x$–dependence of these distributions are depicted at $Q^2=10$GeV$^2$ in fig.1. (The explicit forms of $\delta s(x, Q^2)$ are presented in respective references.) By using the polarized s–quark distribution of each type, together with the polarized and unpolarized parton distribution of BBS parametrization[@BBS] for (i), of Cheng–Lai[@Cheng90] and DFLM parametrizaton[@DFLM] for (ii), and of KMTY[@Kobayakawa] and DO parametrization[@Duke] for (iii), we estimate the $A_{L}^{W^{\mp}}$ at a typical charged lepton energy $E=180$GeV and momentum transfer squared $Q^2=10$GeV$^2$ whose kinematical region can be covered by SMC experiments. We see that $A_{L}^{W^-}$ for the $W^-$ exchange process depends on the behavior of polarized s–quark distributons little because the spin–dependent proton structure functions, $g_1^{W^-}(x, Q^2)$ and $g_3^{W^-}(x, Q^2)$, included in $A_{L}^{W^-}$ are dominantly controlled by the polarized valence u–quark ditribution $\delta u_v(x, Q^2)$ whose magnitude $\Delta u_v$ is much larger than $\Delta\bar s_s$ in the proton. However, the situation is quite different for $g_1^{W^+}(x, Q^2)$ and $g_3^{W^+}(x, Q^2)$ originated from $W^+$ exchanges because $g_1^{W^+}(x, Q^2)$ and $g_3^{W^+}(x, Q^2)$ have no contribution of the polarized valence u–quark distribution. Although $g_1^{W^+}(x, Q^2)$ and $g_3^{W^+}(x, Q^2)$ contain the polarized valence d–quark distribution $\delta d_v(x, Q^2)$, the absolute value of $\Delta d_v$ is quite smaller than that of $\Delta u_v$. Thus, $A_{L}^{W^+}$ is expected to be rather sensitive to the polarized s–quark distribution function compared to the case of $A_{L}^{W^-}$. The $x$–dependence of $xg_1^{W^+}$, $g_3^{W^+}$ and $A_{L}^{W^+}$ are calculated and shown in figs.2, 3 and 4, respectively. From fig.4, one can observe that the behavior of $A_{L}^{W^+}$ significantly depends on the polarized s–quark distribution for not very small $x$ regions and hence we can distinguish the model of polarized s–quark distributions from the data of $A_{L}^{W^+}$. The reader might consider that the difference obtained here could be originated from our procedure of having used the different unpolarized parton distributions for respective models, (i), (ii) and (iii). In order to examine if the results are really meaningful, we have carried out the same calculation using in common the DFLM parametrization for unpolarized distributions as an example. Calculated results are presented by lines with small circles in fig.4. From this result, we can say that the conclusion remains to be unchanged. However, it should be noted that one cannot directly extract the s–quark distribution from $xg_1^{W^+}(x, Q^2)$ and $g_3^{W^+}(x, Q^2)$. This is because, as shown in eq.(\[eqn:dDs\]), the differential cross sections are described by a linear combination of $xg_1^{W^+}(x, Q^2)$ and $g_3^{W^+}(x, Q^2)$ and we cannot measure independently $xg_1^{W^+}(x, Q^2)$ and $g_3^{W^+}(x, Q^2)$ in experiments. It is interesting to recall that the situation is quite different for the case of one–photon exchange processes. Although the differential cross sections are described by the linear combination of $g_1^p(x)$ and $g_2^p(x)$ in that case as well, $g_2^p(x)$ can be kinematically neglected compared to $g_1^p(x)$ and hence one can easily measure $g_1^p(x)$ in experiment. This is not the case for $W^{\pm}$ exchange processes. The information on the polarized s–quark distribution reflects to $A_{L}^{W^+}$ which contains $xg_1^{W^+}(x, Q^2)$ and $g_3^{W^+}(x, Q^2)$. In practice, the differential cross sections for these processes are small because of weakly interacting $W$–boson exchanges: for example, $d\Delta_L\sigma^{W^+}/dx=-0.093$ ($-0.011$, $-0.099$) \[pb\] and $d\sigma^{W^+}/dx=0.62$ ($0.75$, $0.52$) \[pb\] for the type of (iii) ((i), (ii)) at $x=0.1$, $E=180$GeV and $Q^2=10$GeV$^2$. Therefore, from the experimental point of view, we must have high luminosity in order to get informations on the polarized s–quark distribution function $\delta s(x, Q^2)$ inside a proton. Futhermore it might be practically rather difficult to identify the missing events from $\ell^+ + \vec P\to\bar{\nu_{\ell}} + X$. However, we believe that these difficulties can be technically overcome. Another importance for getting $g_1^{W^{\pm}}$ comes from the fact that, below charm threshold, one can uniquely determine the magnitude of individual polarized parton density by combining the data on $g_1^{W^{\pm}}$ with the ones on neutron $\beta$–decay, hyperon $\beta$–decay and the spin–dependent structure function of proton $g_1^p$. The following combinations of individual polarized parton content are well–known, $$\begin{aligned} &&\Delta u-\Delta d=a_3~, \label{eqn:neu}\\ &&\Delta u+\Delta d-2\Delta s=a_8~, \label{eqn:hyp}\\ &&\frac{4}{18}\Delta u+\frac{1}{18}\Delta d+\frac{1}{18}\Delta s -\frac{\alpha_s}{6\pi}\Delta g=a_0^{\gamma}~, \label{eqn:sing}\end{aligned}$$ where $\Delta g$ in eq.(\[eqn:sing\]) represents the amount of the proton spin carried by gluons and is introduced by taking account of the U$_A(1)$ anomaly of QCD[@anomaly]. Although the values of $a_3$, $a_8$ and $a_0^{\gamma}$ are known from the experimental data on neutron $\beta$–decay, hyperon $\beta$–decay and the spin–dependent structure function of proton $g_1^p$, respectively, it is impossible to determine the magnitude of individual content uniquely from these equations alone since there exist four independent variables for three equations. However, if the values of the first moment of $g_1^{W^+}(x)$ and $g_1^{W^-}(x)$ can be obtained experimentally $$\int_0^1 g_1^{W^+}(x)dx+\int_0^1 g_1^{W^-}(x)dx= c_1~\Delta u+c_2~\Delta d+c_3~\Delta s=a_0^{W^+}+a_0^{W^-}~, \label{eqn:Wmp}$$ then, from four independent equations, (\[eqn:neu\]), (\[eqn:hyp\]), (\[eqn:sing\]) and (\[eqn:Wmp\]), $\Delta u$, $\Delta d$, $\Delta s$ and $\Delta g$ can be determined as follows, $$\begin{aligned} &&\Delta u=\frac{(2c_2+c_3)a_3+c_3a_8+2(a_0^{W^+}+a_0^{W^-})} {2(c_1+c_2+c_3)}~, \label{eqn:Du}\\ &&\Delta d=\frac{(-2c_1-c_3)a_3+c_3a_8+2(a_0^{W^+}+a_0^{W^-})} {2(c_1+c_2+c_3)}~, \label{eqn:Dd}\\ &&\Delta s=\frac{(-c_1+c_2)a_3-(c_1+c_2)a_8+2(a_0^{W^+}+a_0^{W^-})} {2(c_1+c_2+c_3)}~, \label{eqn:Ds}\\ &&\Delta g=-\frac{\pi}{3\alpha_s}\frac{1}{2(c_1+c_2+c_3)}\left\{ 3(c_1-3c_2-c_3)a_3\right. \label{eqn:Dg}\\ &&\left. +(c_1+c_2-5c_3)a_8+36(c_1+c_2+c_3)a_0^{\gamma}- 12(a_0^{W^+}+a_0^{W^-})\right\}~.\nonumber\end{aligned}$$ But how can we actually measure $a_0^{W^{\pm}}$ in experiment? As long as we remain in the experiment with the longitudinally polarized proton target, we cannot determine $a_0^{W^{\pm}}$ experimentally. However, if we consider the cross section with the transversely polarized proton target, one can obtain $a_0^{W^{\pm}}$ as described in the following. The formulas of the transversely polarized cross section have been given by Anselmino et al.[@Anselmino] as follows, $$\begin{aligned} \frac{d\Delta_T\sigma^{W^{\pm}}}{dxdyd\phi}&=& 32M_N\frac{\alpha^2}{Q^4}\eta\cos (\psi-\phi) \sqrt{xyM_N\left\{2(1-y)E-xyM_N\right\}}\nonumber\\ &\times&x(1-y) \left(\pm g_1^{W^{\pm}}(x, Q^2)+\frac{1}{2x}~g_3^{W^{\pm}}(x, Q^2)\right)~, \label{eqn:dDs_T}\end{aligned}$$ where $\phi$ is the azimuthal angle of the lepton in the final state, and $\psi$ an angle between the proton spin and $x$–axis in the $xy$ plane orthogonal to the lepton direction ($z$–axis). These angles must be fixed in principle by suitably arranging experimental apparatus. $d\Delta_T\sigma^{W^{\pm}}/dxdyd\phi$ is defined as $$\frac{d\Delta_T\sigma^{W^{\pm}}}{dxdyd\phi}= \frac{d\sigma^{W^{\pm}}_{0\uparrow}}{dxdyd\phi}- \frac{d\sigma^{W^{\pm}}_{0\downarrow}}{dxdyd\phi}~, \label{eqn:dfntrans}$$ where $d\sigma^{W^{\pm}}_{0\uparrow}$ ($d\sigma^{W^{\pm}}_{0\downarrow}$) denotes that the lepton is unpolarized and the proton is transversely polarized with its spin orthogonal to the lepton direction at an angle $\psi$ ($\psi+\pi$) to the $x$–axis. By integrating eq.(\[eqn:dDs\_T\]) over $y$, one can easily derive the following formula, $$\begin{aligned} \frac{d\Delta_T\sigma^{W^+}}{dxd\phi}- \frac{d\Delta_T\sigma^{W^-}}{dxd\phi}&=& C~x^{3/2} \left[\int^1_0 dy\sqrt{y}(1-y)\sqrt{1-y-\frac{xyM_N}{2E}}\right]\nonumber\\ &\times&\left(g_1^{W^+}(x, Q^2)+g_1^{W^-}(x, Q^2)\right)~, \label{eqn:AT}\end{aligned}$$ with $C=32\sqrt{2E}M_N^{3/2}\frac{\alpha^2}{Q^4}\eta\cos(\psi-\phi)$. In eq.(\[eqn:AT\]), the integral in the square bracket depends on $x$ alone and can be written by $f(x)$. Practically, $f(x)$ can be very nicely approximated by the formula, $$f(x)=0.19635~(1-\frac{2.45}{E}x)~, \label{eqn:funcf}$$ which reproduces the exact result with accuracy better than $10^{-4}$ for $E>50$GeV. Then, we can get $$g_1^{W^+}(x)+g_1^{W^-}(x)=\frac{d\Delta_T\sigma^{W^+}/dxd\phi- d\Delta_T\sigma^{W^-}/dxd\phi}{C~x^{3/2}~f(x)}~, \label{eqn:trnsasy}$$ where the right–hand side of eq.(\[eqn:trnsasy\]) can be determined from experiment. Therefore, if we carry out the experiment with the transversely polarized target, we can obtain the sum of $a_0^{W^-}$ and $a_0^{W^+}$ by integrating eq.(\[eqn:trnsasy\]) over $x$, which leads to the unique determination of the magnitude of individual polarized parton densities in a proton. On the contrary, we can predict the values of $a_0^{W^-}$ and $a_0^{W^+}$ by using each type of polarized s–quark density. Some examples are given $a_0^{W^-}=-0.400$ and $a_0^{W^+}=0.798$ for type (i), $-0.282$ and $0.956$ for type (ii), $-0.196$ and $0.943$ for type (iii) at $E=180$GeV and $Q^2=10$GeV$^2$. These predictions should be tested in the forthcoming experiment. In summary, we have discussed the processes sensitive to the polarized s–quark distribution and have found that parity–violating reactions with $W^+$–boson exchange, $\ell^+ + \vec P\to\bar{\nu_{\ell}} + X$, are quite promising for giving us informations on polarized s–quark distribution functions inside a proton. Since the single–spin asymmetry $A_L^{W^+}$ for these processes significantly depends on the behavior of polarized s–quark distributions, one can test the behavior and magnitude of the s–quark polarization by measuring this quantity in experiments. Futhermore, we have shown that the amount of each quark and gluon carrying the proton spin can be uniquely determined if the spin–dependent proton structure functions, $g_1^{W^+}$ and $g_1^{W^-}$, are obtained below charm threshold by carrying out the experiment with the transversely polarized target. Informations on polarized s–quark distributions are decisively important to understand the proton spin strucutre. We hope the present predictions could be tested in the forthcoming experiments. [**Acknowledgements**]{} Two of us (T.M. and T.Y.) would like to thank V. V. Burov and the theory members at JINR, Dubna for their kind hospitality. [1]{} T. P. Cheng and R. F. Dashen, Phys. Rev. Lett. [**26**]{} (1971) 594; T. P. Cheng, Phys. Rev. [**D13**]{} (1976) 2161; C. A. Dominguez and P. Langacker, [*ibid.*]{} [**24**]{} (1981) 190; J. F. Donoghue and C. R. Nappi, Phys. Lett. [**B168**]{} (1986) 105; J. F. Donoghue, Annu. Rev. Nucl. Part. Sci. [**39**]{} (1989) 1; J. Gasser, H. 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[**D41**]{} (1990) 91. K. Kobayakawa, T. Morii, S. Tanaka and T. Yamanishi, Phys. Rev. [**D46**]{} (1992) 2854. M. Diemoz, F. Ferroni, E. Longo and G. Martinelli, Z. Phys. [**C39**]{} (1988) 21. D. W. Duke and J. F. Owens, Phys. Rev. [**D30**]{} (1984) 49. [**Figure captions**]{} Fig. 1: : The $x$–dependence of $x\delta s(x, Q^2)$ at $Q^2=10$GeV$^2$ for various types of polarized s–quark distributions (i)–(iii). (See text.) Type (i) is evolved up to $Q^2=10$GeV$^2$. The dash–dotted, dashed and solid lines indicate the results calculated for types (i), (ii) and (iii), respectively. Fig. 2: : The $x$–dependence of spin–dependent proton structure functions $xg_1^{W^+}$ for partiy–violating reactions with $W^+$–boson exchanges at $Q^2=10$GeV$^2$ for various types of $\Delta s$. Various lines represent the same as in fig.1. Fig. 3: : The spin–dependent proton structure functions $g_3^{W^+}$ as a function of $x$ at $Q^2=10$GeV$^2$. Various lines represent the same as in fig.1. Fig. 4: : The single–spin asymmetry $A_L^{W^+}$ as a function of $x$ at $E=180$GeV and $Q^2=10$GeV$^2$. The dash–dotted, dashed and solid lines correspond to types (i), (ii) and (iii), respectively. The lines added small circles for respective types are the results calculated by using in common the DFLM parametrization for unpolarized distributions.

--- author: - Xiangyu Cao - Alexandre Nicolas - Denny Trimcev - Alberto Rosso bibliography: - 'yield.bib' title: ' Soft modes and strain redistribution in continuous models of amorphous plasticity: the Eshelby paradigm, and beyond?' --- ![(a) Sketch of the macroscopic shear stress response of a disordered solid subject to a quasistatic deformation, with depictions of common deformation protocols. Stress fluctuations are not represented in the sketch. (b) Representations of the new strain variables $e_1$, $e_2$, and $e_3$. In this work we consider pure shear along the $e_2$ direction and $\gamma$ identifies to the average of $e_2$.[]{data-label="fig:Macroscopic_response"}](figure1.png){width="0.8\columnwidth"} Apply a dab of toothpaste onto a toothbrush and slightly tilt the brush. The paste will respond to the small shear stresses $\Sigma$ thus created in its bulk by deforming elastically. In contrast, when you squeeze a toothpaste tube, the stresses in the material exceed a critical yield value $\Sigma_y$, and the paste starts to flow. This “liquid”-like phase under shear is observed not only in pastes and (concentrated) suspensions, but also in other soft solids such as emulsions and foams [@coussot2014yield]. Other disordered materials such as metallic glasses also depart from an elastic behavior under large enough stresses, but then break instead of flowing. In the athermal limit, the change observed at $\Sigma_y$ is a dynamical phase transition known as yielding transition. To study it, a standard experimental protocol consists in slowly[^1] deforming the material and monitoring its macroscopic stress. For small deformations, the response is linear and elastic. For larger ones, the deformation becomes macroscopically irreversible, due to the onset of plasticity. Three distinct plastic responses can be observed, as shown in Fig. \[fig:Macroscopic\_response\]: (i) the stress grows monotonically and saturates at a steady-state value $\Sigma_y$; (ii) the stress overshoots $\Sigma_y$ and, upon reaching $\Sigma_\text{max}>\Sigma_y$, drops rapidly to the stationary value $\Sigma_y$ [@divoux2011stress] (note that it is still unclear if the material fails globally at $\Sigma_\text{max}$ – as in a spinodal transition [@zapperi1997first; @procaccia2017mechanical] – or through a large sequence of finite-size avalanches) (iii) at $\Sigma_\text{max}$, the material breaks and the stress drops to zero. Microscopically, the mechanism underlying the irreversible plastic response is the localised rearrangement of a few particles (droplets in emulsions, bubbles in foams), a process called shear transformation (ST) [@argon1979plastic]. Recently, more detailed investigation has revealed that these ST do not arise randomly in the material, but display spatial correlations [@chikkadi2012shear; @nicolas2014spatiotemporal]. It is now widely believed that these correlations stem from the elastic deformation induced by the ST, which has a peculiar quadrupolar shape, as predicted by Eshelby half a century ago [@eshelby1957determination; @schall2007structural]. This quadrupolar kernel has been observed in atomistic simulations of several model glasses [@maloney2006amorphous; @puosi2014time]. Moreover, the plastic ST instability is preceded by the emergence of localisation in the low-frequency modes of the vibrational spectrum [@tanguy10mode; @manning11soft; @charbonneau16universal]. The localised soft spots tend to coincide with the subsequent ST. Interestingly, the displacement field around a soft spot displays a long-range tail, decaying as $r^{1-d}$, with $d$ the spatial dimension, which is consistent with the quadrupolar shape observed after the plastic event. On the basis of this picture of localised plastic rearrangements, elasto–plastic models (EPM) have proposed to coarse-grain disordered solids into a collection of blocks alternating between an elastic regime and plastic events interacting via a quadrupolar kernel. Following similar endeavours for the study of earthquakes [@chen1991self], these models have succeeded in capturing the presence of strongly correlated dynamics in these systems (avalanches, possible shear bands, etc.) [@baret2002extremal; @budrikis2013avalanche; @nicolas2017deformation; @lin2014scaling; @lin15prl; @gueudre17scaling]. However, a clear connection between the microscopic description and these coarse-grained models is missing. In particular, the universality of the quadrupolar propagator used in EPM may still be questioned and the discreteness of EPM precludes the study of vibrational modes. An alternative approach is provided by continuum models that extend the free energy description of solids beyond the perfect elastic limit. In these models plasticity is introduced by means of a disordered potential which displays many local minima, as explained in Section \[sec:model\]. Such models, possibly pioneered by Kartha and co-workers [@kartha95tweed], were intensively studied by Onuki [@onuki2003plastic] and Jagla [@jagla07shear]. This paper intends to use the continuum approach to bridge the gap between atomistic simulations and discrete EPM [^2], with an emphasis on the initial soft modes and the actual response to ST. Considering two-dimensional (2D) materials subjected to pure shear, we find that the low-frequency modes are always peaked in point-like “soft spots”, where the next ST will take place. This extreme localisation is at variance with short-range depinning models, where soft modes have a finite localisation length which can be tuned by playing with the disorder strength [@cao2017localisation; @Tanguy2004localisation]. A closer analysis shows a halo of finite displacements around the soft spots pointing in the radial direction, with a $1/r$ radial decay and a two-fold azimuthal symmetry (this corresponds to a $1/r^2$ decay with four-fold azimuthal symmetry in the strain field), due to the elastic embedding of the impurities, see Section \[sec:denny\]. Surprisingly, these halos do not always match Eshelby’s solution. Instead, we find a one-parameter continuous family of kernels depending on the distribution of plastic disorder in the system (see Fig. \[fig:displacement\_th\]). Their shapes are rationalised analytically in Section \[sec:single\_impurity\] by calculating the soft mode associated with a point-like plastic impurity at $r=0$ embedded in an incompressible elastic medium. In polar coordinates, the (non-affine) displacement field $u$ reads: $$u_r(r,\theta) \propto \frac{ \cos(2\theta)} {1 + \delta \cos(4\theta)} \,,\, u_\theta = 0\,, \label{eq:u}$$ where $\theta = 0$ is the principal axis of positive stretch, and $\delta=(\mu_3-\mu_2)/(\mu_3+\mu_2)$ quantifies the plasticity-induced anisotropy in the shear moduli $\mu_2$ and $\mu_3$ (associated with the strains $e_2$ and $e_3$, respectively; see Fig \[fig:Macroscopic\_response\]). For $\delta=0$ we recover the standard quadrupolar (Eshelby-like) propagator. When $\delta \to 1$ we find a fracture-like kernel concentrating the deformation along the diagonal directions. This limit is obtained when the plastic potential softens the material to such an extent that the modulus along these directions vanishes (namely $\mu_2 \to 0$ while $\mu_3$ remains finite). To the best of our knowledge, the fracture-like propagator has not been observed yet, but we speculate that it might be seen in carefully aged glasses, in the marginal state that precedes global failure, when extended regions are on the brink of plastic failure. In the case of a single impurity, the soft mode has exactly the same shape as the final strain field induced by the ST, but also closely (or exactly if $\mu_2=\mu_3$) matches the transient strain field during the plastic event, up to renormalisation (Section \[sec:SPE3\]). {width=".9\textwidth"} Field-based models \[sec:model\] ================================ To begin with, we recall how plasticity is introduced in Continuum Mechanics descriptions of disordered solids [@kartha95tweed; @puglisi2000mechanics; @onuki2003plastic; @lookman03ferro; @jagla07shear; @jagla2017non]. In the spirit of the works of Jagla [@jagla07shear], this is achieved by first writing the free energy of an elastic material and then incorporating the plastic disorder in it. Strain variables and linear elasticity -------------------------------------- Even though glassy materials are discrete at the atomic scale, they can be handled as continua as long as one is interested in length scales larger than a few particle diameters [@tsamados2009local]. To linear order, deformations in a continuous medium are quantified by the strain tensor $$\begin{aligned} &\epsilon_{ij} = \frac12(\partial_j u_{i} + \partial_i u_{j}) \text{ with } i,j= 1,2\text{ in 2D.} \label{eq:tensors}\end{aligned}$$ It is convenient to trade off the strain tensor $\epsilon_{ij}$ for the following three strain variables: the volume distortion $e_1 = (\epsilon_{11} + \epsilon_{22})/2$ and the two independent shear strains $e_2 = (\epsilon_{11} - \epsilon_{22})/2$ and $e_3 = \epsilon_{12}$, represented in Fig. \[fig:Macroscopic\_response\](b). Since the $e_i$ derive from the same displacement field, the commutation rule for partial derivatives, *i.e.*, $\partial_{jk} u_{i} = \partial_{kj} u_{i}$, imposes a constraint on these variables, known as St. Venant condition. In terms of the Fourier transforms $ \hat{e}_i (q) = \int e_i(r) e^{{\mathbf{i}}q.r} {\mathrm{d}}^2 r $, this constraint reads $$\begin{aligned} &g:=\sum_{i=1}^3 Q_i \hat{e}_i = 0 \,,\, \label{eq:StV} \\ &(Q_i)_{i=1}^3 = (-{\left\vertq\right\vert}^2, q_1^2 - q_2^2, 2 q_1 q_2) \,, \label{eq:Ds} \end{aligned}$$ where $q_j = -{\mathbf{i}}\partial_j$ are the differential operators in Fourier space. These conditions turn out to be sufficient to prove the existence of the displacement field. With these new variables, the free energy of a uniform linear elastic solid reads $$F = \int_r B e_1^2 + \mu_2 e_2^2 + \mu_3 e_3^2 \label{eq:elastic_free} \,,$$ where $B$ is the bulk modulus and $\mu_2$ and $\mu_3$ are the shear moduli. The incompressible limit corresponds to $B\rightarrow \infty$ while, for an isotropic material, $\mu_2=\mu_3$. Plasticity ---------- If a disordered solid is strongly sheared, it will start to respond plastically by accumulating irretrievable deformation. This irreversible response is accounted for by introducing an anharmonic contribution to the free energy $F$ of eq. \[eq:elastic\_free\] so that $F$ can have multiple local minima, viz., $$F = \int_r \left[B e_1(r)^2 + V_{2,r}[e_2(r)] + \mu_3 e_3(r)^2 \right] dr \label{eq:free} \,,$$ where $V_{2,r}(e_2)= \mu_2 e_2^2 + W_{r}(e_2)$, with typically $ W_{r}(e_2)\leqslant 0$. Here, and in all the following, we will consider that plasticity only develops along the macroscopic shear direction, chosen to be $e_2$. Schematically, when the local strain $e_2$ at, say, $r=0$ is driven to a local maximum of $V_{2,r=0}$, a plastic event begins in which $e_2(0)$ slides into the next potential valley. This local change is elastically coupled to other regions of the material and may trigger other plastic events at $r^\prime \neq 0$, if $V_{j,r^\prime}(e_j)$ is also anharmonic. It is interesting to remark that, because of the shear-softening contribution $W_r$ to $V_{2,r}$, the local tangential shear modulus $\tilde{\mu}_2(0)= \frac{1}{2} V_{2,r=0}''$ vanishes at the onset of the instability, as observed in atomistic simulations. The softening of $\tilde{\mu}_2$ has the same effect on the local stress-strain relation as a shift of the minimum of the potential $V_{2,r=0}$ by an amount $e^{pl}$ in a material where the shear modulus would remain constant ($\tilde{\mu}_2=\mu_2$), *viz.*, $$\begin{aligned} &\sigma(r) = V_{2,r}^\prime[e_2(r)] = 2 \mu_2 [ e_2(r) - e^{pl} ]\,,\, \\ \text{where } & e^{pl} := - \frac{1}{2\mu_2}W_r^\prime[e_2(r)] \,. \label{eq:epldef}\end{aligned}$$ ![ *Illustrations of possible disordered potentials $V_{2,r}(e_2)$ –* (a,b) Interpretation of a plastic rearrangement from the viewpoint of the potential: While the system sits in a linear elastic region (a), the applied macroscopic stress $\Sigma$ tilts the potential, so much so that the system starts exploring the shear-softening region ($W_r<0$) and is finally able to briskly slide into a deeper minimum (b). (c) Illustration of the disordered potential used to derive rules of elasto-plastic models. The local stress variations during a plastic event are highlighted by green arrows and explained in the main text. []{data-label="fig:V2"}](potentials){width=".7\columnwidth"} Dynamics -------- Turning to the dynamics, the evolution of $e_i$ in the overdamped limit (relevant for foams and concentrated emulsions) is obtained by differentiating the total free energy $F_\mathrm{tot}$, viz. $$2 \eta_i \dot{e}_i = - \frac{\delta F_{\text{tot}}}{\delta e_i} \,,\label{eq:EoM}$$ where $\eta_i$ refers to the effective microscopic viscosity associated with the strain $e_i$. A difference with respect to the equation describing the depinning of an elastic line deserves to be underscored here. In the depinning case, the damping acts on the velocities $\dot{u}_i$, due to friction against a fixed substrate (or medium). Here, there is no such substrate and dissipation is due to the non-uniform velocities within the system; thus, only the velocity gradients are damped, as in the Navier-Stokes equation for fluids. In addition to the terms in eq. \[eq:free\], the total free energy $F_{\text{tot}}$, includes two contributions: the driving along $e_2$, $- \Sigma \, \hat{e}_2(0)$, where $ \Sigma$ corresponds to the macroscopic stress, as well as the St Venant constraint (eq. \[eq:StV\]) for all $q\neq0$ enforced by the Lagrange multipliers $\lambda(q)$, viz., $$F_{\text{tot}} = F + \int_{q \neq 0} \hat{\lambda}(q) \hat{g}(q) - \Sigma\, \hat{e}_2(0) \,. \label{Ftot}$$ The St. Venant constraint is naturally obtained by extremising $F_\text{tot}$ with respect to $\lambda(q)$, i.e., solving $\hat{g}(q) = 0$. The system can be driven in two distinct ways: *strain-controlled* protocols consist in controlling the time evolution of the macroscopic strain $\hat{e}_2(0)$ by properly adjusting $\Sigma$. Conversely, *stress-controlled* protocols result from imposing a constant macroscopic stress $\Sigma$ in eq. \[Ftot\] and leaving $\hat{e}_2(0)$ free. Via eqs. -, the macroscopic stress can be expressed as $\Sigma= \frac{1}{V} \int_r \sigma(r) + \eta_2 \dot \gamma$, where $\dot{\gamma}= \frac{2}{V} \dot{\hat{e}}_2(0) $ is the shear rate. If $\Sigma$ exceeds a value $\Sigma_y$ (set by the potentials $V_{2,r}$), the system will flow forever, with an ever–increasing strain $\hat{e}_2(0) \sim \dot{\gamma} t$ in the stationary state. Otherwise, it will reach a new equilibrium. Incompressible limit -------------------- From now on, we will focus on the incompressible limit $B\rightarrow \infty$, but equations for compressible systems are provided in Appendix \[sec:compressible\]. In this limit, $e_1=0$ so the St. Venant constraint yields $$2 q_1 q_2 \hat{e}_3(q) = (q_2^2 - q_1^2) \hat{e}_2(q) \,. \label{eq:StV-incomp}$$ As a consequence, the Fourier modes $\hat{e}_2(q)$ vanish whenever $q_1=0$ or $q_2=0$, with ${\left\vertq\right\vert}>0$. In real space, this means that the integral of $e_2$ along each horizontal or vertical line is constant and equal to the average strain, regardless of the mechanical law. Similarly, the integral of $e_3$ along each $\pm \pi/4$-direction line is also constant. These constraints on the strains are more fundamental than their counterparts for the stresses (or elastic strains), which are largely used in elasto-plastic models. The latter constraints impose that $\sigma_2$ ($\sigma_3$) have constant average along $\pm \pi/4$ lines (horizontal/vertical lines, respectively), where the stress components ${\sigma}_j$’s are in direct correspondence to the strain variables $e_j$, but these constraints are derived under the assumption of mechanical equilibrium. In addition, the displacement field $u_i$, which is easier to interpret than $e_2$, reduces to: $$\hat{u}_1 = {\mathbf{i}}q_1^{-1} \hat{e}_2 \,,\, \hat{u}_2 = -{\mathbf{i}}q_2^{-1} \hat{e}_2 \, \label{eq:integration}$$ if $q_1 q_2\neq 0$ and $\hat{u}_1,\hat{u}_2=0$ otherwise. Note that the zero mode $q=(0,0)$ of $\hat{u}_j$ corresponds to a global translation of the system. Hereafter, we assume isotropic viscosities, which are set to unity, viz., $\eta_2= \eta_3= 1$. Thus, the evolution of $e_2$, eq. \[eq:EoM\], turns into $$\begin{aligned} \label{eq:motion} 2 \dot{e}_2 = & - \frac{\delta F}{\delta e_2} + Q_2 \frac{ Q_2 \frac{\delta F}{\delta e_2} + Q_3 \frac{\delta F}{\delta e_3}}{ Q_2^2 + Q_3^2} + \Sigma \end{aligned}$$ and simplifies to $$2 \dot{e}_2 = -2\mu_2 e_2 -W_r^\prime(e_2) + \Sigma - \mathcal{G}_q [ W_r^\prime(e_2) - 2 \delta \mu e_2 ] , \label{eq:EoM_inc_real}$$ where we have used the shorthands $\delta \mu =\mu_3 - \mu_2$ and $$\mathcal{G}_q=\frac{-(q_1^2 - q_2^2)^2}{q^4} \label{eq:Gq}$$ for $q\neq 0$, and $\mathcal{G}_{q=0} = 0$. For systems with finite inertia, the foregoing overdamped equation can be generalised: $$\begin{aligned} 2 \rho \ddot{e}_2 = -q^2 \left[ \frac{1+\mathcal{G}_q}2 W_r^\prime(e_2) + \mu_2 e_2 - \delta \mu \, \mathcal{G}_q e_2 + \dot{e}_2 \right] \,, \end{aligned}$$ where $\rho$ is the density (see Appendix \[app:underdamped\]). Mathematically, one may notice that, for any given $q \neq 0$, the incompressible St Venant constraint  is satisfied on the line $e_3= -Q_2 Q_3^{-1} e_2$ of the plane $(\hat{e}_2(q),\hat{e}_3(q))$ and that the action of the Lagrange multiplier $\hat{\lambda}(q)$ is equivalent to taking only the tangential component of $\nabla F:= \left[ \delta F / \delta \hat{e}_2, \delta F / \delta \hat{e}_3 \right] $ along this line, via a projection of $\nabla F$ onto the *unit* tangential vector $\boldsymbol{t} := (Q_3/q^2,-Q_2/q^2)$ \[note that $Q_2^2 + Q_3^2 = q^4$ through eq. \]. Therefore, it is incorrect to just express $F$ as a function of $e_2$ on this line, *viz.*, $\tilde{F}(e_2) := F(e_2, -Q_2 Q_3^{-1} e_2)$ and then omit the Lagrange multiplier. Indeed, differentiating $\tilde{F}$ with respect to $e_2$ is equivalent to projecting $\nabla F$ onto the *non-normalised* tangential vector $\boldsymbol{\tilde{t}} = \left[ 1,-Q_2 Q_3^{-1} e_2\right]$. To obtain the correct dynamics without using multipliers, one should parametrise $F$ with the properly rescaled tangential coordinate $$\hat{e} (q):= \left[ \hat{e}_2(q), \hat{e}_3(q) \right] \cdot \boldsymbol{t} = q^2 Q_3^{-1} \hat{e}_2 (q) \,, \label{eq:rescaled_strain}$$ where $\cdot$ denotes the inner product.The rescaled coordinate $\hat{e}$ will prove useful in Sec. \[sec:denny\]. Response to a single plastic event \[sec:single\_impurity\] =========================================================== Although the equations established in the previous section fully define the model once the disorder potential $W_r$ is chosen, solving the full problem is complex. It is thus enlightening to start by considering the simple case where the plastic disorder is confined to an ‘impurity’ of size $a \to 0$ around $r=0$, while the rest of the material is elastic, viz., $$\begin{aligned} V_{2,r}(e_2) =& \mu_2 e_2^2(r) + a^2 \delta_{2D}(r) W(e_2(0)) - \Sigma\, \hat{e}_2(0) \,, \label{eq:V2oneimp}\end{aligned}$$ where $W$ is a disordered potential and $\delta_{2D}(r)$ is the Dirac distribution in 2D, evaluated at position $r$. Note that, for convenience, we have incorporated the driving contribution $- \Sigma\, \hat{e}_2(0)$ to the total free energy $F_\mathrm{tot}$ of eq.  into $V_{2,r}$. Beginning with the stable configuration sketched in Fig. \[fig:V2\](a), the driving gradually tilts the potential until the configuration becomes unstable (Fig. \[fig:V2\](b)). This triggers an instability whereby the local strain $e_2(0,t=0)$ evolves rapidly with time $t$ and modifies the local plastic disorder $W^\prime[e_2(0,t)]$ and plastic strain $e^{pl}(t)=- W^\prime[e_2(0,t)] / 2\mu_2$. Finally, the next stable configuration is attained, after a plastic strain $e^{pl}= e^{pl}_\star$ has been cumulated. In elasto–plastic models, the mechanical equilibration time is neglected everywhere in the material, except in the plastic impurity [@nicolas2017deformation]; thus, the strain field $e_2$ is always an equilibrium configuration for a given plastic strain $e^{pl}$, but this plastic strain may need a finite time to reach its final value $e^{pl}_\star$. In this section, we first derive the equilibrium configurations for the continuous models under study here and then show that, up to a normalisation coefficient, they coincide with the soft modes of the system just before the onset of the instability. Finally, we explore the transient dynamics during the plastic event. During the whole evolution, the dynamics are governed by the equation of motion . For Fourier modes $q \neq 0$ and for a single impurity, this equation reduces to: $$\dot{\hat{e}}_2(q) = \hat{e}_2(q) \left( - \mu_2 + \mathcal{G}_q \delta \mu \right) + ( 1 + \mathcal{G}_q ) \mu_2 a^2 e^{pl} \, \label{eq:EoM2}$$ with $\delta \mu = \mu_3 - \mu_2$. Equilibrium configuration ------------------------- To find the equilibrium configuration, $\dot e_2=0$ is set to zero in eq. . This immediately leads to $$\begin{aligned} \hat{e}_2(q) &= a^2 \, e^{pl}_* \frac {1+\mathcal{G}_q} {1+\mathcal{G}_q\,(1-\mu_3/\mu_2)}\,\mathrm{for}\,q\neq 0\, , \label{eq:e2q} \\ \hat{e}_2(0) &= a^2\, e^{pl}_* + \frac{V\,\Sigma}{2 \mu_2},\end{aligned}$$ where $V$ is the volume of the system and $e^{pl}_*$ must be determined self-consistently by integrating both sides of eq.  and noticing that $\int_q \hat{e}_2(q) dq \propto e_2(0)$ (for explicit results, see Appendix \[sec:C\]). Regarding the $q=0$-mode (derived from eq.  ), i.e., the extensive total strain $\hat{e}_2(0)$, one may remark that it will remain constant in a strain-controlled protocol, in which the macroscopic stress $\Sigma$ will thus decrease by $2\mu_2 a^2\, e^{pl}_* /V$, whereas in a stress-controlled protocol $\hat{e}_2(0)$ will increase by $a^2\, e^{pl}_*$ because of plasticity. For simplicity, the formulae below are given in the case of a strain-controlled protocol with $\hat{e}_2(0)=0$. In real space, the Fourier expression of eq.  translates to $$e_2(r, \theta) = \frac{ C }{r^2} \frac{\delta + \cos(4\theta)}{(1 + \delta \cos(4\theta))^2} + C_0 \delta(r) \,, \label{eq:e2r}$$ where the anisotropy parameter $\delta = \delta \mu/(\mu_3 + \mu_2)$ was introduced below eq. , and the constants $C_0$ and $C$ are given in Appendix \[app:Fourier\] (eq. ), along with details of the derivation. Finally, we can integrate $e_2(r, \theta)$ to obtain the noteworthy formula of eq. \[eq:u\] for the displacement field $u(r,\theta)$, as detailed in Appendix \[sec:ur\]. Let us mention that these calculations (for an incompressible material) can be generalised to the compressible regime ($B<\infty$, see Appendix \[sec:compressible\]) by substituting $\mathcal{G}_q$ in eq.  with $$\label{eq:kernelB} \mathcal{G}_{q}^{B<\infty} = \frac{B}{B + \mu_3} \mathcal{G}_q \,.$$ More general expressions giving the elastic field generated by an *extended* impurity can be found in the literature on the mechanics of anisotropic solids[@yang1976generalized; @kinoshita1971elastic; @dunn1997inclusions], but, being more general, they are also (much) more complex. Connection with elasto–plastic models {#sec:EPM} ------------------------------------- How do the foregoing results compare with the propagator and rules implemented in elasto-plastic models? To address this question, we set $a=1$ and focus on the isotropic ($\mu_2 = \mu_3$) and incompressible ($B=+\infty$) case. The elastic strain $\hat{e}_2^{el}(q) := \hat{e}_2 - e^{pl}_* $ then boils down to $e_*^{pl} \mathcal{G}_q$ by virtue of eq. , where the quadrupolar elastic propagator $\mathcal{G}_q=\frac{-(q_1^2 - q_2^2)^2}{q^4}$ used in EPM is made apparent. This quadrupolar propagator was derived by Picard and co-workers [@picard04elastic; @nicolas2014universal] using a quite different approach, by writing mechanical equilibrium ($\nabla \cdot \boldsymbol\sigma=0$, where $\boldsymbol\sigma$ is the stress tensor) in an incompressible medium with a plastic eigenstrain $e^{pl}_*$ at the origin. It can also be regarded as the point-wise limit of an Eshelby inclusion [@eshelby1957determination], *i.e.*, a circular inclusion that would spontaneously deform into an ellipse in free space. In real space, the propagator $\mathcal{G}_q$ becomes $$\mathcal{G}(r,\theta) = -\mathrm{cos}(4\theta)/\pi r^2 - \delta(r)/ 2 \label{eq:prop_EPM}$$ in polar coordinates \[see eq. \]. The elastic strain $e_2^{el}(r=0)$ at the origin is thus depressed by $e^{pl}_*/2$ to mitigate the local surge of the plastic strain $e^{pl}_*$, and a quadrupolar halo surrounds the plastic event. Bearing the foregoing considerations in mind, it is easy to understand that the rules of elasto-plastic cellular automata [@picard2005slow; @lin14epl; @nicolas2017deformation] ensue from the choice of the piecewise quadratic potential $V_{2,r}$ sketched in Fig. \[fig:V2\] (c) and the neglect of the transient dynamics before mechanical equilibration. Indeed, with this choice, while $e_2(r)$ evolves in a continuous region of $V_{2,r}$, the material is locally Hookean, with $\dot{e}_2(r)=\dot{e}_2^{el}(r)$. Upon reaching a discontinuity of $V_{2,r}$, $e_2(r)$ falls into another quadratic branch of the potential, which corresponds to a finite jump of the local plastic strain by an amount $e_*^{pl}(r)$. Using eq. \[eq:prop\_EPM\], this local distortion generates a quadrupolar halo of elastic stress and a depression of the local elastic stress by $e_*^{pl}(r)/2$, viz., $$\begin{aligned} \sigma(r) & \to \sigma(r) - \mu_2 e^{pl}_*(r) \pm \mu_2 e^{pl}_*(r)/V \, \\ \sigma(r') & \to \sigma(r') + 2 \mu_2 e^{pl}_*(r) \mathcal{G}(r^\prime - r) \pm \mu_2 e^{pl}_*(r)/V \label{eq:CA2}\end{aligned}$$ where the sign $\pm$ is positive (negative) for the stress-controlled (strain-controlled, respectively) protocol. Soft modes {#sec:SPE2} ---------- One of the advantages of the continuous approach under study over discrete elasto–plastic automata is that, instead of describing a sequence of mechanical equilibria, it contains the whole dynamics. This, in particular, gives access to the soft modes, whose study was so far restricted to atomistic simulations [@tanguy10mode; @manning11soft; @charbonneau16universal; @yang2017correlations] and, to a lesser extent, experiments in which the fluctuations in the particle positions can be imaged [@henkes2012extracting]. In order to perform a linear stability analysis, we assume that $$W(e_2) = - \mu_0 e_2^2 + O(e_2^3) \,,\, \mu_0 > 0$$ near the equilibrium position $e_2 = 0$, see Fig. \[fig:V2\] (a). Decomposing the dynamics of $\hat{e}_2(q,t)$ into a linear sum of Laplace modes, viz., $\hat{e}_2(q,t)= \int_0^{\infty} e^{\omega t} \tilde{e}_2(q,\omega) d\omega$ (so that $\Re(\omega) > 0$ refers to *unstable* modes, contrary to the convention of Ref. [@charbonneau16universal; @cao2017localisation]), and inserting into eq. \[eq:EoM\_inc\_real\], we arrive at the following equation: $$\left[ \mu_2 - \delta \mu \mathcal{G}_q + \omega \right] \tilde{e}_2(q, \omega) = \mu_0 (1 + \mathcal{G}_q ) a^2 e_2^o(\omega)\, ,\label{eq:softmodegen}$$ where $e_2^o(\omega)$ is a shorthand for $\tilde{e}_2(r=0, \omega)$. (For inertial systems, an additional term $2 \rho \omega^2 q^{-2}$ is present between the brackets on the left-hand side, as detailed in Appendix \[app:underdamped\]). This equation determines the shape of the dynamical modes. Indeed, if $e_2^o(\omega)=0$, then $\tilde{e}_2(q, \omega)$ must vanish at all wavenumbers, except those which cancel the prefactor on the left-hand side of eq. \[eq:softmodegen\]. Bearing in mind that $\mathcal{G}_q \in [-1,0]$, this cancellation is possible if and only if $\omega \in [-\mu_3,-\mu_2]$; there is thus a continuous range of admissible growth rates $\omega$. On the other hand, if $e_2^o(\omega) \neq 0$, integrating $\tilde{e}_2(q, \omega)$ over $q$ from eq. \[eq:softmodegen\] and noticing that $\int_q \tilde{e}_2(q,\omega) \propto e_2^o(\omega)$ gives the following closure relation for $\omega$: $$e_2^o(\omega) = e_2^o(\omega) \frac{\mu _0}{\mu _2+\omega } \mathcal{I}\left( \frac{\mu _2-\mu _3}{\mu _2+\omega } \right) \,$$ where the integral function $\mathcal{I}(x)$ is made explicit in eq.  of the Appendix. The closure relation is only satisfied for $\omega = \omega_\star$, with $$\omega_\star = -\mu_2+\frac{\mu _0^2}{2 \mu _0 + \mu _3 - \mu _2} > - \mu_2 \,. \label{eq:omega}$$ Since $\omega_\star$ is the maximal growth rate, it is associated with the most unstable mode. Following the same steps as before to compute this eigenmode from eq. , we find that it derives from the following displacement field: $$\begin{aligned} \label{eq:delta'} u_{r,\star} \propto \frac{1}{r} \frac{\cos (2\theta)}{1 + \delta_\star \cos (4\theta) } \,, u_{\theta,\star} = 0 \,. \end{aligned}$$ in polar coordinates. Here $ \delta_\star = \frac{\mu _3-\mu _2}{\mu _2+\mu _3+2 \omega_\star }$ quantifies the anisotropy of the elastic medium, as the parameter $\delta$ defined below eq. . Another consequence of eq.  concerns the situations of marginal stability, when $\omega_\star \to 0$, or $\mu_0 \to \mu_2 + \sqrt{\mu_2 \mu_3}$ by eq. . These situations are on no account mathematical curiosities, but occur whenever a plastic instability is about to take place during the *quasi-static* deformation of disordered solids; this instability is triggered by the marginally stable soft mode [@cao2017localisation]. In this case ($\omega_\star \to 0$), the two anisotropy parameters, $\delta$ and $\delta_\star$, converge and the soft mode (given by eq. \[eq:delta’\]) coincides with the equilibrium configuration that will be reached after the development of this plastic instability, up to a proportionality coefficient. Besides, this mode is not affected by the presence of inertia, which becomes negligible for $\omega \to 0$. Therefore, in quasi-static protocols, we can identify the marginal soft mode and the equilibrium configuration. ![Time evolution of the displacement field $u(r,\theta)$ in response to a single plastic event for a highly anisotropic incompressible material at fixed macroscopic strain $\overline{e_2}=0$, with $\mu_2 = 1/19$ and $\mu_3 = 1$ (so that $\delta = 0.9$, as in Fig. \[fig:displacement\_th\], (c,f)). We chose an anharmonic potential $W(e_2) = -\mu_0 e_2^2 - \frac32 e_2^3 + e_2^4$, where $\mu_0 = \mu_2 + \sqrt{\mu_2 \mu_3}$ so that $e_2 \equiv 0$ is a marginally stable equilibrium. The initial condition is a small number times the soft mode $\hat{e}_2(q) = 0.04 a^2 (1+\mathcal{G}_q)/( \mu_2 + (\mu_2 - \mu_3) \mathcal{G}_q )$ and the system is evolved according to eq.  for a duration $T \approx 150$; the evolutions of $e_2(0,t)$ with time and of $W$ as a function of $e_2(0)$ are plotted in insets (ii) and (i), respectively. The main plot shows snapshots of the normalised displacement field at $t = T/10, 2T/10, \dots, T$, computed from $\hat{e}_2$ using the methods of Appendices \[sec:ur\] and \[app:Fourier\]. The colors represent time in the way indicated by the insets. The final configuration is plotted with dashed line since it overlaps with the initial one.[]{data-label="fig:evolution"}](evolution.pdf){width=".9\columnwidth"} Transient dynamics {#sec:SPE3} ------------------ We close this section by discussing the transient dynamics during the plastic event, i.e., we wonder how the marginal soft mode gradually unfurls and fades into the new equilibrium state, which is of similar shape. In the special case of isotropic elasticity, i.e., $\mu_2 = \mu_3$, one can check, using eq. \[eq:EoM2\], that the shape is conserved during the *whole* dynamics, viz., $\hat{e}_2(q,t) = e^{pl}(t) a^2 (1 + \mathcal{G}_q)$, even if the instability originated in an excited mode ($\omega_\star<0$) and not in the marginal soft mode. Only the normalisation constant evolves (in the strain-controlled protocol with $\gamma = 0$): $$\begin{aligned} \frac{{\mathrm{d}}e^{pl}}{{\mathrm{d}}t} = \mu_2 e^{pl} - \frac12 W'\left(\frac{e^{pl}}{2}\right) \,.\end{aligned}$$ The anisotropic case, $\mu_2 \neq \mu_3$, is more involved and requires to solve the equation of motion eq.  numerically, which can be done efficiently by noticing that $\hat{e}_2(q)$ depends only on $q_2/q_1$. Perturbing a marginally stable equilibrium configuration $e^0_2$ along a soft mode and letting the system relax at fixed macroscopic strain $\gamma = 0$, we observe that the transient between the initial and final stages is short-lived and only weakly deviates from the equilibrium shape given by eq. (\[eq:u\]) (up to normalisation), as shown in Fig. \[fig:evolution\]. In particular, in the strongly anisotropic case $\mu_2 \ll \mu_3$, the equilibrium ‘fracture’-like profile is apparent during the whole evolution and clearly distinct from the quadrupolar shape found in an isotropic medium. In summary, the propagators of eq.  are robust signatures of the elastic medium anisotropy, even during the transient dynamics. Response of a disordered medium with many impurities {#sec:denny} ==================================================== The previous section has shed light on the response of a (possibly anisotropic) elastic medium to a single plastic impurity, with $W_r \propto \delta_{2D}(r)$, and unveiled a continuous family of elastic propagators. Now, we extend the study to fully disordered media, with many plastic impurities (generic $W_r$). In this case, the problem becomes analytically intractable, but deserves to be investigated on account of its relevance for collective effects in disordered solids under shear, including avalanches of rearrangements. A recent analysis of the yielding behaviour of these materials close to the critical point was notably proposed on the basis of a very similar model by Jagla [@jagla2017non]. Incidentally, even the numerical study of these models presents difficulties. In particular, the fluctuating sign of the propagator $\mathcal{G}_q$ imposes to carefully follow the order of the rearrangements. This is in stark contrast with the related, but distinct [@lin2014scaling; @lin14epl], problem of elastic line depinning, where the propagator is non-negative (hence the existence of Middleton theorems [@MiddletonPRL]) and efficient algorithms can be devised [@werner02rough]. To proceed, we will inspect the low-frequency excitations of an equilibrium configuration in this generic case by numerically computing the eigenvectors of the so called dynamical matrix, under the assumption of spatially uncorrelated plastic disorder. Dynamical matrix {#sec:matrix} ---------------- To start with, we write the general equation of motion (\[eq:EoM\_inc\_real\]) in terms of the properly rescaled [^3] strain variable $e= \mathcal{F}^{-1} e_2$ introduced in eq. (\[eq:rescaled\_strain\]), with $ \mathcal{F}= Q_3 / q^2$, $$\dot{e} = -\mu_2 e - \frac{1}{2} \mathcal{F} W_r^\prime\left( \mathcal{F} e \right) + \frac{\Sigma}{2\mathcal{F}} + (1 - \mathcal{F}^2) (\mu_2-\mu_3) e \, , \label{eq:edot_overdamped}$$ where we have used $\mathcal{F}^2 = 1+ \mathcal{G}_q$. A small deviation $\delta e$ away from an equilibrium configuration $e^{(0)}$ thus decays as $\delta \dot{e}(r) = -\sum_{r^\prime} \mathcal{M}_{rr^\prime} \delta e(r^\prime)$ in discrete space, where the dynamical matrix $\mathcal{M}_{rr^\prime}$ reads $$\begin{aligned} \mathcal{M}_{rr^\prime} = \mu_2 \delta_{rr^\prime} - \mathcal{F} \mathcal{D}_{rr^\prime} \mathcal{F} + (\delta_{rr'} - \mathcal{F}^2) (\mu_2-\mu_3)& \, \nonumber \\ \text{with } \mathcal{D}_{rr'} = -\frac{1}{2} \delta_{rr'} W''_r\left(\mathcal{F} e^{(0)}(r)\right) \,.& \label{eq:D}\end{aligned}$$ From now on, we focus on the case $\mu_2 = \mu_3$. Then, $$\mathcal{M}_{rr^\prime} = \mu_2 \delta_{rr^\prime} - \mathcal{F} \mathcal{D}_{rr^\prime} \mathcal{F} \label{eq:Mdef}$$ is the sum of the scalar matrix $\mu_2 \delta_{rr^\prime}$ and a matrix product between $\mathcal{F}$ (diagonal in Fourier space) and $\mathcal{D}_{rr^\prime}$ (diagonal in real space). Accordingly, rescaling the plastic disorder strength as $\mathcal{D}_{rr^\prime} \leadsto k \mathcal{D}_{rr^\prime} $ has no effect on the excitation modes, bar an affine transformation of the eigenvalues $\omega \leadsto k(\omega + \mu_2) - \mu_2$. Once again, the contrast with respect to the equation describing the depinning of an elastic interface should be noted. In that case, the dynamical matrix is a *sum* of a propagator $\mathcal{G}^{\mathrm{(d)}}$ accounting for the elastic couplings within the interface (usually a fractional Laplacian that is diagonal in Fourier space) and a disorder matrix $\mathcal{D}^{\mathrm{(d)}}_{rr^\prime} := \delta_{rr'} W_r^{\mathrm{(d)}\prime\prime}\left(u^{(0)}(r)\right)$ obtained by deriving the disorder potential $W_r^{\mathrm{(d)}}$ with respect to the local [displacement]{} $u(r)$. The disorder strength can, and does, affect the shape of the eigenmodes, in particular, their localisation length [@cao2017localisation]. Random approximation {#sec:anderson} -------------------- ![The Probability density function of the Weibull law $\text{Pdf}(D_r) = \kappa D_r^{\kappa-1} e^{-D_r^\kappa}$ for parameters $\kappa = 1, 2,5$. []{data-label="fig:weibull"}](weibull.pdf){width=".9\columnwidth"} {width="85.00000%"} In light of the foregoing observation that the dynamical matrix has the same eigenvectors as $\mathcal{F} \mathcal{D}_{rr^\prime} \mathcal{F}$, where $\mathcal{D}_{rr'}$ is given by eq. (\[eq:D\]), we are interested in gaining insight into $D_r := - W''_r\left(\mathcal{F} e^{(0)}(r)\right) / 2$. A priori, it will depend on the specific equilibrium configuration under study and the spatial correlations of $W_r$. However, the spatial correlations of the disorder will be discarded here, which will spare us the meticulous search of equilibria. Albeit uncontrolled, this approximation was recently found to preserve key properties of the eigenmodes at the depinning transition [@cao2017localisation]. Indeed, over an ensemble of equilibria, the values taken by $D_r$ at different $r$ are only weakly correlated and they will be considered random. More precisely, we will handle $D_r$ as independent random variables drawn from a Weibull distribution of parameter $\kappa$, which means that $\left(D_r \right)^\kappa$ is exponentially distributed (see Fig. \[fig:weibull\]). This enforces $D_r\geqslant 0$ (because plasticity tends to soften the material). The choice of a Weibull distribution is arbitrary, but it will prove convenient, in that it allows us to tune the dispersion of plastic disorder via the parameter $\kappa$. Considering the limit-cases, when $\kappa \leq 1$, the distribution of $D_r$ is peaked at $D_r = 0$ and, apart from large outliers, $D_r \sim 0$. To the contrary, when $\kappa \gg 1$, a peak around $1$ emerges and the whole distribution concentrates in this peak. These distributions of $D_r$, combined with the formula of eq. \[eq:Mdef\], are used to populate random matrices $\mathcal{M}_{rr^\prime}$ (of size $256^2 \times 256^2$, *i.e.*, $r = (x,y), x,y = -128,\dots,127$ and similarly for $r^\prime$). The lowest-frequency eigenvector (softest mode) of each matrix is found numerically by the power/Lancsoz iteration method and is integrated via eq. \[eq:integration\] to get the associated displacement field. Some representative displacement fields for different values of $\kappa$ are plotted in Fig. \[fig:fits\], with the peak value shifted to the origin, for easier comparison with Fig. \[fig:displacement\_th\]. Two major conclusions can be drawn from the observation of these plots (among many similar plots). First, irrespective of $\kappa$, we always find ‘soft’ modes that are localised, with displacements $u$ and strains $e_2$ clearly peaked at an individual site. In other words, no collective pinning (with a peak region spread over many sites) is seen. At a distance $r$ away from the peak, the radial displacement $u_r$ and the strain $e_2$ decay as a power laws $\propto 1/r$ and $1/r^2$, respectively, while the azimuthal component $u_\theta$ remains very small compared to $u_r$. Secondly, the overall shapes of the displacement fields are strongly reminiscent of the elastic propagators derived in Section \[sec:single\_impurity\] for the single-impurity problem and range from the standard quadrupolar propagator (for small $\kappa$) to the fracture-like propagator (for large $\kappa$). In fact, the theoretical expressions of eq. (\[eq:delta’\]) turn out to fit the numerical displacements quite well, if one is given the freedom to adjust the anisotropy parameter $\delta_\star$ of the elastic medium, although here $\mu_2=\mu_3$. How can we rationalise these findings? If all sites in the system were decoupled, the softest mode would just be a local excitation of the softest site (where $D_r$ is maximal). But, because of the elastic embedding of the site (and, more formally, the St Venant constraint), this excitation is coupled with an elastic deformation of the surroundings, as though it were an impurity. For small $\kappa$, the disorder $D_r$ is close to zero on most sites (but not on the softest one, of course), so the medium is virtually a perfect isotropic solid, hence the standard quadrupolar shape of the mode, that is well captured by an anisotropy parameter $\delta_\star \to 0$. On the other hand, for larger $\kappa$, most sites display considerable plasticity-induced softening, with $D_r \approx 1$ (while $D_r>1$ on the softest site). This global softening along $e_2$ is tantamount to a lowering of the shear modulus $\mu_2$. Indeed, the dynamical matrix of eq. (\[eq:D\]) is not altered by simultaneously reducing $D_r$ by its spatial average $\overline{D_r}$ and lowering $\mu_2$ to $\mu_2-\overline{D_r}$. Therefore, the impurity is effectively coupled with an anisotropic elastic medium characterised by a finite $\delta_\star$ and, in the limit of large anisotropy ($\kappa \gg 1$), a fracture-like propagator can emerge, with $\delta_\star \to 1$. Still, it is noteworthy that, even in this regime where the $D_r$ are narrowly distributed around 1, the observed fields remain dominated by single impurities. Discussion ========== In this work, we have studied an intermediate class of models describing the plastic deformation of disordered solids, that operate at a coarse-grained scale similar to that of elasto–plastic models (EPM) while still describing the non-linear elastic properties of atomistic systems. The models considered focus on the (continuous) strain fields in the material and incorporate them into a Ginzburg-Landau free energy which combines a purely elastic part and a plastic disorder potential. We investigated the equilibrium configurations, the soft modes and the equilibration dynamics associated with this free energy. The last two aspects are lost in EPM, which hop between mechanically equilibrated configurations, following rules that we could clarify. In contrast, the transient dynamics during mechanical equilibration are present in the continuous models under study, whose global relaxation thus depends on the deformation rate.[^4] When plastic disorder is spatially confined in a single impurity, we were able to derive analytically the soft mode and the equilibrium configuration of the system, which coincide (up to a rescaling factor). Our first important result is that the quadrupolar propagator routinely used in EPM is not ubiquitous. Indeed, in the presence of strong elastic anisotropy, we found a new, fracture-like propagator, in which the deformation concentrates along the easy directions. A continuous family of propagators, obeying a simple formula  depending on one anisotropy parameter, interpolates between the quadrupolar propagator and the fracture-like one. Remarkably, these single-impurity calculations keep being relevant in fully disordered systems, whose lowest-frequency excitations were numerically found to localise around the softest (point-like) site, even for weakly dispersed disorder. Moreover, the deformation halo around this soft site replicates the foregoing family of propagators, as the distribution of plastic disorder is varied. The fracture-like propagator is recovered for finite, narrowly distributed plastic disorder. Indeed, the latter softens the material along one shearing direction and thus renders the surroundings of the soft site effectively anisotropic. Accordingly, one may expect to find the fracture-like propagator whenever extended regions of the material collectively soften on the brink of failure, without immediately failing. In the presently studied models, this situation is precluded when the disorder potential is piecewise parabolic with cusps, but should be possible with any smooth potential. Recently, Jagla showed that these two classes of potentials led to different critical exponents at the yielding transition for the flow curve [@jagla2017non], but the shape of the propagator in each case and its possible relevance were not studied. In atomistic simulations and experiments on glasses, non-standard elastic propagators (in the soft mode or the actual stress redistribution) have not been reported either, to the best of our knowledge. Certainly, specific conditions are required to observe collective softening of the material along *one* direction, such as high mechanical homogeneity in the initial configuration, and they may not be met often. In addition, the noise and fluctuations in the numerical and experimental data, notably due to the granularity of the material at the microscale, may complicate the distinction of a new propagator, all the more so as the paradigm resting on Eshelby’s solution is overwhelming. Nevertheless, we may tentatively expect to see it in carefully aged (ultra-stable [@berthier2017origin]) glasses just before their dramatic macroscopic failure; the incipience of a shear band[@nguyen16shearband; @tanguy10mode] may also reflect the presence of collective softening. So we advocate to test fits to the different propagators in future simulations and experiments. Acknowledgments {#acknowledgments .unnumbered} --------------- We thank S. Bouzat and A. B. Kolton for collaboration on related projects, E. Jagla, T. de Geus, P. Le Doussal, A. Tanguy, S. Cazayus-Claverie, and M. Wyart, for insightful discussions. The authors acknowledge support from a Simons Investigatorship, Capital Fund Management Paris and LPTMS (X.C.), a LabEx-ICFP scholarship and CNRS (D.T.), and an ANR grant ANR-16-CE30-0023-01 THERMOLOC (A.R.). *Conflict of interest.* There are no conflicts of interest to declare. Compressible material {#sec:compressible} ===================== In this Appendix, the results derived in the main text for the incompressible case are extended to compressible systems ($B<\infty$). To this end, the equation of motion is generalised to compressible systems, while one still assumes that the viscosities acting on the different strain variables are equal, $\eta=1$ . Extremising the free energy of eq.  with respect to the Lagrange multipliers leads to $\sum_j Q_j \dot{\hat{e}}_j(q) = 0$, for $q \neq 0$. The equation of motion then straightforwardly generalises to $$\label{eq:motiongen} \dot{\hat{e}}_j(q) = -\frac{\delta F}{\delta \hat{e}_j(q)} - Q_j \lambda \,,\, \lambda = -\frac{\sum_{k=1}^3 Q_k \frac{\delta F}{\delta \hat{e}_j(q)} }{\sum_{k=1}^3 Q_k^2 } \,.$$ for $j = 1, 2, 3$ and $q\neq0$. For a single impurity, the above equation can be recast into a matrix form, in terms of the 3-vector $\mathbf{e}(q) =\begin{bmatrix} {\hat{e}}_1(q) & {\hat{e}}_2(q) & {\hat{e}}_3(q) \end{bmatrix}$: $$\begin{aligned} & \dot{\mathbf{e}} = -\mathbf{P} \left( \mathbf{M} \mathbf{e} + \mathbf{v} \right) \label{eq:vectorform}\end{aligned}$$ where $ \mathbf{v}$ is a $3$-vector, and $\mathbf{P}$ and $\mathbf{M}$ are $3\times3$ matrices defined as follows: $$\begin{aligned} & \mathbf{P}_{jk} = \delta_{j,k} - \frac{ Q_j Q_k}{\sum_{\ell=1}^3 Q_\ell^2 }\,,\, \mathbf{M}_{jk} = \delta_{j,k} \mu_k \,,\, \mu_1 := B\,, \nonumber \\ & \mathbf{v} = \begin{bmatrix} 0 & \mu_2 a^2 e^{pl} & 0 \end{bmatrix} \,. \nonumber\end{aligned}$$ At equilibrium, $\dot{\mathbf{e}}$ vanishes in eq. . The strain modes $\mathbf{e}$ that cancel the right-hand side of eq.  and satisfy the St Venant constraint are $$\begin{aligned} \hat{e}_1(q) &= -a^2 \, e^{pl}_* \frac{Q_2}{Q_1} \frac{ \mu_3/(B + \mu_3) }{ {1+\mathcal{G}_q^{B<\infty}\,(1-\mu_3/\mu_2)} } \nonumber \\ \hat{e}_2(q) &= a^2 \, e^{pl}_* \frac {1+\mathcal{G}_q^{B<\infty}} {1+ \mathcal{G}_q^{B<\infty} \,(1-\mu_3/\mu_2)} \label{eq:e2qgen} \\ \hat{e}_3(q) &=- a^2 \, e^{pl}_* \frac{Q_2}{Q_3} \frac{ 1 + \mathcal{G}_q^{B<\infty} - \mu_3/(B + \mu_3) }{ {1+\mathcal{G}_q^{B<\infty}\,(1-\mu_3/\mu_2)} } \nonumber\end{aligned}$$ where $e^{pl}_*$ must be determined self-consistently, as detailed in Appendix \[sec:C\], and $$\mathcal{G}_q^{B<\infty} = \frac{B}{B + \mu_3} \mathcal{G}_q \,.$$ Plastic strain in the equilibrium situation {#sec:C} =========================================== Section \[sec:single\_impurity\] exposed how the shear softening induced by a single plastic impurity at $r=0$, quantified by $e^{pl}(t=0) = - W'(e_2(0,t=0)) / 2 \mu_2$, generates an elastic deformation field $e_2(r,t)$, which deforms the impurity and possibly further softens the material at $r=0$. Equilibrium is reached when $$e^{pl}(t) \to e^{pl}_*= -\frac{1}{2 \mu_2} W'(e_2(0)),$$ where $e_2(0)$ forms part of a mechanically equilibrated strain field $e_2(r)$ (no time dependence), or $\hat{e}_2(q)$ in Fourier space. The Fourier components $\hat{e}_2(q)$ depend on $e^{pl}$ via eq.  (in the incompressible case). Integrating these components over $q$ to get $ e_2(0) \propto \int \hat{e}_2(q) dq$ leads to $$e^{pl}_*= -\frac{1}{2 \mu_2} W'\left(\beta e^{pl}_* + \overline{e_2} \right) \,. \label{eq:plastic1}$$ where $\beta= (1+ \sqrt{\mu_3 / \mu_2})^{-1}$. Note that special attention was paid to the zero-mode $\hat{e}_2(0)$, which is proportional to the average strain $\overline{e_2}$, because its value depends on the deformation protocol, as explained in the main text. These results can be generalised to the compressible case $B<\infty$, where the Fourier components $\hat{e}_2(q)$ obey eq.  instead of eq. \[eq:e2q\]. It turns out that eq.  still holds, provided that one replaces $\beta$ with $$\frac{B}{B+\mu _3} \mathcal{I}(x) + \frac{\mu_3}{(B+\mu _3)\sqrt{1-x}}.$$ Here, we have introduced the shorthand $x=\frac{B \left(\mu _2-\mu _3\right)}{\mu _2 \left(B+\mu _3\right)}$ and the following integral (that can be calculated by Cauchy’s residue theorem for $x<1$), $$\label{eq:integral} \mathcal{I}(x) = \int_{0}^{2\pi} \frac{{\mathrm{d}}\phi}{2 \pi} \frac{\sin (2\phi)^2 }{1 - x \cos(2\phi)^2} = \frac{1}{\sqrt{1-x}+1}.$$ Fourier transforms in polar coordinates \[app:Fourier\] ======================================================= The calculations in the main text heavily rely on the (convenient) use of Fourier transforms (F.T.), notably in polar coordinates, with $$(x,y)= (r \cos \theta, r \sin \theta) \,,\, q = (\rho \cos \phi, \rho \sin \phi).$$ This Appendix collects some results that are useful to derive the real-space expressions from their F.T. First of all, for functions with an $1/r^2$ dependence (such as the elastic strain generated by a plastic impurity), the following property applies: $$\begin{aligned} \cos(n \theta) r^{-2} \stackrel{\text{F.T.}} \longrightarrow 2 \pi {\mathbf{i}}^n\cos(n \phi) n^{-1} \,,\, n = 1,2,\dots\,. \label{eq:Fourier}\end{aligned}$$ *Proof.* To derive this result, we recall the Jacobi–Anger identity: $$e^{{\mathbf{i}}q x} = \sum_{m\in {\mathbb{Z}}} {\mathbf{i}}^m e^{{\mathbf{i}}m (\theta -\phi)} J_m(\rho r)$$ $J_m(y)$ is the Bessel $J$ function. Now let $G(x) = e^{{\mathbf{i}}n \theta} r^{-2}$, with some $n > 0$, then its Fourier transform is $$\begin{aligned} & \hat{G}(q) = \int_{0}^{\infty} r^{-1} {\mathrm{d}}r \int_0^{2\pi} {\mathrm{d}}\theta G(x) e^{{\mathbf{i}}k x}\nonumber \\ = & \int_{0}^{\infty} r^{-1} {\mathrm{d}}r \int {\mathrm{d}}\theta G(\theta) \sum_{m\in {\mathbb{Z}}} {\mathbf{i}}^m e^{{\mathbf{i}}m (\phi - \theta)} J_m(\rho r) \nonumber \\ =& 2 \pi {\mathbf{i}}^{n} e^{{\mathbf{i}}n \phi} \int_{0}^{\infty} r^{-1} {\mathrm{d}}r J_n( \rho r ) = 2 \pi G_n {\mathbf{i}}^{n} e^{{\mathbf{i}}n \phi} n^{-1} $$ where in the last line we used the identity $$\int_{0}^{\infty} r^{-1} {\mathrm{d}}r J_n(r) = n^{-1} \,,\, n = 1, 2, \dots$$ This means that, for $n> 0$, $$e^{{\mathbf{i}}n \theta} r^{-2} \stackrel{\text{F.T.}} \longrightarrow 2 \pi {\mathbf{i}}^{n} n^{-1} e^{{\mathbf{i}}n \phi} \,,\,$$ which proves eq.  by parity considerations $\blacksquare$ Now, we specifically turn to the derivation of the real-space field $e_2(r)$ from its F.T. eq.  $$\hat{e}_2(q) \propto \frac {1+\mathcal{G}_q} {1+x\mathcal{G}_q},$$ where $x=1-\mu_3/\mu_2$ and $\mathcal{G}_q = -\cos(2\phi)^2$ by virtue of eq. . We start by writing the following identity (obtained *e.g.* via Cauchy residues theorem) $$\frac {1+\mathcal{G}_q} {1+x\mathcal{G}_q} = K + 2 K' \sum_{n=1}^\infty \cos(4n \phi) z^{n} \,,\, \label{eq:Cauchy}$$ where $K = (\sqrt{1-x}+1)^{-1}$, $K' = -\sqrt{1-x}/x$ and $z =x / (1 +\sqrt{1-x})^2$. Then we apply an inverse F.T. to eq.  term by term with the help of eq.  and arrive at $$e_2(r) \propto \frac{2}{\pi} \frac{\sqrt{1-x}}{x-2} \frac{\delta + \cos(4\theta)}{(1 + \delta \cos(4\theta))^2} \frac{1}{r^2} + \frac{\delta_{2D}(r)}{1 + \sqrt{1-x}} \,. \label{eq:inverseF}$$ where $\delta = \frac{x}{x-2}$. This result is consistent with eq. , provided that $$C = -\frac{2 a e_*^{pl}}{ \pi} \frac{\sqrt{\mu_2 \mu_3}}{\mu_2+\mu_3} \,,\, C_0 = \frac{a^2 e_*^{pl}}{\sqrt{\mu_3/\mu_2}+1} \,. \label{eq:CC0}$$ Incidentally, the formula for the isotropic case is recovered for $x = 0$, $$1+\mathcal{G}_q = \frac{1}{2} - \frac{1}{2}\cos(4\phi) \stackrel{\text{F.T.}^{-1}}{\longrightarrow} \frac{\delta(r)}{2} - \frac{ \cos(4\theta)} {\pi r^{2}} \,.\label{eq:EshelbyFourier}$$ Displacement field and strain {#sec:ur} ============================= We explicit the relation between displacement field and strain tensor, for a displacement field which points in the radial direction, and whose amplitude $\Vert u \Vert = F(\theta) / r$. The corresponding strain tensor is as follows \[see eq.  and text below\]: $$\label{eq:e2diff} e_1 = 0 \,,\, e_2 = - \frac1{2r^2}\frac{{\mathrm{d}}\left[\sin (2 \theta ) F(\theta ) \right]} {{\mathrm{d}}\theta}\,.$$ So the displacement is incompressible for any $F$. Comparing to eq.  yields a differential equation for $F(\theta)$, whose general solution is:$$F(\theta) = -C \frac{\cos(2\theta)}{1+\delta \cos(4\theta)} + c_1 \csc(2\theta) \,,$$ where $c_1$ is an integral constant which we set to $0$ (since $\csc(2\theta)$ is divergent at $\theta = 0, \pi/2, \pi, 3\pi/2$), leading to eq. . Eq.  was also used to obtain the displacement field in the numerical simulation of transient dynamics in Section \[sec:SPE3\]. Generalisation to underdamped systems {#app:underdamped} ===================================== The main text focuses on the overdamped limit relevant for foams and concentrated emulsions, notably. Here, the results are generalised to all damping regimes. Following the approach of [@lookman03ferro], this is achieved by complementing the Euler-Lagrange equations of motion with a strain-rate-dependent Rayleigh dissipation term $R = \eta \int_r \left[\dot{e}_2^2 + \dot{e}_3^2 \right] = \eta \int_r \dot{e}^2$, where $\eta=1$ is the viscosity and the rescaled strain $e$ was introduced in eq. . The resulting equations read $$\frac{d}{dt} \frac{ \delta \mathcal{L} }{\delta \dot{e}} - \frac{ \delta \mathcal{L} }{\delta e} = - \frac{\delta R}{\delta \dot{e}} \,,$$ where the Lagrangian $$\mathcal{L}= T- F$$ is the difference between the kinetic energy $T=\frac{1}{2}\int_r \rho \dot{u}^2$ and the free energy $$F = \int_r \left[W_{r}[e_2(r)] + \mu_2 e_2(r)^2 + \mu_3 e_3(r)^2 \right] \,.$$ Since displacement is a (non-local) function of the strain field, the kinetic energy $T$ can be expressed in terms of the strain rate, instead of the velocity. Adapting the results of [@lookman03ferro] (Sec. III.A) to an incompressible system, we obtain $$T = 2 \rho \int_q \frac{ |\dot{\hat{e}}|^2 (q) }{q^2} \,.$$ The Lagrange-Rayleigh equations then simplify to $$\begin{aligned} 2 \rho \ddot{e} = - q^2 \left( \frac12 \frac{Q_3}{q^2} \frac{\partial W_{r}}{\partial e_2(r)} + \mu_2 e + \delta \mu \frac{Q_2^2}{q^4} e + \dot{e} \right) \,, \label{eq:e} \end{aligned}$$ where $\delta \mu = \mu_3 - \mu_2$. Note that the overdamped dynamics of eq.  are recovered for $\rho \to 0$. Linearising the above equation and writing $\hat{e}(q,t)= \int_0^{\infty} e^{\omega t} \tilde{e}(q,\omega) d\omega$ yields $$2 \rho \omega^2 \tilde{e} = q^2 \left( \mathcal{F} \mathcal{D} \mathcal{F} + \delta \mu \, \mathcal{G}_q - \mu_2 - \omega \right) \tilde{e} \,, \label{eq:app_EoM}$$ where $\mathcal{F} = Q_3 / q^2$, $\mathcal{G}_q=\mathcal{F}^2-1$, and $\mathcal{D} = -\frac{1}{2} \delta_{rr'} W''_r\left(\mathcal{F} e^{(0)}(r)\right)$ were already defined in Sec. \[sec:denny\]. Only in the marginal case $\omega \to 0$ do the resulting eigenmodes coincide with those of the overdamped dynamical matrix of eq. . Otherwise, the propagation of sound waves affects the transient dynamics. The foregoing statement can be made more explicit in the case of a single impurity, i.e., $W_r =a^2 \delta(r) W$. While the equilibrium configuration does not depend on the damping, the transient dynamics do. In particular, the vibrational modes from eq. , expressed in terms of the original strain variable $e_2$, read $$\tilde{e}_2(q) \propto \frac{1 +\mathcal{G}_q}{2 \rho \omega^2 q^{-2} + \omega + \mu_2 - \delta \mu \, \mathcal{G}_q},$$ which matches the (overdamped) eigenmodes of eq.  only for the marginally stable mode at $\omega = 0$. [^1]: One may regard the deformation as slow if the stress-strain curve does not substantially vary when the driving rate is reduced. [^2]: See [@salman2011minimal] for a related endeavour to connect continuous models with discrete automata in the context of crystal plasticity. [^3]: Using the properly rescaled variable $e$ allows us to obtain a Hermitian dynamical matrix, whereas a non–Hermitian one is obtained if one considers the variable $e_2$. This is reminiscent of the symmetrization transform from the Fokker–Planck to the Schroedinger equation. [^4]: Nevertheless, the possibility to relax to different energy basins *locally*, depending on the deformation rate, is not taken into account here.

--- author: - Setsuko Wada - Takashi Onaka - Issei Yamamura - Yoshitada Murata - 'Alan T. Tokunaga' date: 'Received 6 May 2003 / Accepted 2 June 2003' title: '$^{13}$C isotope effects on infrared bands of quenched carbonaceous composite (QCC)' --- Introduction ============ A wide range of the / ratio has been reported in various celestial objects. The ratio changes due to nucleosynthesis and mixing in the interior of stars. During the first dredge-up on the red giant branch (RGB) the convective envelope reaches regions abundant in that was processed from and the ratio decreases. Observations of RGB stars often show the / ratio of 5–20, even lower than theoretically predicted, suggesting the presence of extra-mixing below the convective envelope (e.g. Gilroy [@gil89]). In the third dredge-up during the asymptotic giant branch (AGB) phase, an increase in / is generally expected, but the ratio could also decrease due to cool bottom processing for low-mass stars (Wasserburg et at. [@was95]; Nollett et al. [@nol03]) or to hot bottom burning for more massive stars (Frost et al. [@fro98]). Observations of five carbon-rich circumstellar envelopes indicate the ratio of 30–65 (Kahane et al. [@kah92]), while ten carbon stars are shown to have the ratio in the range 12–60 in their circumstellar envelopes (Greaves & Holland [@gre97]). Some carbon stars show the ratio of / as low as 3 (Lambert et al. [@lam86]; Ohnaka & Tsuji [@ohn96]; Schöier & Olofsson [@sch00]). The / ratio in post-AGB stars and planetary nebulae (PNe) reflects the cumulative effects of different mixing and nuclear processing events during the entire evolution of their progenitors. Lower limits of the ratio of 3 to 10 have been obtained for several objects in the post-AGB phase (Palla et al. [@pal00]; Greaves & Holland [@gre97]). Recently Josselin & Lébre ([@jos01]) estimated an upper limit of / of 5 for the post-AGB candidate, HD179821, whereas a relatively large ratio of $72 \pm 26$ is reported for another post-AGB star, HD56126 (Bakker & Lambert [@bak98]). Clegg et al. ([@cle97]) found low ratios of 15 and 21 in two PNe. Further low values of the ratio of 2–30 have been reported in recent studies of several PNe (Palla et al. [@pal00]; Balser et al. [@bal02]; Josselin & Bachiller [@jos03]), suggesting that some stars undergo non-standard processing in the stellar interior and a low / can be expected during the late stage of their evolution. The solar system value is 89 (Anders & Grevesse [@and89]). The / ratio also provides key information on the chemical evolution in the Galaxy (for a review, Wilson [@wil99]). Observations of molecules and solid CO$_2$ in interstellar medium indicate that the ratio ranges from 10–100 and increases with the Galactocentric distance. The / ratio is suggested to be about 10–20 in the Galactic center region (Wilson [@wil99]; Boogert et al. [@boo00]; Savage et al. [@sav02]). The Galactic gradient is thought to be built by nucleosynthesis of the Galactic chemical evolution and the suggested ratio of 10–20 in the Galactic center indicates the presence of significant stellar sources of . Interstellar graphite spherules in the Murchison meteorite show a range of the ratio of 7–1330 (Bernatowicz et al. [@ber91]). Some presolar SiC grains show very low / ratios of less than 10 and they are thought to originate from very -rich stars in the AGB phase (Amari et al. [@ama01]). These observations suggest that low / ratio environments are not uncommon in objects in the AGB, post-AGB, and PN phases as well as in some interstellar medium. Carbon-bearing species formed in these environments could thus show non-negligible carbon isotopic effects in their spectrum. A set of emission bands at 3.3, 6.2, 7.6–7.8, 8.6, and 11.2$\mu$m have been observed in various celestial objects and are called the unidentified infrared (UIR) bands. Fainter companion bands are also sometimes seen. The exact nature of the carriers has not yet been understood completely, but it is generally believed that the emitters or emitting atomic groups containing polycyclic aromatic hydrocarbons (PAH) or PAH-like atomic groups of carbonaceous materials, including such as nanodiamond grains, are responsible for the UIR bands (Léger & Puget [@leg84]; Allamandola et al. [@all85]; Sakata et al. [@sak84]; Papoular et al. [@pap89]; Arnoult et al. [@arn00]; Jones & d’Hendecourt [@jon00]). Alternatively, Holmid ([@hol00]) has recently proposed de-excitation of Rydberg matters as possible carriers. The UIR bands have been observed in a wide range of objects, including regions, reflection nebulae, post-AGB stars, and PNe (for a review, see Tokunaga [@tok97]). They have also been commonly seen in the diffuse Galactic emission (Tanaka et al. [@tan96]; Onaka et al. [@ona96]; Mattila et al. [@mat96]; Kahanpää et al. [@kah03]) as well as in external galaxies (e.g. Mattila et al. [@mat99]; Helou et al. [@hel00]; Reach et al. [@rea00]; Lu et al. [@lu03]), indicating that the carriers are a common member of interstellar medium and present in various environments. Carbon-rich objects in the evolutionary stage from post-AGBs to PNe often show the emission bands and thus isotopic effects should be detectable if they arise from carbonaceous materials of low / ratios. Simple calculations of a -benzene molecule suggest that the peak shift can be as much as 0.15$\mu$m for the C$-$C stretching mode (Appendix \[cal\]). Observations of the Infrared Space Observatory (ISO; Kessler et al. [@kes96]) have provided a large database of the UIR band spectra in various objects (e.g. Beintema et al. [@bei96]; Molster et al. [@mol96]; Verstraete et al. [@ver96; @ver01]; Boulanger [@bou98]; Cesarsky et al. [@ces00a; @ces00b]; Uchida et al. [@uch98; @uch00]; Moutou et al. [@mou00]; Honey et al. [@hon01]). Recently Peeters et al. ([@pee02]) have investigated in detail the 6–9$\mu$m spectra of 57 sources taken by the Short Wavelength Spectrometer (SWS; de Graauw et al. [@degr96]) on board the ISO and found that the 6.2, 7.7, and 8.6 $\mu$m UIR bands show appreciable variations particularly for post-AGB stars and PNe. On the other hand, the variations in the 11.2$\mu$m band are relatively modest and those in the 3.3$\mu$m are less pronounced (Tokunaga et al. [@tok91]; Roche et al. [@roc91]; Hony et al. [@hon01]; van Diedenhoven et al. [@van03]). These variations can be interpreted in part by nitrogen substitutions in PAHs and anharmonicity, but not all of the observed aspects of the UIR bands have yet been fully understood (Verstraete et al. [@ver01]; Pech et al. [@pec02]; Peeters et al. [@pee02]). Part of the observed variations could also originate from isotopic effects of the UIR band carriers since the objects that show the variations are mostly post-AGB stars and PNe, in which small / ratios can be expected. In the present paper we investigate isotopic effects on the UIR bands experimentally. We synthesize a laboratory analogue of carbonaceous dust, the quenched carbonaceous composite (QCC; Sakata et al. [@sak84]), with various / ratios from the starting gas of a mixture of H$_4$ and H$_4$. The QCC shows infrared bands similar to the UIR bands and shifts in the band peaks due to the substitution are clearly detected. In Sect.2 we describe the experimental procedure. The results are shown in Sect. 3 and discussed in comparison with observations in Sect. 4. A summary is given in Sect. 5. Experimental ============ The experimental procedure for synthesizing the QCC is described in detail in Sakata et al. ([@sak84]). Methane (CH$_4$), the source gas of the QCC, is decomposed by the imposed microwave radiation and becomes a plasma. Carbonaceous condensates are formed in the injection beam of the plasma by quenching of the gas. Typically two types of the QCC are formed. One is a brown-black material (hereafter called dark-CC), which is collected on a substrate in the main injection beam. It has been shown to consist of a coagulation of carbon-onion-like particles (Wada et al. [@wad99]). The other is a yellow-brownish material (hereafter filmy-QCC), deposited on the surrounding region of the injected plasma gas (Sakata et al. [@sak87]). It is a material rich in organic molecules, such as PAHs. We prepare the starting gas of CH$_4$ with the fraction of 1%, 11%, 25%, 45%, 65% and 99% by mixing 99% methane with the natural isotope abundance (1% ) methane. We assume that the fraction in the QCC is equal to that of the starting gas in the present analysis because the reactions in the present experiments take place at high temperatures. The QCC is collected either on a KBr (for filmy-QCC) or on a BaF$_2$ crystal (for dark-QCC) and the absorption spectra are measured at room temperature. The spectra are taken by a Perkin-Elmer 2000-FTIR spectrometer with the resolution of 4 cm$^{-1}$. The spectra of the dark-QCC are measured after washing it with a small amount of acetone and removing organic molecules in the dark-QCC in a similar manner to previous experiments (Sakata et al. [@sak84]). Results ======= The isotopic substitution gives rise to a shift in the wavelength of vibration modes due to mass effects. With the replacement of with , the infrared band is expected to be shifted to longer wavelengths. Spectra of filmy-QCC -------------------- The filmy-QCC is a coagulation of many kinds of organic molecules. Mass analysis of the gases evaporating from the filmy-QCC shows that they contain various kinds of PAHs. The filmy-QCC also emits red fluorescence under ultraviolet irradiation (Sakata et al. [@sak92]). The spectra of the filmy-QCC with various fractions are shown in Fig. \[fig1\]. They show several absorption features similar to the UIR bands observed in celestial objects (Sakata et al. [@sak84; @sak87; @sak90]). For the filmy-QCC with 1% the bands of the C$-$H stretching characteristic group appear at 3.29, 3.42, 3.51, and 3.53$\mu$m (see also Fig. \[fig2\]). An aromatic C$=$C bond vibration appears at 6.2$\,\mu$m, and aromatic C$-$H out-of-plane bending modes are observed at 11.4, 11.9, and 13.2$\mu$m. A weak feature also appears clearly at 7.6$\mu$m in addition to the bands at 7.0 and 7.3$\mu$m. A very broad band is seen around 8.6$\mu$m, but its peak wavelength cannot be determined accurately because of the weak and broad feature. All the features observed in the samples of various fractions have corresponding features in the 1% sample and the peak wavelengths are shifted to longer wavelengths with the fraction. No new absorption band associated with appears. ![Absorption spectra of filmy-QCC with various fractions. The fractions are (a) 1%, (b) 11%, (c) 25%, (d) 65%, and (e) 99%. Each curve is shifted arbitrarily in the vertical direction.[]{data-label="fig1"}](h4507f1.GIF.eps){width="\hsize"} A large shift ($\Delta\lambda \sim 0.23$$\mu$m between 1% to 99% ) is observed in the 6.2$\mu$m band. The C$-$C mode involves at least two carbon atoms and makes the shift large. The shifts in the peaks arising from C$-$H out-of-plane bending modes are also seen. The 11.39$\mu$m peak at 1% shifts to 11.55$\mu$m at 99% . On the other hand, the shift in the 3.29$\mu$m band of aromatic C$-$H stretching is quite small ($\Delta\lambda \sim 0.014$$\mu$m). Fig. \[fig2\] shows the spectra expanded in the 3$\mu$m region. It is noticeable that the profile of the 3.3$\mu$m band becomes narrower with the fraction. The cause of this change is unknown at present. A small shift is also observed in the 3.4$\mu$m band, which is attributed to an asymmetric vibration of methylene (C$-$H$_2$). ![Spectra of the filmy-QCC in the 3$\mu$m region. The fractions are (a) 1%, (b) 11%, (c) 25%, (d) 65%, and (e) 99%. Each curve is shifted arbitrarily in the vertical direction.[]{data-label="fig2"}](h4507f2.GIF.eps){width="\hsize"} The peak wavelength of the small feature at 7.6$\mu$m is also shifted to longer wavelengths with the fraction ($\Delta\lambda \sim 0.22$$\mu$m). The large shift indicates that the vibration involves more than one carbon atoms, such as C$-$C. The broad bump at 8.6$\mu$m does not shift clearly. Sakata et al. ([@sak87]) assigned the 7.6 and 8.6$\mu$m peaks of the filmy-QCC to vibration modes of a kind of the ketone bond C$=$C$-$C$=$O. The 7.6$\mu$m peak is always observed in the filmy-QCC spectrum although the relative strength changes slightly with individual runs. The exposure to air does not increase the strength of the 7.6$\mu$m band. The present experiment thus suggests that the 7.6$\mu$m component is a molecular product of the plasma gas and is not directly related to oxidation. Spectra of dark-QCC ------------------- High-resolution electron-microscopy reveals that the dark-QCC is comprised of a coagulation of onion-like particles of the diameter of 10–15nm (Wada et al. [@wad99]). It is insoluble in acetone. Even after washing with acetone, the dark-QCC still contains some organic molecules. Mass spectroscopy indicates that they are mainly composed of compact PAHs. The dark-QCC adsorbs organic molecules that are formed together and also contains hydrogen atoms in its structure. C$-$H bonds in both components will give rise to vibration modes in the infrared. Fig. \[fig3\] shows the absorption spectra of the dark-QCC with 1% fraction together with the filmy-QCC spectrum for comparison. The dark-QCC shows a strong continuum component, which decreases with the wavelength. Small peaks are observed at 3.3, 3.42, 5.8, 6.3, 7.0, 7.3, 8.2, 11.4, 12.0, and 13.2$\mu$m. The 3.3, 3.42, 11.4, 12.0, and 13.2$\mu$m bands are attributed to CH vibrations, and the 6.3$\mu$m band is ascribed to an aromatic C$=$C bond vibration. The broad 8.2$\mu$m band becomes strong when it is exposed to air. An increase of the 5.8$\mu$m band is also seen during the exposure to air, suggesting that both of them are associated with C$-$O$-$ and C$=$O bonds formed by oxidation. ![Absorption spectra of (a) dark-QCC and (b) filmy-QCC with the natural isotopic fraction ( = 1.1%).[]{data-label="fig3"}](h4507f3.GIF.eps){width="\hsize"} Fig. \[fig4\] shows the absorption spectra of the dark-QCC with various fractions. The continua were fitted with cubic spline functions and have been subtracted to show the absorption features clearly. As in the filmy-QCC spectra, peak shifts with the fraction are clearly observed and new features without corresponding ones in the 1% sample are not seen in large fraction samples. A large shift ($\Delta\lambda \sim 0.26$$\mu$m) is again observed for the 6.3$\mu$m band. The peak position of the 8.2$\mu$m band cannot be measured accurately because of the wide profile, but the peak is clearly shifted to longer wavelengths with the fraction. The band width also seems to change. Particularly for the 99% sample the 8.2$\mu$m feature becomes very broad. The 3.3$\mu$m band is quite weak in the dark-QCC and the peak position of the 3.3$\mu$m is difficult to determine (see Fig. \[fig5\]). The feature almost fades away in the 99% sample. The change in the profile of the 3.3$\mu$m seen in the filmy-QCC sample is not observed in the dark-QCC. ![Absorption spectra of the dark-QCC with various fractions with the continuum subtracted. The fractions are 1% (a), 11% (b), 25 % (c), 65% (d), and 99% (e). Each curve is shifted arbitrarily in the vertical direction.[]{data-label="fig4"}](h4507f4.GIF.eps){width="\hsize"} ![Spectra of the dark-QCC in the 3$\mu$m region. The fractions are (a) 1%, (b) 11%, (c) 25%, (d) 65%, and (e) 99%. Each curve is shifted arbitrarily in the vertical direction.[]{data-label="fig5"}](h4507f5.GIF.eps){width="\hsize"} Peak shifts of the absorption bands ----------------------------------- {width="14cm"} Fig. \[fig6\] plots the peak wavelength of each band against the fraction. The change of the peak wavelength is roughly linear with the fraction for all the bands plotted. Linear lines obtained by least-squares fits are also shown. The slope of the fitted line is given in Table \[tab1\] together with the shifts estimated from the calculation of C- and -benzene (Appendix \[cal\]). The amount of the shift in the 6.2$\mu$m band is large and even larger than the estimate based on the simple calculation of benzene molecules. The shift of the 11.4$\mu$m band is also larger than the calculation. The shift in the 3.3$\mu$m band of the filmy-QCC is in agreement with the calculation, while that of the dark-QCC is smaller than the calculation. It should be noted, however, that the 3.3$\mu$m band of dark-QCC is weak and the peak position cannot be determined accurately (Fig. \[fig5\]). ----------------- --------------------- ------------------------- ------------------------- -------------------- Peak wavelength Estimated shift between Observed range of of QCC and benzene$^a$ the peak wavelength$^b$ ($\mu$m) filmy-QCC dark-QCC ($\mu$m) ($\mu$m) 3.3 $ 0.014 \pm 0.001 $ $0.0063 \pm 0.0007 ^c$ 0.013 3.288 – 3.297 6.2 $0.231 \pm 0.018$ $0.262 \pm 0.023$ 0.14 6.20 – 6.27 (6.30) 7.6 $ 0.219 \pm 0.010$ — — 7.72 – 7.97$^d$ 8.2 — $0.379 \pm 0.124 ^c$ — ($\sim 8.22$) 11.4 $0.163 \pm 0.007$ $ 0.183 \pm 0.007$ 0.04 11.20 – 11.25 ----------------- --------------------- ------------------------- ------------------------- -------------------- $a$ See Appendix \[cal\]. $b$ The observed range between class $A$ and $B$ objects taken from Tokunaga et al. ([@tok91]), Peeters et al. ([@pee02]), and van Diedenhoven et al. ([@van03]). The peak wavelengths of class $C$ objects are indicated in parentheses (see text). $c$ The slopes have large uncertainties because of the broad feature. $d$ The peak wavelength range of the 7.8$\mu$m subcomponent (see text). \[tab1\] The present experiment shows no new features corresponding to . If the vibration mode is local, there should appear peaks corresponding to and . The spectral resolution is sufficiently high except for the 3.3$\mu$m band, which shows only small shifts, but any appreciable broadening of the feature that indicates a combination of multiple peaks is not seen with the fraction. Thus the peak shift cannot be interpreted in terms of a combination of the components corresponding to and . The present results indicate that the vibration modes in the QCC are not very localized, but involve rather large molecular structures. The peak shift is related to the number of carbon atoms involved in the vibration mode. The larger shift than the calculation suggests that more than two carbon atoms are associated with the 6.2$\mu$m band of the QCC. The large shift in the 11.4$\mu$m band also indicates that this feature originates from CH bonds associated with large benzene structures if it is ascribed to a CH out-of-plane bending mode. The two kinds of the QCC have very different chemical compositions from each other. The spectrum of the dark-QCC shows a continuum component, which indicates the development of sp$^2$ carbon-carbon bonds. The broad bump around 8.2$\mu$m and the strong continuum are the characteristics of carbon-rich particles. The filmy- and dark-QCC have absorption bands at similar wavelengths, but the bands seen in the dark-QCC always peak at longer wavelengths than the filmy-QCC except for the 3.3$\mu$m band with large fractions (Fig. \[fig6\]). The isotopic shift appears similar both in the filmy-QCC and dark-QCC except for the 3.3$\mu$m band. The 3.3$\mu$m band in the dark-QCC is rather weak and thus further examination is needed to understand the different behavior of the 3.3$\mu$m band between the filmy- and dark-QCC. Discussion ========== The infrared spectra of the QCC show several similarities to the observed UIR bands, but they are not exactly the same. The peak wavelength of the 3.3$\mu$m band of the f-QCC is in agreement with observations (Sakata et al. [@sak90]), while the exact peak wavelengths are not at the right positions for some other bands and there are differences in the band profiles. The 11.4$\mu$m band in the QCCs is obviously located at a longer wavelength than observed (11.23–11.25$\mu$m). Some UIR bands, such as the 6.2$\mu$m and 11.2$\mu$m bands, are not symmetric, but skewed to longer wavelengths, whereas most bands in the QCC seem to be more or less symmetric. Therefore a direct comparison of the QCC spectra with observations is not very straightforward since the QCC is probably not the very material that exists in interstellar space. However we believe that the QCC contains many of the structural units corresponding to the material in the interstellar medium because of the spectral similarities of the QCC and the UIR bands. In the following we examine whether or not observed variations of the UIR bands seen in post-AGB stars and PNe can be accounted for by the isotopic shifts observed in the QCC. We concentrate on the relative shifts in the band peaks of the QCC and the absolute peak wavelengths and band profiles are not discussed. The results of the filmy-QCC are compared with the observations for clarity because the difference in the relative shifts between filmy- and dark-QCC is small except for the 3.3$\mu$m band. Peeters et al. ([@pee02]) have investigated the 6–9$\mu$m UIR bands in various objects and found that there are at least three classes, designated as $A$, $B$, and $C$, according to the peak wavelengths of the emission bands. Class $A$ is a major class and includes a wide range of objects, such as regions, non-isolated young stellar objects (YSOs), reflection nebulae, and galaxies. They do not show appreciable variations in the 6.2, 7.7, and 8.6$\mu$m band features. Class $B$ consists of isolated YSOs, post-AGB stars, and PNe. They exhibit quite a large variation in the peak wavelengths of the 6–9$\mu$m features. The peaks are all shifted to longer wavelengths compared to class $A$ objects. The observed ranges of the peak wavelengths are listed in Table \[tab1\]. The apparent variation in the 7.7$\mu$m complex comes partly from the change in the relative strengths of the subcomponents. The peak shift in the 7.8$\mu$m subcomponent is well observed, while that in the 7.6$\mu$m subcomponent is less clear. Table \[tab1\] indicates the observed range of the peak wavelength of the 7.8$\mu$m subcomponent. Class $C$ consists of only two objects (CRL2688 and IRAS13146$-$6243). They show a very different spectrum from the other two classes. They have a band at 6.3$\mu$m but do not show any features around 7.7 and 8.6$\mu$m. Instead they show a broad band at 8.22$\mu$m. Hence class $C$ objects seem to have different kinds of the band carriers from those in classes $A$ and $B$. In the following discussion the peak wavelength of the 6.3$\mu$m band in class $C$ objects is not included in the observed variation since it will probably not be directly related to the isotopic effects. Although there are some internal variations present, the class in the 6–9$\mu$m bands is well defined and almost all class $B$ objects show the peaks shifted to longer wavelengths in all the 6.2, 7.7, and 8.6$\mu$m bands. Compared to the 6–9$\mu$m spectrum, the 3.3 and 11.2$\mu$m UIR bands show small variations (Hony et al. [@hon01]), but the variations in the peak position do exist (e.g. Nagata et al. [@nag88]; Tokunaga et al. [@tok88]; Roche et al. [@roc91]). Tokunaga et al. ([@tok91]) have suggested that there are two types of the 3.3$\mu$m band from a small number of objects. Type 1 is a major class, in which the peak is located at 3.289$\mu$m, while type 2 has the peak at 3.296$\mu$m. They also suggest that type 2 objects have a narrower width than type 1. van Diedenhoven et al. ([@van03]) have extended the investigation of the 3.3 and 11.2$\mu$m UIR bands on the same sample as in Peeters et al. ([@pee02]). They found that there are also two classes in each band, designated as $A_{3.3}$, $B_{3.3}$, $A_{11.2}$, and $B_{11.2}$, respectively. Class $A_{3.3}$ corresponds to type 1 and $B_{3.3}$ to type 2 in Tokunaga et al. ([@tok91]). $A_{3.3}$ and $A_{11.2}$ are major classes, and $B_{3.3}$ and $B_{11.2}$ classes show the bands at 3.3$\mu$m and 11.2$\mu$m shifted to longer wavelengths compared to $A_{3.3}$ and $A_{11.2}$, respectively. The observed ranges of the shifts in the 3.3 and 11.2$\mu$m bands are also shown in Table \[tab1\]. The shift in the 3.3$\mu$m band is quite small and that in the 11.2$\mu$m band is modest. In general objects classified as class $B$ in the 6–9$\mu$m spectrum are also classified as $B_{3.3}$ and $B_{11.2}$, but the correlation is not very good. Objects classified as class $B$ in the 6–9$\mu$m spectrum sometimes do not show detectable shifts in the 3.3 and 11.2$\mu$m bands (e.g. BD +30 3639) and vice versa (e.g. He2$-$113). van Diedenhoven et al. ([@van03]) concluded that the appearance of the 3.3 and 11.2$\mu$m band variations is not tightly correlated with the UIR bands in 6–9$\mu$m. The present results suggest that if / $<10$ the isotopic shifts are detectable particularly in the 6.2 and 7.7$\mu$m bands. A low / ratio of this range is often suggested in post-AGB stars and PNe, and thus part of the peak shifts in these objects could be attributed to the isotopic effects. The shift in the 11.2$\mu$m of a medium degree can also be expected, while the shift in the 3.3$\mu$m should be small. This shift pattern is qualitatively compatible with observations except for the 7.8$\mu$m band. The 7.8$\mu$m subcomponent shows a larger shift in its peak wavelength compared to the QCC sample. The 7.7$\mu$m complex exhibit complicated profile variations and there may be other weak subcomponents present. The observed range of the 3.3$\mu$m, 6.2$\mu$m, and 11.2$\mu$m bands suggest that the objects with the largest shifts should have / $\sim 2$. We have searched in literature for the / ratio of objects that had been observed by ISO/SWS. Table \[tab2\] lists the objects and the SWS observing modes. SWS01 was the full grating scan mode and the spectral resolution depends on the scan speed. SWS06 was the fixed-range grating scan mode and provided the highest spectral resolution in the grating mode of SWS (Leech et al. [@lee02]). We used the highest spectral resolution data as much as possible if there are several observations made for the same object. IRAS23133+6050 is included in the list as a reference of the class $A$ object. The Off-Line Processing data of version 10.1 (OLP 10.1) were obtained from the ISO Archival Data Center and reduced further by the Observers SWS Interactive Analysis Package (OSIA) version 3.0.[^1] The peak wavelength for each band is estimated after the local continuum has been subtracted (see Fig.\[fig7\]). The peak position is slightly dependent on the assumed local continuum. Strong Pf$\delta$ (3.29699$\mu$m) has been removed in the estimation of the peak wavelength of the 3.3$\mu$m band for IRAS23133+6050, NGC7027, and BD+303639. Table \[tab3\] lists the objects with the derived isotope ratio and their observed peak wavelengths. The peak wavelengths are in agreement with previous investigations within the uncertainty (Nagata et al. [@nag88]; Tokunaga et al. [@tok91]; Peeters et al. [@pee02]). The objects are listed in the order of the peak wavelength of the 6.2$\mu$m band. Each band spectrum is normalized at the peak after the continuum has been subtracted and shown in Fig.\[fig7\]. Name Type TDT$^a$ Obs. mode$^b$ ------ ---------- ---------- --------------- $^c$ region 56801906 SWS01 (2) post-AGB 15901777 SWS01 (3) PN 33800505 SWS06 PN 86500540 SWS01 (3) PN 36701824 SWS01 (1) post-AGB 17100101 SWS01 (2) post-AGB 70201801 SWS01 (4) post-AGB 33800604 SWS06 : Objects and the ISO/SWS observation mode $a$ Target Dedicated Time (TDT) of the ISO observation.\ $b$ Number in brackets indicates the scanning speed of the SWS01 mode.\ $c$ Reference object of class $A$.\ \[tab2\] ---------------- ------------ ----------- --------------- ------------ ------------------------------------------------- ----------------- Object / Reference$^b$ 3.3$\mu$m 6.2$\mu$m 7.8$\mu$m 11.2$\mu$m IRAS23133+6050 3.289 6.219 7.795 11.218 IRAS21282+5050 3.290 6.219 7.817 11.245 $>200$, $>32$ 1, 2 NGC7027 3.292 6.222 7.830 11.247 $>31$, $>65$, $>25$, $>11$ 3, 4, 5, 6 BD+30 3639 3.292 6.240 7.850 11.237 $\geq4$ 7 IC5117 3.295: 6.25: 7.82: 11.28: 23, 14 6, 7 HR4049 3.296:$^c$ 6.256 7.880: 11.285 $\sim 1.5^a$ 8 HD44179 3.296 6.267 7.861 11.250 $\geq 22$, $\geq 12.5$ 3, 9 CRL2688 3.297 6.290 – – $>19$, $32^{+10}_{-7}$, $>3$, $20^{+8}_{-6}$, 5 3, 4, 5, 10, 11 ---------------- ------------ ----------- --------------- ------------ ------------------------------------------------- ----------------- $a$ Intensity ratio of CO$_2$ bands (Cami & Yamamura [@cam01])\ $b$ References\ 1: Likkel, et al. ([@lik88]); 2. Balser et al. ([@bal02]); 3: Bakker et al. ([@bak97]); 4: Kahane et al. ([@kah92]); 5: Bachiller et al. ([@bac97]); 6: Josselin & Bachiller ([@jos03]); 7: Palla et al. ([@pal00]); 8: Cami & Yamamura ([@cam01]); 9: Greaves & Holland ([@gre97]); 10: Wannier & Sahai ([@wan87]); 11: Jaminet et al. ([@jam92]).\ $c$ Doubly peaked.\ \[tab3\] Very few data of the carbon isotope abundance are available for post-AGB stars. Observations of isotopic lines of C$_2$, CN, and in several post-AGB stars resulted in negative detection of molecular features of except for one object (Bakker et al. [@bak97; @bak98]). They thus provided only lower limits for the / ratio of $>10-20$. The carbon isotope ratios in a few post-AGB stars and several PNe have been obtained mostly from radio observations of CO molecules. The intensity ratio of to emission is thought to be a good measure for the isotope ratio if both lines are optically thin since the selective photodissociation of should be counterbalanced by the charge exchange reaction (e.g. Mamon et al. [@mam88]). However lines often become optically thick, only lower limits of / being estimated. It should also be noted that observations of CO probe the entire region of the circumstellar envelope. The post-AGB is a transient phase and a rapid change of the abundance is expected to occur in their envelopes. Thus the isotope ratio in the vicinity of the star might be different from that inferred from the CO intensity ratio. {width="14cm"} IRAS21282+5050 is a post-AGB star and classified as class $A$ in the 6–9$\mu$m spectrum and $A_{3.3}$. The 3.3$\mu$m band peaks around 3.288–3.290$\mu$m (e.g. Nagata et al. [@nag88]). Likkel et al. ([@lik88]) obtained that the intensity ratio of / ($J$=1–0) is about 100 and inferred / $\sim 200$, taking account of the optical thickness effect of . Based on the large gradient velocity model analysis, Balser et al. ([@bal02]) obtained / $>32$ from observations of CO $J$=2–1 and $J$=3–2 transitions. The observed peak wavelengths are compatible with the large / ratio. NGC7027 is a bright young PN. It shows the class $A$ 6.2$\mu$m band feature, while the 7.7 and 11.2$\mu$m bands are slightly shifted to longer wavelengths relative to class $A$. Also the 7.8$\mu$m component is stronger than the 7.6$\mu$m component, indicating a signature of the class $B$ object. The 3.3$\mu$m band peak appears at the normal (class $A$) wavelength. This object has been observed by several groups in molecular lines. Because of the optical depth effect of the CO line intensity ratio should be taken as lower limits for the isotope abundance ratio of /. They range from 65 (CO($J$=1–0); Kahane et al. [@kah92]) to a recent value of 11 (CO($J$=2–1); Josselin & Bachiller [@jos03]), suggesting that the / ratio in the CO envelope of NGC7027 is probably not less than 10. Except for the 7.7$\mu$m band appearance the suggested range of / is compatible with the class $A$ classification. BD+303639 is a young PN and classified as class $B$ in the 6–9$\mu$m spectrum. Palla et al. ([@pal00]) reported a tentative detection of ($J=2-1$) and obtain a lower limit of / as $\geq 4$. This object shows medium size band shifts, which can be accounted for by the isotopic shifts with / $\sim 5$. The corresponding shift in the 3.3$\mu$m would be only $\sim$ 0.03$\mu$m, which is also compatible with the observed peak position. The 11.2$\mu$m band shows a large tail in the longer wavelength side. Assuming that CO lines are optically thin, Palla et al. ([@pal00]) derived the / of 14 for IC5117, while Josselin & Bachiller ([@jos03]) indicated the ratio of 23 in this object. IC5117 was not included in the sample of Peeters et al. ([@pee02]). The SWS spectrum is noisy and the peak wavelengths cannot be determined accurately. However it clearly indicates the characteristics of the class $B$ spectrum. The 7.7$\mu$m band complex is dominated by the 7.8$\mu$m component and the 3.3 and 6.2$\mu$m bands peak at longer wavelengths than class $A$. The indicated / ratio seems to be too small to account for the observed spectrum by the isotopic effects, although a qualitative comparison is difficult because of large uncertainties in the peak wavelengths. HR4049 is a very metal-poor post-AGB star and it is difficult to estimate its carbon isotope ratio because of very few metallic lines detected in its spectrum. Recently Cami & Yamamura ([@cam01]) have investigated circumstellar CO$_2$ features in the infrared region and suggested that this star has a very peculiar oxygen isotope ratio, such that and are quite enhanced. They have detected $_2$, , and lines, and the intensity ratios of the to lines were all about 0.6–0.7. Although detailed analysis is required to derive a reliable ratio, this implies that HR4049 is quite rich also in . The positive correlation of / with / seen in several environments (Wannier & Sahai [@wan87]) also suggests a low / ratio. CO$_2$ molecular emission originates from the region near the star and thus the derived ratio should correspond to the region in which the UIR bands are also emitted. The observed UIR bands in HR4049 are all shifted to longer wavelengths relative to normal class $A$ positions, which can be accounted for by isotopic shifts with / $\sim 3$. The small / ratio suggested by the CO$_2$ line emissions is compatible with the observed shifts. HD44179 is a well-studied post-AGB star and very bright in the infrared. Greaves & Holland ([@gre97]) made observations of CO ($J$=2–1) and obtained the peak intensity ratio of $2.2 \pm 0.2$ for /. The central part of the profile appears optically thick and from the wing of the profile they estimate /$\geq 12.5$. The negative detection of suggests / $\geq 22$ (Bakker et al. [@bak97]). The 6–9$\mu$m spectrum of HD44179 belongs to class $B$ and shows one of the largest shifts in the peak wavelengths among the class. HD44179 is also the only one object that shows two subcomponents in the 6.2$\mu$m clearly. The 3.3$\mu$m band of HD44179 is of type 2 and peaks at 3.296$\mu$m. The 11.2$\mu$m band is also shifted to 11.25$\mu$m. These shifts can be accounted for consistently by the isotopic effects if / is $\sim 2$. This required ratio is obviously smaller than the one indicated by CO and observations. A complicated torus-bipolar cone structure resulted from a transition from oxygen-rich to carbon-rich envelope has been suggested in HD44179 (Waters et al. [@wat98]; Men’shchikov et al. [@men02]), indicating that the carbon isotope ratio may also spatially vary. In fact Kerr et al. ([@ker99]) have indicated that the 3.3$\mu$m band peak changes with positions within the nebula and that the longer wavelength peak appears near the star and the interface region of the biconical structure. The emission with a similar strength to has also been detected for the 4.6$\mu$m bands in the nebular region (Waters et al. [@wat98]). Two components seen in the 6.2$\mu$m band in HD44179 may then be ascribed to the band carriers with a small isotope abundance in the outer part and to those with a low / formed in the region close to the star. Unless there is an appreciable increase in the abundance in the vicinity of the star, however, the observed peak shifts in HD44179 cannot be accounted for by isotopic effects alone. High-spatial resolution observations in other UIR bands are interesting to investigate possible spatial variations in their peaks (Song et al. [@son03]). CRL2688 is a carbon-rich post-AGB star which shows a peculiar 6–9$\mu$m spectrum (class $C$). The integrated intensity ratio of to ($J$=2–1) is about 3–4 (Wannier & Sahai [@wan87]; Bachiller et al. [@bac97]). The line is optically thick and thus this gives a lower limit. Wannier & Sahai ([@wan87]) derived the CO/ ratio to be 20 from model analysis on CO ($J$=2–1) observations and Kahane et al. ([@kah92]) obtained the ratio of $32^{+10}_{-7}$ from observations of CS. Jaminet et al. ([@jam92]) have indicated, on the other hand, that the fast wind component has / of 5 from the wing profile of CO ($J$=3–2) emissions. Bakker et al. ([@bak97]) gave a lower limit of 19 based on the negative detection of circumstellar molecules with . Goto et al. ([@got02]) indicated that the 3.3$\mu$m band profile does not show any spatial variations, suggesting that the extended component of the band emission is the scattered light of the emission from the central region. There is no evidence so far for the spatial variation in the band features within the nebula. Since this object may have a different type of the band carriers, the peak position of the 6.2$\mu$m band (in fact it is located at 6.29$\mu$m) should be discussed separately from other objects. The 8.2$\mu$m band seen in the dark-QCC may correspond to the band at 8.22$\mu$m detected in class C objects. It should be noted that the dark-QCC of the natural isotope abundance also has a band at 6.3$\mu$m, shifted to a longer wavelength compared to the filmy-QCC. The 3.3$\mu$m band is seen at the class $B$ position and the 11.2$\mu$m is very weak, if present. Differences in the type of the band carriers themselves may account for the observed shift in the 6.2$\mu$m band as suggested by the difference between filmy- and dark-QCC spectra. As described above, carbon isotope ratios obtained from CO observations of post-AGB stars and PNe are upper limits in most cases or show large scatter because of uncertainties in the observed intensities of faint emissions (see Josselin & Bachiller [@jos03]) and in the correction for the optical depth effects of lines. In addition very few objects in the post-AGB phase with the UIR bands have measured / data, including upper limits. Thus it is difficult at present to directly compare the present results with observations for individual objects. The overall shift pattern of the UIR band peaks is qualitatively in agreement with the isotopic shift pattern except for the 7.7$\mu$m band complex. However, Table \[tab2\] does not indicate a quantitatively clear correlation between / and the peak wavelengths and the suggested abundance in individual objects seems to be slightly too small to account for the observed shifts in terms of the isotopic effects. Observations also indicate that the correlation among the peak wavelengths of the 6–9$\mu$m bands is good, but the peak shifts in the 6–9$\mu$m bands are not tightly correlated with those in the 3.3 and 11.2$\mu$m bands. The UIR band emission is thought to come from the vicinity of the central star, while the CO emission probes the entire region of the circumstellar envelope. Part of the lack of clear correlations and the poor quantitative agreement between the isotope ratio and the peak wavelengths may be attributed to possible spatial variations in the isotope abundance within the object. If the 3.3 and 11.2$\mu$m bands come from regions different (either closer to or farther away from the source) from that of the 6–9$\mu$m bands, the poor correlation of the peak wavelengths among the 3.3, 6–9, and 11.2$\mu$m bands may also be attributed to the spatial variation in the isotope ratio with which each band carrier was formed in the envelope. Most PAHs have band peaks longer than 6.3$\mu$m for a C$=$C vibration contrary to the filmy-QCC and it has been suggested that the substitution of a carbon atom by a hetero-atom, such as nitrogen, oxygen, or silicon, will shift the band to shorter wavelengths as observed in class $A$ objects (Peeters et al. [@pee02]). Then the observed variations are interpreted in terms of the relative abundance of pure-carbon PAHs and substituted PAHs. The hetero-atom substitution, on the other hand, does not make systematic shifts in other bands. The isotopic substitution of has similar effects on the peak wavelength for the 6.2$\mu$m band but in the opposite direction. In addition it makes systematic shifts in the peak wavelengths of other bands and qualitatively accounts for the observed variations correlated with the 6.2$\mu$m band. The peak wavelength of the C$=$C stretching vibration band decreases with the size of PAHs, while the increase of the molecular size does not make systematic shifts in the 7.7$\mu$m band (Peeters et al. [@pee02]). Hetero-atom substitutions and size variations are likely to occur in interstellar medium and the observed variations could be a summation of various processes. The present results indicate that the isotopic effects can contribute to the observed variations for objects with low / ratios. It should also be emphasized that the peak shifts of the UIR bands are observed mostly in post-AGB stars and PNe, which are the objects where small / ratios are expected. It has been suggested that deuterated PAHs (PADs) can be efficiently formed in dense molecular clouds because the large difference in the zero-point energy between H and D containing species leads to preferential formation (fractionation) of deuterated species at low temperatures (e.g. Tielens [@tie97]; Sandford et al. [@san00]). For the carbon isotope case, the zero-point energy difference is not large and the temperature in the circumstellar envelope is much higher then the zero-point energy. Hence isotopic fractionation should not be significant. Instead a rather large abundance is expected to exist in certain environments and carbonaceous dust that is formed in such environments should show isotopic shifts in its band features. Shifts in the band peaks of PADs from those of PAHs are quite large because of the large mass difference in H and D. Contrary to deuterated species, expected shifts for carbonaceous material are small but should still be in the detectable range as shown by the present results. Summary ======= We synthesized the QCC, a laboratory analogue of carbonaceous dust, with various fractions and measured the isotopic shifts in the peak wavelengths of the infrared bands. They all shift to longer wavelengths approximately linearly with the fraction. The fact that no separate peaks originating from -QCC appear indicates that the vibration modes of infrared bands associated with carbon atoms in the QCC should not be very localized, but that they stem from rather large structures containing several carbon atoms. The shifts in the 6.2 and 7.8$\mu$m are quite large ($\Delta \lambda > 0.2$$\mu$m per fraction) and that in the 11.2$\mu$m is modest ($\sim 0.16$$\mu$m), while the shift in the 3.3$\mu$m band is small ($<0.02 \mu$m). The small shift in the 3.3$\mu$m band is in agreement with a simple calculation of benzene molecules and those in the 6.2 and 11.2$\mu$m seem to be larger, also suggesting that a larger number of carbon atoms associated with these bands than benzene. The isotopic shifts obtained in the present experiment are qualitatively in agreement with the shift pattern observed in the peak wavelengths of the UIR bands except for the 7.7$\mu$m band complex. However the observed variations seem to be larger than those inferred from the / ratio and the quantitative agreement between the carbon isotope ratio and the peak wavelengths is not good for individual objects. This may be attributed partly to large uncertainties in the isotope ratios derived from observations, to possible spatial variations in the isotope abundance in the envelope, and to combinations with other effects, such as hetero-atom substitutions. The present results indicate that the isotopic shifts in the peak wavelengths of the UIR bands should be detectable in objects of low / ratios and part of the observed variations in the band peaks can be attributed to the isotopic effects. This work is based in part 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. Part of this work was supported by Grant-in-Aids for Scientific Research from Japan Society for the Promotion of Science. Calculation of isotopic peak shifts of benzene molecule {#cal} ======================================================= In this appendix, we estimate isotopic peak shifts in the vibration modes of a benzene molecule to compare with the isotopic shifts in the corresponding vibration modes of the QCC. Benzene is the simplest PAH and the results should be used as a 0-th order reference in comparison with the isotopic shifts in the QCC. The $\vec{GF}$ matrix method is used to calculate isotopic shifts of a benzene molecule (Wilson et al. [@wil55]). The $\vec{G}$ matrix represents the molecular geometry and the $\vec{F}$ matrix consists of the force constants. The eigenvalue of the $\vec{GF}$ matrix is given by $\lambda=2\pi^2\nu^2$, where $\nu$ is the frequency of the vibration mode. In the calculation of the isotopic shift, only the values of the $\vec{G}$ matrix elements change due to the isotopic substitution and the $\vec{F}$ matrix elements remain unchanged if all carbon atoms are substituted by C. We use the values of the force constants of benzene given by Crawford & Miller ([@cra46; @cra49]). The vibrational modes of infrared active fundamentals in benzene consist of one $A_\mathrm{2u}$ (C-H out-of-plane bending) and three $E_\mathrm{1u}$ (three kinds of in-plane vibrations) modes. The single $\vec{G}$ matrix element of $A_{\rm 2u}$ is written by $$\vec{G}(A_\mathrm{2u}) = (\mu_\mathrm{H} + \mu_\mathrm{C})~ \sigma^2 \label{a1}$$ and those of $E_\mathrm{1u}$ are given by $$\begin{aligned} \lefteqn{\vec{G}(E_\mathrm{1u})} \nonumber \\ & = \left( \begin{array}{ccc} \mu_\mathrm{H}+\mu_\mathrm{C} & -\frac{\sqrt{3}}{2}\mu_\mathrm{C} & \frac{3}{4}\tau\mu_\mathrm{C} \\ & \frac{3}{2}\mu_\mathrm{C} & -\frac{\sqrt{3}}{4}(2\sigma+3\tau)\mu_\mathrm{C} \\ & & \sigma^2\mu_\mathrm{H} +(\sigma^2+\frac{3}{2}\sigma\tau+\frac{9}{8}\tau^2)\mu_\mathrm{C} \\ \end{array} \right), \nonumber \\ \label{a2}\end{aligned}$$ where only the upper right elements are shown for the symmetric $\vec{G}$ matrix. The parameters $\sigma$ and $\tau$ indicate the reciprocals of the C$-$H and C$-$C bond lengths, while $\mu_\mathrm{H}$ and $\mu_\mathrm{C}$ are the reciprocal masses of H and C, respectively. The bond angles are all set as 120$^{\circ}$. From Eqs.(\[a1\]) and (\[a2\]) and the $\vec{F}$ matrix we calculate the eigenvalues of the $\vec{GF}$ matrix and obtain the wavelength ratios of each vibrational mode between -benzene and -benzene. Then the wavelengths of -benzene are scaled to the measured values. The results are listed in Table \[ta\]. Because the wavelength of each mode is different from that in the QCC, we scale the wavelengths of -benzene to the bands measured in the -QCC and estimate the isotopic shifts (Table \[tab1\]). The $\lambda_{20}$ mode is scaled to the 3.3$\mu$m band and the $\lambda_{19}$ mode to the 6.2$\mu$m band. The band at 11.4$\mu$m in the QCC is assigned to a solo or duo C$-$H out-of-plane bending, which does not have a corresponding mode in benzene. We estimate the isotopic shift by simply scaling the $\lambda_{11}$ mode to 11.4$\mu$m because it is also a C$-$H out-of-plane mode of benzene. ---------------- ---------- ---------- mode -benzene -benzene ($\mu$m) ($\mu$m) $\lambda_{20}$ 3.245 3.257 $\lambda_{19}$ 6.812 6.964 $\lambda_{18}$ 9.524 9.709 $\lambda_{11}$ 14.90 14.95 ---------------- ---------- ---------- : Wavelength of the vibration modes of - and -benzene \[ta\] Allamandola, L. J., Tielens, A. G. G. M., & Barker, J. R. 1985, , 290, L25 Allamandola, L. J., Hudgins, D. M., & Sandford, S. A. 1999, , 511, L115 Amari, S., Nittler, L. R., Zinner, E., Lodders, K., & Lews, R. S. 2001, , 559, 463 Anders, E., & Grevesse, N. 1989, Geochim. Cosmochim. Acta, 53, 197 Arnoult, K. M., Wdowiak, T. J., & Beegle, J. W. 2000, , 535, 815 Bachiller, R., Forveille, T., Huggins, P. J., & Cox, P. 1997, , 324, 1123 Bakker, E. J., & Lambert, D. L. 1998, , 508, 387 Bakker E. J., van Dishoeck E. F., Waters L. B. F. 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L., Falgarone, E., et al. 1996, , 315, L337 Verstraete, L., Pech, C., Moutou, C., et al. 2001, , 372, 981 Wada, S., Kaito, C., Kimura, S., Ono, H., & Tokunaga, A. T. 1999, , 345, 259 Wannier, P. G., & Sahai, R. 1987, , 319, 367 Wasserburg, G. J., Boothroyd, A. I., & Sackmann, I.-J. 1995, , 447, L37 Waters, L. B. F. M., Cami, J., de Jong, T., et al. 1998, , 391, 868 Wilson, T. L. 1999, Rep. Prog. Phys., 62, 143 Wilson E. B. Jr., Decius J. C., & Cross, P. C. 1955, “Molecular Vibrations, The Theory of Infrared and Raman Vibrational Spectra”, McGraw-Hill Book Company, Inc. (New York) [^1]: The OSIA is a joint development of the SWS consortium. Contributing institutes are SRON, MPE, KUL and the ESA Astrophysics Division.

= 10000 Recently there has been much interest in the search of unconventional electron behavior deviating from the Fermi liquid picture[@unconv]. Besides this, the other paradigm that is well-established on theoretical grounds is the Luttinger liquid behavior of one-dimensional (1D) electron systems[@sol; @hal]. There have been suggestions that this behavior could be extended to two-dimensional (2D) systems, in the hope that it may explain some of the features of the copper-oxide materials[@and]. However, at least for the Luttinger model, the analytic continuation in the number $D$ of dimensions has shown that the Luttinger liquid behavior is lost as soon as one departs from $D = 1$[@ccm; @arri]. Several authors have also analyzed the possibility that singular interactions could lead to the breakdown of the Fermi liquid picture[@sing]. With regard to real low-dimensional systems, such as carbon nanotubes, the main electron interaction comes actually from the long-range Coulomb potential $V(|{\bf r}|) \sim 1/|{\bf r}| $. This is also the case of the 2D layers in graphite, which have a vanishing density of states at the Fermi level. Quite remarkably, a quasiparticle decay rate linear in energy has been measured experimentally in graphite[@exp], pointing at the marginal Fermi liquid behavior in such 2D layers. Due to the singular Coulomb interaction, the imaginary part of the electron self-energy in the 2D system behaves at weak $g$ coupling like $g^2 \omega $[@expl]. It is crucial, though, the fact that the effective coupling scales at low energy as $g \sim 1/\log (\omega )$. This prevents the logarithmic suppression of the quasiparticle weight, which gets corrected by terms of order $g^2 \log (\omega ) \sim 1/\log (\omega )$[@marg]. In this letter we investigate whether the long-range Coulomb interaction may lead to the breakdown of the Fermi liquid behavior at any dimension between $D = 1$ and 2. The issue is significant for the purpose of comparing with recent experimental observations of power-law behavior of the tunneling conductance in multi-walled nanotubes[@mwnt]. These are systems whose description lies between that of a pure 1D system and the 2D graphite layer. It turns out, for instance, that the critical exponent measured for tunneling into the bulk of the multi-walled nanotubes is $\alpha \approx 0.3$. This value is close to the exponent found for the single-walled nanotubes[@bock; @yao]. However, it is much larger than expected by taking into account the reduction due to screening ($\sim 1/\sqrt{N}$) in a wire with a large number $N$ of subbands, what points towards sensible effects of the long-range Coulomb interaction in the system. We develop the analytic continuation in the number of dimensions having in mind the low-energy modes of metallic nanotubes, which have linear branches crossing at the Fermi level. From this picture, we build at general dimension $D$ a manifold of linear branches in momentum space crossing at a given Fermi point. We consider the hamiltonian $$\begin{aligned} H & = & v_F \int_0^{\Lambda } d p |{\bf p}|^{D-1} \int \frac{d\Omega }{(2\pi )^D} \; \Psi^{+} ({\bf p}) \; \mbox{\boldmath $\sigma \cdot $} {\bf p} \; \Psi ({\bf p}) \nonumber \\ \lefteqn{ + e^2 \int_0^{\Lambda } d p |{\bf p}|^{D-1} \int \frac{d\Omega }{(2\pi )^D} \; \rho ({\bf p}) \; \frac{c(D)}{|{\bf p}|^{D-1}} \; \rho (-{\bf p}) \;\;\;\;\;\; } \label{ham}\end{aligned}$$ where the $\sigma_i $ matrices are defined formally by $ \{ \sigma_i , \sigma_j \} = 2\delta_{ij}$. Here $\rho ({\bf p})$ are density operators made of the electron modes $\Psi ({\bf p})$, and $ c(D)/|{\bf p}|^{D-1} $ corresponds to the Fourier transform of the Coulomb potential in dimension $D$. Its usual logarithmic dependence on $|{\bf p}|$ at $D = 1$ is obtained by taking the 1D limit with $ c(D) = \Gamma ((D-1)/2)/(2\sqrt{\pi})^{3-D}$. The dispersion relation $\varepsilon ({\bf p}) = \pm |{\bf p}|$ is that of Dirac fermions, with a vanishing density of states at the Fermi level above $D = 1$. This ensures that the Coulomb interaction remains unscreened in the analytic continuation. At $D = 2$ we recover the low-energy description of the electronic properties of a graphite layer, dominated by the presence of isolated Fermi points with conical dispersion relation at the corners of the Brillouin zone[@graph]. In the above picture, we are neglecting interactions that mix the two inequivalent Fermi points common to the low-energy spectra of graphite layers and metallic nanotubes. In the latter, such interactions have been considered in Refs. and , with the result that they have smaller relative strength ($\sim 0.1/N$, in terms of the number $N$ of subbands) and remain small down to extremely low energies. More recently, the question has been addressed regarding the interactions in the graphite layer, and it also turns out that phases with broken symmetry cannot be realized, unless the system is doped about half-filling[@nos] or it is in a strong coupling regime[@khves]. We will accomplish a self-consistent solution of the model by looking for fixed-points of the renormalization group transformations implemented by the reduction of the cutoff $\Lambda $[@sh]. As usual, the integration of high-energy modes at that scale leads to the cutoff dependence of the parameters in the low-energy effective theory. We will see that the Fermi velocity $v_F$ grows in general as the cutoff is reduced towards the Fermi point. On the other hand, the electron charge $e$ stays constant as $\Lambda \rightarrow 0$. This comes from the fact that the polarizability $\Pi $ does not show any singular dependence on the high-energy cutoff $\Lambda $ for $D < 3$. The polarizability is then given by $$\Pi ({\bf k}, \omega_k) = b(D) \frac{v_F^{2-D} {\bf k}^2} { | v_F^2 {\bf k}^2 - \omega_k^2 |^{(3-D)/2} }\; ,\;$$ where $b(D) = \frac{2}{ \sqrt{\pi} } \frac{ \Gamma ( (D+1)/2 )^2 \Gamma ( (3-D)/2 ) }{ (2\sqrt{\pi})^D \Gamma (D+1) }$. The dependence of $v_F$ on the cutoff $\Lambda $ implies an incomplete cancellation between self-energy and vertex corrections to the polarizability. The dressed polarizability depends therefore on the effective Fermi velocity $v_F (\Lambda )$. The renormalized value of $v_F$ is determined by fixing it self-consistently to the value obtained in the electron propagator $G$ corrected by the self-energy contribution $$\begin{aligned} \Sigma ({\bf k}, \omega_k) & = & - e^2 \int_0^{\Lambda } d p |{\bf p}|^{D-1} \int \frac{d\Omega }{(2\pi )^D} \int \frac{d \omega_p}{2\pi } \nonumber \\ \lefteqn{ G ({\bf k} - {\bf p}, \omega_k - \omega_p) \frac{-i}{ \frac{|{\bf p}|^{D-1}}{c(D)} + e^2 \Pi ({\bf p}, \omega_p) } . } \label{selfe}\end{aligned}$$ The fixed-points of the renormalization group in the limit $\Lambda \rightarrow 0$ determine the universality class to which the model belongs. At $D = 2$, we are bound to obtain the low-energy fixed-point at vanishing coupling of the model of Dirac fermions with Coulomb interaction[@marg]. On the other hand, at $D = 1$ there has to be presumably a fixed-point corresponding to Luttinger liquid behavior. We note, however, that no solution of the model has been obtained yet without carrying dependence on the transverse scale needed to define the 1D logarithmic potential. Our dimensional regularization overcomes the problem of introducing such external parameter, which prevents a proper scaling behavior of the model[@wang]. At general $D$, the self-energy (\[selfe\]) shows a logarithmic dependence on the cutoff at small frequency $\omega_k$ and small momentum ${\bf k}$. This is the signature of the renormalization of the electron field scale and the Fermi velocity. In the low-energy theory with high-energy modes integrated out, the electron propagator becomes $$\begin{aligned} \frac{1}{G} & = & \frac{1}{G_0} - \Sigma \approx Z^{-1} ( \omega_k - v_F \mbox{\boldmath $\sigma \cdot$}{\bf k}) \;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \nonumber \\ \lefteqn{ - Z^{-1} f(D) \sum_{n=0}^{\infty} (-1)^n g^{n+1} \left( \frac{n(3-D)}{n(3-D)+2} \omega_k \right. } \nonumber \\ \lefteqn{ + \left. \left(1 - \frac{2}{D} \frac{n(3-D)+1}{n(3-D)+2} \right) v_F \mbox{\boldmath $\sigma \cdot$} {\bf k} \right) h_n (D) \log (\Lambda ) , } \label{prop}\end{aligned}$$ where $g = b(D) c(D) e^2 / v_F $, $f(D) = \frac{1}{ 2^D \pi^{(D+1)/2} \Gamma (D/2) b(D) }$, $h_n (D) = \frac{ \Gamma (n(3-D)/2 + 1/2) } { \Gamma (n(3-D)/2 + 1) }$ . The quantity $Z^{1/2}$ represents the scale of the bare electron field compared to that of the renormalized electron field for which $G$ is computed. The renormalized propagator $G$ must be cutoff-independent, as it leads to observable quantities in the quantum theory. This condition is enforced by fixing the dependence of the effective parameters $Z$ and $v_F$ on $\Lambda $ as more states are integrated out from high-energy shells. We get the differential renormalization group equations $$\begin{aligned} \Lambda \frac{d}{d \Lambda} \log Z (\Lambda ) & = & - f(D) \sum_{n=0}^{\infty} (-1)^n g^{n+1} \;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \nonumber \\ \lefteqn{ \frac{n(3-D)}{n(3-D)+2} h_n (D) , } \label{zflow} \end{aligned}$$ $$\begin{aligned} \Lambda \frac{d}{d \Lambda} v_F (\Lambda ) & = & - v_F f(D) \sum_{n=0}^{\infty} (-1)^n g^{n+1} \;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \nonumber \\ \lefteqn{ \left( 1 - \frac{1}{D} \frac{n(3-D)(2-D) + 2}{n(3-D) + 2} \right) h_n (D) . } \label{vflow}\end{aligned}$$ At $D = 2$, the right-hand-side of these equations can be summed up to the functions that have been found previously in the renormalization of the graphite layer[@marg]. Furthermore, they also provide meaningful expressions in the 1D limit. At $D = 1$, the right-hand-side of Eq. (\[vflow\]) vanishes identically as a function of the variable $g $. Therefore, the 1D model has formally a line of fixed-points, as it happens in the case of a short-range interaction. The scaling of the electron wavefunction can be read from the right-hand-side of Eq. (\[zflow\]), which becomes $(2 + g)/(2\sqrt{1 + g}) - 1$ at $D = 1$. This coincides with the anomalous dimension that is found in the solution of the Luttinger model, what provides an independent check of the renormalization group approach to the 1D system. We have therefore a model that interpolates between marginal Fermi liquid behavior, that is known to characterize the 2D model, and non-Fermi liquid behavior at $D = 1$. As the electron charge $e$ is not renormalized for $D < 3$, the scaling of the effective coupling $g = b(D) c(D) e^2 / v_F $ is given after Eq. (\[vflow\]) by $$\begin{aligned} \Lambda \frac{d}{d \Lambda} g (\Lambda ) & = & f(D) \sum_{n=0}^{\infty} (-1)^n g^{n+2} \;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \nonumber \\ \lefteqn{ \left( 1 - \frac{1}{D} \frac{n(3-D)(2-D) + 2}{n(3-D) + 2} \right) h_n (D) . } \label{aflow}\end{aligned}$$ The right-hand-side of Eq. (\[aflow\]) is a monotonous increasing function of $g $, for any dimension between 1 and 2, as observed in Fig. \[one\] . = 6cm = 6cm We find that, away from $D = 1$, there is only one fixed-point of the renormalization group at $g = 0$. The scale dependence of the effective coupling $e^2 /v_F$ is displayed for different values of $D$ in Fig. \[two\], where the flow to the fixed-point is seen. Consequently, the scale $Z$ of the wavefunction is not renormalized to zero in the low-energy limit, and the quasiparticle weight remains finite above $D = 1$. We conclude then that, even in a model that keeps the Coulomb interaction unscreened, the breakdown of the Fermi liquid behavior only takes place formally at $D = 1$. = 8cm = 8cm The subtlety concerning the long-range Coulomb interaction is that the function $c(D)$ diverges in the limit $D \rightarrow 1$. This is actually what transforms the power-law dependence of the potential into a logarithmic dependence at $D = 1$. We observe that the 1D limit and the low-energy limit $\Lambda \rightarrow 0$ do not commute. If we stick to $D = 1$, we obtain a divergent coupling $g $ for the Coulomb interaction as well as a divergent electron scaling dimension. At any dimension slightly above $D = 1$, however, the fixed-point is at $g = 0$, with its corresponding vanishing anomalous dimension. In order to understand whether the 1D model has any stable fixed-point for finite values of $e^2 /v_F$, one can study the model by performing an expansion in powers of $g^{-1}$. The Fermi velocity is renormalized by terms that are analytic near the point $g = \infty$, and that lead to the scaling equation $$\Lambda \frac{d}{d \Lambda} v_F (\Lambda ) = \\ v_F f(D) \left( 3 - D - \frac{2}{D} \right) \frac{ \Gamma (D/2 - 1) }{ \Gamma ((D+1)/2) } \label{inf}$$ up to terms of order $O(g^{-1})$. In the limit $D \rightarrow 1$, $g \rightarrow \infty$, the right-hand-side of Eq. (\[inf\]) vanishes identically. This confirms, on nonperturbative grounds, that the 1D model with the Coulomb interaction has a line of fixed-points covering all values of $e^2 /v_F$. In the vicinity of $D = 1$, the presence of such critical line becomes sensible, and a crossover takes place to a behavior with a sharp reduction of the quasiparticle weight. This can be seen in the renormalization of the electron field scale $Z$, displayed in Fig. \[three\]. For values of $D$ above $\approx 1.2$, we have a clear signature of quasiparticles in the value of $Z$ at low energies. For lower values of $D$, the picture cannot be distinguished from that of a vanishing quasiparticle weight for all practical purposes. The drastic suppression of the electron field scale $Z$ takes place over a variation of only two orders of magnitude in the energy scale. = 6cm = 6cm The above picture allows us to make contact with the experiments carried out in multi-walled nanotubes[@mwnt]. In the proximity of the $D = 1$ fixed-point, the density of states displays an effective power-law behavior, with an increasingly large exponent. Moving to the other side of the crossover, the density of states approaches the well-known behavior of the graphite layer, $n( \varepsilon ) \sim |\varepsilon |$. In Fig. \[four\] we give the representation of the density of states $$n( \varepsilon ) \sim Z( \varepsilon ) |\varepsilon |^{D-1}$$ for several dimensions approaching $D = 1$. = 6cm = 6cm A value $e^2 /(\pi^2 v_F) = 0.5 $ for the bare coupling is appropriate for typical multi-walled nanotubes, as it takes into account the reduction due to the interaction with the inner metallic cylinders[@egger]. We observe that the exponents of $n( \varepsilon )$ at different dimensions are always larger than a lower bound $\alpha \approx 0.26$. This is in agreement with the values measured experimentally. Our analysis stresses the need of an appropriate description of the dimensional crossover between one and two dimensions, showing that the picture of a thick nanotube as an aggregate of 1D channels does not allow to obtain the correct values of the critical exponents. To summarize, we have studied the renormalization of the Coulomb interaction in graphene-based structures. We have made a rigorous characterization of the different behaviors, as we have proceeded by identifying the fixed-points of the theory. We have seen that the Fermi liquid behavior persists formally for any dimension above $D = 1$, as it also happens in the case of a short-range interaction[@ccm]. On the other hand, the proximity to the 1D fixed-point influences strongly the phenomenology of real quasi-onedimensional systems, giving rise to an effective power-law behavior of observables like the tunneling density of states. This is the case of the multi-walled nanotubes, for which we predict a lower bound for the corresponding exponent that turns out to be very close to the value measured experimentally. Financial support from CICyT (Spain) and CAM (Madrid, Spain) through grants PB96/0875 and 07N/0045/98 is gratefully acknowledged. J. Gan and E. Wong, Phys. Rev. Lett. [**71**]{}, 4226 (1993). C. Nayak and F. Wilczek, Nucl. Phys. B [**417**]{}, 359 (1994). S. Chakravarty, R. E. 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--- abstract: | The rotation curve for the IV galactic quadrant, within the solar circle, is derived from the Columbia University - U. de Chile CO(J=1$\to$0) survey of molecular gas. A new sampling, four times denser in longitude than in our previous analysis, is used to compute kinematical parameters that require derivatives w/r to galactocentric radius; the angular velocity $\Omega(R)$, the epicyclic frequency $\kappa(R)$, and the parameters $A(R)$ and $B(R)$ describing, respectively, gas shear and vorticity. The face-on surface density of molecular gas is computed from the CO data in galactocentric radial bins for the subcentral vicinity, the same spectral region used to derive the rotation curve, where the two-fold ambiguity in kinematical distances is minimum. The rate of massive star formation per unit area is derived, for the same radial bins, from the luminosity of IRAS point-like sources with FIR colors of UC H[II]{} regions detected in the CS(J=2$\to$1) line. Massive star formation occurs preferentially in three regions of high molecular gas density, coincident with lines of sight tangent to spiral arms. The molecular gas motion in these arms resembles that of a solid body, characterized by constant angular velocity and by low shear and vorticity. The formation of massive stars in the arms follows the Schmidt law, $\Sigma_{MSFR} \propto [\Sigma_{gas}]^n$, with an index of $n = 1.2 \pm 0.2$. Our results suggest that the large scale kinematics, through shear, regulate global star formation in the Galactic disk. author: - 'A. Luna, L. Bronfman, L. Carrasco, and J. May' title: | Molecular Gas, Kinematics, and OB Star Formation\ in the Spiral Arms of the Southern Milky Way --- INTRODUCTION ============ The rotation curve, describing the circular speed of rotating material as a function of galactocentric radius, is a fundamental tool for the study of the kinematics of our Galaxy. It is best derived, because of interstellar extinction, from observations of atomic and molecular gas in radio and mm wavelengths. The derivation involves determining the [*terminal velocity*]{}, or maximum absolute radial velocity relative to the Sun, toward lines of sight that sample the Galaxy within the solar circle (quadrants I and IV). Such terminal velocities correspond, assuming pure circular motion, to the tangent points to circumferences around the galactic center, named [*subcentral points*]{}. These points subtend a circumference that connects the solar position with the galactic center. A detailed analysis of the rotation curve can reveal important physical characteristics of the rotating material, such as the amount of shear and vorticity at each galactocentric radius. These physical quantities regulate the gravitational stability of a differentially rotating gaseous disk and, consequently, the large scale distribution and properties of star formation in the galactic disk. The first derivation of the rotation curve for the IV galactic quadrant that made use of the CO(J=1$\to$0) line - the best tracer of molecular hydrogen in the interstellar medium - was presented by Alvarez, May, & Bronfman (1990). The spectral data used to determine the terminal velocities were taken from the Columbia - U. de Chile surveys [@grabelsky87; @bronf89], which have a sampling interval of 0$^{\circ}$.125 (roughly the beam size). However, the terminal velocities in @alvarez90 were measured only every 0$^{\circ}$.5 in galactic longitude, due to difficulties involved in the visual examination of a very large number of spectra. A new derivation of the rotation curve, that uses a computer search code to examine all the available spectra ( $\approx$15000), is presented here. The disk kinematic characteristics in the IV galactic quadrant are analyzed in detail, from this new rotation curve. These characteristics, as a function of galactocentric radius, are compared with the molecular gas density and with the local rate of massive star formation. A proper derivation of the spiral pattern of our Galaxy requires knowledge of the distances to the adopted tracers. These distances are also required to compute the masses and luminosities of such tracers. For the gas, kinematical distances can be obtained from radial velocity data of radio line observations, adopting a rotation curve, under the assumption of pure circular motions. For clouds within the solar circle, however, there is a two-fold ambiguity in the kinematic distance, that is difficult to circumvent and has to be resolved in a case-by-case basis. But in the vicinity of the subcentral points such ambiguity is minimal, since at the subcentral points themselves the kinematic distances are univocally defined. It is worth noting that large scale streaming motions in spiral arms, with amplitude of $\sim$10 km/s, which produce deviations from pure rotation, have been observed in a number of regions of the Galaxy (Burton et al. 1988). Streaming motions of such amplitude may introduce uncertainties of up to 5% in the estimation of galactocentric radii when the streaming is along the line of sight. In such unfavorable case, the corresponding uncertainties in the estimated distances, for the section of the Galaxy analyzed here, may go from of 0.6 kpc to 1.7 kpc. In any case, for objects beyond $\sim$3kpc from the Sun, because of optical extinction, kinematical distances are usually the only ones available. Massive stars are formed within aggregates of molecular gas and dust of 10$^5$-10$^6$ solar masses, about 50-100 pc in size, which are commonly known as giant molecular clouds, or GMCs for short. The association between OB stars and the interstellar medium has been established through optical, infrared, and CO observations of GMCs close enough to be largely unaffected by extinction (Orion, Carina, etc). The physical conditions in GMCs control their rates of OB star formation, and are one of the main agents that regulate the evolution of the galactic disk [@evans99]. There is a close relationship between the galactic spiral structure and the formation of GMCs and, hence, with the formation rate of OB stars [@dame86; @solomon86]. Therefore, the GMCs and the regions of OB star formation provide a very good tool to trace the spiral arm pattern of a galaxy. An early description of the Milky Way spiral arm pattern was given by @gg76, who observed the H109$\alpha$ line emitted in H[II]{} regions associated with young massive stars. A four arm spiral pattern for the southern Milky Way was later proposed by @cyh87, using a larger observational database of hydrogen recombination lines (H109$\alpha$ & H110$\alpha$). The four arm spiral pattern is in general agreement with that obtained from H[I]{} and CO large scale observations of the Galaxy [@rob83; @grabelsky87; @bronf88; @alvarez90; @valle02]. Star formation is likely to occur in regions where the gas in the Galactic disk is unstable to the growth of gravitational perturbations. In a classical paper, @schm59 introduced the parametrization of the volume density of star formation and the volume density of gas, relating them through a power law; such parametrization, known as "Schmidt Law”, has been studied observationally [@kennic89; @wong02] and explained on theoretical grounds [@toomre64; @tan00]. A study of the gas stability in the galactic disk must include (a) comparison of the gas density with a critical value above which the gaseous aggregates undergo gravitational collapse [@toomre64; @kennic89] and (b) examination of the gas shear rate, that governs the process of destruction of molecular clouds (e.g. Kenney, Carlstrom, & Young 1993; Wong & Blitz 2002), presumably through the injection of turbulent motions [@maclow04]. The link between massive star formation and kinematical conditions in disks has been studied mostly for external spiral galaxies [@aalto99; @wong02; @bossier03], where the spatial resolution that can be achieved by the observations is not as good as for the Milky Way. The main goal of the present paper is, therefore, to accurately describe the spiral arm structure in the [*subcentral vicinity*]{} of our Galaxy, focusing on the molecular gas kinematics, density, and on the rate of massive star formation, with the hope of contributing to the understanding of the formation and evolution of disk galaxies in general. The analysis is carried out for the IV galactic quadrant, where the spiral structure is more evident [@bronf88] than in the I quadrant. Preliminary work has been presented by @cys83 and, more recently by @aluna01. Section (§2) describes the observational datasets used, the most complete available in their kind. These data are used in section (§3) to derive the rotation curve and analyze the relation between molecular gas kinematics, molecular gas surface density, and massive star formation rate. The validity of Schmidt Law for the Milky Way is analyzed in section (§4), and a summary of the results is given in section §5. OBSERVATIONS ============ The data used to derive the rotation curve and the molecular gas surface density are part of the Columbia-U. Chile $^{12}$CO(J=1$\to$0) surveys. These surveys provide us with the most extensive and homogeneous observational dataset of CO emission in the galactic disk [@grabelsky87; @bronf89; @dame01]. The beam-size of the antenna in the CO line is 8$\arcmin$.8, and an angular sampling of 0$^\circ$.125 was adopted. The surveys cover the entire IV galactic quadrant in longitude, and $\pm 2^\circ$ in latitude about the galactic equator. The velocity resolution is 1.3kms$^{-1}$, and the typical rms noise antenna temperature of the observations is 0.1K. Main beam temperatures, $T_{MB}$, are used throughout the analysis, obtained by dividing the antenna temperature $T^*_A$ by the main beam efficiency, $T_{MB}=T^*_A$/$\eta_{MB}$, with $\eta_{MB}=0.82$ [@bronf89]. Hereinafter we refer to the $^{12}$CO(J=1$\to$0) line as CO. For a detailed description of the observations see @grabelsky87 and @bronf89. The rate of massive star formation is estimated from the integrated FIR luminosity [@Boul88] of IRAS point-like sources, with FIR colors of UC H[II]{} regions [@wyc89], detected in a CS(J=2$\to$1) survey of 1427 sources in the whole Galaxy by Bronfman, Nyman, & May (1996). The CS(J=2$\to$1) emission line requires high molecular gas densities, of $10^4$ - $10^5$cm$^{-3}$, to become excited. Therefore, it constitutes a good tracer of massive star forming regions. The survey used here is the most complete currently available, listing 843 massive star forming regions in the galactic disk; the observed velocity profiles provide a good estimator of kinematical distances to the sources. The observations of the CS(J=2$\to$1) line toward the IV galactic quadrant were obtained with the SEST (Swedish-ESO Sub-millimeter Telescope) at La Silla Observatory, in Chile, with a beam-size of 50”. Typical rms noise in the spectra is 0.1 K, at a velocity resolution of 0.52kms$^{-1}$. The radial velocity coverage is 260kms$^{-1}$, large enough to detect all the sources in the radial velocity range allowed within the solar circle. Hereinafter we refer to the CS(J=2$\to$1) transition as CS , and to the IRAS point-like sources detected in CS as IRAS/CS sources. A number of 66 sources from the @bronfnyman survey are used in the present analysis. This number was complemented with 13 sources, undetected in the original CS survey, yet detected in a new CS survey with three times better sensitivity, completed with the SEST telescope, to be published elsewhere. The present analysis adopts the IAU recommended values for the galactocentric distance of the Sun and for the solar circular velocity, i.e. $R_0=8.5$kpc and $V_0=220$kms$^{-1}$ respectively [@kerrlynde86]. The analysis excludes galactic longitudes from 350$^\circ$ to 360$^\circ$, because the method used to derive the kinematical distance is highly uncertain for that region. Furthermore, the kinematics of the central region of the Milky Way are more complex than those of the galactic disk [@sawada01]. ANALYSIS ======== Derivation of the rotation curve -------------------------------- The rotation curve is derived from [*terminal velocities*]{} of CO spectra at each sampled galactic longitude $l$. Since the emission in the IV quadrant is blue-shifted, the most negative velocity is recorded, as well as the galactic latitude $b$ where it occurs (Fig. 1). To define the terminal velocity, following work by [@sinha78] for H[I]{} and by [@alvarez90] for CO, the half intensity point of the blue-shifted side of the emission peak with the most negative velocity is selected (Fig. 2). To be considered, an emission peak is required to be larger than 4 times the noise temperature $T_{rms}$. The emission lines thus selected have an average peak temperature of $1.8 \pm 0.7$K, and an average HWHM of $3 \pm 0.8$kms$^{-1}$. The rotation curve is obtained from the set of terminal velocities by assigning to every longitude a subcentral point of galactocentric distance $R = R_0 \mid \sin (l) \mid$ and heliocentric distance $D = R_0 \cos (l)$. The rotational velocity is then derived for each subcentral point, and assigned to the galactocentric radius $R = R_0 \mid \sin (l) \mid$, under the assumption of (a) pure circular motions and (b) differential rotation with angular velocity not growing with galactocentric radius [@sofuerubin01]. The rotation curve obtained, shown in Figure 3 ([*top*]{}), is similar to that derived by @alvarez90, but more detailed; the agreement for the longitude range covered in both studies is better than $\pm2$kms$^{-1}$ (rms). A slight difference is apparent in the radial range $R/R_0=$ \[0.56,0.62\], where their analysis yields a systematic shift in velocity of $2$kms$^{-1}$. The distance $Z \equiv R \tan (b)$ of the subcentral point to the galactic plane (Fig. 3 [*bottom*]{}) is also determined, for each longitude, providing a simple tool to examine the $Z$ distribution of the CO emission as a function of galactocentric radius. The results obtained compare well with those by @alvarez90, which adopted the same method, and with those obtained from an axisymmetric analysis of the CO emission in the IV quadrant [@bronf88]. Disk kinematics --------------- The rotation curve contains vast amount of information about the kinematics of the galactic disk [@byt87]. Among the principal parameters characterizing the kinematics, derived here, are $A(R)$ and $B(R)$, which describe the radial trends of shear and vorticity, respectively; these parameters, when evaluated at $R = R_0$, correspond to the well known Oort’s constants. As observed from a non-inertial coordinate frame, e.g. that centered on the Sun (Local Standard of Rest, LSR), the balance between Coriolis, centrifugal, and gravitational forces induce non-circular orbits about the galactic center. These motions can be described as small periodic oscillations superimposed onto circular orbits. In a coordinate frame with its origin in the galactic center and corotating with the solar orbit, these periodic oscillations trace small ellipses called [*epicycles*]{}, with an associated [*epicyclic frequency*]{}, $\kappa$, which roughly accounts for deviations from circular motion. The value of $\kappa(R)$ depends, at each galactocentric radius, on the angular velocity $\Omega(R)$: $$[\kappa(R)]^2=4[\Omega(R)]^2\left[1+\frac{1}{2}\frac{R}{[\Omega(R)]}\left(\frac{d[\Omega(R)]}{dR}\right)_R\right],$$ or, using the Oort’s $A$ and $B$ parameters in their general definition [@byt87], $$A(R)=-\frac{1}{2}R\frac{[d\Omega(R)]}{dR} \hspace{0.3cm} , \hspace{0.3cm} B(R)=-\Omega(R)+A(R),$$ it is possible to write the epicyclic frequency as: $$\kappa(R)=\sqrt{-4[B(R)][A(R)-B(R)]}.$$ A smoothed version of the rotation curve (Fig. 4 [*top*]{}) is used in the calculation, in order to provide continuity for the derivative of $\Omega(R)$ with respect to $R$ . The size of the smoothing box is 0.5kpc, smaller than the typical width of spiral arms. There are three galactocentric radial regions, marked in Fig. 4 [*top*]{}, where $V(R)$ increases monotonically, so that the radial derivative is positive and roughly constant, i.e. the signature of solid body rotation. The kinematical parameters $A(R)$, $B(R)$, $\kappa(R)$, and $\Omega(R)$, as functions of galactocentric radius, are shown in Figure 4 ([*bottom*]{}). The angular velocity, $\Omega(R)$, the epicyclic frequency, $\kappa(R)$, and the parameter $A(R)$, describing the shear, decrease on the average with galactocentric radius. On the contrary, the parameter $B(R)$, describing the vorticity, grows with galactocentric radius. The shear and vorticity present three relative minima, coincident with local maxima of the epicyclic frequency $\kappa(R)$, at radii 0.39, 0.47 and 0.73 $R/R_0$. The parameter $A(R)$, directly proportional to the radial derivative of the angular velocity, tends to zero at these radii, so the angular velocity is almost constant, a characteristic of solid body rotation. The relative maxima in $\kappa(R)$ at these three radii can be interpretated also as evidence for solid body rotation; at a given galactocentric radius, $\kappa$ would be $\sqrt{2}$ times higher for solid body rotation than for a flat rotation curve $V(R) = V_0$ = constant. The results of our analysis are consistent with other work in that the parameters $A(R)$ and $B(R)$ at $R = R_0$, are very close to the values of the Oort’s constants $A_0$ and $B_0$ recommended by the IAU [@kerrlynde86]. The values derived here are obtained from linear fits to $A(R)$ and $B(R)$, within the range 0.83 to $0.97 R/R_0$, extrapolated to $ R/R_0 = 1 $. Our results yield $A_0 = 14.9 \pm 4$ kms$^{-1}$kpc$^{-1}$, and $B_0 = -12.3 \pm 4$ kms$^{-1}$kpc$^{-1}$, in good agreement with those recommended by the IAU, $A_0 = 14.5 \pm 2$ kms$^{-1}$kpc$^{-1}$ and $B_0 = -12.0 \pm 3$ kms$^{-1}$kpc$^{-1}$. In a similar manner a value of $\kappa_0 = 35 \pm 7$kms$^{-1}$kpc$^{-1}$ is obtained for the epicyclic frequency at $ R = R_0$, in agreement with the value of $\kappa_0 = 36 \pm 10$ kms$^{-1}$kpc$^{-1}$ given by [@byt87]. The angular velocity obtained for $ R = R_0$, $\Omega_0 = 27.2 \pm 1$ kms$^{-1}$kpc$^{-1}$, is comparable with the value of $\Omega_p = 26 \pm 4$ kms$^{-1}$kpc$^{-1}$, derived for the spiral pattern angular velocity (Fernández, Figueras, & Torra 2001). Our results therefore suggest that the Sun location is close to the corotation radius (see also discussion in Amaral & Lepine 1997). The subcentral vicinity ----------------------- To study the possible relationships between the kinematical characteristics of the Galaxy, the molecular gas density distribution and the massive star formation rates (MSFR), it is necessary to use consistent data sets. The densities and MSFRs should be estimated, preferentially, for the very same regions for which the kinematics have been inferred. The rotation curve is obtained using information from the subcentral points of the disk, so the densities and MSFRs we wish to examine are evaluated in regions adjacent to the subcentral points. Such procedure advantageously minimizes the two-fold ambiguity in kinematical distance occurring within the solar circle since, at the subcentral points, kinematical distances are univocally defined [@byt87]. Limiting the analysis to a reduced area of the longitude-velocity space avoids also the azimuthal averaging of physical conditions which pertain both to spiral arms and to inter-arm regions, a difficulty that appears in axisymmetric analysis of the entire longitude-velocity space. Azimuthal averaging, over disk annuli, of the gas surface density and star formation rates, allows derivation of the star formation activity threshold in disk galaxies [@myk01; @bossier03]; the detailed information on the spiral structure of the disk, however, is washed out in the azimuthal average. The “[*subcentral vicinity*]{}”, used to inspect the gas physical conditions and MSFR, is defined here, at each galactic longitude, as the velocity span between the terminal velocity and the terminal velocity plus $\Delta V = 15$kms$^{-1}$ (Fig. 5). The value of $15$kms$^{-1}$ is large enough to include all the emission from each CO profile component used to select a terminal velocity. Were the emission to be interpreted as originated by a set of clouds at different velocities, the range $\Delta R $ implied by $\Delta V = 15$kms$^{-1}$ would be 0.5 kpc at $l = 307^\circ$ and 0.2 kpc at $l =342^\circ$, roughly the longitude limits of the analyzed region. These values of $\Delta R $ are small compared to the large scale trends with respect to galactocentric radius and therefore do not affect our conclusions. The properties of the ISM, molecular gas face-on surface density and MSFR per unit area, are averaged within the subcentral vicinity, to evaluate their dependance on galactocentric radius. It is assumed that the gas shares the kinematics of the corresponding subcentral point at each longitude. The molecular gas face-on surface density is computed from the same CO database used to derive the rotation curve, while the MSFR is computed from the FIR luminosity of the IRAS/CS sources. Molecular gas surface density ----------------------------- The derivation of molecular clouds mass from our CO survey data is based on the widely made assumption that the velocity-integrated CO intensity is directly proportional to the total H$_2$ column density in molecular clouds, and is therefore proportional (with a correction to account for helium) to the total molecular gas mass column density. [@byt87]. To evaluate the molecular gas surface density, $\Sigma_{gas}$, the CO emission is binned in galactocentric annuli of thickness 0.05$R/R_0$ and, for each radial bin, integrated over all velocities within the subcentral vicinity and over $[-2^\circ,2^\circ]$ in latitude. The resultant quantity is divided by the sampled face-on area. The proportionality factor $X$ adopted, where N(H$_2) = X W(^{12}$CO), is equal to $1.56 \times 10^{20}$ cm$^{-2}$ (Kkms$^{-1})^{-1}$, [@hunter97]. A correction of 1.36 is used to account for helium (see Murphy & May 1991). The results obtained for the gas surface density in the subcentral vicinity, within the solar circle, are shown in Figure 6 ([*top*]{}). Rate of massive star formation ------------------------------ Massive stars form in the dense cores of giant molecular clouds. The UV radiation from the young stars heats the surrounding dust, which re-radiates the energy principally in the far infrared. Most of these cores are detected as IRAS point-like sources, with FIR colors typical of ultra-compact H[II]{} regions [@wyc89]. The density is high enough for the excitation of the CS line, above $10^4$ - $10^5$cm$^{-3}$ [@bronfnyman]. Following [@kennic98a], the rate of massive star formation as a function of galactocentric radius is computed by adding the FIR luminosity of the IRAS/CS sources that fall within each radial bin. $$MSFR_{FIR}=6.5 \times 10^{-10}\left(\frac{L_{FIR}}{L_{\odot}}\right) [M_\odot yr^{-1}].$$ The FIR luminosity is evaluated as $L_{FIR}=4\pi D^2 F_{FIR}$ [@Boul88], where $D$ is the subcentral distance, $$F_{FIR} = \sum^4_{i =1} \nu F_\nu(i),$$ and $F_\nu(i)$ are the FIR fluxes of the IRAS/CS point-sources in the four IRAS bands, as published in the Point Source Catalog. The galactic FIR face-on surface luminosity of IRAS/CS sources as a function of galactocentric radius, for the subcentral vicinity, is presented in Figure 6 ([*bottom*]{}), compared with the axisymmetric analysis of the complete southern galactic disk by @bronf00. The FIR surface luminosity estimated here for massive star forming regions constitutes only a lower limit, since the emitting regions powered by embedded massive stars can be more extended than the IRAS resolution. But deconvolution of the extended FIR continuum emission can be much more difficult because of superposition of sources along the line of sight. For point-like sources there is no such confusion since all the CS profiles measured have only one velocity component [@bronfnyman] and, therefore, the derivation of kinematical distances is straightforward. The derived MSFR per unit area (Figure 7[*c*]{}) has relative maxima at the same galactocentric radial regions than the molecular gas surface density (Fig. 7[*b*]{}). These radial regions are characterized by solid-body like rotation, as shown in the smoothed version of the rotation curve displayed in (Fig. 7[*a*]{}). Gravitational disk stability ---------------------------- The Toomre criterion for disk stability [@toomre64; @byt87], used here, is described through the $Q$ stability parameter for gas, defined by $$Q(R)=\frac{ \alpha c \kappa(R)}{\pi G \Sigma_{gas}(R)},$$ where $c$ is the velocity dispersion, G is the gravitational constant, $\Sigma_{gas}$ is the gas surface density and $\alpha$ is a dimensionless parameter that accounts for deviations of real disks from the idealized Toomre thin disk, single fluid model (Tan 2000). When $Q < 1$ a gaseous disk is gravitationally unstable. The particular case $Q = 1$ defines a critical surface density value, governed by the Coriolis force represented by $\kappa(R)$; $$\Sigma_{crit}(R)=\frac{\alpha c \kappa(R)}{\pi G}.$$ The Toomre parameter $Q$ can be expressed, hence, as $Q = \Sigma_{crit}/\Sigma_{gas}$. The constant $\alpha$ is usually estimated defining $Q = 1$ where massive star formation ceases along disks of galaxies [@kennic98b; @hunter98; @Pisano2000]. Here a value of $\alpha = 0.08$ is found by defining $Q = 1$ in the galactocentric radial range $R/R_0 = 0.4$ to 0.45, where the MSFR per unit area is close to zero. Following [@kennic98b], a value of 6kms$^{-1}$ is adopted for the velocity dispersion. The dependance of $Q$ on galactocentric radius is shown in Figure 7[*d*]{}. To analyze the case of cloud destruction governed by the shear rate instead of the Coriolis force, Toomre’s criterion has been modified by @elmegreen93. The shear rate is described by the Oort parameter $A(R)$; a new parameter, $Q_{A}$, that evaluates the survival of a cloud, is in this case $$Q_{A}(R)=\frac{ \alpha_A 2.5 cA(R) }{\pi G \Sigma_{gas}(R)},$$ Clouds are sheared to destruction, in differentially rotating disks, for gas surface densities larger than the critical surface density $\Sigma_{crit}^A$ $$\Sigma_{crit}^A(R)=\frac{2.5\alpha_A c A(R)}{\pi G},$$ so that $Q_{A} = \Sigma_{crit}^A/\Sigma_{gas}$. This modified stability criterion takes into account the important role of shear in the destruction of GMCs, and can be used to describe how shear controls the rate of star formation in galactic disks in the regions where the Coriolis force does not. The parameter $\alpha_A$ is found, as $\alpha$, to be $\alpha_{A} = 0.2$. The dependance on galactocentric radius of $Q_{A}$ is shown in Figure 7[*e*]{}. A complementary analysis of the stability of the gas clouds to tidal shear is proposed by @kenney93, who defines a critical tidal surface density $\Sigma_{tide}$. $$\Sigma_{tide}(R)=\frac{\sigma_z(R)[3A(A-B)]^{1/2}}{\pi G},$$ where $\sigma_z$ is the velocity dispersion in the z-direction of the disk. If the gas density is less than $\Sigma_{tide}$, tidal shear will rip apart the clouds. The relative importance of tidal shear and gravitational stability can be expressed by the ratio of $\Sigma_{tide}$ to the critical surface density for gravitational instabilities as defined by Toomre for an ideal disk, $$\Sigma_{grav}=\frac{ c \kappa}{\pi G}.$$ Assuming that $c~\approx~\sigma_z$, the ratio depends almost entirely on the shape of the rotation curve, $$\frac{\Sigma_{tide}}{\Sigma_{grav}}=0.87\left(\frac{-A}{B}\right)^{1/2},$$ and tells whether tidal or gravitational forces determine the disk stability. The dependance of $\Sigma_{tide}$/$\Sigma_{grav}$ in galactocentric radius is shown in Figure 7[*f*]{}. Disk stability, gas density, and massive star formation ------------------------------------------------------- Some of our most relevant results are displayed in Figure 7, which we now describe in more detail. The values plotted and their errors are also listed in Table 1. Panel[*a*]{}, depicts the smoothed rotation curve used to derive the Oort parameters $A(R)$ and $B(R)$. Uncertainties in the rotation curve, of $\pm2$kms$^{-1}$, are dominated by the deviation of the measured subcentral velocities from the smoothed curve. Panel[*b*]{} presents the molecular gas face-on surface density evaluated in galactocentric radial bins of extent $0.05 R/R_0$, with the first bin centered at $R/R_0 = 0.275$. Panel[*c*]{} shows the MSFR, in units of solar masses per year per square pc ($\Sigma_{MSFR}$), averaged in the same galactocentric radial bins as for [*b*]{}. Uncertainties are estimated from Poisson statistics. Panel[*d*]{} displays the gravitational stability parameter $Q$ defined by Toomre; panel[*e*]{} the $Q_A$ parameter defined by @elmegreen93 for the case when shear dominates over Coriolis force; and panel[*f*]{}, finally, the ratio of the tidal to the gravitation surface instabilities (eq. \[11\]). The ratio [*f*]{} is estimated from the parameters $A(R)$ and $B(R)$ derived from observed rotation curve. The gas surface density and the MSFR are higher than average, as apparent from Figure 7, in regions where the rotation is typical of a solid body, roughly indicated by the vertical lines at radii 0.39, 0.47 and $0.73 R/R_0$ (see also Fig. 4). These three regions are very close to the accepted values for the tangent points of spiral arms identified in the literature (e.g. Vallée 2002 and references therein); the 3kpc arm approximately at $0.36 R/R_0$, the Norma spiral arm at $0.51 R/R_0$ and the Crux spiral arm at $0.77 R/R_0$. Figure 7[*d*]{} tells us that although gravitational instabilities are present in the Galactic disk, the disk is self-regulated in the sense that $Q$ is of order 1 everywhere [@tan00]. However, as shown in Figure 7[*e*]{}, where the Coriolis force is replaced by shear as the principal agent of cloud destruction, the three regions coincident with spiral arm tangents are characterized by relative minima. This result is reinforced by the three relative minima in Figure 7[*f*]{}, where the tidal shear is clearly lower than that needed to disrupt the clouds and, therefore, allows them to survive enough so as to collapse gravitationally and ultimately form massive stars. To estimate the influence of the derived quantities and parameters in the process of massive star formation, we examine the correlation between $\Sigma_{MSFR}$ (Panel[*c*]{}) and $\Sigma_{gas}$, $\Sigma_{crit}$/$\Sigma_{gas}$, $\Sigma_{crit}^A$/$\Sigma_{gas}$, $\Sigma_{tide}/\Sigma_{grav}$, $\Omega$ and $\kappa$. A Spearman rank-order correlation coefficient test [@numrecip] is adopted; the ranking and significance of each correlation are listed in Table 2. Massive star formation is clearly correlated with the gas surface density, with a confidence level larger than $90\%$, and less clearly correlated with the angular velocity and the epicyclic frequency, with a confidence level larger than $75\%$ and lower than $85\%$. For the variables $\Sigma_{crit}^A$/$\Sigma_{gas}$ and $\Sigma_{tide}$/$\Sigma_{grav}$, which take into account shear, there is no correlation with respect to the NULL hypothesis that the dependent variables are drawn from a random distribution. These results are discussed in the following section, searching for a possible scenario to explain the observed correlations. DISCUSSION ========== Spiral arms in the Southern Galaxy ---------------------------------- There are regions in the rotation curve (Fig. 7[*a*]{}) that present solid-body like kinematics; these regions are coincident with the loci of known spiral arm tangents. The molecular gas surface density is at least a factor of 2 larger for the arm regions than for the inter-arm regions (Fig. 7[*b*]{}). In those regions where the rotation is typical of a solid body, the rate of massive star formation per unit area, $\Sigma_{MSFR}$, presents relative maxima (Fig. 7[*c*]{}). The relative maxima in the molecular gas surface density and MSFR are coincident with relative minima in the kinematical parameters $\Sigma_{crit}^A$/$\Sigma_{gas}$ and $\Sigma_{tide}$/$\Sigma_{grav}$. Such results, as shown in Figure 6[*b*]{}, cannot be obtained alone from an axisymmetric analysis of the Galactic disk. In those radial regions where the kinematics resembles that of a solid body, i.e., spiral arm tangents, marked with vertical lines in Figure 7, $V/R = \Omega\approx$constant and $-A/B\ll 1$. Stability analysis suggests that molecular gas in regions characterized by solid body rotation is stable ($\Sigma_{gas}\approx\Sigma_{crit}$) or self-regulated [@tan00]; GMCs are not destroyed by tidal disruption ($\Sigma_{tide}\ll\Sigma_{gas}$), and there is an enhancement in $\Sigma_{MSFR}$. In regions that deviate from solid body rotation, the star formation decreases, implying that shearing motions may disrupt/destroy clouds and, therefore, play an important role in regulating large scale star formation in the disk. The destruction of clouds by shearing motions, therefore, is an important parameter controlling the large scale star formation in the disk, particularly in the inter-arm regions where shear dominates gravitational stability. High tidal disruption appears to be the agent for GMC destruction in the inter-arm regions, inhibiting the formation of massive stars, while low tidal disruption facilitates the accumulation of gas in the arms, increasing the gas surface density locally and promoting the higher observed MSFR. Regions of solid body rotation should present less collisions due to the low shearing motions and, therefore, the MSFR could presumably be limited by a process different from shear. The relation between solid body motion and enhanced star formation has been also found for external galaxies. A nearby example of a galaxy in which shear instability may regulate star formation is M33 [@corbelli03; @heyer04]. Using a shear stability criterion, they find that the predicted outer threshold radius for star formation is consistent with the observed drop in the H$\alpha$ surface brightness. Another example is the central region of NGC3504. The molecular gas kinematics are derived by @kenney93, and compared with the rate of star formation obtained from the H$\alpha$ recombination line. The central region of NGC3504 is characterized by solid body kinematics and high rate of star formation; they propose that the behavior of star formation is strongly influenced by the strength of tidal shear, which can help control the star formation rate via cloud destruction. For irregular galaxies, @hunter98 notice that in slowly raising rotation curves, characteristic of this type of galaxies, the stability parameter derived by [@elmegreen93] (eq. 8), provides a good criterion to describe the boundary between cloud survival and disruption. The conclusions of these authors are similar to our results for the Southern Milky Way arms, which show solid body kinematics, i.e. that the SF is strongly influenced by tidal shear, and in particular that tidal shear controls the rate of star formation through cloud destruction. Theoretical work in agreement with the present results has been presented by @RYS87. They use a galactic disk model where the ISM is simulated by a system of particles, representing clouds, which orbit in a one arm spiral-perturbed gravitational field, and include dissipative cloud-cloud collision. Their conclusion is that the distribution of GMCs and star formation is enhanced across the full finite width of the spiral arm and is not restricted to either the preshock or postshock regions. The agreement between their theoretical results and our observations implies that the best tracer for molecular gas spiral structure is the kinematical parameter that compares gravitational stability and cloud shear destruction ($\Sigma_{tide}/\Sigma_{grav}$, Fig. 7[*f*]{}), and is also the simplest one, because it involves only kinematical parameters that can be derived directly from the rotation curve. Spiral arm regions can be identified, therefore, as those where shear disruption of clouds has less influence than gravitational instabilities, so that molecular clouds can pile up, increasing their mass and evolving, until they collapse to form stars that will thereby increase the mean kinetic temperature. Such increment in temperature in the spiral arm regions has been preliminarily measured by @aluna04. Massive star formation and the Schmidt Law ------------------------------------------ There is a good correlation, as shown in section 3.7, between the MSFR per unit area (Fig. 7[*c*]{}) and the molecular gas face-on surface density (Fig. 7[*b*]{}) in the Milky Way. Such kind of correlation, formally known as Schmidt law, has been explored mostly for external galaxies. In this section we evaluate, for the subcentral vicinity of the Milky Way disk, four different enunciations of the Schmidt Law, whose simplest form [@kennic89] is, in our case, $$\Sigma_{MSFR} \propto(\Sigma_{gas})^{n},$$ Previous efforts at deriving the Schmidt Law for the Milky Way include those of @guibert78, who find values of n between 1.3 and 2.0, based on the distribution of OB associations, H II regions, H I, and early CO surveys; and [@bossier03], who using known gaseous and stellar radial profiles, find that the Schmidt Law is satisfied for the Milky Way from R = 4 to 15 kpc. Recently [@krumMckee05], in a general theory of turbulence-regulated star formation in galaxies, predict the star formation rate as a function of galactocentric radius for the Milky Way taking into account the fraction of molecular gas in the form of clouds, and enunciate Schmidt Law in the assumption that the clouds are virialized and supersonically turbulent. A second form of the Schmidt law, which modulates the gas density by a kinematical parameter, in this case the angular velocity $\Omega$ [@kennic89], is given by $$\Sigma_{MSFR}\propto\Sigma_{gas}\Omega.$$ A third derivation of the Schmidt Law [@tan00], can be expressed as $$\Sigma_{MSFR}\propto\Sigma_{gas}^{n}\Omega(1-0.7\beta),$$ with $$\beta \equiv \frac{d [\ln (v_{circ})]}{d [\ln (R)]},$$ where $v_{circ}$ is the circular velocity at a particular galactocentric radius. This version of Schmidt law has the advantage that the proportionality constant can be derived from theory, and includes evaluation of the mean free path for cloud-cloud collision, the fraction of collisions that lead to star formation, and the fraction of gas transformed into stars in each collision. A fourth derivation, of theoretical character, that assumes molecular clouds to be virialized and supersonically turbulent, has been given by @krumMckee05, and is expressed here as $$\Sigma_{MSFR}\propto f_{GMC} \Sigma_{gas}^{0.68}\Omega^{1.32},$$ where $f_{GMC}$ is the fraction of gas in the form of molecular clouds The four enunciations of Schmidt Law are tested (Figure 8) using the values of molecular gas face-on surface density and of the MSFR per unit area obtained here for galactocentric radial bins in the Milky Way. Although most derivations of Schmidt Law include both atomic and molecular gas, our calculations are limited to the molecular gas component of the ISM. @wong02 have found, for a set of seven galaxies, that the azimuthally averaged star formation rate per unit area correlates much better with $\Sigma{_{H_{2}}}$ than with $\Sigma_{HI}$. One should note that they required a large, uncertain correction for extinction to derive the rate of star formation using H$\alpha$ images; however, such correction does not affect their main conclusion, i. e., that considering the total gas density, rather than H$\small{I}$ and H$_{2}$ separately, may obscure underlying physical processes that are essential to star formation. The situation may be presumably different in galaxies where the ISM is predominantly atomic; however, even for M33, with a predominantly atomic ISM, [@heyer04] has shown that the star formation rate per unit area still correlates better with $\Sigma{_{H_{2}}}$ than with $\Sigma_{HI}$. The Schmidt law is roughly satisfied in all four cases analyzed, with a power index close to 1. The dispersion of the data points is large if all radial bins are taken into account; however, the relation becomes much tighter if the radial bins that correspond to interarm regions, shown in Figure 8 as thin squares, are removed. For the simplest relation (Fig. 8[*a*]{}), $\Sigma_{gas}\propto(\Sigma_{MSFR})^n$, the index value $n=1.2 \pm 0.2$ obtained is in good agreement with those derived for extragalactic disks [@wong02]. The relation plotted at Figure 8[*b*]{}, taking into account the kinematics through the angular velocity $\Omega$, with an index value $n=1.0 \pm 0.1$, is in very good agrement with @kennic98b. Because they derive an index from disk-averaged values for a set of galaxies, rather than the variation of star formation rate with surface density within a galaxy, presumably describing quite different processes, our results show an underlaying physical link between $\Sigma_{gas}$ and $\Sigma_{MSFR}$ that dominates both local and global scales in the process of star formation. The relation for the Schmidt law at Figure 8[*c*]{}, dependent on the kinematics, as proposed by @tan00, shows a relation with an index value $n=1.4 \pm 0.2$, consistent with theory. An index of $1.2 \pm 0.1$ is obtained when fitting the relation derived by @krumMckee05, using a value of $f_{GMC}=0.25$ [@dame87]. It is apparent that the simplest form (Fig. 8[*a*]{}) of Schmidt law, that normally describes the averaged properties of disks of galaxies [@kennic89], applies also fairly well to $\Sigma_{MSFR}$ in a scale range which is much smaller, like the radial bins in the Milky Way. The galactic disk kinematics, through shear, additionally modulates the global star formation in the Galactic disk; destroys the clouds in the interarm regions and permits the piling up of gas in the spiral arms, allowing gravity or another agents like compression and turbulence (via supernovas), to collapse the clouds into stars (see recent reviews by @elmegreen02 [@maclow04]. Timescale for the growth of gravitational instabilities -------------------------------------------------------- The gas depletion timescale due to star formation, $\tau_{SF}$, can be crudely estimated from the derived MSFR per unit area, $\Sigma_{MSFR}$, and the molecular gas surface density, $\Sigma_{gas}$, $$\tau_{SF}=\frac{\Sigma_{gas}}{\Sigma_{MSFR}}.$$ The galactocentric trend of $\tau_{SF} $ is shown in Figure 9 ([*top*]{}). Its average value is 10$^{10}$yr, an upper limit since $\Sigma_{MSFR}$ represents a lower limit. The timescale for growth of GMCs through gravitational instabilities can be expressed as $\tau\sim$Q/$\kappa$ [@larson88]. Results here yield timescales of the order $\sim 10^7$yr, with relative minima at the location of the spiral arm tangents (Fig. 9 [*middle*]{}). Following @kenney93, the efficiency of star formation ($\epsilon$) can be estimated via $\tau_{SF}$=$\epsilon^{-1} \tau$ if the timescales for gravitational instabilities ($\tau$) are similar to the timescale for cloud collapse. The values of a few tenths of a percent obtained, however, are rather low as compared with values of a few percent typically reported in the literature. The efficiency for massive star formation, according to @maclow04, can be further expressed as $$\epsilon=\tau_{OB}\frac{\Sigma_{MSFR}}{\Sigma_{gas}}=\frac{\tau_{OB}}{\tau_{SF}}.$$ Adopting a typical lifetime for an OB star region of $\tau_{OB}=10^8$yr and our results for $\Sigma_{MSFR}$ and $\Sigma_{gas}$ we obtain values for $ \epsilon $ of a few percent in the spiral arms, and extremely low values for inter-arm regions, (Fig. 9 [*bottom*]{}). A comparison of the timescales of gas consumption and growth of gravitational instabilities, as a function of galactocentric radius, suggests that fast star formation is present in regions where rapid cloud growth takes place. CONCLUSIONS =========== This work analyzes the correlation between molecular gas kinematical properties, molecular gas surface density, and rate of massive star formation in the IV galactic quadrant, using the most complete data bases available. The analysis is carried out for galactocentric radial bins 0.5 kpc wide, a compromise to avoid arm-interarm confusion while having good statistics. The data used are restricted to the subcentral vicinity, to avoid the two-fold distance ambiguity within the solar circle. The main conclusions are: $\bullet$ The rotation curve obtained is similar to that presented by Alvarez et al. (1990). Since the sampling in longitude is 4 times denser, however, it can be used to calculate, as a function of galactocentric radius, kinematical parameters that require radial derivatives. $\bullet$ The angular velocity, $\Omega(R)$, the epicyclic frequency, $\kappa(R)$, and the parameter $A(R)$ describing the gas shear, tend to decrease with galactocentric radius; the parameter $B(R)$, describing the gas vorticity, tends to grow. The values derived for Oort’s constants $A_0$ and $B_0$, at R = $R_0$, are consistent with those recommended by the IAU. $\bullet$ Shear and vorticity have relative minima, and the epicyclic frequency relative maxima, at radii 0.39, 0.47 and 0.73 $R/R_0$, coincident with the known positions of spiral arm tangent regions. Near these radii the kinematics are characteristic of solid body rotation: $A(R)$, proportional to the radial derivative of the angular velocity, tends to zero, so the angular velocity is roughly constant. The relative maxima in epicyclic frequency are consistent with such scenario since $\kappa$ is $\sqrt{2}$ times higher for solid body rotation than for a flat rotation curve. $\bullet$ Differential rotation and shear are weaker for the spiral arm regions than for the interarm regions. The relative importance of tidal shear w/r to gravitation in the stability of the gas for spiral arm regions, where the rotation curve resembles that of a solid body, is half than for spiral arms, where the rotation curve is nearly flat. $\bullet$ Massive star formation occurs in regions of high molecular gas density, roughly coincident with the three lines of sight tangent to spiral arms. In these arms the formation of massive stars follows the Schmidt law, $\Sigma_{MSFR} \propto [\Sigma_{gas}]^n$, with an index of $n = 1.2 \pm 0.2$. While this law is characteristic of spiral galactic disks, here it applies to much smaller spatial scale. A modified version of Schmidt law, which modulates the gas density by the angular velocity, $\Sigma_{MSFR} \propto\Sigma_{gas}\, \Omega$, describes better the behavior of the gas at this scale, suggesting that the kinematics, through shear, regulate global star formation in the Galactic disk. A.L. and L.C. acknowledge support by CONACYT México through the research grants 89964, 42609 and G28586. L.B., J.M., and A.L. acknowledge support from FONDECYT Grant 1010431 and from the Chilean [*Center for Astrophysics*]{} FONDAP No. 15010003. We thank Drs. E. Brinks, A. Garcia-Barreto, N. Vera and W. Wall for fruitful comments. A.L. thanks INAOE’s Astronomy Department and is also grateful for the use of facilities and the support at the Departmento de Astronomía, Universidad de Chile. We are also grateful to the referee for several helpful comments. Aalto, S., Huttemeister, S., Scoville, N.Z., & Thaddeus, P. 1999, , 522, 165 Alvarez, H., May, J., & Bronfman, L. 1990, , 348, 495 Amaral, L. 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P. 1978, , 69, 227 Sofue, Y., & Rubin, V. 2001, , 39, 137 Tan, J. 2000, , 536, 173 Toomre, A. 1964, , 139, 1217 Vallée, J. 2002, , 566, 261 Wong, T., & Blitz, L. 2002, , 569, 157 Wood, D.O.S., $\&$ Churchwell, E. 1989, , 340, 265 Gal. radii circ. vel. $\Sigma_{gas}$ $\Sigma_{MSFR}$(10$^{-9}$) $\Sigma_{crit}/\Sigma_{gas}$ $\Sigma^{A}_{crit}/\Sigma_{gas}$ $\Sigma_{tide}/\Sigma_{grav}$ ------------ ------------ ---------------------- ------------------------------- ------------------------------ ---------------------------------- ------------------------------- R/R$_0$ kms$^{-1}$ M$_{\odot}$pc$^{-2}$ M$_{\odot}$yr$^{-1}$pc$^{-2}$ $\pm$2 0.325 212.4 3.57 $\pm$0.03 0.23 $\pm$0.03 0.99 $\pm$0.11 1.12 $\pm$0.10 1.03 $\pm$0.12 0.375 209.6 3.93 $\pm$0.03 0.89 $\pm$0.15 1.00 $\pm$0.04 0.43 $\pm$0.10 0.56 $\pm$0.11 0.425 214.2 2.33 $\pm$0.03 0.19 $\pm$0.03 1.07 $\pm$0.14 1.45 $\pm$0.11 1.17 $\pm$0.18 0.475 215.7 3.06 $\pm$0.02 0.75 $\pm$0.07 1.01 $\pm$0.05 0.51 $\pm$0.10 0.61 $\pm$0.11 0.525 223.6 4.12 $\pm$0.02 1.21 $\pm$0.34 0.59 $\pm$0.06 0.58 $\pm$0.06 0.93 $\pm$0.11 0.575 219.3 1.94 $\pm$0.02 0.34 $\pm$0.03 1.09 $\pm$0.11 1.13 $\pm$0.11 0.97 $\pm$0.13 0.625 218.0 1.66 $\pm$0.02 0.28 $\pm$0.04 1.14 $\pm$0.12 1.25 $\pm$0.11 1.00 $\pm$0.15 0.675 214.5 2.19 $\pm$0.01 0.09 $\pm$0.01 0.91 $\pm$0.06 0.65 $\pm$0.09 0.75 $\pm$0.13 0.725 225.3 3.92 $\pm$0.01 0.46 $\pm$0.15 0.63 $\pm$0.01 0.09 $\pm$0.06 0.31 $\pm$0.21 0.775 236.7 2.40 $\pm$0.01 0.18 $\pm$0.04 0.99 $\pm$0.02 0.17 $\pm$0.10 0.34 $\pm$0.20 0.825 234.3 1.68 $\pm$0.01 0.18 $\pm$0.04 0.91 $\pm$0.10 1.03 $\pm$0.09 1.03 $\pm$0.22 0.875 236.1 0.84 $\pm$0.01 0.29 $\pm$0.09 1.69 $\pm$0.20 2.03 $\pm$0.17 1.08 $\pm$0.26 0.925 233.1 0.41 $\pm$0.01 0.30 $\pm$0.06 – – – 0.975 233.6 0.33 $\pm$0.01 0.03 $\pm$0.01 – – – : Values of our derived and measured parameters in the adopted bins at the subcentral vicinity. Our analysis beyond $0.875 R/R_0$ is uncertain and therefore excluded. correlation with $\Sigma_{MSFR}$ $\Sigma_{gas}$ $\Sigma_{crit}/\Sigma_{gas}$ $\Sigma^{A}_{crit}/\Sigma_{gas}$ $\Sigma_{tide}/\Sigma_{grav}$ $\Omega$ $\kappa$ ---------------------------------- ---------------- ------------------------------ ---------------------------------- ------------------------------- ---------- ---------- Rank 0.53 -0.02 -0.24 -0.26 0.41 0.36 confidence level $>$90$\%$ 7$\%$ 55$\%$ 60$\%$ 82% 76% : The Spearman test to explore possible correlations between $\Sigma_{MSFR}$ and $\Sigma_{gas}$, as well as between $\Sigma_{MSFR}$ and kinematical and instability parameters presented in Table 1.

--- address: 'Department of Mathematics, Hokkaido University, Sapporo, 060-0810 Japan' author: - Toshiyuki Akita title: | A formula for the Euler characteristics of\ even dimensional triangulated manifolds --- [^1] A finite simplicial complex $K$ is called an [*Eulerian manifold*]{} (or a [*semi-Eulerian complex*]{} in the literature) if all of maximal faces have the same dimension and, for every nonempty face $\sigma\in K$, $$\chi({\operatorname{Lk}}\sigma)=\chi(S^{\dim K-\dim\sigma-1})$$ holds, where ${\operatorname{Lk}}\sigma$ is the link of $\sigma$ in $K$ and $S^n$ is the $n$-dimensional sphere. Note that $K$ is not necessary connected. Any triangulation of a closed manifold is an Eulerian manifold. More generally, a triangulation of a homology manifold without boundary provides an Eulerian manifold. The purpose of this short note is to prove the following alternative formula for the Euler characteristics of even dimensional Eulerian manifolds. \[main\] Let $K$ be a $2m$-dimensional Eulerian manifold. Then $$\label{eq-main} \chi(K)=\sum_{i=0}^{2m}\left(-\frac{1}{2}\right)^i f_i(K)$$ holds, where $f_i(K)$ is the number of $i$-simplices of $K$. A finite simplicial complex $L$ is called a [*flag complex*]{} if every collection of vertices of $L$ which are pairwise adjacent spans a simplex of $L$. The formula was proved in [@akita] under the additional assumptions that $K$ is a PL-triangulation of a closed $2m$-manifold and is a flag complex. M. W. Davis pointed out that the formula follows from a result in [@davis], provided $K$ is a flag complex (see [*Note added in proof*]{} in [@akita]). Both results follow from the considerations of the Euler characteristics of Coxeter groups. In this note, we deduce the formula from the generalized Dehn-Sommerville equations proved by Klee [@klee]. Let $K$ be a finite $(d-1)$-dimensional simplicial complex and $f_i=f_i(K)$ the number of $i$-simplices of $K$ as before. The $d$-tuple $(f_0,f_1,\dots,f_{d-1})$ is called the [*$f$-vector*]{} of $K$. The [*$f$-polynomial*]{} $f_K(t)$ of $K$ is defined by $$f_K(t)=t^d+f_0t^{d-1}+\cdots+f_{d-2}t+f_{d-1}.$$ Define the [*$h$-polynomial*]{} $h_K(t)$ of $K$, $$h_K(t)=h_0t^d+h_1t^{d-1}+\cdots+h_{d-1}t+h_d,$$ by the rule $h_K(t)=f_K(t-1)$. The $(d+1)$-tuple $(h_0,h_1,\dots,h_d)$ is called the [*$h$-vector*]{} of $K$. The $h$-vector of $K$ satisfies the generalized Dehn-Sommerville equations, as stated below in Theorem \[DS\]. \[DS\] Let $K$ be a $(d-1)$-dimensional Eulerian manifold. Then $$h_{d-i}-h_i=(-1)^i\binom{d}{i}(\chi(K)-\chi(S^{d-1}))$$ holds for all $i$ $(0\leq i\leq d)$. Klee stated the generalized Dehn-Sommerville equations in terms of the $f$-vector rather than the $h$-vector. The formulae quoted in Theorem \[DS\] are equivalent to those in [@klee] and can be found in [@ed]. Theorem \[DS\] was also proved in [@panov] by a quite different method, provided that $K$ is a triangulation of a closed manifold. Now we prove Theorem \[main\]. We have $$h_K(-1)=\sum_{i=0}^{2m+1}(-1)^{2m+1-i}h_i =\sum_{i=0}^{m} (-1)^i (h_{2m+1-i}-h_i).$$ Now Theorem \[DS\] asserts that $$h_{2m+1-i}-h_i=(-1)^i\binom{2m+1}{i}(\chi(K)-2).$$ Hence we obtain $$\label{h-poly} h_K(-1)=(\chi(K)-2)\sum_{i=0}^m\binom{2m+1}{i} =2^{2m}(\chi(K)-2).$$ On the other hand, we have $$\label{f-poly} f_K(-2)=(-2)^{2m+1}+\sum_{i=0}^{2m}(-2)^{2m-i}f_i =2^{2m}\left( -2+\sum_{i=0}^{2m}\left(-\frac{1}{2}\right)^if_i \right).$$ Since $h_K(-1)=f_K(-2)$ by the definition of the $h$-polynomial $h_K(t)$, Theorem \[main\] follows from and . [1]{} T. Akita, Euler characteristics of Coxeter groups, PL-triangulations of closed manifolds, and cohomology of subgroups of Artin groups, J. London Math. Soc. (2) 61 (2000), 721–736. V. M. Buchstaber, T. E. Panov, [*Torus actions and their applications in topology and combinatorics*]{}, University Lecture Series 24, American Mathematical Society, Providence, 2002. R. Charney, M. W. Davis, Reciprocity of growth functions of Coxeter groups, Geom. Dedicata 39 (1991), 373–378. V. Klee, A combinatorial analogue of Poincaré’s duality theorem, Canad. J. Math. 16 (1964), 517–531. E. Swartz, From spheres to manifolds, preprint (2005). [^1]: Partially supported by the Grant-in-Aid for Scientific Research (C) (No.17560054) from the Japan Society for Promotion of Sciences.

--- abstract: 'We introduce an intuitive measure of genuine multipartite entanglement which is based on the well-known concurrence. We show how lower bounds on this measure can be derived that also meet important characteristics of an entanglement measure. These lower bounds are experimentally implementable in a feasible way enabling quantification of multipartite entanglement in a broad variety of cases.' author: - 'Zhi-Hao Ma$^{1}$, Zhi-Hua Chen$^{2}$, Jing-Ling Chen$^{3}$' - 'Christoph Spengler, Andreas Gabriel, Marcus Huber' title: Measure of genuine multipartite entanglement with computable lower bounds --- Introduction ------------ Entanglement is an essential component in quantum information and at the same time a central feature of quantum mechanics [@Horodecki09; @Guhne09]. Its potential applications in quantum information processing vary from quantum cryptography [@Ekert91] and quantum teleportation [@Bennett93] to measurement-based quantum computing [@BRaussendorf01]. The use of entanglement as a resource not only bears the question of how it can be detected, but also how it can be quantified. For this purpose, several entanglement measures have been introduced, one of the most prominent of which is the concurrence [@Wootters98; @Horodecki09; @Guhne09]. However, beyond bipartite qubit systems [@Wootters98] and highly symmetric bipartite qudit states such as isotropic states and Werner states [@Terhal00; @Werner01] there exists no analytic method to compute the concurrence of arbitrary high-dimensional mixed states. For a bipartite pure state $|\psi\rangle$ in a finite-dimensional Hilbert space $\mathcal{H} _1\otimes \mathcal{H}_2=\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}$ the concurrence is defined as [@Mintert05] $C(|\psi\rangle)=\sqrt{2\left(1-\texttt{Tr}\rho_1^2\right)}$ where $\rho_1=\texttt{Tr}_2\rho$ is the reduced density matrix of $\rho={\ensuremath{| \psi \rangle}}{\ensuremath{\langle \psi |}}$. For mixed states $\rho$ the concurrence is generalized via the convex roof construction $C(\rho)=\inf_{\{p_i,|\psi_i\rangle\}} \sum_i p_i C({\ensuremath{| \psi_i \rangle}})$ where the infimum is taken over all possible decompositions of $\rho$, i.e. $\rho=\sum_i p_i |\psi_i\rangle\langle\psi_i|$. This generalization is well-defined, however, as it involves a nontrivial optimization procedure it is not computable in general. The concurrence is a useful measure with respect to a broad variety of tasks in quantum information which exploit entanglement between two parties. However, considering multipartite systems, a generalization of the concurrence is needed that strictly quantifies the amount of genuine multipartite entanglement - the type of entanglement that not only is the key resource of measurement-based quantum computing [@Briegel09] and high-precision metrology [@Giovannetti04] but also plays a central role in biological systems [@Sarovar; @Caruso], quantum phase transitions [@Oliv; @Afshin] and quantum spin chains [@spinchains]. Although many criteria detecting genuine multipartite entanglement have been introduced (see e.g. Refs. [@Huber10; @Huberqic; @HuberDicke; @Krammer; @HuberClass; @Deng09; @Deng10; @Chen10; @Bancal; @Horodeckicrit; @Yucrit; @Hassancrit; @Seevinckcrit; @Uffink; @Collins; @Guehnecrit]), there is still no computable measure quantifying the amount of genuine multipartite entanglement present in a system. There are only few quantities available for pure states (a set of possible measures is given in Ref. [@HHK1]) which, however, are in general incomputable for mixed states and corresponding computable lower bounds have not been found so far. In this paper, we define a generalized concurrence (analogously to a measure proposed for pure states in Ref. [@Milburn]) for systems of arbitrarily many parties as an entanglement measure which distinguishes genuine multipartite entanglement from partial entanglement. As a main result we show that strong lower bounds on this measure can be derived by exploiting close analytic relations between this concurrence and recently introduced detection criteria for genuine multipartite entanglement. Genuine multipartite entanglement --------------------------------- An $n$-partite pure state $|\psi\rangle\in \mathcal{H}_1\otimes \mathcal{H}_2\otimes\cdots\otimes\mathcal{H}_n$ is called biseparable if it can be written as $|\psi\rangle=|\psi_A\rangle \otimes |\psi_B\rangle$, where $|\psi_A\rangle \in \mathcal{H}_{A} = \mathcal{H}_{j_1}\otimes \ldots \otimes \mathcal{H}_{j_k}$ and $|\psi_B\rangle \in \mathcal{H}_{B} = \mathcal{H}_{j_{k+1}}\otimes \ldots \otimes \mathcal{H}_{j_n}$ under any bipartition of the Hilbert space, i.e. a particular order $\{j_1,j_2,\ldots j_{k}|j_{k+1},\cdots j_n \}$ of $\{1,2,\cdots, n\}$ (for example, for a 4-partite state, $\{1,3|2,4\}$ is a partition of $\{1,2,3,4\}$). An $n$-partite mixed state $\rho$ is biseparable if it can be written as a convex combination of biseparable pure states $\rho=\sum\limits_{i}p_i|\psi_i\rangle \langle\psi_i|$, wherein the contained $\{|\psi_i\rangle\}$ can be biseparable with respect to different bipartitions (thus, a mixed biseparable state does not need to be separable w.r.t. any particular bipartition of the Hilbert space). If an $n$-partite state is not biseparable then it is called genuinely $n$-partite entangled.\ If we denote the set of all biseparable states by $\mathcal{S}_2$ and the set of all states by $\mathcal{S}_1$ we can illustrate the convex nested structure of multipartite entanglement (see Fig. \[fig\_convex\]).\ ![Illustration of the convex nested structure of multipartite entanglement. The set of biseparable states $\mathcal{S}_2$ is convexly embedded within the set $\mathcal{S}_1$ of all states ($\mathcal{S}_2 \subset \mathcal{S}_1$).[]{data-label="fig_convex"}](zwiebel.eps) A measure of genuine multipartite entanglement (g.m.e.) $E(\rho)$ should at least satisfy: - $E(\rho)=0 \,\forall\,\rho\in \mathcal{S}_2$ (zero for all biseparable states) - $E(\rho)>0 \,\forall\,\rho\in \mathcal{S}_1$ (detecting all g.m.e. states) - $E(\sum_ip_i\rho_i)\leq \sum_ip_iE(\rho_i)$ (convex) - $E(\Lambda_{LOCC}[\rho])\leq E(\rho)$ (non-increasing under local operations and classical communication)[^1] - $E(U_{local}\rho U^\dagger_{local})= E(\rho)$ (invariant under local unitary transformations) There are of course further possible conditions which are sometimes required (such as e.g. additivity), but this set of conditions constitutes the minimal requirement for any entanglement measure. For a more detailed analysis of such requirements consult e.g. Refs. [@Mintertrep05; @HHK1]. Concurrence for genuine $n$-partite entanglement ------------------------------------------------ Let us now introduce a measure of multipartite entanglement satisfying all necessary conditions (M1-M5) for being a multipartite entanglement measure.\ [**Definition 1.**]{} For $n$-partite pure states ${\ensuremath{| \Psi \rangle}} \in \mathcal{H}_{1}\otimes \mathcal{H}_{2}\otimes\cdots\otimes \mathcal{H}_{n}$, where $dim(\mathcal{H}_{i})=d_{i},i=1,2, \cdots ,n$ we define the gme-concurrence as $$\begin{aligned} \label{gmeconcurrence} C_{gme}({\ensuremath{| \Psi \rangle}}):=\min\limits_{\gamma_i \in \gamma} \sqrt{2(1-{\mbox{Tr}}(\rho^{2}_{A_{\gamma_i}}))}\ ,\end{aligned}$$ where $\gamma=\{\gamma_i\}$ represents the set of all possible bipartitions $\{A_i|B_i\}$ of $\{1,2,\ldots,n\}$. The gme-concurrence can be generalized for mixed states $\rho$ via a convex roof construction, i.e. $$\begin{aligned} C_{gme}(\rho)= \inf_{\{p_i,|\psi_i\rangle\}} \sum_{i}p_{i}C_{gme}({\ensuremath{| \psi_i \rangle}} ) \ , \label{4}\end{aligned}$$ where the infimum is taken over all possible decompositions $\rho=\sum_i p_i {\ensuremath{| \psi_i \rangle}} {\ensuremath{\langle \psi_i |}}$. For example, for a tripartite pure state ${\ensuremath{| \psi \rangle}}\in \mathcal{H}_1 \otimes \mathcal{H}_2 \otimes \mathcal{H}_3$ there are three possible bipartitions $\gamma=\{ \{1|2, 3\}, \{2|1, 3\}, \{3|1, 2\} \}$. Consequently, the gme-concurrence is $C_{gme}(\psi)=\min\{\sqrt{2(1- Tr(\rho_{1}^2))},\sqrt{2(1- Tr(\rho_{2}^2))},\sqrt{2(1- Tr(\rho_{3}^2))}\}$.\ The definition of $C_{gme}(\rho)$ directly implies $C_{gme}(\rho)=0$ for all biseparable states (M1) and $C_{gme}(\rho)>0$ for all genuinely $n$-partite entangled states (M2). Convexity (M3) also follows directly from the fact that any mixture $\lambda\rho_1+(1-\lambda)\rho_2$ of two density matrices $\rho_1$ and $\rho_2$ is at least decomposable into states that attain the individual infima. As the concurrence of any subsystem was proven to be non-increasing under LOCC (see e.g. Ref. [@Mintert05]), the minimum of all possible concurrences will of course still remain non-increasing, thus proving (M4) also holds. Furthermore $\text{Tr}(\rho^2)$ is invariant under local unitary transformations for every reduced density matrix irrespective of the decomposition, which proves that also condition (M5) holds. For pure states the gme-concurrence is closely related to the entanglement of minimum bipartite entropy introduced for pure states in Ref. [@Milburn]. In contrast to the original definition using von Neumann entropies of reduced density matrices, we use linear entropies. In this way we can derive lower bounds even on the convex roof extension which had not been considered before. Lower bounds on the gme-concurrence ----------------------------------- As the computation of any proper entanglement measure is in general an NP-hard problem (see Ref. [@gurvits]), it is crucial for the quantification of entanglement that reliable lower bounds can be derived. These lower bounds should be computationally simple and also experimentally (locally) implementable to be of any use in practical applications. Let us now derive lower bounds on $C_{gme}$ which meet these requirements. Consider inequality II from Ref. [@Huber10], which is satisfied by all biseparable states (such that its violation implies genuine multipartite entanglement) $$\label{ineqII} \underbrace{\sqrt{{\ensuremath{\langle \Phi |}}\rho^{\otimes 2}\Pi_{\{1,2,\cdots,n\}}{\ensuremath{| \Phi \rangle}}}-\sum\limits_{\gamma}\sqrt{{\ensuremath{\langle \Phi |}}\Pi_{\gamma}\rho^{\otimes 2}\Pi_{\gamma}{\ensuremath{| \Phi \rangle}}}}_{=: I[\rho,|\Phi\rangle]}\leq 0\, ,$$ where ${\ensuremath{| \Phi \rangle}}$ is any state separable with respect to the two copy Hilbert spaces and $\Pi_{\{\alpha\}}$ is the cyclic permutation operator acting on the twofold copy Hilbert space in the subsystems defined by $\{\alpha\}$, i.e. exchanging the vectors of the subsystems $\{\alpha\}$ of the first copy with the vectors of the subsystems $\{\alpha\}$ of the second copy. A simple example would be $\Pi_{\{1\}}|\phi_1\phi_2\rangle\otimes|\psi_1\psi_2\rangle=|\psi_1\phi_2\rangle\otimes|\phi_1\psi_2\rangle$.\ For sake of comprehensibility let us show how to derive lower bounds for three qubits and then generalize the result (in the appendix). If we consider a most general 3-qubit pure state in the computational basis $$\begin{aligned} \label{explicit} |\psi\rangle=&&a|000\rangle+b|001\rangle+c|010\rangle+d|011\rangle\nonumber\\&&+e|100\rangle+f|101\rangle+g|110\rangle+h|111\rangle\, ,\end{aligned}$$ the squared concurrences $C^2(\rho_\gamma)=2(1-\text{Tr}(\rho^2_\gamma))$ with respect to the three bipartitions read $$\begin{aligned} C^2(\rho_1)=4|ah-de|^2+F_1\,,\\ C^2(\rho_2)=4|ah-cf|^2+F_2\,,\\ C^2(\rho_3)=4|ah-bg|^2+F_3\,,\end{aligned}$$ where $F_i$ are non-negative functions. The following relations thus hold $$\begin{aligned} C(\rho_1)\geq 2|ah-de|\geq2|ah|-2|de|\,,\\ C(\rho_2)\geq 2|ah-cf|\geq2|ah|-2|cf|\,,\\ C(\rho_3)\geq 2|ah-bg|\geq2|ah|-2|bg|\,,\end{aligned}$$ and finally $$\begin{aligned} &\min\{C(\rho_1),C(\rho_2),C(\rho_3)\} \nonumber \\ \geq &2|ah|-2\max\{|de|,|cf|,|bg|\}\nonumber\\\geq&2|ah|-2(|de|+|cf|+|bg|)\ =: B \,.\end{aligned}$$ Now for any given mixed state the convex roof construction is bounded by $$\begin{aligned} C_{gme}(\rho)\geq\inf_{\{p_i,|\psi_i\rangle\}}\sum_i p_iB_i\,.\end{aligned}$$ For the choice $|\Phi\rangle=|000111\rangle$ and the abbreviation $\rho_{uvwxyz}:=\langle uvw|\rho|xyz\rangle$, inequality (\[ineqII\]) reads $$\begin{aligned} I[\rho,|000111\rangle]=|\rho_{000111}|-\sqrt{\rho_{001001}\rho_{110110}}\nonumber\\-\sqrt{\rho_{010010}\rho_{101101}}-\sqrt{\rho_{100100}\rho_{011011}}\leq 0\,.\end{aligned}$$ Now $$\begin{aligned} 2|\rho_{000111}|\leq\inf_{\{p_i,|\psi_i\rangle\}}\sum_i p_i2|a_ih_i|\,,\end{aligned}$$ due to the triangle inequality and $$\begin{aligned} 2\sqrt{\rho_{001001}\rho_{110110}}\geq\inf_{\{p_i,|\psi_i\rangle\}}\sum_i p_i2|b_ig_i|\,,\end{aligned}$$ holds due to the Cauchy-Schwarz inequality (of course for all the parts of the other bipartitions).\ This leads to a lower bound on the convex roof construction $$\begin{aligned} C_{gme}(\rho)\geq 2I[\rho,|000111\rangle]\,.\end{aligned}$$ As $C_{gme}(\rho)$ is invariant under local unitary transformations, we can infer that indeed every $2I[\rho,|\Phi\rangle]$ constitutes a proper lower bound. By taking into account the set of all vectors $\{|\Phi\rangle\}$ we can thus define a computable lower bound which itself has many favorable properties (satisfying M1, M3, M4 and M5): $$\begin{aligned} C_{gme}(\rho)\geq \max_{|\Phi\rangle}2I[\rho,|\Phi\rangle]\,.\label{lowerbound}\end{aligned}$$ As the lower bound is straightforwardly generalized (the structure of the proof essentially remains the same, see the appendix for details), eq.(\[lowerbound\]) is indeed a proper lower bound on (\[gmeconcurrence\]) for any $n$-partite qudit state.\ ![Contour plot of the lower bound $\max_{|\Phi\rangle}2I[\rho,|\Phi\rangle]$ on the gme-concurrence for the set of three-qubit-states $\rho=c_1 {\ensuremath{| GHZ \rangle}}{\ensuremath{\langle GHZ |}} + c_2 {\ensuremath{| W \rangle}}{\ensuremath{\langle W |}}+\frac{1-c_1-c_2}{8}\mathbbm{1}$ given by convex mixtures of a GHZ state, W state and the maximally mixed state. The greyscale is related to the bound $\max_{|\Phi\rangle}2I[\rho,|\Phi\rangle]$ varying from $0$ to $1$ (where $0$ is white), while the blue region denotes states which are positive under partial transposition with respect to all bipartitions. The optimization over all $\{|\Phi\rangle\}$ was realized using the composite parametrization of the unitary group (see Ref. [@SHH2]).[]{data-label="fig_contourpl"}](contourpl.eps) (0, 0)(0,0) (-0.8,0.2) (0,0)\[\] $1$ (-6.6,6.3) (0,0)\[\] $1$ (-0.8,1.6) (0,0)\[\] $c_1$ (-5.5,6.3) (0,0)\[\] $c_2$ \ **Discussion**\ The detection quality of our obtained bounds on the gme-concurrence is illustrated in Fig. \[fig\_contourpl\] for the family $\rho=c_1 {\ensuremath{| GHZ \rangle}}{\ensuremath{\langle GHZ |}} + c_2 {\ensuremath{| W \rangle}}{\ensuremath{\langle W |}}+\frac{1-c_1-c_2}{8}\mathbbm{1}$ of three-qubit-states, where $$\begin{aligned} {\ensuremath{| GHZ \rangle}} = \frac{1}{\sqrt{2}}({\ensuremath{| 000 \rangle}}+{\ensuremath{| 111 \rangle}}) \quad \textrm{and} \nonumber \\ {\ensuremath{| W \rangle}} = \frac{1}{\sqrt{3}}( {\ensuremath{| 001 \rangle}} + {\ensuremath{| 010 \rangle}} + {\ensuremath{| 100 \rangle}}) \end{aligned}$$ are the well-known genuinely multipartite entangled $GHZ$- and $W$-state, respectively. It can be seen that the bounds are non-zero for a considerable amount of multipartite entangled states, especially in the vicinity of the GHZ-state.\ In fact, our bounds are exact for GHZ-like states, i.e. states of the form ${\ensuremath{| gGHZ \rangle}}=\alpha {\ensuremath{| 0' \rangle}}^{\otimes n}+\beta {\ensuremath{| 1' \rangle}}^{\otimes n}$ wherein ${\ensuremath{| 0' \rangle}}\in\mathcal{H}_i$ and ${\ensuremath{| 1' \rangle}}\in\mathcal{H}_i$ are arbitrary mutually orthogonal vectors. By expanding ${\ensuremath{| gGHZ \rangle}}$ in terms of ${\ensuremath{| 0' \rangle}}$ and ${\ensuremath{| 1' \rangle}}$ analogously to (\[explicit\]) one finds $C(\rho_{A_{\gamma_i}})=2|\alpha \beta|\,\forall\,\gamma_i$, hence $C_{gme}(\rho)=2|\alpha \beta|$. In order to prove the exactness of the bound we choose ${\ensuremath{| \phi \rangle}}={\ensuremath{| 0' \rangle}}^{\otimes n}{\ensuremath{| 1' \rangle}}^{\otimes n}$ for inequality II which then yields $2I\left[{\ensuremath{| gGHZ \rangle}}{\ensuremath{\langle gGHZ |}},{\ensuremath{| \phi \rangle}}\right]=2|\alpha \beta|$. In fact we already know from the results of Ref. [@Huber10], that the inequality will detect a huge amount of genuinely multipartite entangled mixed states in arbitrary high dimensional and multipartite systems. In all of these situations we thus also have a lower bound on the gme-concurrence.\ \ **Experimental Implementation**\ In order to be useful in practice, measures for multipartite entanglement need to be experimentally implementable by means of local observables (since all particles of composite quantum systems may not be available for combined measurements) without resorting to a full quantum state tomography (since the latter requires a vast number of measurements, which is unfeasible in practice). The bound (\[lowerbound\]) satisfies these demands, as for fixed ${\ensuremath{| \Phi \rangle}}$ its computation only requires at most the square root of the number of measurements needed for a full state tomography. Furthermore it can be implemented locally as explicitly shown in [@Huberqic]. In an experimental situation where one aims at producing a certain state ${\ensuremath{| \psi \rangle}}$, it is now possible to choose the corresponding ${\ensuremath{| \phi \rangle}}$ and not only detect the state as being genuinely multipartite entangled, but also have a reliable statement about the amount of multipartite entanglement the state exhibits. Even if the produced state deviates from the desired states, the criteria are astonishingly noise robust (as e.g. analyzed in Ref. [@Huber10]), as for example a GHZ state mixed with white noise is shown to be genuinely multipartite entangled with a white noise resistance of $\approx57\%$.\ \ **Conclusion**\ We introduced a measure of genuine multipartite entanglement, which can be lower bounded by means of one of the currently most powerful detection criteria. These bounds are experimentally implementable and computationally very efficient, allowing to not only detect, but also to quantify genuine multipartite entanglement in an experimental scenario. This has grave implications on applications where genuine multipartite entanglement is a crucial resource (as e.g. in quantum computing [@BRaussendorf01] or cryptopgraphy [@SHH3]) and might allow to give a good estimate of the relevance of genuine multipartite entanglement in other physical systems (as e.g. in biological systems [@Caruso] or quantum spin chains [@spinchains]).\ \ [**Acknowledgement**]{}: A. Gabriel, M. Huber and Ch. Spengler gratefully acknowledge the support of the Austrian FWF (Project P21947N16). J.L. Chen is supported by NSF of China (Grant No.10975075). Z.H. Ma is supported by NSF of China(10901103), partially supported by a grant of science and technology commission of Shanghai Municipality (STCSM, No.09XD1402500).\ \ **Appendix**\ Let us finish by proving the lower bound for the general n-qudit case. For the most general pure n-qudit state $|\psi\rangle=\sum_{i_1,i_2,(\cdots),i_n}c_{i_1,i_2,(\cdots),i_n}|i_1i_2(\cdots)i_n\rangle$ the squared concurrences $C^2(\rho_\gamma)=2(1-\text{Tr}(\rho^2_\gamma))$ with respect arbitrary bipartitions ($\gamma$) always take the form $$\begin{aligned} C^2(\rho_\gamma)=4|c_{00(\cdots)0}c_{11(\cdots)1}-c_{\alpha(\gamma)}c_{\beta(\gamma)}|^2+F_\gamma\,,\\\end{aligned}$$ where $F_\gamma$ are non-negative functions (see e.g. Refs. [@HH2; @HHK1] for details on how to calculate the linear entropies of arbitrary subsystems). For every bipartition $\gamma$ there exists one pair $\alpha(\gamma)$ and $\beta(\gamma)$ that can be retrieved from $$\begin{aligned} \{\alpha(\gamma),\beta(\gamma)\}=\pi_\gamma\{00(\cdots)0,11(\cdots)1\}\,\end{aligned}$$ where $\pi_\gamma$ permutes every number from the subset defined by $\gamma$ from the first half of the joint set with the second. Thus $$\begin{aligned} C(\rho_\gamma)\geq 2|c_{00(\cdots)0}c_{11(\cdots)1}-c_{\alpha(\gamma)}c_{\beta(\gamma)}|\,\end{aligned}$$ will hold also for every $\gamma$. Now for the GME-concurrence we can infer $$\begin{aligned} &\min_\gamma\{C(\rho_\gamma)\} \geq 2|c_{00(\cdots)0}c_{11(\cdots)1}|-(\sum_\gamma|c_{\alpha(\gamma)}c_{\beta(\gamma)}|)\, =: B \,.\end{aligned}$$ Now for any given mixed state the convex roof construction is bounded by $$\begin{aligned} C_{gme}(\rho)\geq\inf_{\{p_i,|\psi_i\rangle\}}\sum_i p_iB_i\,.\end{aligned}$$ For the choice $|\Phi\rangle=|0\rangle^{\otimes n}\otimes|1\rangle^{\otimes n}$, inequality (\[ineqII\]) reads $$\begin{aligned} I[\rho,|\Phi\rangle]=|\rho_{00(\cdots)011(\cdots)1}|-\sum_\gamma\sqrt{\rho_{\alpha(\gamma)\alpha(\gamma)}\rho_{\beta(\gamma)\beta(\gamma)}}\leq 0\,.\end{aligned}$$ Now $$\begin{aligned} 2|\rho_{00(\cdots)011(\cdots)1}|\leq\inf_{\{p_i,|\psi_i\rangle\}}\sum_i p_i2|c^i_{00(\cdots)0}c^i_{11(\cdots)1}|\,,\end{aligned}$$ due to the triangle inequality and $$\begin{aligned} \sqrt{\rho_{\alpha(\gamma)\alpha(\gamma)}\rho_{\beta(\gamma)\beta(\gamma)}}\geq\inf_{\{p_i,|\psi_i\rangle\}}\sum_i p_i2|c_{\alpha(\gamma)}c_{\beta(\gamma)}|\,,\\\end{aligned}$$ due to the Cauchy-Schwarz inequality.\ This leads to a lower bound on the convex roof construction $$\begin{aligned} C_{gme}(\rho)\geq 2I[\rho,|\Phi\rangle]\,.\end{aligned}$$ And again due to the local unitary invariance of $C_{gme}(\rho)$ this proves our lower bound for all $|\Phi\rangle$. 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--- abstract: 'We study the inertia stack of $[\mathcal{M}_{0,n}/S_n]$, the quotient stack of the moduli space of smooth genus $0$ curves with $n$ marked points via the action of the symmetric group $S_n$. Then we see how from this analysis we can obtain a description of the inertia stack of $\mathcal{H}_g$, the moduli stack of hyperelliptic curves of genus $g$. From this, we can compute additively the Chen–Ruan (or orbifold) cohomology of $\mathcal{H}_g$.' address: 'KTH Matematik, Lindstedtsvägen 25, S-10044 Stockholm\' author: - Nicola Pagani title: '**The orbifold cohomology of moduli of hyperelliptic curves**' --- Introduction ============ A hyperelliptic curve of genus $g$ is a smooth algebraic curve that admits a $2:1$ map to $\mathbb{P}^1$, and thus has $2g+2$ branch points. From its very definition, it is clear that the moduli stack of genus $g$ hyperelliptic curves $\mathcal{H}_g$ admits a map onto the moduli stack $[\mathcal{M}_{0,2g+2}/S_{2g+2}]$, which is an isomorphism at the level of coarse moduli spaces. The foundations for moduli of hyperelliptic curves, as well as the precise definition of the previous map, can be found in [@lonsted] (in particular Theorem 5.5). The last decade has seen tremendous improvements in our understanding of the moduli space of hyperelliptic curves $\mathcal{H}_g$. We mention here some of the recent achievements that are relevant to the present work. In the paper [@arsievistoli], $\mathcal{H}_g$ is described as a moduli stack of cyclic covers of the projective line. As a consequence of this description, the authors are able to determine its Picard group. Along these lines, the Picard group of the Deligne-Mumford compactification $\overline{\mathcal{H}}_g$ was computed (see [@cornpic]), and very recently the whole integral Chow ring of $\mathcal{H}_g$ was computed in [@fulghesu] (see also [@edidin], [@gorviv]). In the last years, much effort was also made in studying the automorphism groups of hyperelliptic curves [@shaska1], [@shaska2], [@shaska3], [@shaska4]. In this paper we deal with rational cohomology and Chow group with rational coefficients. From both these points of view, the moduli stacks $\mathcal{H}_g$ are trivial. The triviality of $H^*(\mathcal{H}_g, \mathbb{Q})$ follows from [@kisinlehrer Theorem 2.13], while the triviality of $A^*_{\mathbb{Q}}(\mathcal{H}_g)$ follows from its description as finite quotient of the affine variety $\mathcal{M}_{0,n}$. Still some nontriviality can be measured with rational coefficients, but one has to consider instead the *orbifold cohomology* or the *stringy Chow group*. The orbifold cohomology as a vector space (or Chen–Ruan cohomology) of an orbifold $\mathcal{X}$ is obtained by adding to the usual cycles of $\mathcal{X}$ the cycles of all the *twisted sectors* of $\mathcal{X}$. The twisted sectors are orbifolds that parametrize pairs $(x, g)$ where $x$ is a point of $\mathcal{X}$ and $g \in \operatorname{Aut}(x)$. The new cycles are then given an unconventional degree, which is the sum of their original degree as cycles inside their twisted sector $Y$, plus a rational number (called *age* or *degree shifting number*) that depends on the normal bundle $N_Y \mathcal{X}$. The orbifold cohomology of moduli spaces of curves is studied in [@pagani1], [@pagani2], [@spencer2] (see also the PhD thesis [@paganitesi], [@spencer]). The present work has some nontrivial intersection with [@pagani2] and [@spencer2], since in these two papers in particular the orbifold cohomology and stringy Chow group of $\mathcal{M}_2= \mathcal{H}_2$ are described. The main result of this paper is Theorem \[principale\], where we give for any $g$ a closed formula for the *orbifold Poincaré polynomial* of $\mathcal{H}_g$, that is, a [^1], whose coefficient of $q^i$ corresponds to the dimension of the group $H^{i}$. To achieve this result, we first describe in Section \[sezione2\] the twisted sectors of $[\mathcal{M}_{0,n}/S_n]$ as quotients of certain $\mathcal{M}_{0,k}$ modulo a subgroup of $S_k$. Then, in Section \[sezione3\], we study the twisted sectors of $\mathcal{H}_g$. If $g$ is odd, we see that the twisted sectors of $\mathcal{H}_g$ are simply the twisted sectors of $[\mathcal{M}_{0,2g+2}/S_{2g+2}]$ repeated twice. If $g$ is even, most of the twisted sectors of $\mathcal{H}_g$ correspond to the twisted sectors of $[\mathcal{M}_{0,2g+2}/S_{2g+2}]$, whose distinguished automorphism is not an involution, repeated twice. The remaining few twisted sectors of $\mathcal{H}_g$ are still described as quotients of moduli of genus $0$, pointed curves modulo the action of a certain subgroup of the symmetric group on the marked points. Finally, in Section \[sezione4\] we compute all the degree shifting numbers, and we write the explicit results by recollecting the results of the previous sections. Notation -------- We work over $\mathbb{C}$; cohomologies and Chow groups are taken with rational coefficients. Orbifold for us means smooth Deligne–Mumford stack, and we always work within the category of Deligne–Mumford stacks. If a finite group $G$ acts on a scheme (stack) $X$, $[X/G]$ is the stack quotient and $X/G$ is the quotient as a scheme. We call $\mu_N:= \mathbb{Z}_N^{\vee}$ the group of characters of $\mathbb{Z}_N$, and $\mu_N^*$ the subgroup whose elements are the invertible characters. We make an implicit use of the relative language of schemes. For instance, when no confusion can arise, we speak of a genus $g$ smooth curve, meaning a family of genus $g$ smooth curve over a certain base $S$. Definition of Orbifold Cohomology ================================= In this section we define orbifold cohomology. For a more detailed study of this topic, we address the reader to [@agv2 Section 3] for the various inertia stacks, and to [@agv2 Section 7.1] for the degree shifting number (the original reference is [@chenruan]). What we call orbifold cohomology is the graded vector space underlying the Chen–Ruan cohomology ring (or algebra): the latter is a more refined object that we will not introduce in this work. We introduce the following natural stack associated to a Deligne–Mumford stack $X$, which points to where $X$ fails to be an algebraic space. \[definertia\] ([@agv1 4.4], [@agv2 Definition 3.1.1]) Let $X$ be an algebraic stack. The *inertia stack* $I(X)$ of $X$ is defined as $$I(X) := \coprod_{N \in \mathbb{N}} I_N(X)$$ where $I_N(X)(S)$ is the following groupoid: 1. The objects are pairs $(\xi, \alpha)$, where $\xi$ is an object of $X$ over $S$, and $\alpha: \mu_N \to \operatorname{Aut}(\xi)$ is an injective homomorphism. 2. The morphisms are the morphisms $g: \xi \to \xi'$ of the groupoid $X(S)$, such that $g \cdot \alpha(1)= \alpha'(1) \cdot g$. We also define $I_{TW}(X):= \coprod_{N > 1}I_N(X)$, in such a way that $$I(X)=I_1(X) \coprod I_{TW}(X).$$ The connected components of $I_{TW}(X)$ are called *twisted sectors* of the inertia stack of $X$, or also twisted sectors of $X$. The inertia stack comes with a natural forgetful map $f:I(X) \to X$. We observe that, by our very definition, $I_N(X)$ is an open and closed substack of $I(X)$, but it rarely happens that it is connected. One special case is when $N$ equals to $1$: in this case the map $f$ restricted to $I_1(X)$ induces an isomorphism of the latter with $X$. The connected component $I_1(X)$ will be referred to as the *untwisted sector*. We also observe that given a generator of $\mu_N$, we obtain an isomorphism of $I(X)$ with $I'(X)$, where the latter is defined as the ($2$-)fiber product $X \times_{X \times X} X$ where both morphisms $X \rightarrow X \times X$ are the diagonals. Over more general fields than $\mathbb{C}$, the object we defined in \[definertia\] is usually called *cyclotomic inertia stack*, whereas the *inertia stack* is the above ($2$-)fiber product. We address the reader to [@agv2 Section 3] for more details on the inertia stack and its variants. \[mappaiota\] There is an involution $\iota: I_N(X) \to I_N(X)$, which is induced by the map $\iota': \mu_N \to \mu_N$, that is $\iota'(\zeta):= \zeta^{-1}$. \[liscezza1\] [@agv2 Corollary 3.1.4] Let $X$ be a smooth algebraic stack. Then the stacks $I_N(X)$ (and therefore $I(X)$ itself) are smooth. We now study the behaviour of the inertia stack under arbitrary morphisms of stacks. \[pullinertia\] Let $f: X \to Y$ be a morphism of stacks. We define $f^*(I(Y))$ as the stack that makes the following diagram $2-$cartesian: $$\xymatrix{f^*(I(Y)) \ar[r]^{I(f)} \ar[d] \ar@{}|{\square}[dr] & I(Y) \ar[d] \\ X \ar[r]^{f} & Y }$$ and $I(f)$ as the map that lifts $f$ in the diagram. Obviously, there is an induced map that we call $I'(f)$, which maps $I(X) \to f^*(I(Y))$. We now define the degree shifting number for the twisted sectors of the inertia stack of a smooth stack $X$. With $R{\mu_N}$, we denote the representation ring of $\mu_N$. [@agv2 Section 7.1] Let $\rho:\mu_N \to \mathbb{C}^*$ be a group homomorphism. It is determined by an integer $0 \leq k \leq N-1$ as $\rho( \zeta_N)= \zeta_N^k$. We define a function *age*: $$\textrm{age}(\rho)=k/N.$$ This function extends to a unique group homomorphism: $$\textrm{age}: R \mu_N \to \mathbb{Q}.$$ We now define the age of a twisted sector $Y$ of a smooth stack $X$. Let $f$ be the restriction to $Y$ of the natural forgetful map $I(X) \to X$. \[definitionage\] ([@chenruan Section 3.2], [@agv2 Definition 7.1.1]) Let $Y$ be a twisted sector and $g: \operatorname{Spec}\mathbb{C} \to Y$ a point. Then the pull-back via $f \circ g$ of the tangent sheaf, $(f \circ g)^*(T_X)$, is a representation of $\mu_N$ on a finite dimensional vector space. We define $$a(Y):= \textrm{age}((f \circ g)^*(T_X)).$$ We can then define the orbifold, or Chen–Ruan, degree. \[defcoomorb2\] ([@chenruan Definition 3.2.2]) We define the $d-th$ degree orbifold cohomology group as follows: $$H^d_{CR}(X, \mathbb{Q}):= \bigoplus_i H^{d-2 a(X_i,g_i)}(X_i, \mathbb{Q})$$ where the sum is over all twisted sectors. The *orbifold Poincaré polynomial* of $X$ is $$P^{CR}_{X}(q):= \sum_{i \in \mathbb{Q}^+} \dim \left(H^{i}_{CR}(X) \right) q^i .$$ \[stringychow\] One can also define the stringy Chow group and its unconventional grading in complete analogy with the above definition. See [@agv2 Section 7.3] for this construction. The inertia stack of the configuration of unordered points on the Riemann sphere {#sezione2} ================================================================================ In this section, we study the cohomology of the inertia stack of $[\mathcal{M}_{0,n}/S_n]$ (also known in the literature as $\widetilde{\mathcal{M}}_{0,n}$). For this, it is enough to give a description of the coarse moduli spaces of the twisted sectors of the inertia stack of $[\mathcal{M}_{0,n}/S_n]$. We thus describe the coarse moduli spaces of the twisted sectors of the latter stack as quotients of the kind $\mathcal{M}_{0,k}/S$, for $S$ a certain subgroup of $S_k$. The cohomology of these quotients is well known. The cohomology of $\mathcal{M}_{0,n}$ was first computed as a representation of the symmetric group $S_n$ by Getzler [@getzleroperads 5.6] (see also [@kisinlehrer]). In particular, we shall use the following result. The Poincaré polynomial of $\mathcal{M}_{0,n+2}/S_n$ is $$P^0_{n+2;n,1,1}(q)=\sum_{i=0}^{n-1} q^i .$$ The Poincaré polynomial of $\mathcal{M}_{0,n+2}/S_n \times S_2$ is $$P^0_{n+2;n,2}(q) =\begin{cases} 1 & n=1 \\ \sum_{i=0}^{\lfloor \frac{n-2}{4}\rfloor} q^i+ q^{i+1} & n>1 . \end{cases}$$ It follows from [@kisinlehrer Theorem 2.9]. It will be convenient to have a definition for the set where each invertible character of $\mathbb{Z}_N$ is identified with its inverse: \[identificati\] We define $\widetilde{\mu}_N^*$ as the quotient set $\mu_N^*/\mathbb{Z}_2$ where $\overline{1} (\zeta_N):= \zeta_N^{-1}$. If $N$ is even, we define $\overline{\mu}_N^*$ to be the quotient set $\mu_N^*/\mathbb{Z}_2$ where the action of $\overline{1}$ is defined to be: $\overline{1} (\zeta_N):= -\zeta_N^{-1}$. The following proposition describes the inertia stack of $[\mathcal{M}_{0,n}/S_n]$. \[inertian\] We describe the coarse moduli space of the inertia stack of $[\mathcal{M}_{0,n}/S_n]$. 1. Suppose $N>2$, or $n$ odd. If there exists $a \in \{0,1,2\}$ such that $n=kN+a$: $$I_N([\mathcal{M}_{0,n}/S_n])= \begin{cases} \coprod_{\chi \in \widetilde{\mu}_N^*} \left(\mathcal{M}_{0,k+2}/S_k \times S_2, \chi \right) & a=0,2\\\coprod_{\chi \in {\mu_N^*}} \left(\mathcal{M}_{0,k+2}/S_k , \chi \right) & a=1 \end{cases}$$ and $I_N([\mathcal{M}_{0,n}/S_n])$ is empty otherwise. 2. if $n$ is even, $n=:2g+2$: $$I_2([\mathcal{M}_{0,n}/S_n]) = \left(\mathcal{M}_{0, g+2}/S_g \times S_2,-1 \right) \coprod \left(\mathcal{M}_{0, g+3}/S_{g+1} \times S_2,-1 \right)$$ Let $C$ be a smooth genus $0$ curve and $\alpha$ an automorphism of finite order $N$ of it. From the Riemann–Hurwiz formula, $\alpha$ has at least two fixed points, and if it had more it would be the identity. We can choose coordinates on $C$ in such a way that the two fixed points are $0$ and $\infty$, and $\alpha$ is the multiplication by a primitive $N-$th root of unity. Now let $C'= C / \langle \alpha \rangle$ be the quotient curve. Given a suitable choice of coordinates on $C$ and $C'$, the quotient map $C \to C'$ becomes the map $z \to z^N$. An automorphism $\alpha$ of $C$ is an automorphism of $C$ with $n$ unordered points if the unordered points are chosen in the subsets of $C$ that are invariant under the action of $\langle \alpha \rangle$. The finite subsets of $C$ invariant under $\langle \alpha \rangle$ contain exactly $k N + 2$ points: $k$ orbits of $N$ elements each plus the two points fixed by $\alpha$. Let us first deal with the case when $N>2$. In this case there is at most one choice of $k \in \mathbb{N}^+$ and $a \in \{0,1,2\}$ such that $n=kN +a$. The set of $n=k N+a$ marked points corresponds to a set of $k+a$ marked points on $C'$, where the $a$ points are a subset of the branch divisor. If $a$ is equal to $1$, there is a choice of a point $p$ in $C$ that is the only point that is both in the set of $n$ points, and a fixed point for $\alpha$. Then $\alpha$ determines a $\chi \in \mu_N^*$: the character of the representation of $\alpha$ on $T_p C$. From the set of $k$ unordered points on $C'$ plus two ordered (branch) points and a character $\chi \in \mu_N^*$, one can reconstruct the set of $n=k N+1$ points on $C$ and the automorphism $\alpha$. If $a$ equals $0$ or $2$, there is no such a choice of a distinguished point $p$ in the set of two fixed points of $\alpha$. Then $\alpha$ acts on the set of fixed points, thus determining two inverse characters in $\mu_N^*$. In the same way as before, $\alpha$ then gives an equivalence class in the set $\widetilde{\mu}_N^*$ (see Definition \[identificati\]). From the set of $k$ unordered points on $C'$ plus two unordered (branch) points and an element $\chi \in \widetilde{ \mu}_N^*$, one can reconstruct the set of $n=k N+a$ points on $C$ and the automorphism $\alpha$. If $N$ is equal to $2$, the argument is the same, the only differences being that $\widetilde{\mu}_2^*= \mu_2^*$ and that if $n$ is even there are two ways of choosing $k\in \mathbb{N}^+ ,a \in \{0,1,2\}$ such that $n=2 k+a$. \[stackyremark\] In [@agv2], the authors introduce two notions related to the inertia stack: the *stack of cyclotomic gerbes* ([@agv2 Definition 3.3.6]) and the *rigidified inertia stack* ([@agv2 3.4]), showing in [@agv2 3.4.1] that they are equivalent. By substituting $\mathcal{M}_{0,k+2}/S_k \times S_2$ with the stack quotient $[\mathcal{M}_{0,k+2}/S_k \times S_2]$ (respectively, $\mathcal{M}_{0,k+2}/S_k $ with $[\mathcal{M}_{0,k+2}/S_k]$) one obtains a stacky description of the rigidified inertia stack of $[\mathcal{M}_{0,n}/S_n]$. We have stated the earlier-mentioned proposition in this simplified way because this is enough for our purposes, and in this way we could avoid having to introduce the whole theory of inertia stack and its variants (see [@agv2 Section 3]). The inertia stack of moduli of smooth hyperelliptic curves {#sezione3} ========================================================== In this section we study the inertia stack of $\mathcal{H}_g$. We will implicitly use the fact that any family of hyperelliptic curves has a globally defined hyperelliptic involution, a result that follows from [@lonsted Theorem 5.5]. Let $$f: \mathcal{H}_g \to [\mathcal{M}_{0,2g+2}/S_{2g+2}]= \widetilde{\mathcal{M}}_{0,2g+2}$$ be the map that associates to every hyperelliptic genus $g$ curve, the corresponding genus $0$ curve, together with the degree $2g+2$ étale Cartier divisor $D$ obtained by considering the branch locus of the hyperelliptic involution. This map is well defined on families as a consequence of [@lonsted Theorem 5.5]. Let $C\to C'=C/\langle \tau \rangle$ be a hyperelliptic curve, and $\alpha$ an automorphism of it. Then $\alpha$ induces an automorphism $\alpha^{red}$ of $C'$. If $D$ is the degree $2g+2$ branch divisor of $C \to C'$, then $\alpha^{red}$ induces a bijection on the set of reduced points of $D$. We can thus use the order of $\alpha^{red}$ —instead of the order of $\alpha$—to reindex the components of the inertia stack: Let $I_N^{red}(\mathcal{H}_g)$ be the open and closed substack of $I(\mathcal{H}_g)$ whose objects correspond to pairs $(C, \alpha)$, where $C$ is an object of $\mathcal{H}_g$ and $\alpha^{red}: \mu_N \to \operatorname{Aut}(C/ \tau)$ is an injective homomorphism. For our purposes, it is more convenient to work with $I_N^{red}(\mathcal{H}_g)$ than with the usual $I_N(\mathcal{H}_g)$. Note that of course we have in the end that $$I(\mathcal{H}_g)= \coprod_{N \in \mathbb{N}} I_N(\mathcal{H}_g)= \coprod_{N \in \mathbb{N}} I_N^{red}(\mathcal{H}_g)$$ but with the latter decomposition, we have that the natural map of \[pullinertia\], $I'(f): I(\mathcal{H}_g) \to f^*\left(I([\mathcal{M}_{0,2g+2}/S_{2g+2}])\right)$ induces maps: $$I'(f)_N: I_N^{red}(\mathcal{H}_g) \to f^*\left(I_N([\mathcal{M}_{0,2g+2}/S_{2g+2}])\right) .$$ This is not the case for the standard decomposition of the inertia stacks, since an automorphism of order $N$ of the genus $0$ curve can lift to an automorphism of order $N$ or to an automorphism of order $2N$ on the corresponding hyperelliptic curve. Let $n=2g+2$ be the number of Weierstrass points, and $N=\operatorname{ord}( \alpha^{red})$. It is convenient to write (if possible) $n= kN+a$ for $k \in \mathbb{N}^+, a \in \{0,1,2\}$ (following the results of Section \[sezione2\]). If $N>2$ such a decomposition of $n$ is unique. The number $a$ is the number of Weierstrass points whose image in the quotient via the hyperelliptic involution is a branch point for $\alpha^{red}$. We label each twisted sector by a character. If $a$ is equal to zero, then there are four points in $C$ whose image in $C / \tau$ consists of the two points fixed by $\alpha^{red}$. In this case the automorphism $\alpha$ can: 1. fix the four points; 2. exchange them two-by-two; 3. fix two of them and exchange the other two. In the first two cases, we label the twisted sectors by a pair $(\chi,1)$ or $(\chi,-1)$ respectively. \[inertiahg\] We describe the coarse moduli space of the inertia stack of the moduli stack of smooth hyperelliptic curves of genus $g$: $\mathcal{H}_g$. 1. Suppose $N>2$. If there exists $a \in \{0,1,2\}$ such that $2 g+2=k N+a$: $$I_N^{red}(\mathcal{H}_{g})= \begin{cases} \coprod_{\chi \in \widetilde{\mu}_N^*, \lambda \in{\pm1}} \left(\mathcal{M}_{0,k+2}/S_k \times S_2, (\chi,\lambda) \right)& a=0, \ k \textrm{ even} \\ \coprod_{\chi \in {\mu_N}^*} \left(\mathcal{M}_{0,k+2}/S_k \times S_2, \chi \right) & a=0, \ k \textrm{ odd}\\ \coprod_{\chi \in {\mu_N}^* \sqcup \mu_{2N}^*} \left(\mathcal{M}_{0,k+2}/S_k , \chi \right) & a=1 \\ \coprod_{\chi \in \widetilde{\mu}_{2N}^*} \left(\mathcal{M}_{0,k+2}/S_k \times S_2 , \chi \right) & a=2, \ k \textrm{ even, } N \textrm{even} \\ \coprod_{\chi \in \widetilde{\mu}_{N}^*\sqcup \widetilde{\mu}_{2N}^*} \left(\mathcal{M}_{0,k+2}/S_k \times S_2 , \chi \right) & a=2, \ k \textrm{ even, } N \textrm{odd} \\ \coprod_{\chi \in \overline{\mu}_{2N}^*} \left(\mathcal{M}_{0,k+2}/S_k \times S_2 , \chi \right) & a=2, \ k \textrm{ odd} \end{cases}$$ and $I_N^{red}(\mathcal{H}_g)$ is empty otherwise. 2. if $g$ is odd: $$I_2^{red}(\mathcal{H}_g) = \left(\mathcal{M}_{0, g+2}/S_g \times S_2, \zeta_4 \right) \coprod \left(\mathcal{M}_{0, g+2}/S_g \times S_2,\zeta_4^3 \right)\coprod$$ $$\coprod \left(\mathcal{M}_{0, g+3}/S_{g+1} \times S_2,(-1,1) \right)\coprod \left(\mathcal{M}_{0, g+3}/S_{g+1}\times S_2,(-1,-1) \right)$$ where $\{\zeta_4, \zeta_4^3\} = \overline{\mu}_4^*= \mu_4^*$. 3. if $g$ is even: $$I_2^{red}(\mathcal{H}_g)= \left( \mathcal{M}_{0,g+2}/S_{g},-1 \right)\coprod \left( \mathcal{M}_{0,g+3}/S_{g+1},-1 \right) .$$ First we observe that the morphism of \[pullinertia\]: $$I(f)_N: f^*\left(I_N([\mathcal{M}_{0,2g+2}/S_{2g+2}])\right) \to I_N([\mathcal{M}_{0,2g+2}/S_{2g+2}])$$ is a $\mu_2$-gerbe, and as such it induces an isomorphism at the level of coarse moduli spaces. Let us consider then $$I'(f)_N: I_N^{red}(\mathcal{H}_g) \to f^*\left(I_N([\mathcal{M}_{0,2g+2}/S_{2g+2}])\right).$$ This map is a $2:1$ étale cover because every automorphism of a genus $0$ curve with an invariant smooth effective divisor of degree $2g+2$ can be lifted exactly to two automorphisms of the corresponding hyperelliptic curve. To prove the two points $(1)$ and $(2)$, we prove that this is the trivial cover, and then apply the result of Proposition \[inertian\]. To prove point $(3)$, we show that in the particular case when $N= \operatorname{ord}(\alpha^{red})=2$ and $g$ is even, a lifting of $\alpha^{red}$ corresponds to a choice of a distinguished point $p$ in $D$, the branch divisor of $C \to C'$. Let $C$ be a hyperelliptic curve, $\alpha$ an automorphism of it and $\tau$ the hyperelliptic involution. We have the two projections on the quotient: $$\xymatrix{ C \ar[r]^{\hspace{-0.3cm}\pi} & C/\langle \tau \rangle \ar[r]^{\hspace{-0.3cm}p_N} & C/ \langle \tau, \alpha^{red} \rangle}.$$ After choosing suitable coordinates on $C/\langle\tau \rangle \cong \mathbb{P}^1$ and $C/ \langle \tau, \alpha^{red} \rangle \cong \mathbb{P}^1$, the map $p_N$ is simply the map $z \to z^N$. Let $R$ be the set of ramification points of $p_N$. The number of points in $R$ that are branch points for $\pi$ is then $a$, by its very definition. Now we study separately the three cases $a=0,1,2$. , then a hyperelliptic curve $C$ that admits an automorphism $\alpha$ of reduced order $N$ can be written as $$y^2=(x^N- \alpha_1) (x^N- \alpha_2) \ldots (x^N- \alpha_k)$$ with the automorphism $\alpha$: $$\begin{cases}x \to \zeta_N^i x\\ y \to \pm y. \end{cases}$$ Exchanging the coordinates $0, \infty$, the action of $\alpha$ becomes: $$\begin{cases}x \to \zeta_N^{-i} x\\ y \to \pm (\zeta_N)^{i(g+1)} y. \end{cases}$$ If $k$ is odd, then $\alpha$ fixes two of the points in $\pi^{-1}(R)$ and exchanges the other two. The action of $\alpha$ on the two fixed fibers determines the same character in $\mu_N^*$. If $k$ is even, then $\alpha$ can either fix the four points of $\pi^{-1}(R)$ or exchange them two-by-two. In both the cases, the action of $\alpha$ or $\alpha \tau$ on the four fixed points determines an element of $\widetilde{\mu}_N^*$. , then a hyperelliptic curve $C$ that admits an automorphism $\alpha$ of reduced order $N$ can be written as $$y^2=x (x^N- \alpha_1) (x^N- \alpha_2) \ldots (x^N- \alpha_k)$$ with the automorphism $\alpha$: $$\begin{cases}x \to \zeta_N^i x\\ y \to \pm \zeta_{2N}^i y. \end{cases}$$ In this case, if we call $p$ the point in $R$ that is also a branch point of $\pi$, then the action of $\alpha$ on $T_{\pi^{-1}(p)}$ determines a well-defined element of $\mu_N^*$ or $\mu_{2N}^*$. , then a hyperelliptic curve $C$ that admits an automorphism $\alpha$ of reduced order $N$ can be written as $$y^2=x (x^N- \alpha_1) (x^N- \alpha_2) \ldots (x^N- \alpha_k)$$ with the automorphism $\alpha$: $$\begin{cases}x \to \zeta_N^i x\\ y \to \pm \zeta_{2N}^i y. \end{cases}$$ Exchanging the coordinates $0, \infty$, the action of $\alpha$ becomes $$\begin{cases}x \to \zeta_N^{-i} x\\ y \to \pm \zeta_{2N}^{-i} (\zeta_N)^{i g} y. \end{cases}$$ In this case, the action of $\alpha$ fixes the two points in $\pi^{-1}(R)$. Then $\alpha$ induces a well-defined element of $\widetilde{\mu}_{2N}^*$ when $k$ is even and $N$ is even, of $\widetilde{\mu}_N^* \sqcup \widetilde{\mu}_{2N}^*$ when $k$ is even and $N$ is odd, and of $\overline{\mu}_{2N}^*$ when $k$ is odd (and therefore $N$ is even). Now for the point $(2)$, it is enough to check that our separate study in the different cases $a=0,1,2$ carries on also when $N= \operatorname{ord}(\alpha^{red})=2$, if $g$ is odd. The two remaining cases are when $g$ is even, $N=2$; therefore $a$ is equal to zero (then $k=g+1$ is odd), or $a$ is equal to two (then $k=g$ is also even). We have that $\mu_2^*= \widetilde{\mu}_4^*= \widetilde{\mu}_2^*$. In these cases, the action of $\alpha$ on $\pi^{-1}(R)$ distinguishes the two points of $R$. For example, if $a=2$, $k$ even, then $\alpha$ acts on the two points of $\pi^{-1}(R)$, on one of them with character $\zeta_4$ and on the other with character $\zeta_4^3$. In these cases therefore, the two $2:1$ étale covers, at the level of coarse moduli spaces, are the two quotient maps: $$\mathcal{M}_{0,g+2}/S_g \to \mathcal{M}_{0,g+2}/S_g \times S_2 \quad \textrm{and} \quad \mathcal{M}_{0,g+3}/S_{g+1} \to \mathcal{M}_{0,g+3}/S_{g+1} \times S_2.$$ The earlier-mentioned theorem could be restated as a stack-theoretic description of the rigidified inertia stack of $\mathcal{H}_g$ (cfr. Remark \[stackyremark\]), by substituting each occurrence of a quotient $\mathcal{M}_{0,k+2}/S$ ($S$ a subgroup of the symmetric group $S_n$), with the stack quotient $[\mathcal{M}_{0,k+2}/S]$. The Orbifold cohomology of smooth hyperelliptic curves {#sezione4} ====================================================== Here we compute the orbifold Poincaré polynomial for moduli of smooth hyperelliptic curves (see Definition \[defcoomorb2\]). Let us fix a hyperelliptic curve $C$ of genus $g$. A basis for the cotangent space $(T_C \mathcal{H}_g)^{\vee}$ is given by $$\left(\frac{d X}{Y}\right)^2 \quad X\left(\frac{d X}{Y}\right)^2 \quad \ldots \quad X^{2g-2} \left(\frac{d X}{Y}\right)^2$$ If $\alpha$ is an automorphism of $C$, it is straightforward to compute its action on each element of such a basis. What we have done so far gives us the possibility of writing a closed formula for $P_{\mathcal{H}_g}^{CR}$ for fixed $g\geq 2$. \[principale\] The orbifold Poincaré polynomial of moduli of smooth hyperelliptic curves is given by the formula $$P^{CR}_{\mathcal{H}_g}(q)=\sum_{(k,N,i) \in A_{2g+2}} q^{a_g(i,N)} P^0_{k+2;k,2}(q)+ \sum_{(k,N,i) \in A_{2g+1}} 2 q^{b_g(i,N)} P^0_{k+2;k,1,1}(q)+$$ $$+\sum_{(k,N,i) \in A_{2g}} q^{b_g(i,N)} P^0_{k+2;k,2}(q) +2+\begin{cases} q^{\frac{g-1}{2}} P^0_{g+3;g+1,1,1}(q)+ q^{\frac{g}{2}} P^0_{g+2;g,1,1}(q)& \textrm{if } g \textrm{ is even}\\2q^{\frac{g-1}{2}} P^0_{g+3;g+1,2}(q)+2 q^{\frac{g}{2}} P^0_{g+2;g,2}(q) & \textrm{if } g \textrm{ is odd}, \end{cases}$$ where the sets of indices are defined as $$A_{n}:=\left\{(k,N,i) \in \mathbb{N}^2 \times \mathbb{Z}_N^*| \ N>2, \ k N= n \right\}$$ and the exponents are $$a_g(i,N):= 2\left( 2g-1- \sum_{j=1}^{2g-1} \left\{ \frac{i (j+1)}{N}\right\} \right)$$ $$b_g(i,N):= 2 \left( 2g-1 -\sum_{j=1}^{2g-1} \left\{ \frac{i j}{N}\right\} \right).$$ By substituting all the Poincaré polynomials on the right-hand side of Theorem \[principale\] with $1$, one gets the whose coefficients in degree $i \in \mathbb{Q}^{\geq 0}$ are the dimensions of the stringy Chow group of degree $i$ (cf. Remark \[stringychow\]). This is so because all the twisted sectors of $\mathcal{H}_g$ have trivial Chow group, since their coarse moduli spaces are quotients of affine sets. In particular, we can write closed formulas for the total dimensions of the orbifold cohomology of $\mathcal{H}_g$. Let us define $$h_{CR}^h(g):= \dim H^*_{CR}(\mathcal{H}_g).$$ We denote with $\phi$ the Euler totient function. Then we can give a corollary of Theorem \[inertiahg\]: The following explicit formulas for the function just introduced hold: 1. If $g$ is even, $n=2g+2$: $$h_{CR}^g(n)= 3+ 2g + 2 \sum_{N>2| \ n=kN+1} k \phi(N) + 2 \sum_{N>2| \ n=kN, \textrm{or} \ n=kN+2} \lfloor \frac{k-2}{4}\rfloor \phi(N) .$$ 2. If $g$ is odd, $n=2g+2$: $$h_{CR}^g(n)= 2+ 4\left( \lfloor \frac{n-2}{4} \rfloor +\lfloor \frac{n-1}{4} \rfloor \right) + 2 \sum_{N>2| \ n=kN+1} k \phi(N) + 2 \sum_{N>2| \ n=kN, \textrm{or} \ n=kN+2} \lfloor \frac{k-2}{4}\rfloor \phi(N) .$$ Future directions ================= A series of natural questions arise as a consequence of this work. 1. One can address the question of studying the orbifold cohomology of the Deligne–Mumford compactification $\overline{\mathcal{H}}_g$. Using the theory of admissible double covers, it is straightforward to extend Proposition \[inertian\] and Theorem \[inertiahg\] to describe (respectively) the coarse moduli spaces of the closures of the inertia stacks of $[{\mathcal{M}}_{0,n}/S_n]$ and ${\mathcal{H}}_g$ inside the inertia stacks of $[\overline{\mathcal{M}}_{0,n}/S_n]$ and $\overline{\mathcal{H}}_g$ (a similar study is performed for $\mathcal{M}_3$ in [@paganitommasi Section 4]). It is an interesting combinatorial problem to describe and study all the remaining twisted sectors of $[\overline{\mathcal{M}}_{0,n}/S_n]$ and $\overline{\mathcal{H}}_g$. 2. Can one give a reasonably understandable description of the ring structure, thus computing the Chen–Ruan cohomology ring of $\mathcal{H}_g$ and $\overline{\mathcal{H}}_g$? This would require a good description of the so-called second inertia stack, or of the moduli stack of $3$-pointed stable maps[^2]: $$\overline{\mathcal{M}}_{0,3}(\mathcal{H}_g, 0), \quad \overline{\mathcal{M}}_{0,3}(\overline{\mathcal{H}}_g, 0),$$ in the same spirit of Theorem \[inertiahg\]. See [@pagani1] and [@pagani2] for a description of the Chen–Ruan cohomology ring of $\overline{\mathcal{M}}_{1,n}$ and $\overline{\mathcal{M}}_{2,n}$. 3. Another interesting geometric question is to study the inertia stack for the moduli stacks of $k$-gonal curves, $k>2$. In the case of the moduli stacks of *cyclic* $k$-gonal covers, one can adopt the same strategy of the present paper and solve the problem in two steps: studying the twisted sectors of the quotient stack of $\mathcal{M}_{0,n}$ by a certain subgroup of $S_n$ (in analogy with \[inertian\]), and studying a $\mu_k$-torsor over each such twisted sector (in analogy with \[inertiahg\]).  \ <span style="font-variant:small-caps;">Acknowledgments</span> The author is grateful to Gilberto Bini, Torsten Ekedahl, Carel Faber and Barbara Fantechi, for useful discussions and help. The author is grateful to the anonymous referees for useful suggestions and remarks. <span style="font-variant:small-caps;">Funding</span> This project was supported by the Wallenberg foundation, and took place at the Kungliga Tekniska Högskolan and was partly supported by [prin]{} “Geometria delle varietà algebriche e dei loro spazi di moduli”, by Istituto Nazionale di Alta Matematica. 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--- abstract: | We study a crime hotspot model suggested by Short-Bertozzi-Brantingham [@sbb]. The aim of this work is to establish rigorously the formation of hotspots in this model representing concentrations of criminal activity. More precisely, for the one-dimensional system, we rigorously prove the existence of steady states with multiple spikes of the following types: \(i) Multiple spikes of arbitrary number having the same amplitude (symmetric spikes), \(ii) Multiple spikes having different amplitude for the case of one large and one small spike (asymmetric spikes). We use an approach based on Liapunov-Schmidt reduction and extend it to the quasilinear crime hotspot model. Some novel results that allow us to carry out the Liapunov-Schmidt reduction are: (i) approximation of the quasilinear crime hotspot system on the large scale by the semilinear Schnakenberg model, (ii) estimate of the spatial dependence of the second component on the small scale which is dominated by the quasilinear part of the system. The paper concludes with an extension to the anisotropic case. author: - 'Henri Berestycki [^1]' - 'Juncheng Wei [^2]' - 'Matthias Winter [^3]' title: Existence of Symmetric and Asymmetric Spikes for a Crime Hotspot Model --- [**Key words:**]{} crime model, reaction-diffusion systems, multiple spikes, symmetric and asymmetric, quasilinear chemotaxis system, Schnakenberg model, Liapunov-Schmidt reduction [**AMS subject classification:**]{} Primary 35J25, 35 B45; Secondary 36J47, 91D25 Introduction: The statement of the problem ========================================== Pattern forming reaction-diffusion systems have been and are applied to many phenomena in the natural sciences. Recent works have also started to use such systems to describe macroscopic social phenomena. In this direction, Short, Bertozzi and Brantingham [@sbb] have proposed a system of non-linear parabolic partial differential equations to describe the formation of hotspots of criminal activity. Their equations are derived from an agent-based lattice model that incorporates the movement of criminals and a given scalar field representing the “attractiveness of crime”. The system in one dimension reads as follows: $$\begin{aligned} \nonumber A_{t} & =\varepsilon^{2}A_{xx}-A+\rho A+A_{0} (x), \ \mbox{in} \ (-L, L),\\ \rho_{t} & =D (\rho_{x}-2\frac{\rho}{A}A_{x})_{x}-\rho A+\gamma (x), \ \mbox{in} \ (-L, L). \label{sysoriginal}\end{aligned}$$ Here $A$ is the “attractiveness of crime” and $\rho$ denotes the density of criminals. The rate at which crimes occur is given by $\rho A$. When this rate increases, the number of criminals is reduced while the attractiveness increases. The second feature is related to the well documented occurrence of repeat offenses. The positive function $A_0 (x)$ is the intrinsic (static) attractiveness which is stationary in time but possibly variable in space. The positive function $\gamma (x)$ is the source term representing the introduction rate of offenders (per unit area). For the precise meanings of the functions $A_0 (x)$ and $\gamma (x)$, we refer to [@sbb; @sbbt; @soptbbc] and the references therein. This paper is concerned with the mathematical analysis of the one-dimensional version of this system. Let us describe our approach. Setting $$v=\frac{\rho}{A^{2}},$$ the system is transformed into $$\begin{aligned} \nonumber A_{t} & =\varepsilon^{2}A_{xx}-A+vA^{3}+A_{0} (x) \ \mbox{in} \ (-L, L),\\ (A^2v)_{t} & =D\left( A^{2}v_{x}\right) _{x}-vA^{3}+\gamma (x) \ \mbox{in} \ (-L, L). \label{sysdyn}\end{aligned}$$ We always consider Neumann boundary conditions $$A_x(-L)=A_x(L)=\rho_x(-L)=\rho_x(L)=v_{x}(-L)=v_x(L)=0.$$ Note that $v$ is well-defined and positive if $A$ and $\rho$ are both positive. The parameter $0<{\varepsilon}^2$ represents nearest neighbor interactions in a lattice model for the attractiveness. We assume that it is very small which corresponds to the temporal dependence of attractiveness dominating its spatial dependence. This models the case of attractiveness propagating rather slowly, i.e. much slower than individual criminals. It is a realistic assumption if the criminal spatial profile remains largely unchanged, or, in other words, if the relative crime-intensity does only change very slowly. This appears to be a reasonable assumption since it typically takes decades for dangerous neighborhoods, i.e. those attracting criminals, to evolve into safe ones and vice versa. Roughly speaking, a $k$ spike solution $(A, v)$ to (\[sysdyn\]) is such that the component $A$ has exactly $k$ local maximum points. In this paper, we address the issue of existence of steady states with multiple spikes in the following two cases: Symmetric spikes (same amplitudes) or asymmetric spikes (different amplitudes). Our approach is by rigorous nonlinear analysis. We apply Liapunov-Schmidt reduction to this quasilinear system. In this approach, to establish the existence of spikes, we derive the following new results: - Approximation of the crime hotspot system on the large scale of order one by the semi-linear Schnakenberg model (see Section 3, in particular equation (\[approx\])), - Estimate of the spatial dependence of the second component on the small scale of order ${\varepsilon}$, dominated by the quasilinear part of the system (see Section 6, in particular inequalities (\[estw3\]) – (\[estw1\])). We remark that asymmetric multiple spike steady states (of $k_1$ small and $k_2$ large spikes) are an intermediate state between two different symmetric multiple spike steady states of $k_1+k_2$ spikes (for which all spikes are fully developed) and $k_2$ spikes (for which the small spikes are gone). These rigorous results shed light on the formation of hotspots for the idealized model of criminal activity introduced in [@sbb]. Let us now comment on previous works. As far as we know, there are three mathematical works related to the crime model (\[sysdyn\]). Short, Bertozzi and Brantingham [@sbb] proposed this model based on mean field considerations. They have also performed a weakly nonlinear analysis on (\[sysoriginal\]) about the constant solution $$(A, \rho)= \left(\gamma +A_0, \frac{\gamma}{\gamma+A_0} \right)$$ assuming that both $A_0 (x)$ and $\gamma (x)$ are homogeneous. Rodriguez and Bertozzi have further shown local existence and uniqueness of solutions [@rb1]. In [@ccm], Cantrell, Cosner and Manasevich have given a rigorous proof of the bifurcations from this constant steady state. On the other hand, in the isotropic case, Kolokolnikov, Ward and Wei [@kww1] have studied existence and stability of multiple symmetric and asymmetric spikes for (\[sysdyn\]) using formal matched asymptotics. They derived qualitative results on competition instabilities and Hopf bifurcation and gave some extensions to two-space dimensions. The present paper provides rigorous justification for many of the results in [@kww1] and also derives some extensions. In particular, we establish here the following three new results: first, we reduce the quasilinear chemotaxis problems to a Schnakenberg type reaction-diffusion system and prove the existence of symmetric $k$ spikes. Second, this paper gives the first rigorous proof of the existence of asymmetric spikes in the isotropic case. Third, we study the pinning effect in an inhomogeneous setting $A_0 (x)$ and $\gamma (x)$. The stability of these spikes is an interesting issue which should be addressed in the future. We should mention that another model of criminality has been proposed and analyzed by Berestycki and Nadal [@bn]. In a forthcoming paper [@bw], we shall study the existence and stability of hotspots (spikes) in this system as well. It is quite interesting to observe that both models admit hotspot (spike) solutions. The structure of this paper is as follows. We formally construct a one-spike solution in Section 2 in which we state our main results. In Section 3 we show how to approximate the crime hotspot model by the Schnakenberg model. Section 4 is devoted to the computation of the amplitudes and positions of the spikes to leading order. Nondegeneracy conditions are derived in Section 5. These are required for the existence proof, given in Sections 6–8. In Section 6 we introduce and study the approximate solutions. In Section 7 we apply Liapunov-Schmidt reduction to this problem. Lastly, we solve the reduced problem in Section 8 and conclude the existence proof. In Section 9 we extend the proof of single spike solution to the case when both $A_0 (x)$ and $\gamma (x)$ are allowed to be inhomogeneous. Finally, in Section 10 we discuss our results and their significance and mention possible future work and open problems. Steady state: Formal argument for leading order and main results ================================================================ Before stating the main results, we first construct a time-independent spike on the interval $[-L,L]$ located at some point $x_0$. The construction here is carried out using classical matched asymptotic expansions. In the inner region, we assume that $v$ is a constant $v_0$ in leading order: $$v(x)\sim v_{0},\ \ \ |x-x_0|\ll 1.$$ Then, if $0<{\varepsilon}\ll 1$, the equation for $A$ becomes $$\varepsilon^2 A^{''}-A+ v_0 A^3 + A_0 (x)=0.$$ Rescaling $$A(x)=v_{0}^{-1/2} \hat{A}(y),\ \ \ \ y=\frac{x-x_0}{\varepsilon},$$ we get $$\hat{A}_{yy}-\hat{A}+ \hat{A}^3 + A_0 (x_0+\epsilon y) v_0^{1/2}=0.$$ We assume that $v_0 \to 0$ as ${\varepsilon}\to 0$. Then, at leading order, $\hat{A} (y) \sim w(y)$, where $w$ is the unique (even) solution of the following ODE $$w_{yy}-w+w^{3}=0$$ so that$$w(y)=\sqrt{2}\operatorname*{sech}\left( y\right) .$$ In the outer region, we assume that$$vA^{3}\ll1,\ \ \ \frac{x}{\varepsilon}\gg1$$ so that$$A\sim A_{0} (x).$$ We also assume that $D=\frac{\hat{D}}{{\varepsilon}^2}$, where $\hat{D}$ is a positive constant, and we estimate $$\int_{-L}^{L}vA^{3}\,dx\sim v_{0}^{-1/2}\varepsilon\int_{-\infty}^{\infty}w^{3}dy.$$ Integrating the second equation in (\[sysdyn\]), we then have $$\begin{gathered} \nonumber v_{0}^{-1/2}\varepsilon\int_{-\infty}^{\infty}w^{3}dy\sim\int_{-L}^L \gamma (x) dx,\\ v_{0}\sim\frac{\left( \int_{-\infty}^{\infty}w^{3}dy\right) ^{2}}{ \left(\int_{-L}^L \gamma (x) dx \right)^{2}}\varepsilon^{2}.\label{sys0}$$ We remark that $\int w^{3}\,dy=\int w\,dy=\sqrt{2}\pi$ so that$$v_{0}\sim\frac{2 \pi^{2}}{\left( \int_{-L}^L \gamma (x) \right) ^{2}} \varepsilon^{2}.$$ In particular, we obtain$$A(x)\sim\left\{ \begin{array} [c]{c}A_0(x)+ \dfrac{\sqrt{2} \int_{-L}^L \gamma (x) dx }{\varepsilon\pi}\,w\left(\frac{x-x_0}{\varepsilon} \right),\ \ \ x=O\left( \varepsilon\right), \\[3mm] A_{0}(x),\ \ \ \ x\gg O(\varepsilon). \end{array} \right. \label{Aunif}$$ Now we state our main theorems on the existence of multi-spike steady states for system (\[sysdyn\]). We discuss two cases. In the case of isotropic coefficients $A_0 (x) \equiv$ Constant, $\gamma (x)\equiv$ Constant, we will consider two types of solutions: \(i) Multiple spikes of arbitrary number having the same amplitude (symmetric spikes). \(ii) Multiple spikes having different amplitude for the case of one large and one small spike (asymmetric spikes). In the case of anisotropic coefficients $A_0 (x)$ and $\gamma (x)$, we will consider the existence of single spike solution. Our first result concerns the existence of multiple spikes of arbitrary number having the same amplitude (symmetric spikes). \[existencesym\] Assume that $D=\frac{\hat{D}}{{\varepsilon}^2}$ for some fixed $\hat{D}>0$ and $$\label{aniso} A_0(x)\equiv A_0, \gamma (x) \equiv \bar{A}-A_0 \ \ \mbox{where}\ \bar{A}>A_0.$$ Then, provided ${\varepsilon}>0$ is small enough, problem (\[sysdyn\]) has a $K$-spike steady state $(A_{\varepsilon},v_{\varepsilon})$ which satisfies the following properties: $$\label{aep} A_{{\varepsilon}}(x)= A_0+\frac{1}{{\varepsilon}}\sum_{j=1}^K \frac{1}{\sqrt{v_j^{\varepsilon}}} w \left(\frac{x-t_j^{\varepsilon}}{{\varepsilon}}\right)+O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right),$$ $$\label{hep} v_{\varepsilon}(t_i^{\varepsilon})={\varepsilon}^2 v_i^{\varepsilon},\quad i=1,\ldots,K,$$ where $$\label{tep0} t_i^{\varepsilon}\to t_i^0,\quad i=1,\ldots,K$$ with $$\label{limpos} t_i^0=\frac{2i-1-K}{K}\,L,\,i=1,\ldots,K$$ and $$\label{vep0} v_i^{\varepsilon}= v_i^0\left(1+O\left({\varepsilon}\log\frac{1}{{\varepsilon}} \right)\right) ,\quad i=1,\ldots,K$$ with $$\label{limamp} v_i^0=\frac{\pi^2 K^2}{2 (\bar{A}-A_0)^2 L^2},\quad i=1,\ldots,K.$$ [[ Note that in (\[aep\]) a two-term expansion of the solution $A_{\varepsilon}$ is given, where for each spike the term $\frac{1}{\sqrt{v_j^{\varepsilon}}} w \left(\frac{x-t_j^{\varepsilon}}{{\varepsilon}}\right)$ of order $O(\frac{1}{{\varepsilon}})$ is the leading term in the inner solution and the term $A_0$ of order $O(1)$ is the leading term in the outer solution. By using the operator $T[\hat{A}]$ defined in (3.12) this two-term expansion carries over to $\hat{v}$ as well. The same remark applies to (\[aep1\]) and (\[aep200\]). The two-term expansion agrees with that in [@kww1]. ]{} ]{} The next result is about asymmetric two-spikes. \[existenceas\] Under the same assumption as in Theorem \[existencesym\], with $D=\frac{\hat{D}}{{\varepsilon}^2}$ for some fixed $\hat{D}>0$ and suppose moreover that $$\frac{2\sqrt{\pi}(\hat{D} A_0^2)^{1/4}}{(\bar{A}-A_0)^{3/4}L} \leq 1, \label{condc}$$ and $$\frac{2\sqrt{\pi}(\hat{D} A_0^2)^{1/4}}{(\bar{A}-A_0)^{3/4}L} \not= \frac{2}{\sqrt{5}}. \label{condbc}$$ Then, for ${\varepsilon}>0$ small enough, problem (\[sysdyn\]) has an asymmetric $2$-spike steady state $(A_{\varepsilon},v_{\varepsilon})$ which satisfies the following properties: $$\label{aep1} A_{{\varepsilon}}(x)=A_0+\frac{1}{{\varepsilon}} \left( \sum_{j=1}^2 \frac{1}{\sqrt{v_i^{\varepsilon}}} w \left(\frac{x-t_i^{\varepsilon}}{{\varepsilon}}\right) +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right) \right),$$ $$\label{hep1} v_{\varepsilon}(t_i^{\varepsilon})={\varepsilon}^2 v_i^{\varepsilon},\quad i=1,\ldots,K,$$ where $t_i^{\varepsilon}$ and $v_i^{\varepsilon}$ satisfy (\[tep0\]) and (\[vep0\]), respectively. The limiting amplitudes $v_i^0$ and positions $t_i^0$ are given as solutions of (\[amp1\]) and (\[amp6\]). Condition (\[condbc\]) is a kind of nondegeneracy condition. Note that in the case of asymmetric spikes we explicitly characterize the points of non-degeneracy. The last theorem is about the existence of single spike solution in the anisotropic case \[existenceani\] Assume that ${\varepsilon}>0$ is small enough and $D=\frac{\hat{D}}{{\varepsilon}^2}$ for some fixed $\hat{D}>0$. Then, problem (\[sysdyn\]) has a single spike steady state $(A_{\varepsilon},v_{\varepsilon})$ which satisfies the following properties: $$\label{aep200} A_{{\varepsilon}}(x)= A_0 (x)+\frac{1}{{\varepsilon}} \frac{1}{\sqrt{v_0^{\varepsilon}}} w \left(\frac{x-t_0^{\varepsilon}}{{\varepsilon}}\right)+O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right),$$ $$\label{hep200} v_{\varepsilon}(t_0^{\varepsilon})={\varepsilon}^2 v_0^{\varepsilon},$$ where $$\label{tep000} t_0^{\varepsilon}\to t_0, \ \int_{-L}^{t_0} \gamma (x) dx = \int_{t_0}^L \gamma (x) dx$$ and $$\label{vep000} v_0^{\varepsilon}= \frac{ 2\pi^2}{ (\int_{-L}^L \gamma (x) dx)^2} \left(1+ O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)\right).$$ We notice that in the anisotropic case, the single spike location is only determined by the function $\int_{-L}^{x} \gamma (t) dt$ and $A_0 (x)$ has no effect at all. Note also that the location $t_0$ is uniquely determined by the condition $$\int_{-L}^{t_0} \gamma (x) dx= \frac{1}{2} \int_{-L}^L \gamma (x) dx.$$ With more computations, it is possible to construct multiple asymmetric spikes in the isotropic case, and also multiple spikes in the anisotropic case. Since the statements and computations are complicated, we will not present them here. We refer to [@ww-pre] for some results in this direction. Scaling and approximation by the Schnakenberg model =================================================== We will use the following notation for the domain and the rescaled domain, respectively: $$\Omega= (-L, L),\quad {\Omega}_{{\varepsilon}}=\left(-\frac{L}{{\varepsilon}},\frac{L}{{\varepsilon}}\right).$$ This section is devoted to the reduction of the system (\[sysdyn\]) to a particular Schnakenberg type reaction diffusion equation in which no chemotaxis appears. Based on the computations in Section 2, we rescale the solution and the second diffusion coefficient as follows: $$A=A_0 (x)+\frac{1}{\epsilon} \hat{A},\quad v=\epsilon^2 \hat{v},\ \ \ D=\frac{\hat{D}}{\epsilon^2}.$$ Then the steady-state problem becomes $$\begin{aligned} \nonumber 0 & =\varepsilon^{2}\hat{A}_{xx}-\hat{A}+\hat{v} (\epsilon A_0 +\hat{A})^{3} +\varepsilon^3 A_0^{''},\ x\in{\Omega}, \\ 0 & =\hat{D} \left( \left(A_0 (x)+\frac{1}{\epsilon} \hat{A}\right)^2 \hat{v}_{x}\right) _{x}-\frac{1}{\epsilon} \hat{v} (\epsilon A_0(x) +\hat{A})^{3}+\gamma (x),\ x\in{\Omega}. \label{sys1}\end{aligned}$$ We will consider the case when ${\varepsilon}\ll 1$ and $\hat{D}$ is constant, with Neumann boundary conditions. A key observation of this paper is that the solutions of problem (\[sys1\]) are very close to the solutions of the Schnakenberg model $$\begin{aligned} \nonumber 0 & =\varepsilon^{2}\hat{A}_{xx}-\hat{A}+\hat{v} (\epsilon A_0 +\hat{A})^{3} +\varepsilon^3 A_0^{''},\ x\in{\Omega},\\ 0 & =\hat{D} \left( A_0^2 \hat{v}_{x}\right) _{x}-\frac{1}{\epsilon} \hat{v} (\epsilon A_0 +\hat{A})^{3}+\gamma (x),\ x\in{\Omega}, \label{schnak1}\end{aligned}$$ with Neumann boundary conditions. To see this, we first consider the following linear problem: $$\label{vequ} \left\{ \begin{array}{l} \hat{D} (a(x) v_x)_x=f(x),\quad -L<x<L, \\[3mm] v_x(-L)=v_x(L)=0, \end{array} \right.$$ where $a\in C^1(-L,L)$, $a(x)\geq c>0$ for all $x\in(-L,L)$ and $f\in L^1(-L,L)$. We compute $$a(x) v_x(x)=\int_{-L}^x \frac{1}{ \hat{D}} f(t)\,dt.$$ So $$\label{v1} v_x (x)=\frac{1}{ a(x)} \int_{-L}^x \frac{1}{ \hat{D}} f(t)\,dt$$ and hence $$\label{ss2} v(x)- v(-L)= \int_{-L}^x \frac{1}{ \hat{D} a(s)} \int_{-L}^s f(t) dt\,ds$$ which can be rewritten as $$\label{ss3} v(x)- v(-L)=\frac{1}{\hat{D}} \int_{-L}^x K_a (x, s) f(s)\, ds,$$ where $$K_a (x, s)=\int_s^x \frac{1}{a (t)} \,dt.$$ [We note that the kernel $K_a (x, s)$ is an even (odd) function if $a(x)$ is an odd (even) function. More precisely, if $a(x)=\pm a(-x)$, then $$\label{ksym} K_a (-x, -s)=\mp K_a (x,s).$$ ]{} [We note that $v$ is an even (odd) function if $f$ is even and $a$ is even (odd). More precisely, using $$\int_{-L}^0 f(t)\,dt=\frac{1}{2}\int_{-L}^L f(t)\,dt=0 \quad \mbox{if $f$ is even},$$ we compute $$v_x(x)=\frac{1}{a(x)}\int_0^x \frac{1}{\hat{D}}f(t)\,dt.$$ Integration yields $$v(x)-v(0)=\int_0^x \frac{1}{\hat{D}a(s)}\int_0^s f(t)\,dt\,ds$$$$= \frac{1}{\hat{D}} \int_{0}^x K_a(x,s)f(s)\,ds$$ and $$v(-x)-v(0)=\frac{1}{\hat{D}} \int_{0}^{-x} K_a(-x,s)f(s)\,ds$$ $$=-\frac{1}{\hat{D}} \int_{0}^{x} K_a(-x,-s)f(-s)\,ds$$ $$=\pm\frac{1}{\hat{D}} \int_{0}^{x} K_a(x,s)f(s)\,ds$$ $$=\pm(v(x)-v(0))$$ if $a$ is an even (odd) function using (\[ksym\]). Similarly, if $f$ is odd and $a$ is odd (even), then $v$ is an even (odd) function. ]{} Integrating (\[vequ\]), we derive the necessary condition $$\label{intconstr} \int_{-L}^L f(x)\,dx=0.$$ Note that on the other hand $v$ defined by (\[vequ\]) satisfies the boundary conditions $v_x(-L)=v_x(L)=0$ provided that (\[intconstr\]) holds. This follows from (\[v1\]). Let us now consider $a(x)= \left(A_0+ \frac{\gamma }{\epsilon} w(\frac{x}{\epsilon})\right)^2$, where $ w>0$ and $ w(y)\sim e^{-|y|}$ as $|y|\to\infty$. Then we claim that $$\label{ss4} K_{a} (x,s)= K_{A_0^2} (x,s)+ O(\epsilon |s-x|)+ O\left( \left|[s,x] \cap \left(-2\epsilon \log\frac{1}{\epsilon}, 2\epsilon \log\frac{1}{\epsilon}\right)\right|\right).$$ Note that (\[ss4\]) is an $L^\infty$ estimate for $K_a(x,s)$. In fact, we have $$\int_{s}^x \frac{1}{(A_0+\frac{1}{\epsilon} w)^2}\,dt = \int_s^x \frac{1}{A_0^2} dx+ \int_{s}^x \left[\frac{1}{(A_0+\frac{1}{\epsilon} w)^2}-\frac{1}{A_0^2} \right]\,dt,$$ where $$\int_{s}^x \left[\frac{1}{A_0^2}-\frac{1}{(A_0+\frac{1}{\epsilon} w)^2}\right]\,dt= \epsilon \int_{\frac{s}{\epsilon}}^{\frac{x}{\epsilon}} \frac{ 2\epsilon A_0 w+w^2}{ (\epsilon A_0+w)^2}\,dy$$ $$= \epsilon \int_{ [s/\epsilon, x/\epsilon] \cap \left\{ |y|>2 \log \frac{1}{\epsilon}\right\} } \ldots \,dy + \epsilon \int_{ [s/\epsilon, x/\epsilon] \cap \left\{ |y|<2 \log \frac{1}{\epsilon}\right\} }\ldots \,dy.$$ The first term is $O(\epsilon |x-s|)$ since $w=O({\varepsilon}^2)$ and so $ \frac{2\epsilon A_0 w+w^2}{ (\epsilon A_0+w)^2 }=O(\epsilon)$. For the second term, observing that $ \frac{2\epsilon A_0 w+w^2}{ (\epsilon A_0+w)^2 }=O(1)$ we derive (\[ss4\]). All these estimates are in the $L^\infty$ norm. Thus, $v$ satisfies: $$\label{ss5} v(x)- v(-L)=\frac{1}{\hat{D}} \int_{-L}^x K_{A_0^2} (x, s) f(s) ds +O\left(\epsilon \int_{-L}^x \left(|x-s|+\log\frac{1}{{\varepsilon}}\right)\, |f(s)|\, ds\right).$$ [The estimates (\[ss4\]) and (\[ss5\]) also hold if $$a (x)= \left(A_0 +\frac{\gamma}{\epsilon} \left(w\left(\frac{x-x_0}{\epsilon}\right) + \phi\right) \right)^2,$$ where $ \phi (x)$ satisfies $ |\phi (x)|\leq C \epsilon \max ( e^{- \frac{|x-x_0|}{2\epsilon}}, \sqrt{\epsilon})$. This is the class of functions that we will work with. This is also the motivation for our choice of the norm $\| \cdot \|_{*}$ (defined in (\[normdef\])). ]{} Therefore, we can approximate steady states for the crime hotspot model by the Schnakenberg model as follows: Given $\hat{A}>0$, let $\hat{v}=T[\hat{A}]$ be the unique solution of the following linear problem: $$\left\{ \begin{array}{l} \hat{D} \left( (A_0 +\frac{1}{\epsilon} \hat{A})^2 \hat{v}_{x}\right) _{x}-\frac{1}{\epsilon} \hat{v} (\epsilon A_0 +\hat{A})^{3}+\gamma (x)=0, \quad -L<x<L, \\[3mm] \hat{v}_{x}(-L)=\hat{v}_{x}(L)=0. \end{array} \right. \label{ta}$$ Then, by the maximum principle, the solution $T[\hat{A}]$ is positive. By the previous computations and remarks, if $ \hat{A}= w+\phi$ with $ |\phi |\leq C \epsilon \max ( e^{-\frac{|x-x_0|}{2\epsilon}}, \sqrt{\epsilon})$, it follows that $$\label{approx} T[\hat{A}]= v^{0} +O\left(\epsilon \log \frac{1}{\epsilon}\right) \quad\mbox{ in }H^2(-L,L),$$ where $v^0$ satisfies $$\left\{ \begin{array}{l} \hat{D} \left( A_0^2 v^0_{x}\right) _{x}-\frac{1}{\epsilon} v^0 (\epsilon A_0 +\hat{A})^{3}+\gamma (x)=0, \quad -L<x<L, \\[3mm] v^0_{x}(-L)=v^0_{x}(L)=0. \end{array} \right.$$ We adapt an approach based on Liapunov-Schmidt reduction which has been applied to the semilinear Schnakenberg model in [@iww2] and extend it to the quasilinear crime hotspot model. This method has also been used to study spikes for the one-dimensional Gierer-Meinhardt system in [@ww; @ww-pre] as well as two-dimensional Schnakenberg model in [@ww13]. We refer to the survey paper [@wei-survey] and the book [@ww-book] for references. Multiple asymmetric spikes for the one-dimensional Schnakenberg model have been considered using matched asymptotics in [@iww]. Existence and stability of localized patterns for the crime hotspot model have been studied by matched asymptotics in [@kww1] and results on competition instabilities and Hopf bifurcation have been shown including some extensions to two space dimensions. We remark that another approach for studying multiple spikes in one-dimensional reaction-diffusion system is the geometric singular perturbation theory in dynamical systems. For results and methods in this direction we refer to [@dgk; @dkp] and the references therein. Computation of the amplitudes and positions of the spikes ========================================================= In this section, we study (\[sys1\]) in the isotropic case (\[aniso\]). In particular, we compute the amplitudes and positions to leading order. We consider symmetric multi-spike solutions with any number of spikes and asymmetric multi-spike solutions with one small and large spike. We first write down the system for the amplitudes in case of a general number $K$ of spikes, where we have either $K$ spikes of the same amplitude or $k_1$ small and $k_2$ large spikes with $k_1+k_2=K$. We will first solve this system in the case of symmetric spikes. Then we will choose $k_1=k_2=1$ and solve this system in this special case of asymmetric spikes. Integrating the right hand side of the second equation in (\[sys1\]), we compute for $v_j=\lim_{{\varepsilon}\to 0} \hat{v}_{\varepsilon}(t_j^{\varepsilon})$: $$\label{amp1} \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}}= (\bar{A}-A_0)2L.$$ Solving the second equation in (\[sys1\]), using (\[ss2\]) in combination with the approximation (\[approx\]), we get $$\hat{v}_{\varepsilon}(x)- \hat{v}_{\varepsilon}(-L)= \int_{-L}^x \frac{1}{ \hat{D} (A_0+\frac{1}{{\varepsilon}}\hat{A}_{\varepsilon})^2} \int_{-L}^s \left( \frac{1}{{\varepsilon}}\hat{v}_{\varepsilon}({\varepsilon}A_0+\hat{A}_{\varepsilon})^3-\bar{A}+A_0 \right) dt\,ds$$$$=\int_{-L}^x \frac{1}{\hat{D} A_0^2} \int_{-L}^s \left( \frac{1}{{\varepsilon}}\hat{v}_{\varepsilon}({\varepsilon}A_0+\hat{A}_{\varepsilon})^3-\bar{A}+A_0 \right) dt\,ds +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)$$$$\int_{-L}^x \frac{1}{\hat{D} A_0^2} \left[ \sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{\hat{v}_{\varepsilon}(t_j^{\varepsilon})}} H(s-t_j^{\varepsilon})- \left(\sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{\hat{v}_{\varepsilon}(t_j^{\varepsilon})}} \right) \frac{s+L}{2L} \right] \,ds +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)$$$$= \frac{1}{\hat{D} A_0^2} \left[ \sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{\hat{v}_{\varepsilon}(t_j^{\varepsilon})}} (x-t_j^{\varepsilon})- \left(\sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{\hat{v}_{\varepsilon}(t_j^{\varepsilon})}}\right) \frac{(x+L)^2}{4L} \right] +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right),$$ where $$H(x)= \left\{ \begin{array}{l} 1 \quad \mbox{ if } x\geq 0, \\[3mm] 0 \quad \mbox{ if }x<0. \end{array} \right.$$ Taking the limit ${\varepsilon}\to 0$ and setting $x=t_i=\lim_{{\varepsilon}\to 0}t_i^{\varepsilon}$, we derive $$v_i= \frac{1}{\hat{D} A_0^2} \left[ \sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}(t_i-t_j)- \left(\sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}}\right) \frac{(t_i+L)^2}{4L} \right]+C_1 $$ $$\label{amp2} =\frac{1}{\hat{D}A_0^2}\left[ \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}}\frac{1}{2} |t_i-t_j| -\left(\sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}}\right) \frac{1}{4L}t_i^2 \right]+C_2$$ for some real constants $C_1,\,C_2$ independent of $i$, where the last identity in (\[amp2\]) uses (\[amp3\]) which we now explain. We use an assumption on the position of spikes that can be stated as follows: $$\label{fi} F_i(t_1^0,t_2^0,\ldots,t_K^0):= \frac{1}{2} \frac{\sqrt{2}\pi}{\sqrt{v_i}} +\sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}- \left( \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}} \right) \frac{t_i+L}{2L} =0,\quad i=1,\ldots,K.$$ Note that (\[fi\]) will be derived later on in Section 8 below (see equation (\[formulaf\])). We re-write (\[fi\]) and compute $$\frac{1}{2\sqrt{v_i}} +\sum_{j=1}^{i-1} \frac{1}{\sqrt{v_j}} -\left(\sum_{j=1}^{K} \frac{1}{\sqrt{v_j}}\right) \frac{t_i+L}{2L}$$ $$\label{amp3} =\sum_{j=1}^{i-1} \frac{1}{2\sqrt{v_j}} - \sum_{j=i+1}^{K} \frac{1}{2\sqrt{v_j}} - \left( \sum_{j=1}^{K} \frac{1}{\sqrt{v_j}} \right) \frac{t_i}{2L}=0.$$ From (\[amp2\]) and (\[amp3\]), we derive $$\label{amp5} v_{i+1}-v_i=\frac{\pi}{\sqrt{2} \hat{D}A_0^2} (t_{i+1}-t_i)\frac{1}{2} \left(\frac{1}{\sqrt{v_i}}- \frac{1}{\sqrt{v_{i+1}}} \right).$$ In the next two subsection we now solve these equations for the amplitudes of the spikes in the cases of both symmetric and asymmetric spikes. Symmetric spikes ---------------- We first consider the case of symmetric spikes, where $v_i=v$ is independent of $i=1,\ldots,K$, and compute the amplitude $v$ and the positions $t_i$. From (\[amp1\]), we get $$v=\frac{\pi^2 K^2}{2(\bar{A}-A_0)^2 L^2}.$$ From (\[amp5\]), the positions are $t_i=\left(-1+\frac{2i-1}{K}\right)L,\quad i=1,\ldots,K.$ The proof of the existence of multiple symmetric spikes follows from the construction of a single spike in the interval $\left(-\frac{L}{K},\frac{L}{K}\right)$. The proof of the existence of a single spike uses the implicit function theorem in the space of even functions, for which the Liapunov-Schmidt reduction method is not needed. This proof can easily be obtained by specializing the proof given for multiple asymmetric spikes given below. In this case, the proof can thus be simplified. Therefore we omit the details. Asymmetric spikes ----------------- Combining (\[amp3\]) and (\[amp5\]), we get $$v_iv_{i+1}=\frac{\pi}{\sqrt{2}\hat{D}A_0^2} \frac{L}{2} \frac{1}{\sum_{j=1}^K \frac{1}{\sqrt{v_j}}}.$$ This implies that there are only two different amplitudes which we denote by $v_s\leq v_l$ appearing $k_1$ and $k_2$ times, respectively. Hence we get $$\label{amp6} v_sv_l=\frac{\pi}{\sqrt{2}\hat{D}A_0^2} \frac{L}{2} \frac{1}{ \frac{k_1}{\sqrt{v_s}} + \frac{k_2}{\sqrt{v_l}} }.$$ Multiplying (\[amp3\]) by $\frac{t_i}{2}$ and subtracting (\[amp2\]) from the result we get $$v_i-C=\frac{\pi}{\sqrt{2}\hat{D}A_0^2} \sum_{j=1}^k\frac{t_j}{\sqrt{v_j}}\mbox{sgn}(t_j-t_i) +\frac{1}{\hat{D}A_0^2} \left(\sum_{j=1}^k\frac{\pi}{\sqrt{2}}\frac{1}{\sqrt{v_j}}\right) \frac{t_i^2}{2L},$$ where $$\mbox{sgn}(\alpha)=\left\{ \begin{array}{ll} +1 & \mbox{ if }\alpha>0, \\[2mm] 0 & \mbox{ if }\alpha=0, \\[2mm] -1 & \mbox{ if }\alpha<0. \end{array} \right.$$ Next we determine $v_s,\,v_l$ from (\[amp1\]) and (\[amp6\]). Substituting (\[amp1\]) into (\[amp6\]), we get $$v_l=\frac{1}{v_s}\frac{\pi^2}{\hat{D}A_0^2}\frac{1}{4} \frac{1}{\bar{A}-A_0}.$$ Plugging this equation into (\[amp1\]) gives $$C\left(z+\frac{1}{z}\right)=1,$$ where $$C =\frac{\sqrt{\pi}(\hat{D}A_0^2)^{1/4}}{(\bar{A}-A_0)^{3/4}L} \sqrt{k_1k_2}, \quad z= \frac{\sqrt{2v_s}(\hat{D}A_0^2)^{1/4}(\bar{A}-A_0)^{1/4}}{\sqrt{\pi}} \sqrt{\frac{k_2}{k_1}}.$$ To determine a solution, we need to satisfy the necessary condition $2C<1$ which can be summarized as $$\frac{2\sqrt{\pi}(\hat{D}A_0^2)^{1/4}}{(\bar{A}-A_0)^{3/4}L} \sqrt{k_1k_2}<1.$$ The second necessary condition is given by $v_s<v_l$ which is equivalent to $z< \sqrt{\frac{k_2}{k_1}}$. This implies the following cases: [**Case (i):**]{} $k_2\leq k_1$. If $$C\left(\sqrt{\frac{k_2}{k_1}}+\sqrt{\frac{k_1}{k_2}}\right) < 1$$ then there exists exactly one solution with $z\leq \sqrt{\frac{k_2}{k_1}}$. On the other hand, if $$C\left(\sqrt{\frac{k_2}{k_1}}+\sqrt{\frac{k_1}{k_2}}\right)> 1$$ then there exists no solution with $z<\sqrt{\frac{k_2}{k_1}}$. [**Case (ii):**]{} $k_2> k_1$. If $2C>1$ there is no solution. If $2C<1$ and $$C\left(\sqrt{\frac{k_2}{k_1}}+\sqrt{\frac{k_1}{k_2}}\right)< 1$$ then there exists exactly one solution with $z\leq \sqrt{\frac{k_2}{k_1}}$. If $2C<1$ and $$C\left(\sqrt{\frac{k_2}{k_1}}+\sqrt{\frac{k_1}{k_2}}\right)> 1$$ then there exist exactly two solutions with $z< \sqrt{\frac{k_2}{k_1}}$. [**Special case:**]{} $k_1=k_2=1$. Finally, we consider the special case $k_1=k_2=1$ which belongs to Case (i) in the previous classification and we have the following results: If $2C< 1$ then there is one solution with $z< 1. $ If $2C> 1$ then there is no solution with $z> 1. $ Existence and nondegeneracy conditions ====================================== We now describe a general scheme of Liapunov-Schmidt reduction. (We refer to the survey paper [@wei-survey] for more details.) Essentially this method divides the problem of solving nonlinear elliptic equations (and systems) into two steps. In the first step, the problem is solved up to multipliers of approximate kernels. In the second step one solves algebraic equations in terms of finding zeroes of the multipliers. In this section, we linearize (\[sys1\]) around the approximate solution and derive the linearized operator as well as its nondegeneracy conditions, i.e. conditions such that the resulting linear operator is uniformly invertible. Linearizing (\[sys1\]) around the solution, we get: $$\begin{aligned} \nonumber 0 & =\varepsilon^{2}\phi_{xx}-\phi+3\hat{v}({\varepsilon}A_0+\hat{A})^2\phi+\psi ({\varepsilon}A_0+\hat{A})^3\ \ \mbox{ in }{\Omega}, \\[3mm] \nonumber 0 & =\hat{D} \left( \left(A_0+\frac{1}{{\varepsilon}}\hat{A}\right)^2 \psi_{x}\right)_x +\hat{D} \left(2\left(A_0+\frac{1}{{\varepsilon}}\hat{A}\right) \frac{1}{{\varepsilon}}\phi \hat{v}_{x} \right)_x \\[3mm] & -\frac{3}{\epsilon} \hat{v}({\varepsilon}A_0+\hat{A})^2\phi -\frac{1}{\epsilon} \psi ({\varepsilon}A_0+\hat{A})^3 \ \ \mbox{ in }{\Omega}\label{evp}$$ with Neumann boundary conditions $$\phi_x(-L)=\phi_x(L)=\psi_x(-L)=\psi_x(L)=0.$$ Note that for the second equation of (\[evp\]) we have the necessary condition $$\int_{-L}^L \left(-\frac{3}{\epsilon} \hat{v}({\varepsilon}A_0+\hat{A})^2\phi -\frac{1}{\epsilon} \psi ({\varepsilon}A_0+\hat{A})^3 \right)\,dx=0$$ which follows by integrating the equation and using the Neumann boundary conditions for $\psi$ and $v$. In the limit ${\varepsilon}\to 0$ we get $$\sum_{j=1}^K \left[\psi_j \frac{\sqrt{2}\pi}{v_j^{3/2}} +3\int_{-\infty}^\infty w^2\phi_j\,dy \right]=0,$$ where $\psi_j=\psi(x_j)$. The second equation of (\[evp\]) can be solved as follows, using formula (\[ss2\]) and estimate (\[approx\]): $$\psi(x)- \psi(-L)= \int_{-L}^x \frac{1}{ \hat{D} (A_0+\frac{1}{{\varepsilon}}\hat{A})^2} \int_{-L}^s \Bigg[ -\hat{D} \left( 2\left(A_0+\frac{1}{{\varepsilon}}\hat{A}\right) \frac{1}{{\varepsilon}}\phi \hat{v}_{x} \right)_x$$$$+\frac{3}{{\varepsilon}}\hat{v}({\varepsilon}A_0+\hat{A})^2\phi +\frac{1}{{\varepsilon}}\psi({\varepsilon}A_0+\hat{A})^3 \Bigg] dt\,ds$$$$=\int_{-L}^x \frac{1}{\hat{D} A_0^2} \int_{-L}^s \left( \frac{3}{{\varepsilon}}\hat{v}({\varepsilon}A_0+\hat{A})^2\phi +\frac{1}{{\varepsilon}}\psi({\varepsilon}A_0+\hat{A})^3 \right) dt\,ds +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right).$$ Note that the contributions from the term $$\hat{D} \left( 2\left(A_0+\frac{1}{{\varepsilon}}\hat{A}\right) \frac{1}{{\varepsilon}}\phi \hat{v}_{x} \right)_x$$ can be estimated by $O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)$ since $\phi$ vanishes in the outer expansion. In the limit ${\varepsilon}\to 0$, we get $$\psi(x)- \psi(-L)= \int_{-L}^x \frac{1}{\hat{D} A_0^2} \left[ \sum_{j=1}^{i-1} \psi_j \frac{\sqrt{2}\pi}{v_j^{3/2}}H(s-t_j) + 3\sum_{j=1}^{i-1} \int_{-\infty}^\infty w^2\phi_j\,dy H(s-t_j) \right] \,ds $$ $$=\frac{1}{\hat{D} A_0^2} \left[\sum_{j=1}^{i-1} \psi_j \frac{\sqrt{2}\pi}{v_j^{3/2}}(x-t_j) + 3\sum_{j=1}^{i-1} \int_{-\infty}^\infty w^2\phi_j\,dy\, (x-t_j)\right], \label{limeig}$$ where $t_{i-1}< x\leq t_{i}$. From now on, we consider the case of two spikes having different amplitudes (asymmetric spikes). Using the notation $$\Phi= \left(\begin{array}{l}\phi_1 \\[2mm] \phi_2 \end{array} \right),\quad \Psi= \left(\begin{array}{l}\psi_1 \\[2mm] \psi_2 \end{array} \right),\quad \omega= \left(\begin{array}{l}\omega_1 \\[2mm] \psi_2 \end{array} \right) = \left(\begin{array}{l}-\frac{3}{\hat{D}A_0^2}\int w^2\phi_1\,dy \\[2mm] -\frac{3}{\hat{D}A_0^2}\int w^2\phi_2\,dy \end{array} \right),$$ we can rewrite (\[limeig\]) for $x=t_i$ as follows: $$({\cal B}+{\cal C})\Psi=\boldmath{\omega},$$ where $${\cal C}=\frac{\sqrt{2}\pi}{\hat{D}A_0^2} \left( \begin{array}{cc} \displaystyle \frac{1}{v_s^{3/2}} & 0 \\[3mm] \displaystyle 0 & \displaystyle\frac{1}{v_l^{3/2}} \end{array} \right), \quad {\cal B}=\frac{1}{d_2} \left( \begin{array}{rr} 1 & -1\\ -1 & 1 \end{array} \right), \quad d_2=t_2-t_1.$$ Using ${\cal E}={\cal C}({\cal B}+{\cal C})^{-1}$, we get the following system of nonlocal eigenvalue problems (NLEPs) $$L\Phi:=\Phi_{yy}-\Phi+3w^2\Phi-3w^3\frac{\int w^2{\cal E}\Phi\,dy}{\int w^3\,dy}. \label{vecnlep}$$ Diagonalizing the matrix $\cal E$, we know from [@wei99; @wz] that (\[vecnlep\]) has a nontrivial solution iff ${\cal E}$ has eigenvalue $\lambda_e=\frac{2}{3}$. Thus it remains to compute the matrix ${\cal E}$ and its eigenvalues. We get $${\cal E}^{-1}=({\cal B}+{\cal C}){\cal C}^{-1}= {\cal B}{\cal C}^{-1}+I$$ $$=\frac{\hat{D}A_0^2}{\sqrt{2}\pi d_2} \left( \begin{array}{rr} v_s^{3/2} & -v_l^{3/2} \\ -v_s^{3/2} & v_l^{3/2} \end{array} \right) +I.$$ Then ${\cal E}^{-1}$ has the eigenvector $v_{m,1}=\frac{1}{\sqrt{2}}(1,-1)^T$ with eigenvalue $e_{m,1}=\frac{\hat{D}A_0^2}{\sqrt{2}\pi d_2} (v_s^{3/2}+v_l^{3/2})+1$ and the eigenvector $v_{m,2}=\frac{1}{\sqrt{v_s^3+v_l^3}}(v_l^{3/2},v_s^{3/2})^T$ with eigenvalue $e_{m,2}=1\not=\frac{3}{2}$. For nondegeneracy, the condition $e_{m,1}\not=\frac{3}{2}$ has to be satisfied, which is equivalent to $$\frac{\hat{D}A_0^2}{\sqrt{2}\pi d_2} (v_s^{3/2}+v_l^{3/2})\not= \frac{1}{2}.$$ Using the formulas for $d_2,\,v_s,\,v_l$, we compute $$\frac{\hat{D}A_0^2}{\sqrt{2}\pi d_2}(v_s^{3/2}+v_l^{3/2})$$$$=\frac{\hat{D}A_0^2}{\sqrt{2}\pi L}\left((\sqrt{v_s}+\sqrt{v_l})^3-3(\sqrt{v_s}v_l+v_s\sqrt{v_l})\right)$$$$=\frac{\hat{D}A_0^2}{\sqrt{2}\pi L}\left(\frac{(\bar{A}-A_0)^{3/2} L^3}{2\sqrt{2}(\hat{D}A_0^2)^{3/2}} -3 \frac{\pi L}{2\sqrt{2}\hat{D}A_0^2}\right)$$$$=\frac{(\bar{A}-A_0)^{3/2} L^2}{4\pi\sqrt{\hat{D}A_0^2}}-\frac{3}{4}\not=\frac{1}{2}.$$ This implies the condition $$\frac{2\sqrt{\pi}(\hat{D}A_0^2)^{1/4}}{(\bar{A}-A_0)^{3/4}L}\not= \frac{2}{\sqrt{5}}.$$ We have to exclude this point from our existence result Theorem \[existenceas\]. This is why we impose the condition (\[condbc\]) in Theorem \[existenceas\], which amounts to a nondegeneracy condition. If this condition is violated, we expect small eigenvalues to occur and it is an open question to know whether there will be spikes in this case. Approximate solutions ===================== For simplicity, we set $L=1$. In this section and the following we consider the case of general $K=1,2,\ldots$ since it does not cause any extra difficulty here, even in the case of asymmetric spikes. Let $ -1<t_1^0 < \cdots < t_j^0 < \cdots t_K^0 <1$ be $K$ points and let $v_j>0$ be $K$ amplitudes satisfying the assumptions (\[amp1\]), (\[amp2\]) and (\[amp3\]). Let $${\bf t}^0=(t_{1}^0,\ldots,t_{K}^0).$$ We first construct an approximate solution to (\[sys1\]) which concentrates near these prescribed $K$ points. Then we will rigorously construct an exact solution which is given by a small perturbation of this approximate solution. Let $-1<t_1<\cdots<t_j<\cdots<t_K<1$ be $K$ points such that $ {\bf t} = (t_1, \ldots, t_K) \in B_{{\varepsilon}^{3/4}} ( {\bf t}^0)$. Set $$\label{app11} w_j (x)= w \left( \frac{x-t_j}{{\varepsilon}} \right),$$ and $$\label{r0} r_0 =\frac{1}{10} \left( \min \left(t_1^0 +1, 1-t_K^0, \frac{1}{2}\min_{i \not = j} |t_i^0 -t_j^0|\right)\right).$$ Let $\chi: {\mathbb{R}}\to [0, 1]$ be a smooth cut-off function such that $\chi(x)=1$ for $|x|<1$ and $\chi(x)=0$ for $|x|>2$. We now define the approximate solution as $$\label{app1} \tilde{w}_j(x)= w_j (x) \chi\left(\frac{x-t_j}{r_0}\right).$$ It is easy to see that $\tilde{w}_j (x)$ satisfies $$\label{a11} {\varepsilon}^2 \tilde{w}_j^{''} - \tilde{w}_j + \tilde{w}_j^3 =\mbox{e.s.t.}$$ in $L^2(-1,1)$, where e.s.t. denotes an exponentially small term. Let $$\label{vector} \hat{A}=w_{{\varepsilon},{\bf t}}(x)= \sum_{j=1}^K \frac{1}{\sqrt{v_j^{\varepsilon}}} \tilde{w_j}(x),\quad \mbox{ where } v_j^{\varepsilon}=T[w_{{\varepsilon},{\bf t}}](t_j^{\varepsilon}),$$ $$\label{v} \hat{v}=T[w_{{\varepsilon},{\bf t}}], $$ where $T[A]$ is defined by (\[ta\]) and ${\bf t} \in B_{{\varepsilon}^{3/4}} ( {\bf t}^0)$. Then by (\[approx\]) we have $$\label{tauiia} v_i^{\varepsilon}:= T[\hat{A}] (t_i^{\varepsilon})= \lim_{{\varepsilon}\to 0} T[w_{{\varepsilon},{\bf t}}] (t_i^{\varepsilon}) +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right).$$ \[tauii\] Now let $x=t_i+{\varepsilon}y$. We find for $\hat{A}=w_{{\varepsilon}, {\bf t}}$: $$T[\hat{A}](t_i+{\varepsilon}y)-T[\hat{A}](t_i) =$$$$=\int_{t_i}^{t_i+{\varepsilon}y} \frac{1}{ \hat{D} \left(A_0+\frac{1}{{\varepsilon}}\hat{A}\right)^2} \int_{-L}^s \left( \frac{1}{{\varepsilon}}\hat{v}({\varepsilon}A_0+\hat{A})^3-\bar{A}+A_0 \right) dt\,ds$$$$={\varepsilon}^2 \int_{0}^y \frac{1}{ \hat{D} \left(A_0+\frac{1}{{\varepsilon}}w(\bar{s})\right)^2} \int_{0}^{\bar{s}} \frac{1}{{\varepsilon}} \frac{(w(\bar{t}))^3}{\sqrt{v_i^{\varepsilon}}} d\bar{t}\,d\bar{s}$$$$+{\varepsilon}^2 \int_{0}^y \frac{1}{ \hat{D} \left(A_0+\frac{1}{{\varepsilon}}w(\bar{s})\right)^2} \,d\bar{s} \frac{1}{{\varepsilon}} \left[ \int_{-\infty}^{0} \frac{(w(\bar{t}))^3}{\sqrt{v_i}}\,d\bar{t} +\sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}H(t_i-t_j)- \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}} \frac{t_i+L}{2L} \right]$$$$\times \left(1+O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)\right) $$$$= {\varepsilon}P_i^{\varepsilon}(y) +{\varepsilon}\int_{0}^y \frac{1}{ \hat{D} \left(A_0+\frac{1}{{\varepsilon}}w(\bar{s})\right)^2} \,d\bar{s} \left[ \frac{1}{2} \frac{\sqrt{2}\pi}{\sqrt{v_i}} +\sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}- \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}} \frac{t_i+L}{2L} \right]$$$$\times\left(1+O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)\right), \label{tax}$$ in $L^2({\Omega}_{\varepsilon})$, where $$P_i^{\varepsilon}(y)=\int_{0}^y \frac{1}{ \hat{D} \left(A_0+\frac{1}{{\varepsilon}}w(\bar{s})\right)^2} \int_{0}^{\bar{s}} \frac{(w(\bar{t}))^3}{\sqrt{v_i^{\varepsilon}}} d\bar{t}\,d\bar{s}$$ using (\[amp1\]). Note that $P_i^{\varepsilon}$ is an even function and the second term is an odd function in $y$. We now derive the following estimate for all $y\geq 0$: $$(T[w_{{\varepsilon}, {\bf t}}](t_i+{\varepsilon}y)-T[w_{{\varepsilon}, {\bf t}}](t_i))\,w_{{\varepsilon}, {\bf t}}^3 \leq C {\varepsilon}\int_0^y \frac{1}{\hat{D}\left(A_0+\frac{1}{{\varepsilon}} w(\bar{s})\right)^2}\,d\bar{s}\, w^3(y)$$$$\leq \frac{C}{\hat{D}} {\varepsilon}\int_0^y {\varepsilon}^2 \frac{1}{w^2(\bar{s})}\,d\bar{s}\,w^3(y)$$$$\leq \frac{C}{\hat{D}} {\varepsilon}^3 \int_0^y e^{2\bar{s}} \,d\bar{s} \,e^{-3y}$$$$\leq\frac{C}{\hat{D}} {\varepsilon}^3 e^{-y}.$$ For $y<0$ there is an obvious modification of this estimate. Further, it can be extended to cover both the cases when $w_{{\varepsilon}, {\bf t}}^3$ is replaced by ${\varepsilon}w_{{\varepsilon}, {\bf t}}^2$ or ${\varepsilon}^2 w_{{\varepsilon}, {\bf t}}$, respectively, giving the same upper bound in either case. Now if we define the following norm $$\| f\|_{**}= \| f \|_{L^2 (\Omega_\epsilon)} + \sup_{ - \frac{L}{\epsilon} <y <\frac{L}{\epsilon} } [\max(\min_{i} e^{-\frac{1}{2} |y-\frac{t_i}{\epsilon}| }, \sqrt{\epsilon})]^{-1} | f(y)|$$ then by the decay of $w_{{\varepsilon}, {\bf t}}$ and the definition of the norm, we infer that $$\|(T[w_{{\varepsilon}, {\bf t}}](t_i+{\varepsilon}y)-T[w_{{\varepsilon}, {\bf t}}](t_i))\,w_{{\varepsilon}, {\bf t}}^3\|_{**}= O({\varepsilon}^{5/2}), \label{estw3}$$ $$\|(T[w_{{\varepsilon}, {\bf t}}](t_i+{\varepsilon}y)-T[w_{{\varepsilon}, {\bf t}}](t_i))\,{\varepsilon}w_{{\varepsilon}, {\bf t}}^2\|_{**}= O({\varepsilon}^{5/2}) \label{estw2},$$ $$\|(T[w_{{\varepsilon}, {\bf t}}](t_i+{\varepsilon}y)-T[w_{{\varepsilon}, {\bf t}}](t_i))\,{\varepsilon}^2 w_{{\varepsilon}, {\bf t}}\|_{**}= O({\varepsilon}^{5/2}). \label{estw1}$$ Let us now define $$\label{sa} S_{\varepsilon}[\hat{A}] := {\varepsilon}^{2}\hat{A}_{xx}-\hat{A}+T[\hat{A}] (\varepsilon A_0+\hat{A})^{3}$$ where $T[\hat{A}]$ is defined in (\[ta\]). Next we set $\hat{A}=w_{{\varepsilon}, {\bf t}}$ and compute $S_{\varepsilon}[w_{{\varepsilon}, {\bf t}}]$. In fact, $$S_{\varepsilon}[w_{{\varepsilon}, {\bf t}}]= {\varepsilon}^{2} ({w_{{\varepsilon}, {\bf t}}})_{xx}-w_{{\varepsilon}, {\bf t}}+T[w_{{\varepsilon}, {\bf t}}]\,({\varepsilon}A_0+ w_{{\varepsilon}, {\bf t}})^{3}$$$$={\varepsilon}^2 ({w_{{\varepsilon}, {\bf t}}})_{xx}-w_{{\varepsilon}, {\bf t}}+T[w_{{\varepsilon}, {\bf t}}](t_i)\,w_{{\varepsilon}, {\bf t}}^{3}$$$$+T[w_{{\varepsilon}, {\bf t}}]({\varepsilon}^3 A_0^3+3{\varepsilon}^2 A_0^2 w_{{\varepsilon}, {\bf t}} + 3 {\varepsilon}A_0 w_{{\varepsilon}, {\bf t}}^2)$$$$+\left(T[w_{{\varepsilon}, {\bf t}}](t_i+{\varepsilon}y)-T[w_{{\varepsilon}, {\bf t}}](t_i)\right)\, w_{{\varepsilon}, {\bf t}}^{3}$$ $$\label{defe} =:E_1+E_2+E_3.$$ We compute $$E_1= \sum_{i=1}^K \frac{1}{\sqrt{v_{i}^{\varepsilon}}} \left( \tilde{w}_i^{''} -\tilde{w}_i+ \frac{T[w_{{\varepsilon}, {\bf t}}](t_i)}{v_i} \tilde{w}_{i}^3 \right) =\mbox{e.s.t.}$$ in $L_2({\Omega}_{\varepsilon}) $ since $v_i^{\varepsilon}=T[w_{{\varepsilon}, {\bf t}}](t_i)$. Further, we get $$E_2=O({\varepsilon})$$ in $L^2({\Omega}_{\varepsilon}) $ since $T[w_{{\varepsilon}, {\bf t}}]$ is bounded in $L^\infty({\Omega}_{\varepsilon})$ and $w_{{\varepsilon}, {\bf t}}$ is bounded in $L^2({\Omega}_{\varepsilon}) $. We also notice that actually $ E_2= O({\varepsilon}e^{-\min_i (|y-\frac{t_i}{{\varepsilon}}|)})$. Lastly, we derive $$E_3= \sum_{i=1}^K \frac{1}{(v_{i}^{{\varepsilon}})^{3/2}} \left( T[w_{{\varepsilon}, {\bf t}}](t_i+{\varepsilon}y)- T[w_{{\varepsilon}, {\bf t}}](t_i) \right) \tilde{w}_{{\varepsilon}, i}^3 =O({\varepsilon}^3)$$ in $L^2({\Omega}_{\varepsilon})$ by (\[estw3\]). Combining these estimates, we conclude that $$\label{estsa} \|S [ w_{{\varepsilon}, {\bf t}}]\|_{**} = O({\varepsilon}).$$ [The estimate (\[estsa\]) shows that our choice of approximate solution given in (\[vector\]) and (\[v\]) is suitable. This will enable us in the next two sections to rigorously construct a steady state which is very close to the approximate solution. ]{} Liapunov-Schmidt Reduction ========================== In this section, we use Liapunov-Schmidt reduction to solve the problem $$\label{ls} S_{\varepsilon}[ w_{{\varepsilon}, {\bf t}} + v] = \sum_{i=1}^K \beta_i \frac{d\tilde{w}_i}{dx}$$ for real constants $ \beta_i$ and a function $ v\in H^2(-\frac{1}{{\varepsilon}}, \frac{1}{{\varepsilon}})$ which is small in the corresponding norm (to be defined later), where $ \tilde{w}_i$ is given by (\[app1\]) and $w_{{\varepsilon},{\bf t}}$ by (\[vector\]). This is the first step in the Liapunov-Schmidt reduction method. We shall follow the general procedure used in [@ww]. To this end, we need to study the linearized operator $$\tilde{L}_{{\varepsilon}, {\bf t}}: H^2 (\Omega_{\varepsilon}) \to L^2(\Omega_{\varepsilon})$$ given by $$\label{lept} \tilde{L}_{{\varepsilon}, {\bf t}} := S_{\epsilon}^{'} [w_{{\varepsilon}, {\bf t}}]\phi = {\varepsilon}^2\Delta\phi-\phi +T[w_{{\varepsilon}, {\bf t}}] 3({\varepsilon}A_0+w_{{\varepsilon}, {\bf t}})^{2} \phi +(T'[w_{{\varepsilon}, {\bf t}}]\phi) ({\varepsilon}A_0+w_{{\varepsilon}, {\bf t}})^{3},$$ where for $\hat{A}=w_{{\varepsilon}, {\bf t}}$ and a given function $\phi\in L^2({\Omega})$ we define $T^{'}[\hat{A}]\phi$ to be the unique solution of $$\label{tap} \left\{ \begin{array}{ll} \hat{D} \left( (A_0 +\frac{1}{{\varepsilon}} \hat{A})^2 (T'[\hat{A}]\phi)_{x} \right)_{x} -\frac{1}{{\varepsilon}} (T'[\hat{A}]\phi) ({\varepsilon}A_0 +\hat{A})^{3} -\frac{1}{{\varepsilon}} T[\hat{A}]3 ({\varepsilon}A_0 +\hat{A})^{2}\phi =0,\\ \null \hfill\quad\text{for}\quad -L<x<L, \\[3mm] (T'[\hat{A}]\phi)_{x}(-L)=(T'[\hat{A}]\phi)_{x}(L)=0. \end{array} \right.$$ The norm for the error function $\phi$ is defined as follows $$\label{normdef} \| \phi \|_{*}= \| \phi \|_{H^2 (\Omega_{\varepsilon})} + \sup_{ - \frac{L}{{\varepsilon}} <y <\frac{L}{{\varepsilon}} } [\max (\min_{i} e^{-\frac{1}{2}|y-\frac{t_i}{{\varepsilon}}| }, \sqrt{{\varepsilon}})]^{-1} | \phi (y)|.$$ We define the approximate kernel and co-kernel, respectively, as follows: $${\cal K}_{{\varepsilon}, {\bf t}} := \mbox{span} \ \left\{ \frac{d\tilde{w}_i}{dx} \Bigg| i=1,\ldots,K\right\} \subset H^2 ({\Omega}_{\varepsilon}),$$ $${\cal C}_{{\varepsilon}, {\bf t}} := \mbox{span} \ \left\{ \frac{d \tilde{w}_i}{dx} \Bigg| i=1,\ldots,K\right\} \subset L^2 ({\Omega}_{\varepsilon}).$$ From (\[vecnlep\]) we recall the definition of the following system of NLEPs : $$L\Phi:=\Phi_{yy}-\Phi+3w^2\Phi-3w^3\frac{\int w^2{\cal E}\Phi\,dy}{\int w^3\,dy}, \label{linop1}$$ where $$\Phi= \left( \begin{array}{l} \phi_1\\ \phi_2 \\ \vdots \\ \phi_K \end{array} \right) \in(H^2({\mathbb{R}}))^K.$$ By Lemma 3.3 of [@ww] we know that $$L: (X_0\oplus \cdots \oplus X_0)^\perp \cap (H^2({\mathbb{R}}))^K \to (X_0\oplus \cdots \oplus X_0)^\perp \cap (L^2({\mathbb{R}}))^K$$ is invertible and its inverse is bounded. We will show that this system is the limit of the operator $\tilde{L}_{{\varepsilon}, {\bf t}}$ (defined in (\[lept\])) as ${\varepsilon}\to 0$. We also introduce the projection $\pi_{{\varepsilon}, {\bf t}}^\perp: L^2({\Omega}_{\varepsilon})\to {\cal C}^\perp_{{\varepsilon}, {\bf t}}$ and study the operator $L_{{\varepsilon}, {\bf t}}:= \pi_{{\varepsilon}, {\bf t}}^\perp\circ \tilde{L}_{{\varepsilon}, {\bf t}}$. By letting ${\varepsilon}\to 0$, we will show that $L_{{\varepsilon}, {\bf t}}:\, {\cal K}_{{\varepsilon}, {\bf t}}^\perp \to {\cal C}_{{\varepsilon}, {\bf t}}^\perp$ is invertible and its inverse is uniformly bounded provided ${\varepsilon}$ is small enough. This statement is contained in the following proposition. \[A\] \[mainprop\] There exist positive constants $\bar{{\varepsilon}},\,\bar{\delta}, \lambda$ such that for all ${\varepsilon}\in(0,\bar{{\varepsilon}})$, ${\bf t} \in{\Omega}^K$ with $ \min(| 1+t_1|, |1-t_K|, \min_{i \not =j}|t_i-t_j|)>\bar{\delta}$, $$\label{normesta} \|L_{{\varepsilon}, {\bf t} } \phi \|_{**}\geq \lambda \|\phi\|_{*}.$$ Furthermore, the map $$L_{{\varepsilon},{\bf t}}= \pi_{{\varepsilon},{\bf t}}^\perp\circ \tilde{L}_{{\varepsilon},{\bf t}}:\, {\cal K}_{{\varepsilon},{\bf t}}^\perp \to {\cal C}_{{\varepsilon},{\bf t}}^\perp$$ is surjective. [**Proof of Proposition \[mainprop\]:**]{} This proof uses the method of Liapunov-Schmidt reduction following for example the approach in [@iww2], [@ww] and [@ww13]. Suppose that (\[normesta\]) is false. Then there exist sequences $\{{\varepsilon}_k\},\,\{{\bf t}^k\},\,\{\phi^k\}$ with ${\varepsilon}_k\to 0$, ${\bf t}^k\in{\Omega}^K$, $\min(| 1+t_1^k|, |1-t_K^k|, \min_{i \not =j}|t_i^k-t_j^k|)>\bar{\delta}$, $\phi^k=\phi_{{\varepsilon}_k}\in K_{{\varepsilon}_k,{\bf t}^k}^\perp$, $k=1,2,\ldots$ such that $$\begin{aligned} &&\| L_{{\varepsilon}_k,{{\bf t}^k }} \phi^k\|_{**} \to 0 \label{lpt}\qquad\mbox{as }k\to\infty,\\ &&\| \phi^k\|_{*}=1,{\hspace{1cm}}k=1,\,2,\,\ldots\,. \label{onnorm}\end{aligned}$$ We define $\phi_{{\varepsilon},i}$, $i=1,2,\ldots,K$ and $\phi_{{\varepsilon},K+1}$ as follows: $$\label{phei} \phi_{{\varepsilon},i}(x)=\phi_{\varepsilon}(x) \chi\left(\frac{x-t_i}{r_0}\right),\quad x\in{\Omega},$$ $$\phi_{{\varepsilon},K+1}(x)=\phi_{{\varepsilon}}(x)-\sum_{i=1}^K \phi_{{\varepsilon},i}(x),\quad x\in{\Omega}.$$ At first (after rescaling) the functions $\phi_{{\varepsilon},i}$ are only defined on ${\Omega}_{\varepsilon}$. However, by a standard result they can be extended to ${\mathbb{R}}$ such that their norm in $H^2({\mathbb{R}})$ is still bounded by a constant independent of ${\varepsilon}$ and $\bf t$ for ${\varepsilon}$ small enough. In the following we will deal with this extension. For simplicity of notation we keep the same notation for the extension. Since for $i=1,2,\ldots,K$ each sequence $\{\phi_i^k\}:=\{\phi_{{\varepsilon}_k,i}\}$ ($k=1,2,\ldots$) is bounded in $H^2_{loc}({\mathbb{R}})$ it converges weakly to a limit in $H^2_{loc}({\mathbb{R}})$, and therefore also strongly in $L^2_{loc}({\mathbb{R}})$ and $L^{\infty}_{loc}({\mathbb{R}})$. Denoting these limits by $\phi_i$, then $\phi=\left(\begin{array}{c}\phi_1 \\ \phi_2\\ \vdots \\ \phi_K\end{array}\right)$ solves the system $L \phi=0.$ By Lemma 3.3 of [@ww], it follows that $\phi\in \mbox{Ker}(L)= X_0\oplus \cdots \oplus X_0$. Since $\phi^k\in K_{{\varepsilon}_k,t_k}^\perp$, taking $k\to\infty$, we get $\phi\in \mbox{Ker}(L)^\perp$. Therefore, we have $\phi=0$. By elliptic estimates we get $\|\phi_{{\varepsilon}_k,i}\|_{H^2({\mathbb{R}})} \to 0$ as $k\to\infty$ for $i=1,2,\ldots,K$. Further, $\phi_{{\varepsilon},K+1}\to \phi_{K+1}$ in $H^2({\mathbb{R}})$, where $\Phi_{K+1}$ satisfies $$(\phi_{K+1})_{yy}-\phi_{K+1}=0 \quad\mbox{ in }{\mathbb{R}}.$$ Therefore, we conclude that $\phi_{K+1}=0$ and $\|\phi_{K+1}^k\|_{H^2({\mathbb{R}})} \to 0$ as $k\to \infty$. Once we have $ \| \phi_i \|_{H^2 ({\mathbb{R}})} \to 0$, the maximum principle implies that $ \|\phi_i \|_{*} \to 0$ since the operator $L_{{\varepsilon}, {\bf t}}$ essentially behaves like $ \phi_i^{''} - \phi_i$ for $ |x-t_i|>>{\varepsilon}$. This contradicts the assumption that $\|\phi^k\|_{*}=1$. To complete the proof of Proposition \[A\], we just need to show that the conjugate operator to $L_{{\varepsilon}, {\bf t}}$ (denoted by $ L_{{\varepsilon}, {\bf t}}^*$) is injective from ${\cal K}_{{\varepsilon}, {\bf t}}^\perp $ to ${\cal C}_{{\varepsilon}, {\bf t}}^\perp$. Note that $ L_{{\varepsilon},{\bf t}}^*\phi=\pi_{{\varepsilon},{\bf t}}\circ \tilde{L}_{{\varepsilon},{\bf t}}^*\phi$ with $$\tilde{L}_{{\varepsilon},{\bf t}}^*\phi:= {\varepsilon}^2\Delta\phi-\phi +T[w_{{\varepsilon}, {\bf t}}] 3({\varepsilon}A_0+w_{{\varepsilon}, {\bf t}})^{2} \phi +(T'[w_{{\varepsilon}, {\bf t}}]\phi ({\varepsilon}A_0+w_{{\varepsilon}, {\bf t}})^{3}).$$ The proof for $ L_{{\varepsilon}, {\bf t}}^*$ follows the same lines as the proof for $ L_{{\varepsilon}, {\bf t}}$ and is therefore omitted. Here also the nondegeneracy condition (\[condbc\]) is required. For further technical details we refer to [@ww]. [$\Box$]{} Now we are in the position to solve the equation $${\cal \pi}_{{\varepsilon}, {\bf t}}^\perp \circ S_{{\varepsilon}} ( w_{{\varepsilon}, {\bf t}} + \phi) =0. \label{solve}$$ Since $ L_{{\varepsilon}, {\bf t}}|_{K_{{\varepsilon}, {\bf t}}^{\perp}}$ is invertible (call the inverse $L^{-1}_{{\varepsilon}, {\bf t}}$) ,we can rewrite this equation as $$\phi=-(L_{{\varepsilon}, {\bf t}}^{-1} \circ {\cal \pi}_{{\varepsilon},{\bf t}}^\perp \circ S_{{\varepsilon}} (w_{{\varepsilon}, {\bf t}}) ) - ( L_{{\varepsilon}, {\bf t}}^{-1}\circ {\cal \pi}_{{\varepsilon}, {\bf t}}^\perp\circ N_{{\varepsilon}, {\bf t}}(\phi))\equiv M_{{\varepsilon}, {\bf t}}(\phi), \label{fix}$$ where $$\label{nep} N_{{\varepsilon},{\bf t}}(\phi)=S_{{\varepsilon}}(w_{{\varepsilon}, {\bf t}} +\phi ) -S_{{\varepsilon}}(w_{{\varepsilon}, {\bf t}}) -S_{{\varepsilon}}^{'} (w_{{\varepsilon}, {\bf t}}) \phi$$ and the operator $M_{{\varepsilon}, {\bf t}}$ has been defined by (\[fix\]) for $\phi\in H^2 (\Omega_{{\varepsilon}})$. The strategy of the proof is to show that the operator $M_{{\varepsilon}, {\bf t}}$ is a contraction on $$B_{{\varepsilon},\delta}\equiv\{\phi\in H^2(\Omega_{{\varepsilon}}) \,:\, \|\phi\|_{*}<\delta\}$$ if ${\varepsilon}$ is small enough and $\delta$ is suitably chosen. By (\[estsa\]) and Proposition \[A\] we have that $$\|M_{{\varepsilon}, {\bf t}}(\phi)\|_{*} \leq\lambda^{-1} \left(\|{\cal \pi}_{{\varepsilon}, {\bf t}}^\perp \circ N_{{\varepsilon}, {\bf t}}(\phi) \|_{**} +\left\|{\cal \pi}_{{\varepsilon}, {\bf t}}^\perp \circ S_{\varepsilon}( w_{{\varepsilon}, {\bf t}} )\right\|_{**}\right)$$$$\leq \lambda^{-1}C_0( c(\delta)\delta +{\varepsilon}),$$ where $\lambda>0$ is independent of $\delta>0$, ${\varepsilon}>0$ and $c(\delta)\to 0$ as $\delta\to 0$. Similarly, we show that $$\|M_{{\varepsilon},{\bf t}}(\phi)-M_{{\varepsilon},{\bf t}}(\phi^{'})\|_{*} \leq \lambda^{-1}C_0( c(\delta)\delta)\|\phi-\phi^{'}\|_{*},$$ where $c(\delta)\to 0$ as $\delta\to 0$. Choosing $\delta= C_3{\varepsilon}\mbox{ for $\lambda^{-1}C_0<C_3$}$ and taking ${\varepsilon}$ small enough, then $M_{{\varepsilon}, {\bf t}}$ maps $B_{{\varepsilon},\delta}$ into $B_{{\varepsilon},\delta}$ and it is a contraction mapping in $B_{{\varepsilon}, \delta}$. The existence of a fixed point $\phi_{{\varepsilon}, {\bf t}}$ now follows from the standard contraction mapping principle and $\phi_{{\varepsilon}, {\bf t}}$ is a solution of (\[fix\]). [$\Box$]{} We have thus proved There exist $\overline{{\varepsilon}}>0$ $\overline{\delta}>0$ such that for every pair of ${\varepsilon}, {\bf t}$ with $0<{\varepsilon}<\overline{{\varepsilon}}$ and ${\bf t}\in{\Omega}^K$, $1+t_1>\overline{\delta}$, $1-t_K>\overline{\delta}$, $\frac{1}{2}|t_i-t_j|> \overline{\delta}$ there is a unique $\phi_{{\varepsilon}, {\bf t}}\in K_{{\varepsilon}, {\bf t}}^{\perp}$ satisfying $S_{{\varepsilon}}(w_{{\varepsilon}, {\bf t}} + \phi_{{\varepsilon},{\bf t}} ) \in {\cal C}_{{\varepsilon}, {\bf t}}$. Furthermore, the following estimate holds $$\|\phi_{{\varepsilon}, {\bf t}}\|_{*}\leq C_3{\varepsilon}. \label{estphi}$$ \[lem34\] In the next section we determine the positions of the spikes so that the resulting steady state is an exact solution of the original problem. The reduced problem =================== In this section we solve the reduced problem and complete the proof of the existence result for asymmetric spikes in Theorem \[existenceas\]. By Lemma \[lem34\], for every $ {\bf t} \in B_{{\varepsilon}^{3/4}} ({\bf t}^0)$, there exists a unique $\phi_{{\varepsilon}, {\bf t} } \in {\cal K}_{{\varepsilon}, {\bf t}}^\perp$, solution of $$\label{see} S [ w_{{\varepsilon}, {\bf t}} + \phi_{{\varepsilon}, {\bf t} } ] = v_{{\varepsilon}, {\bf t}} \in {\cal C}_{{\varepsilon}, {\bf t}}.$$ The idea here is to find ${\bf t}^{\varepsilon}=(t_1^{\varepsilon},\ldots,t_K^{\varepsilon})$ near ${\bf t}^0$ such that also $$\label{see2} S [w_{{\varepsilon}, {\bf t}^{\varepsilon}} + \phi_{{\varepsilon}, {\bf t}^{\varepsilon}} ] \perp {\cal C}_{{\varepsilon}, {\bf t}^{\varepsilon}}$$ (and therefore $S [w_{{\varepsilon}, {\bf t}^{\varepsilon}} + \phi_{{\varepsilon}, {\bf t}^{\varepsilon}} ]=0$). To this end, we let $$W_{\epsilon,i}( {\bf t}):=\frac{v_i}{{\varepsilon}^2} \int_{-1 }^1 S [w_{{\varepsilon}, {\bf t}} +\phi_{{\varepsilon}, {\bf t}}] \frac{d \tilde{w}_i}{dx} \, dx ,$$ $$W_{\epsilon}({\bf t}):=(W_{\epsilon,1}({\bf t}),..., W_{\epsilon,K}({\bf t}) ) : B_{ {\varepsilon}^{3/4}} ({\bf t}^0) \to {\mathbb{R}}^K.$$ Then $W_{\epsilon}({\bf t})$ is a map which is continuous in ${\bf t}$ and our problem is reduced to finding a zero of the vector field $W_{\varepsilon}({\bf t})$. Let us now calculate $W_{\epsilon}({\bf t})$ as follows: $$W_{{\varepsilon}, i} ( {\bf t})= \frac{v_i}{{\varepsilon}^2} \int_{-1 }^1 S [ w_{{\varepsilon}, {\bf t}} + \phi_{{\varepsilon}, {\bf t}}] \frac{d \tilde{w}_i}{dx}$$ $$= \frac{v_i}{{\varepsilon}^2} \int_{-1 }^1 S [ w_{{\varepsilon}, {\bf t}}] \frac{d \tilde{w}_i}{dx}$$ $$+ \frac{v_i}{{\varepsilon}^2} \int_{-1 }^1 S_{\varepsilon}^{'}[w_{{\varepsilon}, {\bf t}}] \phi_{{\varepsilon}, {\bf t}} \frac{d \tilde{w}_i}{dx}$$ $$+ \frac{v_i}{{\varepsilon}^2} \int_{-L}^L N_{\varepsilon}(\phi_{{\varepsilon}, {\bf t}}) \frac{d \tilde{w}_i}{dx}$$ $$= :I_1 + I_2+ I_3,$$ where $I_1, I_2$ and $I_3$ are defined in an obvious way in the last equality. We will now compute these three integral terms as $\epsilon \to 0$. The result will be that $I_1$ is the leading term and $I_2 $ and $ I_3$ are $O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)$. For $I_1$, we have $$I_1= \frac{v_i}{{\varepsilon}^2} \int_{-L}^L (E_1+E_2+E_3) \frac{d\tilde{w}_i}{dx} \, dx = \frac{v_i}{{\varepsilon}^2} \int_{-L}^L E_3 \frac{d\tilde{w}_i}{dx} \, dx +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right),$$ where $E_1,\,E_2,\,E_3$ are defined in (\[defe\]). For $E_1$ this estimate is obvious. For $E_2$, we use the decomposition $$T[w_{{\varepsilon}, {\bf t}}]({\varepsilon}^3 A_0^3+3{\varepsilon}^2 A_0^2 w_{{\varepsilon}, {\bf t}} + 3 {\varepsilon}A_0 w_{{\varepsilon}, {\bf t}}^2)$$$$=T[w_{{\varepsilon}, {\bf t}}]{\varepsilon}^3 A_0^3$$$$+ T[w_{{\varepsilon}, {\bf t}}](t_i)(3{\varepsilon}^2 A_0^2 w_{{\varepsilon}, {\bf t}} + 3 {\varepsilon}A_0 w_{{\varepsilon}, {\bf t}}^2)$$$$+ (T[w_{{\varepsilon}, {\bf t}}](t_i+{\varepsilon}y)- T[w_{{\varepsilon}, {\bf t}}](t_i))(3{\varepsilon}^2 A_0^2 w_{{\varepsilon}, {\bf t}} + 3 {\varepsilon}A_0 w_{{\varepsilon}, {\bf t}}^2).$$ Then we can estimate the first part directly, the second part using the fact that it is an even function in $y$ and the third part using the estimates (\[estw2\]) and (\[estw1\]). From (\[estw3\]), we derive $$\frac{v_i}{{\varepsilon}^2} \int_{-L}^L E_3 \frac{d\tilde{w}_i}{dx} \,dx$$ $$= \frac{v_i}{{\varepsilon}^2} \int_{-L/{\varepsilon}}^{L/{\varepsilon}} P_i(y)w^3 (y) \frac{w^{'} (y)}{\sqrt{v_i}} \,dy$$$$+\frac{v_i}{{\varepsilon}^2} \left[ \frac{1}{2} \frac{\sqrt{2}\pi}{\sqrt{v_i}} +\sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}- \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}} \frac{t_i+L}{2L} \right]$$$$\times \int_{-L/{\varepsilon}}^{L/{\varepsilon}} \int_0^y \frac{1}{\hat{D}\left(A_0+\frac{1}{{\varepsilon}}\hat{A}(\bar{s})\right)^2}\,d\bar{s} ({\varepsilon}A_0+\hat{A})^3 \frac{w^{'} (y)}{\sqrt{v_i}} \chi_i \,dy$$$$+O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)$$ $$= \mbox{e.s.t.}- \frac{1}{\hat{D}} \left[ \frac{1}{2} \frac{\sqrt{2}\pi}{\sqrt{v_i}} +\sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}- \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}} \frac{t_i+L}{2L} \right] +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right),$$ where $\chi_i(x)=\chi\left(\frac{x-t_i}{r_0}\right)$. Here we have used the fact that $P_i(y)$ is an even function and have computed the following integral $$\int_{-L/{\varepsilon}}^{L/{\varepsilon}} \int_0^y \frac{1}{\hat{D}\left(A_0+\frac{1}{{\varepsilon}}\hat{A}(t_i+{\varepsilon}\bar{s})\right)^2}\,d\bar{s} ({\varepsilon}A_0+\hat{A})^3 \frac{w^{'} (y)}{\sqrt{v_i}} \chi_i(t_i+{\varepsilon}y) \,dy$$$$=\int_{-L/{\varepsilon}}^{L/{\varepsilon}} \int_0^y \frac{1}{\hat{D}\left(A_0+\frac{1}{{\varepsilon}}\hat{A}(t_i+{\varepsilon}\bar{s})\right)^2}\,d\bar{s} \frac{1}{4}\frac{d}{dy}({\varepsilon}A_0+\hat{A})^4 \chi_i(t_i+{\varepsilon}y) \,dy+O\left({\varepsilon}^3\log\frac{1}{{\varepsilon}}\right)$$$$=-\frac{{\varepsilon}^2}{\hat{D}} \int_{-L/{\varepsilon}}^{L/{\varepsilon}} \frac{1}{4}\left({\varepsilon}A_0+\hat{A}(t_i+{\varepsilon}y)\right)^2 \,dy+O\left({\varepsilon}^3\log\frac{1}{{\varepsilon}}\right)$$$$=-\frac{{\varepsilon}^2}{\hat{D}}\int_{{\mathbb{R}}}\frac{(w(y))^2}{4v_i}\,dy+O\left({\varepsilon}^3\log\frac{1}{{\varepsilon}}\right)$$$$=-\frac{{\varepsilon}^2}{\hat{D}v_i} \int_{{\mathbb{R}}} \frac{1}{2\cosh^2 y}\,dy +O\left({\varepsilon}^3\log\frac{1}{{\varepsilon}}\right)$$$$=-\frac{{\varepsilon}^2}{ \hat{D}v_i}+O\left({\varepsilon}^3\log\frac{1}{{\varepsilon}}\right).$$ In summary, we have $$\label{esti1} I_1=-\frac{1}{\hat{D}} \left[ \frac{1}{2} \frac{\sqrt{2}\pi}{\sqrt{v_i}} +\sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}- \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}} \frac{t_i+L}{2L} \right] +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right).$$ For $I_2$, we calculate $$I_2= \frac{v_i}{{\varepsilon}^2} \int_{-1 }^1 S^{'}[w_{{\varepsilon}, {\bf t}}] (\phi_{{\varepsilon}, {\bf t}}) \frac{d\tilde{w}_i}{dx}$$ $$= \frac{v_i}{{\varepsilon}^2} \int_{-L}^L \left[ {\varepsilon}^2\Delta\phi_{{\varepsilon}, {\bf t}}-\phi_{{\varepsilon}, {\bf t}} +T[w_{{\varepsilon}, {\bf t}}] 3({\varepsilon}A_0+w_{{\varepsilon}, {\bf t}})^{2} \phi_{{\varepsilon}, {\bf t}} +(T'[w_{{\varepsilon}, {\bf t}}]\phi_{{\varepsilon}, {\bf t}}) ({\varepsilon}A_0+w_{{\varepsilon}, {\bf t}})^{3} \right] \frac{d\tilde{w}_i}{dx}$$$$= \frac{v_i}{{\varepsilon}^2} \int_{-L}^L \left[ {\varepsilon}^2\Delta \frac{d\tilde{w}_i}{dx} - \frac{d\tilde{w}_i}{dx} + 3\tilde{w}_{i}^{2} \frac{T[w_{{\varepsilon}, {\bf t}}](t_i)}{v_i}d\frac{\tilde{w}_i}{dx} \right] \phi_{{\varepsilon}, {\bf t}}\,dx$$$$+ \frac{v_i}{{\varepsilon}^2} \int_{-L}^L \frac{T[w_{{\varepsilon}, {\bf t}}](x)-T[w_{{\varepsilon}, {\bf t}}](t_i)}{v_{i}} \,3\tilde{w}_{i}^{2} \phi_{{\varepsilon}, {\bf t}} \frac{d\tilde{w}_i}{dx} \,dx$$$$+\frac{v_i}{{\varepsilon}^2} \int_{-L}^L T[w_{{\varepsilon}, {\bf t}}](t_i)3\left({\varepsilon}^2 A_0^2+2{\varepsilon}A_0\frac{\tilde{w}_i}{\sqrt{v_i}}\right) \phi_{{\varepsilon}, {\bf t}} \frac{d\tilde{w}_i}{dx} \,dx$$$$+\frac{v_i}{{\varepsilon}^2} \int_{-L}^L (T[w_{{\varepsilon}, {\bf t}}](x)-T[w_{{\varepsilon}, {\bf t}}](t_i))3\left({\varepsilon}^2 A_0^2+2{\varepsilon}A_0\frac{\tilde{w}_i}{\sqrt{v_i}}\right) \phi_{{\varepsilon}, {\bf t}} \frac{d\tilde{w}_i}{dx}$$$$+ \frac{v_i}{{\varepsilon}^2} \int_{-L}^L (T'[w_{{\varepsilon}, {\bf t}}]\phi_{{\varepsilon}, {\bf t}}) {\varepsilon}^3A_0^3 \frac{d\tilde{w}_i}{dx} \,dx$$$$+ \frac{v_i}{{\varepsilon}^2} \int_{-L}^L (T'[w_{{\varepsilon}, {\bf t}}]\phi_{{\varepsilon}, {\bf t}})(t_i) \left( 3{\varepsilon}^2 A_0^2 \frac{\tilde{w}_i}{\sqrt{v_i}}+3 {\varepsilon}A_0 \frac{\tilde{w}_{i}^{2}}{v_i}+\frac{\tilde{w}_{i}^{2}}{v_i^{3/2}} \right) \frac{d\tilde{w}_i}{dx} \,dx$$ $$+ \frac{v_i}{{\varepsilon}^2} \int_{-L}^L [ (T'[w_{{\varepsilon}, {\bf t}}]\phi_{{\varepsilon}, {\bf t}})(x) - (T'[w_{{\varepsilon}, {\bf t}}]\phi_{{\varepsilon}, {\bf t}})(t_i) ]$$$$\times \left( 3{\varepsilon}^2 A_0^3 \frac{\tilde{w}_i}{\sqrt{v_i}}+3 {\varepsilon}A_0 \frac{\tilde{w}_{i}^{2}}{v_i}+\frac{\tilde{w}_{i}^{2}}{v_i^{3/2}} \right) \frac{d\tilde{w}_i}{dx} \,dx$$$$=I_2^1+I_2^2+I_2^3+I_2^4+I_2^5+I_2^6+I_2^7.$$ With obvious notations, we now show that each one of the seven terms is $O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right)$ as $\epsilon \to 0$. For $I_2^1$, it follows from $T[w_{{\varepsilon}, {\bf t}}]=v_i$, while for $I_2^2$, we use (\[estw3\]) and the fact that $$\left\|\frac{\phi_{{\varepsilon},{\bf t}}}{w_{{\varepsilon},{\bf t}}}\right\|_{L^\infty({\Omega}_{\varepsilon})}\leq C{\varepsilon}. \label{estphiw}$$ For $I_2^3$, we use $\|T[w_{{\varepsilon},{\bf t}}]\|_{L^\infty({\Omega}_{\varepsilon})}=O(1)$ and the fact that $\phi_{{\varepsilon},{\bf t}}$ is an even function. For $I_2^4$, we use (\[estw2\]), (\[estw1\]) and (\[estphiw\]). For $I_2^5$, the estimate is derived from $\|T'[w_{{\varepsilon},{\bf t}}]\phi_{{\varepsilon}, {\bf t}}\|_{L^\infty({\Omega}_{\varepsilon})}=O({\varepsilon})$. For $I_2^6$, we use $(T'[w_{{\varepsilon},{\bf t}}]\phi_{{\varepsilon}, {\bf t}})(t_i)=O({\varepsilon})$ and the fact that $\tilde{w}_i$ is even. Lastly, for $I_2^7$, we use estimates similar to (\[estw3\]), (\[estw2\]), (\[estw1\]) with $T'[w_{{\varepsilon},{\bf t}}]\phi_{{\varepsilon}, {\bf t}}$ instead of $T[w_{{\varepsilon},{\bf t}}]$ and the inequality $$\left\|\frac{T'[w_{{\varepsilon},{\bf t}}]\phi_{{\varepsilon}, {\bf t}}}{T[w_{{\varepsilon},{\bf t}}]}\right\|_{L^\infty({\Omega}_{\varepsilon})}\leq C{\varepsilon}.$$ By arguments similar to the ones for $I_2$, we derive $$\label{i33} I_3 = O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right) \quad\mbox{ in } L^2({\Omega}_{\varepsilon}).$$ Combining the estimates for $I_1$, $I_2$ and $I_3$, we have $$W_{{\varepsilon},i} ({\bf t})= -\frac{1}{\hat{D}} \left[ \frac{1}{2} \frac{\sqrt{2}\pi}{\sqrt{v_i}} +\sum_{j=1}^{i-1} \frac{\sqrt{2}\pi}{\sqrt{v_j}}- \sum_{j=1}^K \frac{\sqrt{2}\pi}{\sqrt{v_j}} \frac{t_i+L}{2L} \right] +O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right).$$ $$\label{formulaf} = -\frac{1}{\hat{D}} F_i ({\bf t}) + O\left({\varepsilon}\log\frac{1}{{\varepsilon}}\right),$$ where $F_i ({\bf t})$ was defined in (\[fi\]). By assumption (\[amp3\]), we have $ F ({\bf t}^0) =0$. Next we show that $$\det ( \nabla_{{\bf t}^0} F ({\bf t}^0)) \not = 0$$ in the case of two spikes with amplitudes $v_1=v_s<v_l=v_2$. We compute $$\nabla_{{\bf t}} F ({\bf t})= D_{\bf t} F+(D_{v} F) (D_{\bf t} v),$$ where $$D_{\bf t} F =-\left(\sum_{j=1}^2 \frac{\sqrt{2}\pi}{\sqrt{v_j}}\frac{1}{2L}\right){\cal I},$$$$D_{v} F=\frac{\sqrt{2}\pi}{4} \left( \begin{array}{cc} 0 & v_2^{-3/2} \\[3mm] -v_1^{-3/2} & 0 \end{array} \right),$$$$D_{\bf t} v= \frac{\pi}{\sqrt{2}\hat{D} A_0^2} \, \frac{1}{\left(\left(\frac{v_2}{v_1}\right)^{3/2}+1-\frac{\pi L}{\sqrt{2}\hat{D}A_0^2 v_1^{3/2}}\right)} \left( \begin{array}{cc} v_1^{-1/2} & v_2^{-1/2} \\[3mm] -v_1^{-2} v_2^{3/2} & -v_1^{-3/2} v_2 \end{array} \right).$$ This implies, using (\[amp1\]) and (\[amp6\]), that $$\nabla_{{\bf t}^0} F ({\bf t}^0)= \frac{\pi^2}{4 \hat{D} A_0^2}\, \frac{1}{\left(\left(\frac{v_2}{v_1}\right)^{3/2}+1-\frac{\pi L}{\sqrt{2}\hat{D}A_0^2 v_1^{3/2}} \right)}$$$$\times \left( \begin{array}{cc} -v_1^{-5/2}v_2^{1/2} -v_1^{-1}v_2^{-1}+v_1^{-2}+2v_1^{-3/2}v_2^{-1/2} & -v_1^{-3/2}v_2^{-1/2} \\[3mm] -v_1^{-2} & -v_1^{-5/2}v_2^{1/2} -v_1^{-1}v_2^{-1}+2v_1^{-2}+v_1^{-3/2}v_2^{-1/2} \end{array} \right).$$ Next, we compute $$\det ( \nabla_{{\bf t}^0} F ({\bf t}^0))= \frac{\pi^4}{16 (\hat{D} A_0^2)^2}\,\frac{1}{v_1^2v_2^2}\, \frac{1}{\left(\alpha^3-2\alpha^2-2\alpha+1 \right)^2}$$$$\times \left( (\alpha^3-\alpha^2-2\alpha+1)(\alpha^3-2\alpha^2-\alpha+1)-\alpha^3 \right)$$ $$= \frac{\pi^4}{16 (\hat{D} A_0^2)^2}\, \frac{1}{v_1^2v_2^2} \,\frac{\alpha^3-\alpha^2-\alpha+1}{\alpha^3-2\alpha^2-2\alpha+1},$$ where $\alpha=\sqrt{\frac{v_2}{v_1}}$. Therefore, we have $\det ( \nabla_{{\bf t}^0} F ({\bf t}^0))\not = 0$, except for two specific positive values of $\alpha$: $\alpha=1$ (the bifurcation point of asymmetric from symmetric spikes which is not included in Theorem \[existenceas\]) and $\alpha=\frac{1+\sqrt{5}}{2}$ (corresponding to the eigenvalue $e_{m,1}=\frac{3}{2}$ in Section 5 which has been excluded from Theorem \[existenceas\]). Thus, under the conditions of Theorem \[existenceas\], we get $$W_{\varepsilon}({\bf t}) = -\frac{1}{\hat{D}} \nabla_{{\bf t}^0} F ({\bf t}^0)({\bf t}-{\bf t}^0) + O\left(|{\bf t} -{\bf t}^0|^2+{\varepsilon}\log\frac{1}{{\varepsilon}}\right).$$ Since $W_{\varepsilon}({\bf t})$ is continuous in $\bf t $, standard degree theory [@danclec] implies that for $\epsilon$ small enough and $\delta$ suitable chosen there exist ${\bf t^{\epsilon}}\in B_\delta({\bf t}^0)$ such that $W_{\epsilon}({\bf t^{\epsilon}})=0$ and ${\bf t^{\varepsilon}} \to {{\bf t}^0}$. For further technical details of the argument, we refer to [@ww13]. [$\Box$]{} Thus we have proved the following proposition. \[redprob\] For $\epsilon$ small enough, there exist points ${\bf t}^{\epsilon}$ with ${ \bf t}^{\epsilon}\to { \bf t}^0$ such that $W_{\epsilon}({\bf t}^{\varepsilon})=0$. \[conver\] Finally, we complete the proof of Theorems \[existencesym\] and \[existenceas\]. [**Proof of Theorem \[existenceas\]:**]{} By Proposition \[redprob\], there exist ${\bf t}^{\varepsilon}\to {\bf t}^0$ such that $ W_{\varepsilon}({\bf t}^{\varepsilon}) =0$. In other words, $S [w_{{\varepsilon}, {\bf t}^{\varepsilon}}+\phi_{{\varepsilon}, {\bf t}^{\varepsilon}}] =0$. Let $\hat{A}_{\varepsilon}= w_{{\varepsilon}, {\bf t}^{\varepsilon}} +\phi_{{\varepsilon}, {\bf t}^{\varepsilon}},\, \hat{v}_{\varepsilon}= T[w_{{\varepsilon}, {\bf t}^{\varepsilon}} + \phi_{{\varepsilon}, {\bf t}^{\varepsilon}} ]$. By the maximum principle, we conclude that $\hat{A}_{{\varepsilon}} >0,\, \hat{v}_{\varepsilon}>0$. Moreover $(\hat{A}_{\varepsilon}, \hat{v}_{\varepsilon})$ satisfies all the properties of Theorem \[existenceas\]. [$\Box$]{} [**Proof of Theorem \[existencesym\]:**]{} To prove Theorem \[existencesym\], we first construct a single spike in the interval $\left(-\frac{L}{K},\frac{L}{K}\right)$ as above. Then we continue the single spike periodically to a function in the interval $(-L,L)$ and get a symmetric multiple spike in the interval $(-L,L)$. [$\Box$]{} Proof of Theorem \[existenceani\] ================================= The proof of Theorem \[existenceani\] goes exactly as that of Theorem \[existenceas\]. First, let us derive the location of the single spikes formally: in the first equation in (\[sys1\]) the term $\varepsilon^3 A_0^{''}$ is very small and can be omitted in the computations. Thus we may assume that $$\label{n100} \hat{A} \sim \xi^{-1/2} w\left(\frac{x-t_0}{\varepsilon}\right), \ v(t^0)= \xi$$ Substituting the above expressions into the second equation of (\[sys1\]) and noting that $ \hat{v} \frac{1}{\varepsilon} (\varepsilon A_0 +\hat{A})^3 \sim \xi^{-1/2} (\int w^3) \delta_{t_0}$, we see that $\hat{v}$ satisfies in leading order $$\hat{D} ( A_0^2 \hat{v}_x)_{x} - \xi^{-1/2} \left(\int_{\mathbb{R}}w^3\,dy\right) \delta_{t_0} + \gamma (x)=0.$$ Solving the above equation, we then obtain $$\label{eqn100} \hat{v}_x (t_0 -)= - \frac{1}{\hat{D} (A_0 (t_0))^2} \int_{-L}^{t_0} \gamma (x) dx, \ \ \hat{v}_x (t_0 +)= \frac{1}{\hat{D} (A_0 (t_0))^2} \int_{t_0}^{L} \gamma (x) dx.$$ Substituting (\[n100\]) into the first equation of (\[sys1\]) and rescaling $x= t_0+\varepsilon y$, we deduce that the error becomes $$\varepsilon ( \hat{v}_x (t_0-) y^{-} + \hat{v}_x (t_0+) y^{+}) w^3 (y) + \varepsilon A_0(t_0) 3 w^2 +O(\varepsilon^2),$$ where $y^{-}=\min(y,0)$ and $y^{+}=\max(y,0)$. from which we conclude that a necessary condition for the existence of a spike at $t_0$ is that $$\int_{\mathbb{R}}\left[\varepsilon ( \hat{v}_x (t_0-) y^{-} + \hat{v}_x (t_0+) y^{+}) w^3 (y) + \varepsilon A_0(t_0) 3w^2 +O(\varepsilon^2)\right] w^{'} (y)\,dy=0$$ whence $$\hat{v}_x (t_0-)+\hat{v}_x (t_0+)=0$$ which is equivalent to (\[tep000\]). It turns out that (\[tep000\]) is also sufficient, since the derivative of $\int_{-L}^{t_0} \gamma (x) dx -\int_{t_0}^L \gamma (x) dx$ with respect to $t_0$ is $\gamma (t_0)$ which is strictly positive. The rest of the proof goes exactly as in the proof of Theorem \[existenceas\]. We omit the details. Discussion ========== In this article we have provided a rigorous mathematical analysis of the formation of spikes in the model of Short, Bertozzi and Brantingham [@sbb]. Thus, we have shown that this model naturally leads to the formation of criminality hot-spots. The existence of such hotspots is one of the main stylized facts about criminality. It is observed for an array of criminal activity types. Hot-spots are extensively reported and discussed in the criminology literature. We refer for example to the articles [@je] and [@bb] as well as to the references therein. Now, the fact that a mathematical model yields such hotspots can be viewed as passing one benchmark of validity. The findings in our paper provide such a test for the Short, Bertozzi and Brantingham model [@sbb]. Furthermore, the rigorous analysis carried here sheds light on the mechanisms for the formation of hotspots in this model and the way it quantitatively depends on the parameters. This type of analysis can then be applied to study issues such as the reduction of hotspots by crime prevention strategies or optimal use of resources to this effect. One of the goals is to understand when policing strategies actually reduce criminal activity and when they merely displace hot-spots to new areas. In this paper we have proved three main new results. First, we showed that we can reduce the quasilinear chemotaxis problems to a Schnakenberg type reaction-diffusion system and derived the existence of symmetric $k$ spikes. Next, we established the existence of asymmetric spikes in the isotropic case. Lastly, we have studied the pinning effect by inhomogeneous media $A_0 (x)$ and $\gamma (x)$. The stability of these spikes is an interesting open issue. In [@kww1] spikes in two space dimensions are considered by formal matched asymptotics. Our approach of rigorous justification can be extended to that case in a radially symmetric setting, i.e. if the domain is a disk and we construct a single spike located at the centre. We remark that in [@kww1] it is assumed that in the outer region (away from the spikes) the system in leading order is semi-linear which allows an extension of the results for the Schnakenberg model to this case. In [@kww1], for the inner region, a numerical computation by solving a core problem yields the profile of the spike. An alternative approach to the problem in one space dimension would be to write it as a first-order semi-linear ODE system and then apply standard methods, e.g. dynamical system methods for the problem on the real line. This approach becomes cumbersome when we impose Neumann boundary conditions and we add inhomogeneity. We nevertheless refer to a recent paper [@HDK], where the dynamical systems approach is used to construct traveling wave solutions of a quasi-linear reaction-diffusion-mechanical system. We remark that there are very few results concerning the analysis of spikes in quasi-linear reaction diffusion systems. As far as we know, there are two such types of systems. The first one is the chemotaxis system of Keller-Segel type. We refer to [@HP] for the background of chemotaxis models and [@KW2] for the analysis of spikes to these systems. The other one is the Shigesada-Kawasaki-Teramoto model of species segregation ([@SKT]). For the analysis of spikes in a cross-diffusion system, we refer to [@KW1], [@LN] and [@WX]. A family of related models for the diffusion of criminality has been proposed in [@bn]. We analyze the formation of hot spots in this class of models in our forthcoming work [@bw]. The equations in [@bn] also envision the possibility of non-local diffusion. Indeed, social influence can be exercised at long range and it is natural to consider descriptions that take long range diffusion into account. Such a non-local system arising in [@bn] reads: $$\label{crime_non-local} \begin{cases} &s_t(x,t) ={\mathcal L} s(x,t) -s(x,t)+s_b+\alpha(x) u(x,t)\\ &u_t(x,t) = \Lambda(s) - u(x,t). \end{cases}$$ The case when ${\mathcal L}=\Delta $ is a local diffusion operator provides the framework of the study in [@brr]. Here, ${\mathcal L}$ can also be a non-local operator such as the fractional Laplace operator or a general non-local interaction term: $${\mathcal L} s (x,t)= \int J(x,y) (s(y,t) - s(x,t) ) dy.$$ Observe that the steady states reduce to a single non-local equation: $$\label{eqs-stat} - {\mathcal L} s = s_b (x) -s +\alpha(x) \Lambda (s).$$ We note that the interaction between non-local diffusion and the mechanism for the formation of spikes is completely open. In particular, the description of the formation of spikes in (\[crime\_non-local\]) and (\[eqs-stat\]) are open problems. We expect that the decay of the kernel may come into play for the formation of spikes. [**Acknowledgment.**]{} The research of Henri Berestycki leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.321186 - ReaDi -Reaction-Diffusion Equations, Propagation and Modelling. Part of this work was done while Henri Berestycki was visiting the Department of Mathematics at University of Chicago. He was also supported by an NSF FRG grant DMS - 1065979, “Emerging issues in the Sciences Involving Non-standard Diffusion”. Juncheng Wei was supported by a GRF grant from RGC of Hong Kong and a NSERC Grant from Canada. Matthias Winter thanks the Department of Mathematics of The Chinese University of Hong Kong for its kind hospitality. 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Wei, Existence and Stability of Spikes for the Gierer-Meinhardt System, in [*HANDBOOK OF DIFFERENTIAL EQUATIONS, Stationary partial differential equations,*]{} volume 5 (M. Chipot ed.), Elsevier, pp. 489-581. J. Wei and M. Winter, Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in $\mathbf R^1$, [*Methods Appl. Anal.*]{} 14 (2007), 119–163. J. Wei and M. Winter, On the Gierer-Meinhardt system with precursors, [*Discr. Cont. Dyn. Syst. A, Special issue for Prof. Mimura’s 65th Birthday*]{} 25 (2009), no.1, 363-398. J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, [*J. Math. Biol.*]{} 57 (2008), 53–89. J. Wei and M. Winter, [*Mathematical Aspects of Pattern Formation in Biological Systems*]{}, Applied Mathematical Sciences 189, Springer, London, 2014. J. Wei and L. Zhang, On a nonlocal eigenvalue problem, [*Ann. Scuola Norm. Sup. Pisa Cl. Sci.*]{} 30 (2001), 41–61. 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--- abstract: 'We construct stationary finitely dependent colorings of the cycle which are analogous to the colorings of the integers recently constructed by Holroyd and Liggett. These colorings can be described by a simple necklace insertion procedure, and also in terms of an Eden growth model on a tree. Using these descriptions we obtain simpler and more direct proofs of the characterizations of the 1- and 2-color marginals.' address: - 'Alexander E. Holroyd' - 'Tom Hutchcroft, Department of Mathematics, University of British Columbia' - 'Avi Levy, Department of Mathematics, University of Washington' author: - 'Alexander E. Holroyd' - Tom Hutchcroft - Avi Levy bibliography: - 'coloring.bib' date: 28 July 2017 title: Finitely dependent cycle coloring --- Introduction ============ A random process indexed by the vertex set of a graph is $k$**-dependent** if its restrictions to any two sets of vertices at graph distance greater than $k$ are independent of each other. A process is **finitely dependent** if it is $k$-dependent for some finite $k$. For several decades it was not known whether every stationary finitely dependent process on $\Z$ is a **block factor** of an i.i.d. process, that is, the image of an i.i.d. process under a finite range function that commutes with translations. This question was raised by Ibragimov and Linnik [@ibragimov1965independent] in 1965 and resolved in the negative by Burton, Goulet, and Meester [@burton1993] in 1993. Recently Holroyd and Liggett [@HL] proved that *proper coloring* distinguishes between block factors and finitely dependent processes: block factor proper colorings of $\Z$ do not exist, but finitely dependent stationary proper colorings do. In fact, these colorings fit into a more general family constructed subsequently in [@hhlMalCol]. See also [@H; @HL2]. The finitely dependent colorings of [@HL] have short but mysterious descriptions. An interesting feature of these colorings is that the supports of proper subsets of the colors can each be expressed as block factors of i.i.d. [@HL Theorem 4], even though the coloring as a whole cannot [@HL Proposition 2]. Moreover, their descriptions as block factors are remarkably simple and explicit. However, the proofs used to obtain these descriptions in [@HL] were in some cases quite involved. Here we introduce a more canonical construction via colorings of the $n$-cycle, which can be expressed in terms of a necklace insertion process akin to those in [@mallowsSheppNecklace; @nakataNecklace], or in terms of the classical Eden growth model on a 3-regular tree. Using this new construction, we are able to obtain simpler and more direct proofs of the statements concerning subsets of colors mentioned above. The necklace insertion process is as follows. Suppose we have a necklace of colored beads. Start with 3 beads with uniformly random distinct colors from $\{1,\ldots,q\}$. At each step, pick a uniformly random gap between consecutive beads and insert a bead with uniformly random color differing from those of the two neighbors. After $n-1$ steps we have a coloring $(C_1,\ldots,C_{n+2})$ of the $(n+2)$-cycle. Here is a different description of the above process, which is easily seen to be equivalent (see Section \[sec:index\] for details). Consider a planar embedding of the 3-regular tree $\mathbb T$ (with a distinguished root vertex), together with its planar dual $\mathbb D$, which is an infinite-degree triangulation (see Figure \[fig:cycleColor\]). For a vertex $v$ of $\mathbb T$, let $\Delta(v)$ be the set of three vertices in $\mathbb D$ incident to the face dual to $v$. We consider the Eden growth model [@eden1961two] on $\mathbb T$, which is a random growing sequence of clusters $T_1,T_2,\ldots$ defined as follows. The initial cluster $T_1$ consists of the root vertex. Given $T_n$, we choose a vertex uniformly at random from those adjacent to but not lying in $T_n$, and add it to $T_n$ to form the new cluster $T_{n+1}$. Let $D_n$ be the subgraph of $\mathbb{D}$ induced by the vertex set $\bigcup_{v \in T_n} \Delta(v)$. The graph $D_n$ inherits a planar embedding from $\mathbb D$, in which there is one outer face containing all of its $n+2$ vertices, and all other faces are triangles. Let $q\geq 3$ be an integer. Conditional on $D_n$, choose a uniformly random proper $q$-coloring of $D_n$ and, independently, a uniform vertex $u$ of $D_n$. Let $C_1,\ldots,C_{n+2}$ be the sequence of colors of the vertices of the outer face in clockwise order starting from $u$. ![\[fig:cycleColor\] Two versions of the construction of the 2-dependent 3-coloring of the cycle, from a cluster of the Eden model on a 3-regular tree. On the left we uniformly properly 3-color the planar map comprising the dual triangles of the vertices of the cluster. The coloring is read clockwise around the outer face. On the right, we may alternatively fix a uniform proper coloring of the infinite dual map in advance. The coloring is read around the outer boudary of the cluster.](tree2.pdf "fig:"){width="49.00000%"} ![\[fig:cycleColor\] Two versions of the construction of the 2-dependent 3-coloring of the cycle, from a cluster of the Eden model on a 3-regular tree. On the left we uniformly properly 3-color the planar map comprising the dual triangles of the vertices of the cluster. The coloring is read clockwise around the outer face. On the right, we may alternatively fix a uniform proper coloring of the infinite dual map in advance. The coloring is read around the outer boudary of the cluster.](tree.pdf "fig:"){width="49.00000%"} \[thm:main\] Fix $n\geq 1$ and $(k,q)\in \{(1,4),(2,3)\}$. The sequence $(C_1,\ldots,C_{n+2})$ constructed above is a $k$-dependent proper $q$-coloring of the $(n+2)$-cycle. The coloring is symmetric in law under rotations and reflections of the cycle and permutations of the colors. Moreover, the sequence $(C_1,\ldots,C_{n+2-k})$ is equal in law to $(X_1,\ldots,X_{n+2-k})$ where $(X_i)_{i\in\Z}$ is the stationary $k$-dependent $q$-coloring of $\Z$ constructed in [@HL]. A third description of our construction is as follows. It is possible to define the uniformly random proper $q$-coloring of the infinite graph $\mathbb D$ (by consistency). When $q=3$ this coloring is simply a uniform choice from the $3!$ proper $q$-colorings of $\mathbb D$, corresponding to the permutations of the colors. (On the other hand when $q=4$, there are uncountably many such colorings.) We can now choose the Eden model cluster $T_n$ independently of the coloring of $\mathbb D$. The coloring of Theorem \[thm:main\] then arises as the sequence of colors appearing in clockwise order around the outer boundary of $T_n$. See Figure \[fig:cycleColor\]. We next address the one- and two-color marginals of the colorings. \[thm:onetwo\] Let $I$ be either $\Z$, or $\{1,\ldots,n\}$ for $n\geq 3$ (interpreted as the vertex set of a cycle). Let $X=(X_i)_{i\in I}$ be the 1-dependent 4-coloring and let $Y=(Y_i)_{i\in I}$ be the 2-dependent 3-coloring of $I$ (arising from [@HL] in the $\Z$ case, or from Theorem \[thm:main\] in the cycle case). 1. The process $(\mathbbm{1}[X_i\in\{1,2\}])_{i\in I}$ is equal in law to $(\mathbbm{1}[U_i>U_{i+1}])_{i\in I}$, where $(U_i)_{i\in I}$ are i.i.d. uniform on $[0,1]$. 2. The process $(\mathbbm{1}[Y_i=1])_{i\in I}$ is equal in law to $(\mathbbm{1}[U_{i-1}<U_i>U_{i+1}])_{i\in I}$, where $(U_i)_{i\in I}$ are i.i.d. uniform on $[0,1]$. 3. The process $(\mathbbm{1}[X_i=1])_{i\in I}$ is equal in law to $(\mathbbm{1}[B_i>B_{i+1}])_{i\in I}$, where $(B_i)_{i\in I}$ are i.i.d. taking values $0,1$ with equal probabilities. All indices are interpreted modulo $n$ in the cycle case. Color symmetry (Theorem \[thm:main\]) of course implies that any other colors may be substituted for $1$ and $2$ in (i)–(iii) above. The $\Z$ case of this theorem was proved in [@HL Theorem 4], but the proof was quite complicated and mysterious. We will deduce the $\Z$ case from the cycle case, which in turn will be proved by very simple and direct methods. The idea will be to construct the various cycle-indexed processes via Markovian necklace insertion procedures like the one already mentioned, and to couple them to each other. The insertion procedure corresponding to part (ii) was studied earlier in [@mallowsSheppNecklace; @nakataNecklace]. The fact that this procedure results in the peak set of a random permutation of the cycle (as follows from our proof of (ii) above) gives simpler proofs of many of the results of those papers. Notwithstanding the simplicity of our proof of Theorem \[thm:onetwo\], these characterizations of marginal processes remain rather mysterious. In particular, we do not know whether a similar characterization exists for the following variant. Consider the function $\iota:\{1,2,3,4\}\to\{1,2,\Star\}$ given by $$\iota(1)=1;\quad \iota(2)=2; \quad \iota(3)=\iota(4)=\Star.$$ Letting $(X_i)_{i\in\Z}$ be the $1$-dependent $4$-coloring of $\Z$ from [@HL], we do not know whether the process $(\iota(X_i))_{i\in \Z}$ has a simple description similar to those in Theorem \[thm:onetwo\]. More precisely, a process $(Z_i)_{i\in\Z}$ is called $r$-block-factor of an i.i.d. process $(U_i)_{i\in\Z}$ if $Z_i=f(U_{i},\ldots,U_{i+r-1})$ for all $i$ and a fixed function $f$. Every $(k+1)$-block-factor of an i.i.d. process is $k$-dependent (but no coloring is a block factor of an i.i.d. process [@HL Proposition 2]). We do not know whether $(\iota(X_i))_{i\in \Z}$ is a block factor of an i.i.d. process. Since the process is $1$-dependent, one might expect it to be a 2-block factor (by analogy with Theorem \[thm:onetwo\]). We show that this is not the case. \[onetwostar\] Let $(X_i)_{i\in\Z}$ be the $1$-dependent $4$-coloring of $\Z$ from [@HL]. The process $(\iota(X_i))_{i\in\Z}$ cannot be expressed as a $2$-block-factor of an i.i.d. process. In Section \[sec:rec\] we give an alternative combinatorial construction of the colorings in terms of recurrences. We prove Theorem \[thm:main\] in Section \[sec:index\] after showing that the coloring arising from the Eden model coincides with the combinatorial construction. Theorem \[thm:onetwo\] is proved in Section \[sec:markov\] and Theorem \[onetwostar\] in Section \[sec:block\]. Recurrences {#sec:rec} =========== For a positive integer $m$, we write $[m]:=\{1,\ldots,m\}$. Fix a positive integer $q$, representing the number of colors. For an integer $n\geq 0$, a **word** $x$ of length $n$ is an element of $[q]^n$, which we write as $x=x_1x_2\cdots x_n$. For words $x$ and $y$ we write $xy$ for their concatenation. The word $x$ is **proper** if $x_i\not=x_{i+1}$ for all $i\in [n-1]$. The word $x$ is **cyclically proper** if it is proper and in addition $x_n\not=x_1$. For $i\in [n]$ and $x\in [q]^n$ we denote the word $x_1x_2\cdots x_{i-1}x_{i+1}\cdots x_n$ by $\hat x_i$. The empty word (i.e., the unique element of $[q]^0$) is denoted by $\emptyset$. Recursively define the function $B^{\circ}$ on $[q]^n$ via $$\label{eq:rec} B^{\circ}(x)=\mathbbm{1}[\text{$x$ is cyclically proper}]\sum_{i=1}^nB^{\circ}(\hat x_i),\qquad n\geq 1,\ x\in [q]^n,$$ and $B^{\circ}(\emptyset)=1$. Let $Z^\circ(n,q):=\sum_{y\in[q]^n}B^\circ(y)$. It is easily verified that $Z^\circ(n,q)>0$ for all $q\geq 3$. For all such integers $q$ and for all $n\geq 0$, let $\Cycle_{n,q}$ denote the probability measure on $[q]^n$ given by $$\label{eq:cmarg} \Cycle_{n,q}(\{x\})=\frac{B^\circ(x)}{Z^\circ(n,q)}.$$ \[lem:shift\] For $n\geq 0$ and $q\geq 3$, the measure $\Cycle_{n,q}$ is invariant under cyclic shifts. When $n=0$ there is nothing to prove. Fix $n\geq 1$ and $q$. We prove, by induction on $n$, that for all words $x\in[q]^n$ the cyclic shift $\hat x_1x_1$ of $x$ satisfies $$\label{eq:shift} B^\circ(\hat x_1 x_1)=B^\circ(x).$$ The base case $n=1$ is apparent. Now suppose that holds for all words of length $n-1$ and let $x$ be a word of length $n$. Observe that $x$ is cyclically proper if and only if the word $y:=\hat x_1 x_1$ is cyclically proper. If neither $x$ nor $y$ has this property, then $B^\circ(x)=B^\circ(y)=0$ and holds trivially. Otherwise, $$\label{eq:xyrec} B^\circ(x)=\sum_{i=1}^nB^\circ(\hat x_i)\qquad\text{ and }\qquad B^\circ(y)=\sum_{i=1}^nB^\circ(\hat y_i).$$ Observe that for all $i\in[n]$, the word $\hat y_i$ is a cyclic shift of $\hat x_{i+1}$ (where indices are interpreted modulo $n$). Thus $B^\circ(\hat y_i)=B^\circ(\hat x_{i+1})$ by the inductive hypothesis. It follows that the two sums in are rearrangements of one another, and therefore $B^\circ(x)=B^\circ(y)$. The following is a useful reformulation of the recurrence . It can be interpreted as an instance of the Möbius inversion formula [@stanley1997enumerative Section 3.7]. \[lem:mrec\] For all $n,q\geq 1$ and for all words $x\in[q]^n$, we have that $$\label{eq:mobrec} B^\circ(x)=\sum_{i=1}^nB^\circ(\hat x_i)-2\sum_{i=1}^{n}\mathbbm{1}[x_i=x_{i+1}]B^\circ(\hat x_i),$$ where indices are interpreted modulo $n$. Fix the word $x$. If $x$ is cyclically proper, the latter term on the right side of vanishes and the result reduces to the defining recurrence for $B^\circ$. If $x$ is not cyclically proper, the left side of vanishes and it remains to verify that $$\label{eq:smobrec} \sum_{i=1}^nB^\circ(\hat x_i)=2\sum_{i=1}^{n}\mathbbm{1}[x_i=x_{i+1}]B^\circ(\hat x_i).$$ Fix an index $j\in[n]$ such that $x_j=x_{j+1}$. Then for all $i\in [n]\setminus \{j,j+1\}$, the word $\hat x_i$ is not cyclically proper. Hence the equality we wish to verify is equivalent to $$B^\circ(\hat x_j)+B^\circ(\hat x_{j+1})=2B^\circ(\hat x_j),$$ which follows since $\hat x_j=\hat x_{j+1}$. For $(k,q)\in\{(1,4),(2,3)\}$, the stationary $k$-dependent $q$-coloring $(X_i)_{i\in \Z}$ constructed in [@HL] is characterized as follows. Recursively define the function $\vec B$ on $[q]^n$ via $$\label{eq:vrec} \vec B(x)=\mathbbm{1}[x\text{ is proper}]\sum_{i=1}^n \vec B(\hat x_i),\qquad n\geq 1,\ x\in [q]^n,$$ and $\vec B(\emptyset)=1$. Let $\vec Z(n,q)=\sum_{y\in[q]^n}\vec B(y)$. Then for all $n\geq 0$ and for all $x\in [q]^n$, $$\label{eq:vmarg} \mathbb P\bigl((X_1,\ldots,X_n)=x\bigr)=\frac{\vec B(x)}{\vec Z(n,q)}.$$ In the following lemma and its proof, we use $\star$ to denote a dummy variable that is summed over $[q]$. For example, given a function $f\colon [q]^m\to \mathbb R$ and a word $x$, we write $f(x\star^m)$ as a shorthand for $\sum_{y\in [q]^m}$ $f(xy)$. \[lem:rest\] Fix integers $(k,q)\in \{(1,4),(2,3)\}$ and $n\geq 0$. Then for all words $x\in[q]^n$, $$\label{eq:resteq} B^\circ(x\star^k)=Z^\circ(k,q)\vec B(x).$$ We establish by induction on $n$. When $n=0$ or when $x$ is non-proper it holds trivially. Now suppose that holds for all words of length $n-1$ and that $x$ is proper. For any word $y\in [q]^k$, we have by Lemma \[lem:mrec\] that $$\label{eq:rbc} B^\circ(xy)=\sum_{i=1}^{n+k}B^\circ(\hat{xy}_i)-2\mathbbm{1}[x_n=y_1]B^\circ(\hat x_ny)-2\sum_{i=1}^k\mathbbm{1}[y_i=y_{i+1}]B^\circ(x\hat y_i),$$ where we have set $y_{k+1}=x_1$ (this occurs in the final summand of the second term). The term $2\mathbbm{1}[x_n=y_1]B^\circ(\hat x_ny)$ in the right side of the above equation is seen to equal $2\mathbbm{1}[x_n=y_1]B^\circ(x\hat y_1)$, and thus $$\label{eq:rbc2} B^\circ(xy)=\sum_{i=1}^{n+k}B^\circ(\hat{xy}_i)-2\mathbbm{1}[x_n=y_1]B^\circ(x\hat y_1)-2\sum_{i=1}^k\mathbbm{1}[y_i=y_{i+1}]B^\circ(x\hat y_i).$$ Next we sum over $y\in [q]^k$ to obtain that $$\begin{aligned} B^\circ(x\star^k)&=\sum_{i=1}^{n+k}\sum_{y\in[q]^k}B^\circ(\hat{xy}_i)-2(k+1)B^\circ(x\star^{k-1})\\ &=\sum_{i=1}^nB^\circ(\hat x_i\star^k)+qkB^\circ(x\star^{k-1})-2(k+1)B^\circ(x\star^{k-1}), \end{aligned}$$ where the coefficient $qk$ in the second term counts the number of choices of which index of $y$ to delete together with the deleted value, and the coefficient $2(k+1)$ in the third term arises from the $k+1$ indicator functions appearing in . Since $qk=2(k+1)$ for both pairs $(k,q)\in\{(1,4),(2,3)\}$, it therefore follows that $$B^\circ(x\star^k)=\sum_{i=1}^nB^\circ(\hat x_i\star^k).$$ Substituting the inductive hypothesis into this equation, using , and recalling that $x$ is proper yields . \[lem:partition\] For all $n\geq 2$ and for all $q\geq 2$, we have that $$Z^\circ(n,q)=n!\, q(q-1)(q-2)^{n-2}.$$ The proof is by induction on $n$. The base case $n=2$ is trivial to verify. Now suppose that $Z^\circ(n,q)=n!\,q(q-1)(q-2)^n$, where $n\geq 2$. Summing over all words yields that $$\label{eq:zc} Z^\circ(n+1,q)=\sum_{i=1}^{n+1}\sum_{x\in[q]^{n+1}}\mathbbm{1}[x\text{ is cyclically proper}]B^\circ(\hat x_i).$$ For all $i\in[n+1]$, it is easy to see that $$\label{eq:zc2} \sum_{x\in[q]^{n+1}}\mathbbm{1}[x\text{ is cyclically proper}]B^\circ(\hat x_i)=(q-2)Z^\circ(n,q).$$ Combining , , and the inductive hypothesis yields the result. Indexing {#sec:index} ======== We introduce a general procedure for constructing cycle-indexed processes by random insertion. We will apply this first to the coloring itself; later we will apply it to other processes in order to prove Theorem \[onetwostar\]. To establish our remaining results we consider the following dynamical probabilistic insertion scheme. Let $n\geq 3$. Given a random sequence $X^{(n)}=(X_1,\ldots,X_n)$, we will choose a uniformly random index $I\in[n]$ (which will be independent of $X^{(n)}$) and insert a new element $Z$ (which can depend on $I$ and $X^{(n)}$) just before element $X_I$ to give $$\label{eq:insert} (X_1,\ldots,X_{I-1},Z,X_I,\ldots,X_n).$$ Here an indexing issue arises. We want to interpret the above sequence as a labelling of the $(n+1)$-cycle, but we would like the random insertion location to be uniform *after* the insertion has taken place as well as before it. Simply defining $X^{(n+1)}$ to equal does not achieve this (because the inserted element $Z$ can never end up at location $n+1$, for instance). Since all the random sequences that we consider are invariant under rotations of the cycle, we instead apply a random rotation after the insertion step. That is, we let $R$ be uniformly random in $[n+1]$ independent of all previous choices, and set $X^{(n+1)}$ to be the image of under the map $\tau_R$, where $\tau_r$ is the rotation $$\tau_r\colon(x_1,\ldots,x_{n+1})\mapsto (x_{r+1},\ldots,x_{r+n+1}),$$ where the indices are interpreted modulo $n+1$. In the following lemma we show that the above insertion procedure yields a coupling of the measures $(\Cycle_n)_{n\geq 3}$. \[lem:coupling\] Fix $q\geq 1$. For each $n\geq 3$, let $X^{(n)}=(X_1,\ldots,X_n)$ be random with law $\Cycle_{n,q}$. Let $I\in[n]$ be uniformly random and independent of $X^{(n)}$. Conditional on $I$ and $X^{(n)}$, let $Z$ be a uniformly random element of the set $[q]\setminus \{X_{I-1},X_I\}$ (where indices are interpreted modulo $n$). Finally, let $R\in[n+1]$ be uniformly random and independent of $I, X^{(n)}, $ and $Z$. Then $$\tau_R(X_1,\ldots,X_{I-1},Z,X_I,\ldots,X_n)\eqd X^{(n+1)}.$$ We show that for all words $x\in[q]^{n+1}$, $$\label{eq:insCyc} \mathbb P\bigl(\tau_R(X_1,\ldots,X_{I-1},Z,X_I,\ldots,X_n)=x\bigr)=\mathbb P\bigl(X^{(n+1)}=x\bigr).$$ Indeed, the probability appearing on the left side of can be written as $$\sum_{r=1}^{n+1}\mathbb P(R=r)\sum_{i=1}^n\mathbb P(I=i)\mathbb P\bigl((X_1,\ldots,X_{i-1},Z,X_i,\ldots,X_n)=\tau_{-r}x\bigr),$$ which by the definitions of $R,I,Z,$ and $\Cycle_{n,q}$ in equals $$\label{eq:insCyc3} \frac{1}{n(n+1)}\sum_{r=1}^{n+1}\sum_{i=1}^n\frac{\mathbbm{1}[x\text{ is cyclically proper}]}{q-2}\frac{B^\circ(\hat{\tau_{-r}x}_i)}{Z^\circ(n,q)}.$$ The word $\hat{\tau_{-r}x}_i$ is a cyclic shift of $\hat{x}_{i-r}$, where indices are interpreted modulo $n+1$. Thus by Lemma \[lem:shift\], we have that equals $$\label{eq:insCyc4} \frac{\mathbbm{1}[x\text{ is cyclically proper}]}{n(n+1)(q-2)Z^\circ(n,q)}\sum_{r=1}^{n+1}\sum_{i=1}^nB^\circ(\hat{x}_{i-r}).$$ For each $j\in[n+1]$, there are $n$ pairs $(r,i)\in[n+1]\times[n]$ for which $i-r\equiv j\mod{(n+1)}$. Combining this observation with Lemma \[lem:partition\] yields that equals $$\label{eq:insCyc5} \frac{\mathbbm{1}[x\text{ is cyclically proper}]}{Z^\circ(n+1,q)}\sum_{j=1}^{n+1}B^\circ(\hat x_j).$$ By and the definition of $\Cycle_{n+1,q}$, the right side of equals that of . We prove by induction on $n\geq 1$ that the coloring $(C_1,\ldots,C_{n+2})$ arising from the Eden model has law $\Cycle_{n+2,q}$. This is clear for $n=1$, when both colorings are uniformly random. Next suppose this holds for $n$. Recall that $(C_1,\ldots,C_{n+2})$ is constructed from an Eden model cluster $T_n$, a uniform proper $q$-coloring of its dual triangulation $D_n$, and an independent uniform vertex $u$ of $D_n$. One obtains a coupling with the corresponding objects at time $n+1$ as follows. Choose a uniform triangle in $\mathbb D$ sharing an edge with the outer boundary of $D_n$, append it to $D_n$ to form $D_{n+1}$, choose a uniformly random non-clashing color for the new vertex, and finally select an independent uniform vertex of $D_{n+1}$. By Lemma \[lem:coupling\] it follows that the coloring of the $(n+3)$-cycle obtained from these data has law $\Cycle_{n+3,q}$, completing the induction. Combining the result of the previous paragraph with , , and Lemma \[lem:rest\], yields the restriction property. Applying [@HL Theorem 1] yields $k$-dependence. Markov chains {#sec:markov} ============= We give simple conceptual proofs that the 1- and 2-color marginals of the 3- and 4-colorings on the cycle are as claimed. Before giving the details, we explain the structure of the proof. For fixed $q\in\{3,4\}$, let $$X^{(n)}=(X_1,\ldots,X_n)=\bigl(X_1^{(n)},\ldots,X_n^{(n)}\bigr)$$ be a coloring with law $\Cycle_{n,q}$. Recall that we are interested in the indicator of the support of one or two of the colors, i.e. the binary vector $J^{(n)}=(J_1,\ldots,J_n)$ where $$J_i=J_i^{(n)}:=\mathbbm{1}[X_i^{(n)}\in A],$$ and where $A$ is $\{1\}$ or $\{1,2\}$. Our goal is to show for each $n$ that $J^{(n)}$ is equal in law to a certain ‘target’ binary vector $Q^{(n)}$ that is defined in terms of random permutations or binary strings. We will couple the random vectors $J^{(n)}$ for $n\geq 3$ by considering them as the states of a Markov chain. We will specify the Markov transition rule from $J^{(n)}$ to $J^{(n+1)}$. Then we will construct another Markov chain whose state at time $n$ is equal in law to the target random vector $Q^{(n)}$. Finally we will show that even though their descriptions differ, the two Markov transition rules coincide. In fact, we will couple the colorings $X^{(n)}$ themselves via a Markov chain indexed by $n$. The vector $J^{(n)}$ is a deterministic function, $F$ say, of $X^{(n)}$. In general, applying a function to the state of a Markov chain does not yield another Markov chain. However, this *will* hold in our case. Specifically, it will follow because the conditional law of $F(X^{(n+1)})$ given $X^{(n)}=x$ will be identical for all $x$ having the same image under $F$. The same remarks will apply to the target laws also: the random vector $Q^{(n)}$ can be expressed as a deterministic function of a random permutation or a binary string. We will give a natural Markov chain on permutations or binary strings which yields a Markov chain for $Q^{(n)}$. All the Markov chains we consider will involve random insertions in the cycle. These insertions take the form described in the previous section. Namely, given $X^{(n)}=(X_1,\ldots,X_n)$, we will choose a uniformly random index $I\in\{1,\ldots,n\}$ (independently of $X^{(n)}$), insert a new element just before element $X_I$ (which can depend on $I$ and $X^{(n)}$), and apply a uniformly random rotation (independently of $I$ and $X^{(n)}$). We now proceed with the details of the proofs. We show that the locations of colors 1 and 2 in the 4-coloring of the $n$-cycle have the same joint law as the descent set of a uniform permutation of $n$-cycle for all $n\geq 3$. This is easy to verify directly when $n=3$, in which case both random sets are uniformly distributed over subsets of size 1 or 2. To prove the result for other $n$, we show that the Markov transition rules are equivalent. Let $X^{(n)}$ be random and $\Cycle_{n,4}$-distributed. Define the function $F$ as follows: $$F(x_1,\ldots,x_n):=\bigl(\mathbbm{1}[x_i\in\{1,2\}]\bigr)_{i=1}^n.$$ Set $J^{(n)}:=F(X^{(n)})$ to be the binary vector indicating the locations of colors 1 and 2. The Markov transition rule for $J^{(n)}$ obtained by applying the function $F$ to the transition rule in Lemma \[lem:coupling\] is the following. Conditional on $J^{(n)}$, sample $(I,B)$ uniformly from $[n]\times \{0,1\}$, let $$Z=\begin{cases} 1-J_I,& J_{I-1}=J_I\\ B,& J_{I-1}\not=J_I \end{cases},$$ and take $J^{(n+1)}$ to be the image of $(J_1,\ldots,J_{I-1},Z,J_I,\ldots,J_n)$ under an independent uniformly random rotation. This transition rule is equivalent to choosing a random insertion location, querying the bit of one of the two neighbors at random, and inserting the complement of the queried bit. Let $\Pi^{(n)}=(\Pi_1,\ldots,\Pi_n)\in[n]^n$ be a uniformly random permutation. That is, $\Pi^{(n)}$ is distributed uniformly over the set of words in $[n]^n$ in which each symbol appears exactly once. Let $G$ be the function $$G(x_1,\ldots,x_n):=\bigl(\mathbbm{1}[x_i>x_{i+1}]\bigr)_{i=1}^n,$$ and set $Q^{(n)}:=G(\Pi^{(n)})$. Before describing the transition rule for $Q^{(n)}$, we describe one for $\Pi^{(n)}$. Given $\Pi^{(n)}$, let $I\in [n]$ be an independent uniformly random index and let $\Pi^{(n+1)}$ be the result of applying a uniformly random rotation to one of the two tuples $$(\Pi_1,\ldots,\Pi_{I-1},n+1,\Pi_I,\ldots,\Pi_n)\text{ or }(\Pi_1+1,\ldots,\Pi_{I-1}+1,1,\Pi_I+1,\ldots,\Pi_n+1),$$ chosen uniformly at random and independently of the rotation and $I$. It is easily seen that the law of $\Pi^{(n+1)}$ is uniform on the set of permutations. The Markov transition rule for $Q^{(n)}$ obtained by applying the function $G$ to the transition rule of $\Pi^{(n)}$ is the following. Conditional on $Q^{(n)}$, sample $(I,B)$ uniformly from $[n]\times \{0,1\}$ and take $Q^{(n+1)}$ to be the image of $(Q_1,\ldots,Q_{I-2},B,1-B,Q_I,\ldots,Q_n)$ under an independent uniformly random rotation. This transition rule is equivalent to choosing a random symbol and replacing it with one of the strings $01$ or $10$, chosen uniformly at random and independently of $Q^{(n)}$ and $I$. The transition rules for $J^{(n)}$ and $Q^{(n)}$ coincide, as both are equivalent to the following. Conditional on $J^{(n)}=Q^{(n)}=x$, let $U$ be chosen uniformly at random from the set $$\bigl\{i-\tfrac14\bigr\}_{i=1}^n\cup\bigl\{i+\tfrac14\bigr\}_{i=1}^n$$ and let $V$ denote the integer closest to $U$. Insert the bit $1-x_V$ between positions $\lfloor U\rfloor$ and $\lceil U\rceil$ of $x$. Since $J^{(3)}$ and $Q^{(3)}$ have the same law and the Markov transition rules for $J^{(n)}$ and $Q^{(n)}$ agree for all $n\geq 3$, it follows that $J^{(n)}$ and $Q^{(n)}$ are equal in law. We show that for all $n\geq 3$, the locations of color 1 in the 3-coloring of the $n$-cycle have the same joint law as the peak set of a uniformly random permutation of the $n$-cycle. This is easy to verify directly when $n=3$, in which case both random sets are uniformly distributed over the singletons. Let $X^{(n)}$ be random and $\Cycle_{n,3}$-distributed, let $F$ be the function $$F(x_1,\ldots,x_n):=\bigl(\mathbbm{1}[x_i=1]\bigr)_{i=1}^n,$$ and let $J^{(n)}:=F(X^{(n)})$. Take $\Pi^{(n)}=(\Pi_1,\ldots,\Pi_n)$ to be a uniformly random permutation and let $G$ be the function $$G(x_1,\ldots,x_n):=\bigl(\mathbbm{1}[x_{i-1}<x_i>x_{i+1}]\bigr)_{i=1}^n,$$ with indices interpreted modulo $n$. Set $Q^{(n)}:=G(\Pi^{(n)})$. The Markov transition rule for $J^{(n)}$ obtained by applying the function $F$ to the transition rule in Lemma \[lem:coupling\] is the following. Conditional on $J^{(n)}$, sample $I$ uniformly from $[n]$, let $$Z=\begin{cases} 1,& J_{I-1}=J_I=0\\ 0,& \text{otherwise} \end{cases},$$ and take $J^{(n+1)}$ to be the image of $(J_1,\ldots,J_{I-1},Z,J_I,\ldots,J_n)$ under an independent uniformly random rotation. Before describing the transition rule for $Q^{(n)}$, we describe one for $\Pi^{(n)}$. Given $\Pi^{(n)}$, let $I\in[n]$ be an independent uniformly random index and let $\Pi^{(n+1)}$ be the result of applying a uniformly random rotation to $$(\Pi_1,\ldots,\Pi_{I-1},n+1,\Pi_I,\ldots,\Pi_n).$$ It is easily seen that the law of $\Pi^{(n+1)}$ is uniform on the set of permutations. The Markov transition rule for $Q^{(n)}$ obtained by applying the function $G$ to the transition rule of $\Pi^{(n)}$ is the following. Conditional on $Q^{(n)}$, sample $I$ uniformly from $[n]$ and take $Q^{(n+1)}$ to be the image of $(Q_1,\ldots,Q_{I-2},0,1,0,Q_{I+1},\ldots,Q_n)$ under an independent uniformly random rotation. We show that the transition rules for $J^{(n)}$ and $Q^{(n)}$ coincide. The state space consists of binary strings that lack adjacent ones. We identify any such a string, $x$, with a partition of the set $\bigl\{i+\tfrac12\bigr\}_{i=1}^n$, by regarding the ones as endpoints of blocks. Both transition rules are equivalent to the following. Conditional on $J^{(n)}=Q^{(n)}=x$, let $U$ be uniformly random element of the set $\bigl\{i+\tfrac12\bigr\}_{i=1}^n$. If all of $U-1$, $U$, and $U+1$ belong to the same block of $x$, split the block at $U$ by inserting a one. Otherwise, grow the length of the block containing $U$ by inserting a zero. Since $J^{(3)}$ and $Q^{(3)}$ have the same law and the Markov transition rules for $J^{(n)}$ and $Q^{(n)}$ agree for all $n\geq 3$, it follows that $J^{(n)}$ and $Q^{(n)}$ are equal in law. We use the following term in the proof of Theorem \[thm:onetwo\](iii). A **hard-core** process on a graph $G$ is a process $(J_v)_{v\in V}$ such that each $J_v$ takes values in $\{0,1\}$, and almost surely we do not have $J_u=J_v=1$ for adjacent vertices $u,v$. Both process are hard-core and 1-dependent. As shown in [@HL] (in the proof of Lemma 23), the law of a 1-dependent hard-core process on any graph is determined by its one-vertex marginals. Since this equals $\tfrac14$ for both processes, it follows that they are equal in law. We remark that Theorem \[thm:onetwo\](iii) may also be proven via an argument involving Markov transition rules, similar to the proofs of Theorem \[thm:onetwo\](i) and (ii), which we now sketch. The transition rule for the ones process inserts a random bit between two zeros, and otherwise inserts a zero. In terms of the half-integer blocks picture, you grow the block if the insertion location is at an endpoint, and otherwise you flip a coin to decide if to split or grow a block. The $\mathbbm{1}[B_i>B_{i+1}]$ process can also be interpreted in this way. Block factor {#sec:block} ============ Recall that $\iota:{1,2,3,4}\to\{1,2,\Star\}$ is the function that fixes $1$ and $2$ but maps $3$ and $4$ to $\Star$. Recall that a process $X=(X_i)_{i\in\Z}$ is said to be an $r$-block-factor of $U=(U_i)_{i\in\Z}$ if $X_i=f(U_{i},\ldots,U_{i+r-1})$ for all $i$ and a fixed function $f$. In this section we prove Theorem \[onetwostar\]. The proof uses the following lemma which characterizes $2$-block-factor hard-core processes. \[rectangle\] Suppose that a hard-core process $Z=(Z_i)_{i\in\Z}$ is a $2$-block-factor of an i.i.d. process $U$. Then, almost surely $$Z_i=\mathbbm{1}\bigl[(U_i,U_{i+1})\in S\bigr],\quad i\in\Z,$$ for some sets $S$ and $A$ satisfying $$S\subseteq A\times A^C.$$ Moreover, we have $\E Z_0\leq \tfrac14$, with equality if and only if $\P(U_0\in A)=\tfrac12$ and $S=A\times A^C$ up to null sets. Above and in the following proofs, we say that two sets are equal up to null sets if their symmetric difference is a null set. Similarly, we write $E\subseteq F$ up to a null set if $E\setminus F$ is null. Note that the inequality $\E Z_0\leq \tfrac14$ holds even for stationary $1$-dependent hard-core processes; see e.g. [@HL Corollary 17]. Without loss of generality, suppose that the $U_i$ are uniformly distributed on $[0,1]$. By definition, any $\{0,1\}$-valued $2$-block-factor $Z$ is of the form $Z_i=\mathbbm{1}[(U_i,U_{i+1})\in S]$ for some $S\subseteq [0,1]^2$. Our task is to show that $S$ has the claimed form. Define $$A:=\Bigl\{u_0\in[0,1]: \P\bigl[ (u_0,U_1)\in S \bigr]>0 \Bigr\}.$$ First note that $S\subseteq A\times[0,1]$ up to a null set by Fubini’s theorem. (Indeed, if $\P(U_0\notin A)>0$ then $\P[(U_0,U_1)\in S\mid U_0\notin A]=0$; thus $\P[(U_0,U_1)\in S,\; U_0\notin A]=0$.) We claim that also $S\subseteq [0,1]\times A^C$ up to a null set. If not, then the event $E:=\{(U_0,U_1)\in S,\; U_1\in A\}$ has positive probability. But $U_2$ is independent of $E$. Therefore, using the definition of $A$, we have $\P[(U_1,U_2)\in S\mid E]>0$. Hence, $$\P(Z_0=Z_1=1)\geq \P\bigl[E,\; (U_1,U_2)\in S\bigr]>0,$$ which contradicts $Z$ being hard-core, thus proving the claim. The two inclusions proved above imply that $S\subseteq A\times A^C$ up to a null set. Adjusting $S$ by a null set if necessary, the first assertion of the lemma follows. To prove the second assertion, note that $$\E Z_0=\P\bigl[(U_i,U_{i+1})\in S\bigr]\leq \P\bigl[(U_i,U_{i+1})\in A\times A^C\bigr]=a(1-a)\leq \tfrac14,$$ where $a:=\P(U_0\in A)$. Equality throughout implies the stated conditions. Let $X$ be the $1$-dependent $4$-coloring of $\Z$, let $W_i=\iota(X_i)$, and suppose for a contradiction that $W$ is a $2$-block factor of the i.i.d. process $U$. Without loss of generality, suppose that the $U_i$ are uniformly distributed on $[0,1]$. Let $Y_i=\mathbbm{1}[X_i=1]$ and $Z_i=\mathbbm{1}[X_i=2]$. Since $Y$ and $Z$ are functions of $W$, they are also $2$-block-factors of $U$. By color symmetry of the 1-dependent 4-coloring (Theorem \[thm:main\] or [@HL Theorem 4(i)]) or by the inequality in Lemma \[rectangle\] we have $\E Y_0=\E Z_0=\tfrac14$, and so Lemma \[rectangle\] implies that there exist sets $A,B\subset[0,1]$ each of measure $\tfrac12$ such that $$Y_i=\mathbbm{1}\bigl[(U_i,U_{i+1})\in A\times A^C\bigr]; \qquad Z_i=\mathbbm{1}\bigl[(U_i,U_{i+1})\in B\times B^C\bigr]$$ almost surely. We claim that $A=B^C$ up to null sets. Indeed, $A\cap B$ has positive measure if and only if $A^C\cap B^C$ does. But in that case, $$\P(Y_0=Z_0=1)\geq \P(U_0\in A\cap B)\,\P(U_1\in A^C\cap B^C)>0,$$ which is impossible. Thus, up to null sets, $W_i=\Star$ if and only if $(U_i,U_{i+1})\in A^2\cup(A^C)^2$. But this gives the wrong law for the process of $\Star$’s. For instance, $$\begin{aligned} \P(W_0=W_1=\Star)&=\P\bigl[(U_0,U_1,U_2)\in A^3\cup(A^C)^3\bigr]=\tfrac14,\\ \intertext{whereas Theorem~\ref{thm:onetwo}(iii) gives} \P(W_0=W_1=\Star)&=\P\bigl[X_0,X_1\in\{3,4\}\bigr]=\P(U_0<U_1<U_2)=\tfrac16. \quad\qedhere \end{aligned}$$ Open problems ============= 1. For $(k,q)\in\{(1,5),(2,4),(3,3)\}$ and for all larger $k$ and $q$, is there a color-symmetric $k$-dependent $q$-coloring of the $n$-cycle for all $n\geq 3$? In particular, is there an analogue of the colorings in [@hhlMalCol] on the cycle? 2. Let $\iota\colon \{1,2,3,4\}\to\{1,2,\Star\}$ fix $1,2$ and map $3,4$ to $\Star$. Let $X=(X_i)_{i\in\Z}$ be the 1-dependent 4-coloring of [@HL]. Is the process $(\iota(X_i))_{i\in\Z}$ a block factor of an i.i.d. process? Acknowledgements {#sec:ack .unnumbered} ================ AL and TH were supported by internships at Microsoft Research while portions of this work were completed. TH was also supported by a Microsoft Research PhD fellowship.

--- abstract: | We propose the general construction formula of shape-color primitives by using partial differentials of each color channel in this paper. By using all kinds of shape-color primitives, shape-color differential moment invariants can be constructed very easily, which are invariant to the shape affine and color affine transforms. 50 instances of SCDMIs are obtained finally. In experiments, several commonly used color descriptors and SCDMIs are used in image classification and retrieval of color images, respectively. By comparing the experimental results, we find that SCDMIs get better results. -color primitives, affine transform, partial differential, shape-color differential moment invariants author: - 'Hanlin Mo$^{1}$$^{(}$$^{)}$$^{,}$[^1]' - 'Shirui Li$^{1}$' - 'You Hao$^{1}$' - 'Hua Li$^{1}$' title: 'Shape-Color Differential Moment Invariants under Affine Transformations' --- Introduction ============ Image classification and retrieval for color images are two hotspots in pattern recognition. How to extract effective features, which are robust to color variations caused by the changes in the outdoor environment and geometric deformations caused by viewpoint changes, is the key issue. The classical approach is to construct invariant features for color images. Moment invariants are widely used invariant features. Moment invariants were first proposed by Hu[@1] in 1962. He defined geometric moments and constructed 7 geometric moment invariants which were invariant under the similarity transform(rotation, scale and translation). Researchers applied Hu moments to many fields of pattern recognition and achieved good results[@2; @3]. Nearly 30 years later, Flusser et al.[@4] constructed the affine moment invariants (AMIs) which are invariant under the affine transform. The geometric deformations of an object, which are caused by the viewpoint changes, can be represented by the projective transforms. However, general projective transforms are complex nonlinear transformations. So, it’s difficult to construct projective moment invariants. When the distance between the camera and the object is much larger than the size of the object itself, the geometric deformations can be approximated by the affine transform. AMIs have been used in many practical applications, such as character recognition[@5] and expression recognition[@6]. In order to obtain more AMIs, researchers designed all kinds of methods. Suk et al.[@7] proposed graph method which can be used to construct AMIs of arbitrary orders and degrees. Xu et al.[@8] proposed the concept of geometric primitives, including distance, area and volume. AMIs can be constructed by using various geometric primitives. This method made the construction of moment invariants have intuitive geometric meaning. The above-mentioned moment invariants are all designed for gray images. With the popularity of color images, the moment invariants for color images began to appear gradually. Researchers wanted to construct moment invariants which are not only invariant under the geometric deformations but also invariant under the changes of color space. Geusebroek et al.\[9\] proved that the affine transform model was the best linear model to simulate changes in color resulting from changes in the outdoor environment. Mindru et al.[@10] proposed moment invariants which were invariant under the shape affine transform and the color diagonal-offset transform. The invariants were constructed by using the related concepts of Lie group. Some complex partial differential equations had to be solved. Thus, the number of them was limited and difficult to be generalized. Also, Suk et al.[@11] put forward affine moment invariants for color images by combining all color channels. But this approach was not intuitive and did not work well for the color affine transform. To solve these problems, Gong et al.[@12; @13; @14] constructed the color primitive by using the concept of geometric primitive proposed in [@8]. Combining the color primitive with some shape primitives, moment invariants that are invariant under the shape affine and color affine transforms can be constructed easily, which were named shape-color affine moment invariants(SCAMIs). In [@14], they obtained 25 SCAMIs which satisfied the independency of the functions. However, we find that a large number of SCAMIs with simple structures and good properties are missed in [@14]. In this paper, we propose the general construction formula of shape-color primitives by using partial differentials of each color channel. Then, we use two kinds of shape-color primitives to construct shape-color differential moment invariants(SCDMIs), which are invariant under the shape affine and color affine transforms. We find that the construction formula of SCAMIs proposed in [@14] is a special case of our method. Finally, commonly used image descriptors and SCDMIs are used for image classification and retrieval of color images, respectively. By comparing the experimental results, we find that SCDMIs proposed in this paper get better results. Related Work ============ In order to construct image features which are robust to color variations and geometric deformations, researchers have made various attempts. Among them, SCAMIs proposed in [@14] are worthy of special attention. SCAMIs are invariant under the shape affine and color affine transforms. Two kinds of affine transforms are defined by $$\left( \begin{array}{c} x^{'}\\ y^{'}\\ \end{array} \right) =SA \cdot \left( \begin{array}{c} x\\ y\\ \end{array} \right) +ST = \left( \begin{array}{cc} \alpha_{1}& \alpha_{2}\\ \beta_{1}& \beta_{2}\\ \end{array} \right)\cdot \left( \begin{array}{c} x\\ y\\ \end{array} \right) + \left( \begin{array}{c} O_{x}\\ O_{y}\\ \end{array} \right)$$ $$\left( \begin{array}{c} R^{'}(x,y)\\ G^{'}(x,y)\\ B^{'}(x,y)\\ \end{array} \right) =CA \cdot \left( \begin{array}{c} R(x,y)\\ G(x,y)\\ B(x,y)\\ \end{array} \right) +CT = \left( \begin{array}{ccc} \ a_{1}&a_{2}&a_{2}\\ \ b_{1}&b_{2}&b_{2}\\ \ c_{1}&c_{2}&c_{2}\\ \end{array} \right)\cdot \left( \begin{array}{c} R(x,y)\\ G(x,y)\\ B(x,y)\\ \end{array} \right) + \left( \begin{array}{c} O_{R}\\ O_{G}\\ O_{B}\\ \end{array} \right)$$ where SA and CA are nonsingular matrices. For the color image $I(R(x,y),G(x,y),B(x,y))$, let $(x_{p},y_{p}),(x_{q},y_{q}),(x_{r},y_{r})$ be three arbitrary points in the domain of $I$. The shape primitive and the color primitive are defined by $$S(p,q)= \left| \begin{array}{cc} (x_{p}-\bar{x})&(x_{q}-\bar{x})\\ (y_{p}-\bar{y})&(y_{q}-\bar{y})\\ \end{array} \right|$$ $$\begin{split} &C(p,q,r)= \left| \begin{array}{ccc} (R(x_{p},y_{p})-\bar{R})&(R(x_{q},y_{q})-\bar{R})&(R(x_{r},y_{r})-\bar{R})\\ (G(x_{p},y_{p})-\bar{G})&(G(x_{q},y_{q})-\bar{G})&(G(x_{r},y_{r})-\bar{G})\\ (B(x_{p},y_{p})-\bar{B})&(B(x_{q},y_{q})-\bar{B})&(B(x_{r},y_{r})-\bar{B})\\ \end{array} \right| \end{split}$$ where $\bar{A}$ represents the mean value of $A$, $A \in \{x,y,R,G,B\}$. Then, using Eq.(3) and (4), the shape core can be defined by $$sCore(n,m;d_{1},d_{2},...,d_{n})=\underbrace{S(1,2)S(k,l)...S(r,n)}_m$$ where n and m represent that the sCore is the product of m shape primitives which are constructed by N points $(x_{1},y_{1}),(x_{2},y_{2}),...,(x_{n},y_{n})$. $k<l$, $r<n$, $k,l,r \in \left\{1,2,...n\right\}$. $d_{i}$ represents the number of point $(x_{i},y_{i})$ in all shape primitives, $i=1,2,...,n$. Similarly, the color core can be defined by $$cCore(N,M;D_{1},D_{2},...,D_{N})=\underbrace{C(1,2,3)C(G,K,L)...C(P,Q,N)}_M$$ Where N and M represent that the cCore is the product of M color primitives which are constructed by N points $(x_{1},y_{1}),(x_{2},y_{2}),...,(x_{N},y_{N})$. $G<K<L$, $P<Q<N$, $G,K,L,P,Q \in \left\{1,2,...N\right\}$. $D_{i}$ represents the number of point $(x_{i},y_{i})$ in all color primitives, $i=1,2,...,N$. Suppose the color image $I(R(x,y),G(x,y),B(x,y))$ is transformed into the image $I^{'}(R^{'}(x^{'},y^{'}),G^{'}(x^{'},y^{'}),B^{'}(x^{'},y^{'}))$ by two transformations defined by Eq.(1) and (2), $(x^{'}_{p},y^{'}_{p}),(x^{'}_{q},y^{'}_{q}),(x^{'}_{r},y^{'}_{r})$ in $I^{'}$ are the corresponding points of $(x_{p},y_{p}),(x_{q},y_{q}),(x_{r},y_{r})$ in $I$. Gong et al.[@14] have proved $$S^{'}(p,q)=|SA| \cdot S(p,q)$$ $$C^{'}(p,q,r)=|CA| \cdot C(p,q,r)$$ Further results can be concluded $$sCore^{'}(n,m;d_{1},d_{2},...,d_{n})=|SA|^{m}\cdot sCore(n,m;d_{1},d_{2},...,d_{n})$$ $$cCore^{'}(N,M;D_{1},D_{2},...,D_{N})=|CA|^{M}\cdot cCore(N,M;D_{1},D_{2},...,D_{N})$$ Therefore, the SCAMIs are constructed by $$\begin{split} &SCAMIs(n,m,N,M;d_{1},...,d_{n};D_{1},...,D_{N})\\&= \frac {I(sCore(n,m,d_{1},...,d_{n})\cdot cCore(N,M,D_{1},...,D_{N}))}{I(sCore(1,0))^{max(n+N)+m-\frac{3M}{2}}\cdot I(cCore(3,2;2,2,2))^{\frac{M}{2}}}\\ \end{split}$$ Then there is a relation $$\begin{split} SCAMIs^{'}(n,m,N,M;d_{1},...,d_{n};D_{1},...,D_{N})&\\=SCAMIs(n,m,N,M;d_{1},...,d_{n};D_{1},...,D_{N}) \end{split}$$ It must be said that $\max\{n,N\}$, $\max\limits_{i}\{d_{i}\}$ and $\max\limits_{i}\{D_{i}\}$ are named the degree, the shape order and the color order of SCAMIs, respectively. In fact, Eq.(11) can be expressed as polynomial of shape-color moment. This moment was first proposed in [@16] and defined by $$SCM_{pq\alpha \beta \gamma}=\iint (x-\bar{x})^{p}(y-\bar{y})^{q}(R(x,y)-\bar{R})^{\alpha}(G(x,y)-\bar{G})^{\beta}(B(x,y)-\bar{B})^{\gamma}dxdy$$ Gong et al.[@14] proposed that they constructed all SCAMIs of which degrees $\leqslant 4$, shape orders $\leqslant 4$ and color orders $\leqslant 2$. They obtained 24 SCAMIs which are functional independencies using the method proposed by Brown [@17]. However, we will point out in the Section 3 that they omitted many simple and well-behaved SCAMIs. The Construction Framework of SCDMIs ==================================== In this section, we introduce the general definitions of shape-color differential moment and shape-color primitive, firstly. Then, using the shape-color primitive, the shape-color core can be constructed. Finally, according to Eq.(11) and the shape-color core, we obtain the general construction formula of SCDMIs. Also, 50 instances of SCDMIs are given for experiments in the Section 4. The Definition of The General Shape-Color Moment ------------------------------------------------ **Definition 1.** Suppose the color image $I(R(x,y), G(x,y), B(x,y))$ have the k-order partial derivatives $(k=0,1,2,...)$. The general shape-color differential moment is defined by $$\begin{split} SCM^{k}_{pq\alpha \beta \gamma}=\iint &(x-\bar{x})^{p}(y-\bar{y})^{q}(R^{(k)}(x,y)-\bar{R}^{\delta(k)})^{\alpha}(G^{(k)}(x,y)-\bar{G}^{\delta(k)})^{\beta}\\&(B^{(k)}(x,y)-\bar{B}^{\delta(k)})^{\gamma}dxdy\\ \end{split}$$ where $(R^{(k)}(x,y)$, $G^{(k)}(x,y), B^{(k)}(x,y))$ represent the k-order partial derivatives of $(R(x,y), G(x,y), B(x,y))$. $\bar{R}, \bar{G}, \bar{B}$ represent the mean values of $R, G, B$. $\delta(k)$ is the impact function which is defined by $$\delta(k)=\left\{ \begin{aligned} 1 &~& (k=0) \\ 0 &~& (k\neq 0)\\ \end{aligned} \right.$$ We can find that Eq.(14) and Eq.(13) are identical, when $k=0$. Therefore, the shape-color moment is a special case of the general shape-color differential moment. The Construction of The General Shape-Color Primitive ----------------------------------------------------- **Definition 2.** Suppose the color image $I(R(x,y), G(x,y), B(x,y))$ have the k-order partial derivatives $(k=0,1,2,...)$. $(x_{p},y_{p}),(x_{q},y_{q}),(x_{r},y_{r})$ are three arbitrary points in the domain of $I$. The general shape-color primitive is defined by $$\begin{split} SCP_{k}(p,q,r)= \left| \begin{array}{ccc} F^{k}_{R}(x_{p},y_{p})&F^{k}_{R}(x_{q},y_{q})&F^{k}_{R}(x_{r},y_{r})\\ F^{k}_{G}(x_{p},y_{p})&F^{k}_{G}(x_{q},y_{q})&F^{k}_{G}(x_{r},y_{r})\\ F^{k}_{B}(x_{p},y_{p})&F^{k}_{B}(x_{q},y_{q})&F^{k}_{B}(x_{r},y_{r})\\ \end{array} \right| \end{split}$$ where $$F^{k}_{C}(x,y)=\sum_{i=0}^{k}\binom{k}{i}(x-\bar{x})^{i}(y-\bar{y})^{k-i}\frac{\partial^{k}C(x,y)}{\partial x^{i}\partial y^{k-i}}, ~~~C \in \{R,G,B\}.$$ We can find that $C(p,q,r)$ defined by Eq.(4) is a special case of $SCP_{k}(p,q,r)$, when $k=0$. The Construction of The General Shape-Color Core ------------------------------------------------ **Definition 3.** Using Definition 2, the general shape-color core is defined by $$scCore_{k}(N,M;D_{1},D_{2},...,D_{N})=\underbrace{SCP_{k}(1,2,3)SCP_{k}(G,K,L)...SCP_{k}(P,Q,N)}_M$$ Where $k=1,2,...$, N and M represent that the $scCore_{k}$ is the product of M shape-color primitives constructed by N points $(x_{1},y_{1}),(x_{2},y_{2}),...,(x_{N},y_{N})$. $G<K<L$, $P<Q<N$, $G,K,L,P,Q \in \left\{1,2,...N\right\}$. $D_{i}$ represents the number of point $(x_{i},y_{i})$ in all shape-color primitives, $i=1,2,...,N$. Obviously, $cCore(N,M;D_{1},D_{2},...,D_{N})$ defined by Eq.(6) is a special case of $scCore_{k}(N,M;D_{1},D_{2},...,D_{N})$, when $k=0$. The Construction of SCDMIs -------------------------- **Theorem 1.** Let the color image $I(R(x,y),G(x,y),B(x,y))$ be transformed into the image $I^{'}(R^{'}(x^{'},y^{'}),G^{'}(x^{'},y^{'}),B^{'}(x^{'},y^{'}))$ by Eq.(1) and Eq.(2), $(x_{p}^{'},y_{p}^{'}),(x_{q}^{'},y_{q}^{'}),(x_{r}^{'},y_{r}^{'})$ in $I^{'}$ are corresponding points of $(x_{p},y_{p}),(x_{q},y_{q}),(x_{r},y_{r})$ in $I$, respectively. Suppose that $R(x,y), G(x,y), B(x,y), R^{'}(x^{'},y^{'}), G^{'}(x^{'},y^{'}), B^{'}(x^{'},y^{'})$ have the k-order partial derivatives $(k=0,1,2,...)$. Then there is a relation $$SCP_{k}^{'}(p,q,r)=|CA| \cdot SCP_{k}(p,q,r)$$ where $$\begin{split} SCP_{k}^{'}(p,q,r)= \left| \begin{array}{ccc} F^{k}_{R^{'}}(x_{p}^{'},y_{p}^{'})&F^{k}_{R^{'}}(x_{q}^{'},y_{q}^{'})&F^{k}_{R^{'}}(x_{r}^{'},y_{r}^{'})\\ F^{k}_{G^{'}}(x_{p}^{'},y_{p}^{'})&F^{k}_{G^{'}}(x_{q}^{'},y_{q}^{'})&F^{k}_{G^{'}}(x_{r}^{'},y_{r}^{'})\\ F^{k}_{B^{'}}(x_{p}^{'},y_{p}^{'})&F^{k}_{B^{'}}(x_{q}^{'},y_{q}^{'})&F^{k}_{B^{'}}(x_{r}^{'},y_{r}^{'})\\ \end{array} \right| \end{split}$$ Further, the following relation can be obtained $$scCore_{k}^{'}(N,M;D_{1},D_{2},...,D_{N})=|CA|^{M} \cdot scCore_{k}(N,M;D_{1},D_{2},...,D_{N})$$ where $$scCore_{k}^{'}(N,M;D_{1},D_{2},...,D_{N})=\underbrace{SCP_{k}^{'}(1,2,3)SCP_{k}^{'}(G,K,L)...SCP_{k}^{'}(P,Q,N)}_M$$ By using Maple2015, the proof of Theorem 1 is obvious. We can find that Eq.(10) is a special case of Eq.(20), when $k=0$. So, when we replace $cCore(N,M;\\D_{1},D_{2},...,D_{N})$ in Eq.(11) with $scCore_{k}(N,M;D_{1},D_{2},...,D_{N})$, Eq.(12) is still tenable. Now, we can define SCDMIs. **Theorem 2.** $$\begin{split} &SCDMIs_{k}(n,m,N,M;d_{1},...,d_{n};D_{1},...,D_{N})\\&= \frac {I(sCore(n,m,d_{1},...,d_{n})\cdot scCore_{k}(N,M,D_{1},...,D_{N}))}{I(sCore(1,0))^{max(n+N)+m-\frac{3M}{2}}\cdot I(scCore_{k}(3,2;2,2,2))^{\frac{M}{2}}}\\ \end{split}$$ Then there is a relation $$\begin{split} SCDMIs_{k}^{'}(n,m,N,M;d_{1},...,d_{n};D_{1},...,D_{N})&\\=SCDMIs_{k}(n,m,N,M;d_{1},...,d_{n};D_{1},...,D_{N})\ \end{split}$$ where $$\begin{split} &SCDMIs_{k}^{'}(n,m,N,M;d_{1},...,d_{n};D_{1},...,D_{N})\\&= \frac {I(sCore^{'}(n,m,d_{1},...,d_{n})\cdot scCore_{k}^{'}(N,M,D_{1},...,D_{N}))}{I(sCore^{'}(1,0))^{max(n+N)+m-\frac{3M}{2}}\cdot I(scCore_{k}^{'}(3,2;2,2,2))^{\frac{M}{2}}}\\ \end{split}$$ Eq.(23) can be expressed as polynomial of $SCM^{k}_{pq\alpha \beta \gamma}$. Eq.(11) is a special case of Eq.(23) when $k=0$. The proof of Eq.(24) is exactly the same as that of Eq.(12) proposed in [@14]. The Instances of SCDMIs ----------------------- We can use Eq.(24) to construct instances of $SCDMIs$ by setting different k values. However, color images are discrete data, the partial derivatives of each order can’t be accurately calculated. With the elevation of order, the error will be more and more, which will greatly affect the stability of SCDMIs. So, we set $k=0,1$ in this paper. When $k=0$, $SCDMIs_{0}$ are equivalent to SCAMIs. We construct $SCDMIs_{0}$ of which degrees $\leqslant 4$, shape orders $\leqslant 4$ and color orders $\leqslant 1$. Gong et al. [@14] thought that in order to obtain the $SCDMIs_{0}$ of which degrees $\leqslant 4$, shape orders $\leqslant 4$ and color orders $\leqslant 1$, $scCore_{0}(3,1;1,1,1)$ must be $C(1,2,3)$. This judgment is wrong. In fact, $scCore_{0}(3,1;1,1,1)$ can be $C(1,2,3)$, $C(1,2,4)$, $C(1,3,4)$ and $C(2,3,4)$. Thus, lots of $SCDMIs_{0}$ were missed in [@14]. By correcting this shortcoming, we get 25 $SCDMIs_{0}$ that satisfy the independence of the function by using the method proposed by [@17]. At the same time, when $k=1$, $SCP_{1}(p,q,r)$ is defined by $$SCP_{1}(p,q,r)= \left| \begin{array}{ccc} F^{1}_{R}(x_{p},y_{p})&F^{1}_{R}(x_{q},y_{q})&F^{1}_{R}(x_{r},y_{r})\\ F^{1}_{G}(x_{p},y_{p})&F^{1}_{G}(x_{q},y_{q})&F^{1}_{G}(x_{r},y_{r})\\ F^{1}_{B}(x_{p},y_{p})&F^{1}_{B}(x_{q},y_{q})&F^{1}_{B}(x_{r},y_{r})\\ \end{array} \right|$$ where $$F^{1}_{C}(x,y)=(x-\bar{x})\frac{\partial C}{\partial x}+(y-\bar{y})\frac{\partial C}{\partial y} ~~~C \in \{R,G,B\}.$$ By replacing $C(1,2,3)$, $C(1,2,4)$, $C(1,3,4)$ and $C(2,3,4)$ with $SCP_{1}(1,2,3)$, $SCP_{1}(1,2,4)$, $SCP_{1}(1,3,4)$ and $SCP_{1}(2,3,4)$, 25 $SCDMIs_{1}$ can be obtained. Therefore, we can construct the feature vector SCDMI50, which is defined by $$SCDMI50=[SCDMIs_{0}^{1},...,SCDMIs_{0}^{25},SCDMIs_{1}^{1},...,SCDMIs_{1}^{25}]$$ The construction methods of 50 instances are shown in Table 1. [lll]{} Name &   scCore &    sCore\ $SCDMIs_{0}^{1}/SCDMIs_{1}^{1}$ &   $C(1,2,3)/SCP_{1}(1,2,3)$ &  $(x_{1}y_{2}-x_{2}y_{1})(x_{1}y_{3}-x_{3}y_{1})^{2}$\ $SCDMIs_{0}^{2}/SCDMIs_{1}^{2}$ &   $C(1,2,3)/SCP_{1}(1,2,3)$ &  $(x_{1}y_{2}-x_{2}y_{1})(x_{1}y_{3}-x_{3}y_{1})^{3}$\ $SCDMIs_{0}^{3}/SCDMIs_{1}^{3}$ &   $C(1,2,3)/SCP_{1}(1,2,3)$ &  $(x_{1}y_{2}-x_{2}y_{1})(x_{1}y_{3}-x_{3}y_{1})(x_{2}y_{3}-x_{3}y_{2})$\ $SCDMIs_{0}^{4}/SCDMIs_{1}^{4}$ &   $C(1,2,3)/SCP_{1}(1,2,3)$ &  $(x_{1}y_{2}-x_{2}y_{1})(x_{1}y_{3}-x_{3}y_{1})(x_{2}y_{3}-x_{3}y_{2})^{3}$\ $SCDMIs_{0}^{5}/SCDMIs_{1}^{5}$ &   $C(1,2,3)/SCP_{1}(1,2,3)$ &  $(x_{1}y_{2}-x_{2}y_{1})^{2}(x_{1}y_{3}-x_{3}y_{1})^{2}(x_{2}y_{3}-x_{3}y_{2})$\ $SCDMIs_{0}^{6}/SCDMIs_{1}^{6}$ &   $C(1,2,4)/SCP_{1}(1,2,4)$ &  $(x_{1}y_{2}-x_{2}y_{1})(x_{2}y_{3}-x_{3}y_{2})(x_{3}y_{4}-x_{4}y_{3})$\ $SCDMIs_{0}^{7}/SCDMIs_{1}^{7}$ &   $C(1,2,4)/SCP_{1}(1,2,4)$ &  $(x_{1}y_{2}-x_{2}y_{1})(x_{2}y_{3}-x_{3}y_{2})(x_{3}y_{4}-x_{4}y_{3})^{3}$\ $SCDMIs_{0}^{8}/SCDMIs_{1}^{8}$ &   $C(1,3,4)/SCP_{1}(1,3,4)$ &  $(x_{1}y_{2}-x_{2}y_{1})(x_{2}y_{3}-x_{3}y_{2})(x_{3}y_{4}-x_{4}y_{3})^{3}$\ $SCDMIs_{0}^{9}/SCDMIs_{1}^{9}$ &   $C(1,2,3)/SCP_{1}(1,2,3)$ &  $(x_{1}y_{2}-x_{2}y_{1})^{2}(x_{2}y_{3}-x_{3}y_{2})(x_{3}y_{4}-x_{4}y_{2})$\ $SCDMIs_{0}^{10}/SCDMIs_{1}^{10}$ &  $C(1,2,4)/SCP_{1}(1,2,4)$ &  $(x_{1}y_{2}-x_{2}y_{1})^{2}(x_{2}y_{3}-x_{3}y_{2})(x_{3}y_{4}-x_{4}y_{3})^{3}$\ $SCDMIs_{0}^{11}/SCDMIs_{1}^{11}$ &  $C(2,3,4)/SCP_{1}(2,3,4)$ &  $(x_{1}y_{2}-x_{2}y_{1})^{2}(x_{2}y_{3}-x_{3}y_{2})(x_{3}y_{4}-x_{4}y_{3})^{3}$\ $SCDMIs_{0}^{12}/SCDMIs_{1}^{12}$ &  $C(1,2,4)/SCP_{1}(1,2,4)$ &  $(x_{1}y_{2}-x_{2}y_{1})^{3}(x_{2}y_{3}-x_{3}y_{2})(x_{3}y_{4}-x_{4}y_{3})^{3}$\ $SCDMIs_{0}^{13}/SCDMIs_{1}^{13}$ &  $C(1,2,3)/SCP_{1}(1,2,3)$ &  $(x_{1}y_{2}-x_{2}y_{1})(x_{2}y_{3}-x_{3}y_{2})^{2}(x_{3}y_{4}-x_{4}y_{3})^{2}$\ $SCDMIs_{0}^{14}/SCDMIs_{1}^{14}$ &  $C(1,2,4)/SCP_{1}(1,2,4)$ &  $(x_{1}y_{2}-x_{2}y_{1})(x_{2}y_{3}-x_{3}y_{2})^{2}(x_{3}y_{4}-x_{4}y_{3})^{2}$\ $SCDMIs_{0}^{15}/SCDMIs_{1}^{15}$ &  $C(1,2,4)/SCP_{1}(1,2,4)$ &  $(x_{1}y_{2}-x_{2}y_{1})(x_{2}y_{3}-x_{3}y_{2})^{3}(x_{3}y_{4}-x_{4}y_{3})$\ $SCDMIs_{0}^{16}/SCDMIs_{1}^{16}$ &  $C(1,3,4)/SCP_{1}(1,3,4)$ &   ---------------------------------------------------------------------------------- $(x_{1}y_{2}-x_{2}y_{1})(x_{2}y_{3}-x_{3}y_{2})(x_{3}y_{4}-x_{4}y_{3})^{2}\cdot$ $(x_{4}y_{1}-x_{1}y_{4})$ ---------------------------------------------------------------------------------- : The construction methods of $SCDMI50$ \ $SCDMIs_{0}^{17}/SCDMIs_{1}^{17}$ &  $C(1,2,3)/SCP_{1}(1,2,3)$ &   -------------------------------------------------------------------------------------- $(x_{1}y_{2}-x_{2}y_{1})^{2}(x_{2}y_{3}-x_{3}y_{2})(x_{3}y_{4}-x_{4}y_{3})^{3}\cdot$ $(x_{4}y_{1}-x_{1}y_{4})$ -------------------------------------------------------------------------------------- : The construction methods of $SCDMI50$ \ $SCDMIs_{0}^{18}/SCDMIs_{1}^{18}$ &  $C(1,2,4)/SCP_{1}(1,2,4)$ &  $(x_{1}y_{2}-x_{2}y_{1})(x_{1}y_{3}-x_{3}y_{1})(x_{1}y_{4}-x_{4}y_{1})$\ $SCDMIs_{0}^{19}/SCDMIs_{1}^{19}$ &  $C(1,2,4)/SCP_{1}(1,2,4)$ &   ------------------------------------------------------------------------------ $(x_{1}y_{2}-x_{2}y_{1})(x_{1}y_{3}-x_{3}y_{1})(x_{1}y_{4}-x_{4}y_{1})\cdot$ $(x_{3}y_{4}-x_{4}y_{3})^{3}$ ------------------------------------------------------------------------------ : The construction methods of $SCDMI50$ \ $SCDMIs_{0}^{20}/SCDMIs_{1}^{20}$ &  $C(2,3,4)/SCP_{1}(2,3,4)$ &   ---------------------------------------------------------------------------------- $(x_{1}y_{2}-x_{2}y_{1})(x_{1}y_{3}-x_{3}y_{1})^{2}(x_{1}y_{4}-x_{4}y_{1})\cdot$ $(x_{3}y_{4}-x_{4}y_{3})^{2}$ ---------------------------------------------------------------------------------- : The construction methods of $SCDMI50$ \ $SCDMIs_{0}^{21}/SCDMIs_{1}^{21}$ &  $C(1,2,3)/SCP_{1}(1,2,3)$ &   ---------------------------------------------------------------------------------- $(x_{1}y_{2}-x_{2}y_{1})^{2}(x_{1}y_{3}-x_{3}y_{1})(x_{1}y_{4}-x_{4}y_{1})\cdot$ $(x_{3}y_{4}-x_{4}y_{3})$ ---------------------------------------------------------------------------------- : The construction methods of $SCDMI50$ \ $SCDMIs_{0}^{22}/SCDMIs_{1}^{22}$ &  $C(1,2,3)/SCP_{1}(1,2,3)$ &   ---------------------------------------------------------------------------------- $(x_{1}y_{2}-x_{2}y_{1})^{2}(x_{1}y_{3}-x_{3}y_{1})(x_{1}y_{4}-x_{4}y_{1})\cdot$ $(x_{3}y_{4}-x_{4}y_{3})^{3}$ ---------------------------------------------------------------------------------- : The construction methods of $SCDMI50$ \ $SCDMIs_{0}^{23}/SCDMIs_{1}^{23}$ &  $C(1,2,4)/SCP_{1}(1,2,4)$ &   ---------------------------------------------------------------------------------- $(x_{1}y_{2}-x_{2}y_{1})^{2}(x_{1}y_{3}-x_{3}y_{1})(x_{1}y_{4}-x_{4}y_{1})\cdot$ $(x_{3}y_{4}-x_{4}y_{3})^{3}$ ---------------------------------------------------------------------------------- : The construction methods of $SCDMI50$ \ $SCDMIs_{0}^{24}/SCDMIs_{1}^{24}$ &  $C(1,2,4)/SCP_{1}(1,2,4)$ &   ---------------------------------------------------------------------------------- $(x_{1}y_{2}-x_{2}y_{1})(x_{2}y_{3}-x_{3}y_{2})(x_{3}y_{4}-x_{4}y_{3})^{2}\cdot$ $(x_{4}y_{1}-x_{1}y_{4})(x_{2}y_{4}-x_{4}y_{2})$ ---------------------------------------------------------------------------------- : The construction methods of $SCDMI50$ \ $SCDMIs_{0}^{25}/SCDMIs_{1}^{25}$ &  $C(1,2,4)/SCP_{1}(1,2,4)$ &   -------------------------------------------------------------------------------------- $(x_{1}y_{2}-x_{2}y_{1})^{2}(x_{2}y_{3}-x_{3}y_{2})(x_{3}y_{4}-x_{4}y_{3})^{2}\cdot$ $(x_{4}y_{1}-x_{1}y_{4})^{2}(x_{2}y_{4}-x_{4}y_{2})$ -------------------------------------------------------------------------------------- : The construction methods of $SCDMI50$ \ In order to more clearly explain that SCDMIs can be expanded into the polynomial of $SCM^{k}_{pq\alpha \beta \gamma}$, we give the shape-color moment polynomial of $SCMIs_{0}^{3}$ which is represented by $$\begin{split} SCDMI^{3}_{0}=&\{6SCM^{0}_{02001}SCM^{0}_{11010}SCM^{0}_{20100}-6SCM^{0}_{02001}SCM^{0}_{11100}SCM^{0}_{20010}\\&-6SCM^{0}_{02010}SCM^{0}_{11001}SCM^{0}_{20100}+6SCM^{0}_{02010}SCM^{0}_{11100}SCM^{0}_{20001}\\&+6SCM^{0}_{02100}SCM^{0}_{11001}SCM^{0}_{20010}-6SCM^{0}_{02100}SCM^{0}_{11010}SCM^{0}_{20001}\}\\&/\{6SCM^{0}_{00002} SCM^{0}_{00020} SCM^{0}_{00200}-6 SCM^{0}_{00002} (SCM^{0}_{00110})^{2}\\&-6 (SCM^{0}_{00011})^{2} SCM^{0}_{00200}+12 SCM^{0}_{00011} SCM^{0}_{00101} SCM^{0}_{00110}\\&-6 SCM^{0}_{00020} (SCM^{0}_{00101})^{2}\}\\ \end{split}$$ Experimental Results ==================== In this section, some experiments are designed to evaluate the performance of SCDMI50. Firstly, we verify the stability and discriminability of SCDMI50 by using synthetic images. Then, some retrieval experiments based on real image databases are performed. Also, we chose some commonly used image descriptors for comparison. It is worth noting that we have to choose the method to calculate the first order partial differentials of $R(x,y)$, $G(x,y)$ and $B(x,y)$. The 5 points difference formulas are used for approximating the first partial derivatives of discrete color images, which are defined by $$\begin{split} &\frac {\partial C(x,y)}{\partial x}=C(x-2,y)-8C(x-1,y)+8C(x+1,y)-C(x+2,y)\\ &\frac {\partial C(x,y)}{\partial y}=C(x,y-2)-8C(x,y-1)+8C(x,y+1)-C(x,y+2)\\ \end{split}$$ where $C \in \{R,G,B\}$. We choose this method because it guarantees the computational accuracy of the first partial differential to a certain extent and also maintains a relatively fast calculation speed. The Stability and Discriminability of SCDMI50 --------------------------------------------- We select 50 different kinds of butterfly images, which are shown in the Fig.(1.a). Then, 5 shape affine transforms and 4 color affine transforms are applied to each image. So, one image can get 20 transformed versions which are shown in the Fig.(1.b). Thus, we obtain the database containing 1000 images. $10\%$ images are used as the training data and the rest make up the testing data. For comparison with $SCDMI50$, we chose some commonly used features of gray images or color images. - **Hu moments**, which were composed of 7 invariants under the shape similarity transform, proposed in [@1]. Hu moments were designed for gray images. - **AMIs** which were composed of 17 invariants under the shape affine transform proposed in [@18]. AMIs were designed for gray images. - **RGhistogram**, which consisted of 60-dimensional features and was scale-invariant in color space, proposed in [@19]. - **Transformed color distribution**, which consisted of 60-dimensional features, was proposed in [@19]. Transformed color distribution was scale-invariant and sift-invariant in color space. - **Color moments** consisted of the first, second and third geometric moments of each channel of the color image, which were proposed in [@15]. Color moments are sift-invariant in color space. - **GPSOs** were invariant under the shape affine transforms and the color diagonal-offset transform, which were proposed in [@10]. GPSOs consisted of 21 moment invariants. Subsequently, image classification is performed on the butterfly database by using different kind of features. The Chi-Square distance is selected to measure the similarity of two images. Finally, we plot the classification accuracies obtained by using different features, which are shown in the Table 2. Descriptor   Accuracy   Descriptor   Accuracy -------------------------------- -------------- --------------- ------------ SCDMI50   **98.67%**   GPSOs   78.56% AMIs   50.11%   Hu moments   25.00% Color moments   74.77%   RGhistogram   80.56% Transformed color distribution   95.78% : The classification accuracies obtained by using different features On the one hand, we can find that the result obtained by using $SCDMI50$ is better than those obtained by using other features. And, color information is very important for the classification. On the other hand, the Table 2 show that $SCDMI50$ have good stability and distinguishability for the shape affine and color affine transforms, and demonstrate that the construction formula of SCDMIs designed in the Section 3 is correct. Image Retrieval on Real Image Database -------------------------------------- In order to further illustrate the performance of $SCDMI50$ better than the commonly used features listed in the Section 4.1, the database COIL-100 proposed in [@20] is chosen for our experiment. COIL-100 contains 7202 images of 100 categories, each of which has 72 images taken from different angles. We choose 30 classes, each class contains 5 images. For each image, 6 color affine transformations are applied. Thus, 900 images of 30 categories are obtained, each of categories contains 30 images which are shown in the Fig.(2). ![Each of categories contains 30 images[]{data-label="1"}](5.jpg "fig:"){width="100mm" height="50mm"}\ Similar to the Section 4.1, the image retrieval experiment is performed on this database. The Precision-Recall curves are shown in the Fig.(3). ![Precision-Recall curves obtained by using different features on the COIL-100 database[]{data-label="1"}](6.jpg "fig:"){width="100mm" height="60mm"}\ Obviously, the result obtained by using SCDMI50 is far superior to those obtained by using other features. This is because that traditional image features are constructed by using only color information or shape information. However, we use two kinds of information while constructing SCDMI50. In addition, traditional image descriptors are only stable for simple changes in color space, such as the diagonal-offset transforms. When the color space changes drastically, they are less robust. Finally, in order to observe the performance of $SCDMIs_{0}$ and $SCDMIs_{1}$ clearly, we compare the retrieval results of $SCDMI50$, $SCDMI_{0}25$ and $SCDMI_{1}25$ which are shown in the Fig.(4). Among them, $SCDMI_{0}25$ and $SCDMI_{1}25$ are defined by $$SCDMI_{0}25=[SCDMIs_{0}^{1},SCDMIs_{0}^{2},...,SCDMIs_{0}^{25}]$$ $$SCDMI_{1}25=[SCDMIs_{1}^{1},SCDMIs_{1}^{2},...,SCDMIs_{1}^{25}]$$ ![Precision-Recall curves obtained by using $SCDMI50$, $SCDMI_{0}25$ and $SCDMI_{1}25$[]{data-label="1"}](7.jpg "fig:"){width="105mm" height="60mm"}\ Because of the error caused by the inaccuracy of partial derivative calculation, the result obtained by using $SCDMI_{1}25$ slightly worse than the result obtained $SCDMI_{0}25$, but far better than the results obtained by other commonly used features. Meanwhile, $SCDMI_{1}25$ increase the number of invariants which have simple structures and good properties. We combine $SCDMI_{1}25$ and $SCDMI_{0}25$ to get $SCDMI50$ which achieve the best retrieval result in our experiment. Conclusion ========== In this paper, we proposed a kind of shape-color differential moment invariants (SCDMIs) of color images, which are invariant under the shape affine and color affine transforms, by using partial derivatives of each color channel. It is obvious that all SCAMIs proposed in [@14] are the special cases of SCDMIs when $k=0$. Then, we correct a mistake in [@14] and obtain 50 instances of SCDMIs, which have simple structures and good properties. Finally, several commonly used image descriptors and SCDMIs are used in image retrieval of color images, respectively. By comparing the experimental results, we find that SCDMIs get better results. [14]{} Hu, M.K.: Visual pattern recognition by moment invariants. IRE Trans. Inf. Theory 8(2), 179-187 (1962) Zhang, Y.D., Wang, S.H., Sun, P., Phillips, P.: Pathological brain detection based on wavelet entropy and Hu moment invariants. 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--- abstract: 'We consider multiple time scales systems of stochastic differential equations with small noise in random environments. We prove a quenched large deviations principle with explicit characterization of the action functional. The random medium is assumed to be stationary and ergodic. In the course of the proof we also prove related quenched ergodic theorems for controlled diffusion processes in random environments that are of independent interest. The proof relies entirely on probabilistic arguments, allowing to obtain detailed information on how the rare event occurs. We derive a control, equivalently a change of measure, that leads to the large deviations lower bound. This information on the change of measure can motivate the design of asymptotically efficient Monte Carlo importance sampling schemes for multiscale systems in random environments.' address: | Department of Mathematics & Statistics\ Boston University\ Boston, MA 02215 author: - Konstantinos Spiliopoulos title: Quenched Large Deviations for Multiscale Diffusion Processes in Random Environments --- Introduction {#S:Intro} ============ Let $0<\varepsilon,\delta\ll 1$ and consider the process $\left(X^{\epsilon}, Y^{\epsilon}\right)=\left\{\left(X^{\epsilon}_{t}, Y^{\epsilon}_{t}\right), t\in[0,T]\right\}$ taking values in the space $\mathbb{R}^{m}\times\mathbb{R}^{d-m}$ that satisfies the system of stochastic differential equation (SDE’s) $$\begin{aligned} dX^{\epsilon}_{t}&=&\left[ \frac{\epsilon}{\delta}b\left(Y^{\epsilon}_{t},\gamma\right)+c\left( X^{\epsilon}_{t},Y^{\epsilon}_{t},\gamma\right)\right] dt+\sqrt{\epsilon}\sigma\left( X^{\epsilon}_{t},Y^{\epsilon}_{t},\gamma\right) dW_{t},\nonumber\\ dY^{\epsilon}_{t}&=&\frac{1}{\delta}\left[ \frac{\epsilon}{\delta}f\left(Y^{\epsilon}_{t},\gamma\right) +g\left( X^{\epsilon}_{t},Y^{\epsilon}_{t},\gamma\right)\right] dt+\frac{\sqrt{\epsilon}}{\delta}\left[ \tau_{1}\left( Y^{\epsilon}_{t},\gamma\right) dW_{t}+\tau_{2}\left(Y^{\epsilon}_{t},\gamma\right)dB_{t}\right], \label{Eq:Main}\\ X^{\epsilon}_{0}&=&x_{0},\hspace{0.2cm}Y^{\epsilon}_{0}=y_{0}\nonumber\end{aligned}$$ where $\delta=\delta(\epsilon)\downarrow0$ such that $\epsilon/\delta\uparrow\infty$ as $\epsilon\downarrow0$. Here, $(W_{t}, B_{t})$ is a $2\kappa-$dimensional standard Wiener process. We assume that for each fixed $x\in\mathbb{R}^{m}$, $b(\cdot,\gamma), c(x,\cdot,\gamma),\sigma(x,\cdot,\gamma),f(\cdot,\gamma)$, $g(x,\cdot,\gamma), \tau_{1}(\cdot,\gamma)$ and $\tau_{2}(\cdot,\gamma)$ are stationary and ergodic random fields. We denote by $\gamma\in\Gamma$ the element of the related probability space. If we want to emphasize the dependence on the initial point and on the random medium, we shall write $\left(X^{\epsilon,(x_{0},y_{0}),\gamma}, Y^{\epsilon,(x_{0},y_{0}),\gamma}\right)$ for the solution to (\[Eq:Main\]). The system (\[Eq:Main\]) can be interpreted as a small-noise perturbation of dynamical systems with multiple scales. The slow component is $X$ and the fast component is $Y$. We study the regime where the homogenization parameter goes faster to zero than the strength of the noise does. The goal of this paper is to obtain the quenched large deviations principle associated to the component $X$, that is associated with the slow motion. The case of large deviations for such systems in periodic media for all possible interactions between $\epsilon$ and $\delta$, i.e., $\epsilon/\delta\rightarrow 0, c\in(0,\infty)$ or $\infty$, was studied in [@Spiliopoulos2013], see also [@Baldi; @DupuisSpiliopoulos; @FS]. In [@Spiliopoulos2013] (see also [@DupuisSpiliopoulosWang]), it was assumed that the coefficients are periodic with respect to the $y-$variable and based on the derived large deviations principle, asymptotically efficient importance sampling Monte Carlo methods for estimating rare event probabilities were obtained. In the current paper, we focus on quenched (i.e. almost sure with respect to the random environment) large deviations for the case $\epsilon/\delta\uparrow\infty$ and the situation is more complex when compared to the periodic case since the coefficients are now random fields themselves and the fast motion does not take values in a compact space. We treat the large deviations problem via the lens of the weak convergence framework, [@DupuisEllis], using entirely probabilistic arguments. This framework transforms the large deviations problem to convergence of a stochastic control problem. The current work is certainly related to the literature in random homogenization, see [@KomorowskiLandimOlla2012; @KosyginaRezakhanlouVaradhan; @Kozlov1979; @Kozlov1989; @LionsSouganidis2006; @Olla1994; @OllaSiri2004; @Osada1983; @Osada1987; @PapanicolaouVaradhan1982; @Papaanicolaou1994; @Rhodes2009a]. Our work is most closely related to [@KosyginaRezakhanlouVaradhan; @LionsSouganidis2006], where stochastic homogenization for Hamilton-Jacobi-Bellman (HJB) equations was studied. The authors in [@KosyginaRezakhanlouVaradhan; @LionsSouganidis2006] consider the case $\delta=\epsilon$ with the fast motion being $Y=X/\delta$ and with the coefficients $b=f=0$ in a general Hamiltonian setting. In both papers the authors briefly discuss large deviations for diffusions (i.e., when the Hamiltonian is quadratic) and the action functional is given as the Legendre-Fenchel transform of the effective Hamiltonian and the case studied there is $\delta=\epsilon$. Moreover, in [@Kushner1; @VeretennikovSPA2000] the large deviations principle for systems like (\[Eq:Main\]) is considered in the case $\epsilon=\delta$ with the coefficients $b=f=0$. In [@Kushner1; @VeretennikovSPA2000] the coefficients are deterministic (i.e., not random fields as in our case) and stability type conditions for the fast process $Y$ are assumed in order to guarantee ergodicity. Lastly, related annealed homogenization results (i.e. on average and not almost sure with respect to the medium) for uncontrolled multiscale diffusions as in (\[Eq:Main\]) in the case $\epsilon=1$, $\delta\downarrow 0$ and $Y=X/\delta$ have been recently obtained in [@Rhodes2009a]. Under different assumptions on the structure of the coefficients, the opposite case to ours where $\epsilon/\delta\downarrow 0$ has been partially considered in [@DupuisSpiliopoulos; @FS; @Souganidis1999; @Spiliopoulos2013]. In contrast to most of the aforementioned literature, in this paper, we study the case $\epsilon/\delta\uparrow \infty$ and we use entirely probabilistic arguments. Because $\epsilon/\delta\uparrow \infty$, we also need to consider the additional effect of the macroscopic problem (i.e., what is called cell problem in the periodic homogenization literature) due to the highly oscillating term $\frac{\epsilon}{\delta}\int_{0}^{T}b\left(Y^{\epsilon}_{t},\gamma\right)dt$. We use entirely probabilistic arguments and because the homogenization parameter goes faster to zero that the strength of the noise does, we are able to derive an explicit characterization of the quenched large deviations principle and detailed information on the change of measure leading to its proof, Theorem \[T:MainTheorem3\]. Due to the presence of the highly oscillatory term $\frac{\epsilon}{\delta}\int_{0}^{T}b\left(Y^{\epsilon}_{t},\gamma\right)dt$, the change of measure in question depends on the macroscopic problem and we determine this dependence explicitly. Additionally, in the course of the proof, we obtain quenched (i.e., almost sure with respect to the random environment) ergodic theorems for uncontrolled and controlled random diffusion processes that may be of independent interest, Theorem \[T:MainTheorem1\] and Appendix \[S:QuenchedErgodicTheorems\]. It is of interest to note that for the purposes of proving the Laplace principle, which is equivalent to the large deviations principle, one can constrain the variational problem associated with the stochastic control representation of exponential functionals to a class of $L^{2}$ controls with specific dependence on $\delta,\epsilon$, Lemma \[L:RestrictingTheControl\]. Partial motivation for this work comes from chemical physics, molecular dynamics and climate modeling, e.g., [@EVMaidaTimofeyev2001; @DupuisSpiliopoulosWang2; @SchutteWalterHartmannHuisinga2005; @Zwanzig], where one is often interested in simplified models that preserve the large deviation properties of the system in the case where $\delta\ll\epsilon$, i.e., in the case where $\delta$ is orders of magnitude smaller than $\epsilon$. Other related models where the regime of interest is $\epsilon/\delta\uparrow \infty$ have been considered in [@Baldi; @DupuisSpiliopoulos; @DupuisSpiliopoulosWang; @FengFouqueKumar; @FS; @HorieIshii; @Spiliopoulos2013]. When rare events are of interest, then large deviations theory comes into play. As mentioned before, we are able to derive an explicit characterization of the quenched large deviations principle, Theorem \[T:MainTheorem3\]. The explicit form of the derived large deviations action functional and of the control achieving the large deviations bound give useful information which can be used to design provably efficient importance sampling schemes for estimation of related rare event probabilities. In the case of a periodic fast motion, the design of large deviations inspired efficient Monte Carlo importance sampling schemes was investigated in [@DupuisSpiliopoulosWang; @DupuisSpiliopoulosWang2; @Spiliopoulos2013]. The paper [@DupuisSpiliopoulosWang] also includes importance sampling numerical simulations in the case of diffusion moving in a random multiscale environment in dimension one. In the present paper, we focus on rigorously developing the large deviations theory and the design of asymptotically efficient importance sampling schemes in random environments is addressed in [@Spiliopoulos2014b]. The rest of the paper is organized as follows. In Section \[S:AssumptionMainResult\] we set-up notation, state our assumptions and review known results from the literature on random homogenization that will be useful for our purposes. In Section \[S:MainResults\] we state our main results. Sections \[S:ProofLLN\], \[S:LDPrelaxedForm\] and \[S:ProofTheoremExplicitLDP\] contain the proofs of the main results of the paper, i.e., quenched homogenization results for pairs of controlled diffusions and occupation measures in random environments and the large deviations principle with the explicit characterization of the action functional. The Appendix \[S:QuenchedErgodicTheorems\] contains the proofs of the necessary quenched ergodic theorems for controlled diffusion processes in random environments. Assumptions, notation and review of useful known results {#S:AssumptionMainResult} ======================================================== In this section we setup notation and pose the main assumptions of the paper. In this section, and for the convenience of the reader, we also review well known results from the literature on random homogenization that will be useful for our purposes. The content of this section is classical. We start by describing the properties of the random medium. Let $\left(\Gamma, \mathcal{G},\nu\right)$ be the probability space of the random medium and as in [@JikovKozlovOleinik1994], a group of measure-preserving transformations $\{\tau_{y},y\in\mathbb{R}^{d}\}$ acting ergodically on $\Gamma$. \[Def:medium\]We assume that the following hold. 1. [$\tau_{y}$ preserves the measure, namely $\forall y\in\mathbb{R}^{d-m}$ and $\forall A\in\mathcal{G}$ we have $\nu(\tau_{y}A)=\nu(A)$.]{} 2. [The action of $\{\tau_{y}: y\in\mathbb{R}^{d-m}\}$ is ergodic, that is if $A=\tau_{y}A$ for every $y\in\mathbb{R}^{d}$ then $\nu(A)=0$ or $1$.]{} 3. [ For every measurable function $f$ on $\left( \Gamma, \mathcal{G}, \nu\right) $, the function $(y,\gamma)\mapsto f(\tau_{y}\gamma)$ is measurable on $\left( \mathbb{R}^{d-m}\times\Gamma, \mathbb{B}(\mathbb{R}^{d-m})\otimes\mathcal{G}\right) $.]{} For $\tilde{\phi}\in L^{2}(\Gamma)$ (i.e., a square integrable function in $\Gamma$), we define the operator $T_{y}\tilde{\phi}(\gamma)=\tilde{\phi}(\tau_{y}\gamma)$. It is known, e.g. [@Olla1994], that $T_{y}$ forms a strongly continuous group of unitary maps in $L^{2}(\Gamma)$. Moreover, if the limit exists, the infinitesimal generator $D_{i}$ of $T_{y}$ in the direction $i$ is defined by $$D_{i}\tilde{\phi}=\lim_{h\downarrow0}\frac{T_{he_{i}}\tilde{\phi}-\tilde{\phi}}{h}. \label{Eq:defofD}$$ and is a closed and densely defined generator. Next, for $\tilde{\phi}\in L^{2}(\Gamma)$, we define $\phi(y,\gamma)=\tilde{\phi}(\tau_{y}\gamma)$. This definition guarantees that $\phi$ will be a stationary and ergodic random field on $\mathbb{R}^{d-m}$. Similarly, for a measurable function $\tilde{\phi}:\mathbb{R}^{m}\times\Gamma\mapsto\mathbb{R}^{m}$ we consider the (locally) stationary random field $(x,y) \mapsto \tilde{\phi}(x,\tau_{y}\gamma)=\phi(x,y,\gamma)$. We follow this procedure to define the random fields $b,c,\sigma,f,g,\tau_{1},\tau_{2}$ that play the role of the coefficients of (\[Eq:Main\]), which then guarantees that they are ergodic and stationary random fields. In particular, we start with $L^{2}(\Gamma)$ functions $\tilde{b}(\gamma),\tilde{c}(x,\gamma),\tilde{\sigma}(x,\gamma),\tilde{f}(\gamma),\tilde{g}(x,\gamma)$, $\tilde{\tau}_{1}(\gamma),\tilde{\tau}_{2}(\gamma)$ and we define the coefficients of (\[Eq:Main\]) via the relations $b(y,\gamma)=\tilde{b}(\tau_{y}\gamma),c(x,y,\gamma)=\tilde{c}(x,\tau_{y}\gamma),\sigma(x,y,\gamma)=\tilde{\sigma}(x,\tau_{y}\gamma),f(y,\gamma)=\tilde{f}(\tau_{y}\gamma), g(x,y,\gamma)=\tilde{g}(x,\tau_{y}\gamma),\tau_{1}(y,\gamma)=\tilde{\tau}_{1}(\tau_{y}\gamma)$ and $\tau_{2}(y,\gamma)=\tilde{\tau}_{2}(\tau_{y}\gamma)$. The main assumption for the coefficients of (\[Eq:Main\]) is as follows. \[A:Assumption1\] 1. The functions $b(y,\gamma),c(x,y,\gamma),\sigma(x,y,\gamma), f(y,\gamma), g(x,y,\gamma), \tau_{1}(y,\gamma)$ and $\tau_{2}(y,\gamma)$ are $C^{1}(\mathbb{R}^{d-m})$ in $y$ and $C^{1}(\mathbb{R}^{m})$ in $x$ with all partial derivatives continuous and globally bounded in $x$ and $y $. 2. For every fixed $\gamma\in\Gamma$, the diffusion matrices $\sigma\sigma^{T}$ and $\tau_{1}\tau_{1}^{T}+\tau_{2}\tau_{2}^{T}$ are uniformly nondegenerate. It is known that under Condition \[A:Assumption1\], there exists a filtered probability space $(\Omega,\mathcal{F},\mathfrak{F}_{t},\mathbb{P})$ such that for every given initial point $(x_{0},y_{0})\in\mathbb{R}^{m}\times\mathbb{R}^{d-m}$, for every $\gamma\in\Gamma$ and for every $\epsilon,\delta>0$ there exists a strong Markov process $\left(X^{\epsilon}_{t},Y^{\epsilon}_{t}, t\geq 0\right)$ satisfying (\[Eq:Main\]). However, if we define a probability measure $\mathcal{P}=\nu\otimes\mathbb{P}$ on the product space $\Gamma\times\Omega$, then when considered on the probability space $(\Gamma\times\Omega,\mathcal{G}\otimes \mathcal{F},\mathcal{P})$, $\{\left(X_{t}^{\epsilon}, Y^{\epsilon}_{t}\right),t\geq0\}$ is not a Markov process. From the previous discussion it is easy to see that the periodic case is a special case of the previous setup. Indeed, we can consider the periodic case with period $1$, $\Gamma$ to be the unit torus and $\nu$ to be Lebesgue measure on $\Gamma$. For every $\gamma\in\Gamma$, the shift operators $\tau_{y}\gamma =(y+\gamma)\mod 1$ and we have $\phi(y,\gamma)=\tilde{\phi}(y+\gamma)$ for a periodic function $\tilde{\phi}$ with period $1$. For every $\gamma\in\Gamma$, we define next the operator $$\mathcal{L}^{\gamma}=f(y,\gamma)\nabla_{y}\cdot+\text{\emph{tr}}\left[\left( \tau_{1}(y,\gamma)\tau^{T}_{1}(y,\gamma)+\tau_{2}(y,\gamma)\tau^{T}_{2}(y,\gamma)\right)\nabla_{y}\nabla_{y}\cdot\right] \label{Def:OperatorFastProcess}$$ and we let $Y_{t}^{\gamma}$ to be the corresponding Markov process. It follows from [@PapanicolaouVaradhan1982; @Osada1983; @Olla1994], that we can associate the canonical process on $\Gamma$ defined by the environment $\gamma$, which is a Markov process on $\Gamma$ with continuous transition probability densities with respect to $d$-dimensional Lebesgue measure, e.g., [@Olla1994]. In particular, we let $$\begin{aligned} \gamma_{t} & =\tau_{Y_{t}^{\gamma}}\gamma\label{Eq:EnvironmentProcess}\\ \gamma_{0} & =\tau_{y_{0}}\gamma\nonumber\end{aligned}$$ \[Def:OperatorFastProcess2\] We denote the infinitesimal generator of the Markov process $\gamma_{t}$ by $$\tilde{L}=\tilde{f}(\gamma)D\cdot+\text{\emph{tr}}\left[ \left( \tilde{\tau }_{1}(\gamma)\tilde{\tau}_{1}^{T}(\gamma)+\tilde{\tau }_{2}(\gamma)\tilde{\tau}_{2}^{T}(\gamma)\right)D^{2}\cdot\right] ,$$ where $D$ was defined in (\[Eq:defofD\]). Following [@Osada1983], we assume the following condition on the structure of the operator defined in Definition \[Def:OperatorFastProcess2\]. This condition allows to have a closed form for the unique ergodic invariant measure for the environment process $\{\gamma_{t}\}_{t\geq0}$, Proposition \[P:NewMeasureRandomCase\]. \[A:Assumption2\] We can write the operator $\tilde{L}$ in the following generalized divergence form $$\tilde{L}=\frac{1}{\tilde{m}(\gamma)}\left[ \sum_{i,j}D_{i}\left( \tilde {a}\tilde{a}_{i,j}^{T}(\gamma)D_{j}\cdot\right) +\sum_{j}\tilde{\beta}_{j}(\gamma)D_{j}\cdot\right]$$ where $\tilde{\beta}_{j}=\tilde{m}\tilde{f}_{j}-\sum_{i}D_{i}\left( \left(\tilde{\tau}_{1}\tilde{\tau}_{1}^{T}+\tilde{\tau}_{2}\tilde{\tau}_{2}^{T}\right)_{i,j}\tilde{m}\right) $ and $\tilde{a}\tilde{a}_{i,j}^{T}= \left(\tilde{\tau}_{1}\tilde{\tau}_{1}^{T}+\tilde{\tau}_{2}\tilde{\tau}_{2}^{T}\right)_{i,j}\tilde{m}$. We assume that $\tilde{m}(\gamma)$ is bounded from below and from above with probability $1$, that there exist smooth $\tilde{d}_{i,j}(\gamma)$ such that $\tilde{\beta}_{j}=\sum_{j}D_{j}\tilde{d}_{i,j}$ with $|\tilde{d}_{i,j}|\leq M$ for some $M<\infty$ almost surely and $$\text{div }\tilde{\beta}=0\text{ in distribution},\quad\text{i.e.,}\int_{\Gamma}\sum_{j=1}^{d}\tilde{\beta}_{j}(\gamma)D_{j}\tilde{\phi}(\gamma)\nu(d\gamma)=0,\quad\forall\tilde{\phi}\in\mathcal{H}^{1},$$ where the Sobolev space $\mathcal{H}^{1}=\mathcal{H}^{1}(\nu)$ is the Hilbert space equipped with the inner product $$(\tilde{f},\tilde{g})_{1}=\sum_{i=1}^{d}(D_{i}\tilde {f},D_{i}\tilde{g}).$$ A trivial example that satisfies Condition \[A:Assumption2\] is the gradient case. Let $\tilde{f}(\gamma)=-D \tilde{Q}(\gamma)$ and $\tilde{\tau}_{1}(\gamma)=\sqrt{2D}=\text{constant}$ and $\tilde{\tau}_{2}(\gamma)=0$. Then, we have that $\tilde{m}(\gamma)=\exp[-\tilde{Q}(\gamma)/D]$ and $\tilde{\beta}_{j}=0$ for all $1\leq j\leq d$. Moreover, if $\tilde{m}=1$ and $\tilde{d}_{i,j}$ are constants then the operator is of divergence form. Next, we recall some classical results from random homogenization. \[[@Osada1983] and Theorem 2.1 in [@Olla1994]\]\[P:NewMeasureRandomCase\] Assume Conditions \[A:Assumption1\] and \[A:Assumption2\]. Define a measure on $(\Gamma,\mathcal{G})$ by$$\pi(d\gamma)\doteq\frac{\tilde{m}(\gamma)}{\mathrm{E}^{\nu}\tilde{m}(\cdot )}\nu(d\gamma).$$ Then $\pi$ is the unique ergodic invariant measure for the environment process $\{\gamma_{t}\}_{t\geq0}$. We will denote by $\mathrm{E}^{\nu}$ and by $\mathrm{E}^{\pi}$ the expectation operator with respect to the measures $\nu$ and $\pi$ respectively. We remark here that since $\tilde{m}$ is bounded from above and from below, $\mathcal{H}^{1}(\nu)$ and $\mathcal{H}^{1}(\pi)$ are equivalent. We also need to introduce the macroscopic problem, known as cell problem in the periodic homogenization literature or corrector in the homogenization literature in general. This is needed in order to address the situation $\tilde{b}\neq0$. For every $\rho>0$, we consider the solution to the auxiliary problem on $\Gamma$. $$\rho\tilde{\chi}_{\rho}-\tilde{L}\tilde{\chi}_{\rho}=\tilde{b}. \label{Eq:RandomCellProblem}$$ Let us review some well known facts related to the solution to this auxiliary problem, e.g., see [@Olla1994; @KomorowskiLandimOlla2012]. By Lax-Milgram lemma, equation (\[Eq:RandomCellProblem\]) has a unique weak solution in the abstract Sobolev space $\mathcal{H}^{1}$. Moreover, letting $\mathcal{R}_{\rho}\tilde {h}(\gamma)=\int_{0}^{\infty}e^{-\rho t}\mathrm{E}_{\gamma}\tilde{h}(\gamma_{t})dt$, for every $\tilde{h}\in L^{2}(\Gamma)$, we have $$\tilde{\chi}_{\rho}(\cdot)=\mathcal{R}_{\rho}\tilde{b}(\cdot),$$ As in [@Osada1983; @PapanicolaouVaradhan1982], there is a constant $K$ that is independent of $\rho$ such that $$\rho\mathrm{E}^{\pi}\left[ \tilde{\chi}_{\rho}(\cdot)\right] ^{2}+\mathrm{E}^{\pi}\left[ D\tilde{\chi}_{\rho}(\cdot)\right] ^{2}\leq K$$ By Proposition 2.6 in [@Olla1994] we then get that $\tilde{\chi}_{\rho}$ has an $\mathcal{H}^{1}$ strong limit, i.e., there exists a $\tilde{\chi}_{0}\in\mathcal{H}^{1}(\pi)$ such that $$\lim_{\rho\downarrow0}\left\Vert \tilde{\chi}_{\rho}(\cdot)-\tilde{\chi}_{0}(\cdot)\right\Vert _{1}=0$$ and that $$\lim_{\rho\downarrow0}\rho\mathrm{E}^{\pi}\left[ \tilde{\chi}_{\rho}(\cdot)\right] ^{2}=0.$$ This implies that $D\tilde{\chi}_{\rho}\in L^{2}(\pi)$ and that it has a $L^{2}(\pi)$ strong limit, i.e., there exists a $\tilde{\xi}\in L^{2}(\pi)$ such that $$\lim_{\rho\downarrow0}\left\Vert D\tilde{\chi}_{\rho}-\tilde{\xi}\right\Vert _{L^{2}}^{2}=0$$ In addition, since $\tilde{b}$ is bounded under Condition \[A:Assumption1\], $\tilde{\chi}_{\rho}$ is also bounded. This follows because the resolvent operator $\mathcal{R}_{\rho}$ corresponding to the operator $\rho I-\mathcal{L}$ is associated to a $L^{\infty}(\Gamma)$ contraction semigroup, see Section 2.2 of [@Olla1994]. Moreover, as in Proposition 3.2. of [@Osada1983], we have that for almost all $\gamma\in\Gamma$ $$\delta \chi_{0}(y/\delta,\gamma)\rightarrow 0, \text{ as }\delta\downarrow 0, \text{ a.s. } y\in\mathcal{Y}.$$ Main results {#S:MainResults} ============ In this section we present the statement of the main results of the paper. In preparation for stating the large deviations theorem, we first recall the concept of a Laplace principle. \[Def:LaplacePrinciple\] Let $\{X^{\epsilon},\epsilon>0\}$ be a family of random variables taking values in a Polish space $\mathcal{S}$ and let $I$ be a rate function on $\mathcal{S}$. We say that $\{X^{\epsilon},\epsilon>0\}$ satisfies the Laplace principle with rate function $I$ if for every bounded and continuous function $h:\mathcal{S}\rightarrow\mathbb{R}$ $$\lim_{\epsilon\downarrow0}-\epsilon\ln\mathbb{E}\left[ \exp\left\{ -\frac{h(X^{\epsilon})}{\epsilon}\right\} \right] =\inf_{x\in\mathcal{S}}\left[ I(x)+h(x)\right]. \label{Eq:LaplacePrinciple}$$ If the level sets of the rate function (equivalently action functional) are compact, then the Laplace principle is equivalent to the corresponding large deviations principle with the same rate function (Theorems 2.2.1 and 2.2.3 in [@DupuisEllis]). In order to establish the quenched Laplace principle, we make use of the representation theorem for functionals of the form $\mathbb{E}\left[e^{-\frac{1}{\epsilon}h\left(X^{\epsilon,\gamma}\right)}\right]$ in terms of a stochastic control problem. Such representations were first derived in [@BoueDupuis]. Let $\mathcal{A}$ be the set of all $\mathfrak{F}_{s}-$progressively measurable $n$-dimensional processes $u\doteq\{u(s),0\leq s\leq T\}$ satisfying $$\mathbb{E}\int_{0}^{T}\left\Vert u(s)\right\Vert ^{2}ds<\infty, \label{A:AdmissibleControls}$$ In the present case, let $Z(\cdot)=(W(\cdot),B(\cdot))$ and $n=2k$. Then, for the given $\gamma\in\Gamma$ we have the representation $$-\epsilon\ln\mathbb{E}_{x_{0}, y_{0}} \left[ \exp\left\{ -\frac{h(X^{\epsilon})}{\epsilon}\right\} \right] =\inf_{u\in\mathcal{A}}\mathbb{E}_{x_{0},y_{0}}\left[ \frac{1}{2}\int_{0}^{T}\left[\left\Vert u_{1}(s)\right\Vert ^{2}+\left\Vert u_{2}(s)\right\Vert ^{2}\right]ds+h(\bar {X}^{\epsilon})\right] \label{Eq:VariationalRepresentation}$$ where the pair $(\bar{X}^{\epsilon},\bar{Y}^{\epsilon})$ is the unique strong solution to $$\begin{aligned} d\bar{X}^{\epsilon}_{t}&=&\left[ \frac{\epsilon}{\delta}b\left(\bar{Y}^{\epsilon}_{t},\gamma\right)+c\left( \bar{X}^{\epsilon}_{t},\bar{Y}^{\epsilon}_{t},\gamma\right)+\sigma\left( \bar{X}_{t}^{\epsilon},\bar{Y}_{t}^{\epsilon},\gamma\right) u_{1}(t)\right] dt+\sqrt{\epsilon}\sigma\left( \bar{X}^{\epsilon}_{t},\bar{Y}^{\epsilon}_{t},\gamma\right) dW_{t}, \nonumber\\ d\bar{Y}^{\epsilon}_{t}&=&\frac{1}{\delta}\left[ \frac{\epsilon}{\delta}f\left(\bar{Y}^{\epsilon}_{t},\gamma\right) +g\left( \bar{X}^{\epsilon}_{t} ,\bar{Y}^{\epsilon}_{t},\gamma\right)+\tau_{1}\left(\bar{Y}^{\epsilon}_{t},\gamma\right)u_{1}(t)+ \tau_{2}\left(\bar{Y}^{\epsilon}_{t},\gamma\right)u_{2}(t)\right] dt\nonumber\\ & &\hspace{5cm}+\frac{\sqrt{\epsilon}}{\delta}\left[ \tau_{1}\left(\bar{Y}^{\epsilon}_{t},\gamma\right) dW_{t}+\tau_{2}\left(\bar{Y}^{\epsilon}_{t},\gamma\right)dB_{t}\right],\label{Eq:Main2}\\ \bar{X}^{\epsilon}_{0}&=&x_{0},\hspace{0.2cm}\bar{Y}^{\epsilon}_{0}=y_{0}\nonumber\end{aligned}$$ This representation implies that in order to derive the Laplace principle for $\{X^{\epsilon}\}$, it is enough to study the limit of the right hand side of the variational representation (\[Eq:VariationalRepresentation\]). The first step in doing so is to consider the weak limit of the slow motion $\bar{X}^{\epsilon}$ of the controlled couple (\[Eq:Main2\]). Fix $\gamma\in\Gamma$ and let us define for notational convenience $\mathcal{Z}=\mathbb{R}^{\kappa}$ and $\mathcal{Y}=\mathbb{R}^{d-m}$. Due to the involved controls, it is convenient to introduce the following occupation measure. Let $\Delta=\Delta(\epsilon )\downarrow0$ as $\epsilon\downarrow0$ that will be chosen later on and is used to exploit a time-scale separation. Let $A_{1},A_{2},B,\Theta$ be Borel sets of $\mathcal{Z},\mathcal{Z},\Gamma,[0,T]$ respectively. Let $u^{\epsilon}_{i}\in A_{i}, i=1,2$ and let $(\bar{X}^{\epsilon },\bar{Y}^{\epsilon})$ solve (\[Eq:Main2\]) with $u^{\epsilon}_{i}$ in place of $u_{i}$. We associate with $(\bar{X}^{\epsilon},\bar{Y}^{\epsilon})$ and $u^{\epsilon}_{i}$ a family of occupation measures $\mathrm{P}^{\epsilon,\Delta,\gamma}$ defined by $$\mathrm{P}^{\epsilon,\Delta,\gamma}(A_{1}\times A_{2}\times B\times\Theta)=\int_{\Theta}\left[ \frac{1}{\Delta}\int_{t}^{t+\Delta}1_{A_{1}}(u_{1}^{\epsilon}(s))1_{A_{2}}(u_{2}^{\epsilon}(s))1_{B}\left( \tau_{\bar{Y}_{s}^{\epsilon}}\gamma\right) ds\right] dt, \label{Def:OccupationMeasures2}$$ assuming that $u_{i}^{\epsilon}(t)=0$ for $i=1,2$ if $t>T$. Next, we introduce the notion of a viable pair, see also [@DupuisSpiliopoulos]. Such a notion will allow us to characterize the limiting behavior of the pair $\left(\bar{X}^{\epsilon,\gamma}, \mathrm{P}^{\epsilon,\Delta,\gamma}\right)$. \[D:ViablePair\] Define the function in $L^{2}(\Gamma)$ $$\tilde{\lambda}(x,\gamma,z_{1},z_{2}) = \tilde{c}(x,\gamma)+\tilde{\xi}(\gamma) \tilde{g}(x,\gamma) +\tilde{\sigma}(x,\gamma)z_{1}+\tilde{\xi}(\gamma)\left(\tilde{\tau}_{1}(\gamma)z_{1}+\tilde{\tau}_{2}(\gamma)z_{2}\right)$$ where $\tilde{\xi}$ is the $L^{2}$ limit of $D\tilde{\chi}_{\rho}$ as $\rho\downarrow 0$ that is defined in Section \[S:AssumptionMainResult\]. Consider the operator $\tilde{L}$ defined in Definition \[Def:OperatorFastProcess2\]. We say that a pair $(\psi,\mathrm{P})\in\mathcal{C}\left( [0,T];\mathbb{R}^{m}\right) \times\mathcal{P}\left( \mathcal{Z}\times\mathcal{Z}\times\Gamma\times\lbrack0,T]\right) $ is viable with respect to $(\tilde{\lambda},\tilde{L})$ and we write $(\psi,\mathrm{P})\in\mathcal{V}$, if the following hold. - [ The function $\psi$ is absolutely continuous and $\mathrm{P}$ is square integrable in the sense that $\int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma\times [0,T]}|z|^{2}\mathrm{P}(dz_{1}dz_{2} d\gamma dt)<\infty$.]{} - [For all $t\in[0,T]$, $\mathrm{P}\left( \mathcal{Z}\times\mathcal{Z}\times\Gamma \times[0,t]\right) =t$. Thus, $\mathrm{P}$ can be decomposed as $\mathrm{P}(dz_{1}dz_{2} d\gamma dt)=\mathrm{P}_{t}(dz_{1}dz_{2} d\gamma)dt$ such that $\mathrm{P}_{t}(\mathcal{Z}\times\mathcal{Z}\times\Gamma)=1$. ]{} - [For all $t\in\lbrack0,T]$, $\left(\psi,\mathrm{P}\right)$ satisfy the ODE $$\psi_{t}=x_{0}+\int_{0}^{t}\left[ \int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma}\tilde{\lambda}(\psi_{s},\gamma,z_{1},z_{2})\mathrm{P}_{s}(dz_{1}dz_{2}d\gamma)\right] ds. \label{Eq:ViablePair1}$$ and for a given $\mathrm{P}$, there is a unique well defined $\psi$ satisfying (\[Eq:ViablePair1\]). ]{} - [For a.e. $t\in\lbrack0,T]$, $$\int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma}\tilde{L}\tilde{f}(\gamma)\mathrm{P}_{t}(dz_{1}dz_{2}d\gamma)=0 \label{Eq:ViablePair2}$$ for all $\tilde{f}\in\mathcal{D}(\tilde{L})$.]{} For notational convenience later on, let us also define $$\tilde{\lambda}_{\rho}(x,\gamma,z_{1},z_{2}) = \tilde{c}(x,\gamma)+D\tilde{\chi}_{\rho}(\gamma) \tilde{g}(x,\gamma) +\tilde{\sigma}(x,\gamma)z_{1}+D\tilde{\chi}_{\rho}(\gamma)\left(\tilde{\tau}_{1}(\gamma)z_{1}+\tilde{\tau}_{2}(\gamma)z_{2}\right)$$ Now, that we have defined the notion of a viable pair we are ready to present the law of large numbers results for controlled pairs $\left(\bar{X}^{\epsilon,\gamma},\mathrm{P}^{\epsilon,\Delta,\gamma}\right)$. \[T:MainTheorem1\] Assume Conditions \[A:Assumption1\] and \[A:Assumption2\]. Fix the initial point $(x_{0},y_{0})\in\mathbb{R}^{m}\times \mathcal{Y}$ and consider a family $\{u^{\epsilon}=(u^{\epsilon}_{1},u^{\epsilon}_{2}),\epsilon>0\}$ of controls (that may depend on $\gamma$) in $\mathcal{A}$ satisfying a.s. with respect to $\gamma\in\Gamma$, the bound \[Eq:UniformlySquareIntegrableControlsAdditionalb\] and $$\sup_{\epsilon>0}\mathbb{E}\int_{0}^{T}\left[\left\Vert u_{1}^{\epsilon}(s)\right\Vert ^{2}+\left\Vert u_{2}^{\epsilon}(s)\right\Vert ^{2}\right]ds<\infty\label{Eq:Ubound}$$ Then the family $\{(\bar{X}^{\epsilon,\gamma },\mathrm{P}^{\epsilon,\Delta,\gamma}),\epsilon>0\}$ is tight almost surely with respect to $\gamma\in\Gamma$. Given any subsequence of $\{(\bar{X}^{\epsilon},\mathrm{P}^{\epsilon,\Delta}),\epsilon>0\}$, there exists a subsubsequence that converges in distribution with limit $(\bar{X},\mathrm{P})$ almost surely with respect to $\gamma\in\Gamma$. With probability $1$, the limit point $(\bar{X},\mathrm{P})\in\mathcal{V}$, according to Definition \[D:ViablePair\]. Next, we are ready to state the quenched Laplace principle for $\left\{X^{\epsilon},\epsilon>0\right\}$. \[T:MainTheorem2\] Let $\{\left(X^{\epsilon},Y^{\epsilon}\right),\epsilon>0\}$ be, for fixed $\gamma\in\Gamma$, the unique strong solution to (\[Eq:Main\]) and assume that $\epsilon/\delta\uparrow \infty$. We assume that Conditions \[A:Assumption1\] and \[A:Assumption2\] hold. Define $$S(\phi)=\inf_{(\phi,\mathrm{P})\in\mathcal{V}}\left[ \frac{1}{2}\int_{\mathcal{Z}\times\mathcal{Z}\times\mathcal{Y}\times\lbrack 0,T]}\left[\left\Vert z_{1}\right\Vert ^{2}+\left\Vert z_{2}\right\Vert ^{2}\right]\mathrm{P}(dz_{1}dz_{2}dydt)\right] , \label{Eq:GeneralRateFunction}$$ with the convention that the infimum over the empty set is $\infty$. Then, we have 1. The level sets of $S$ are compact. In particular, for each $s<\infty$, the set $$\Phi_{s}=\{\phi\in\mathcal{C}([0,T];\mathbb{R}^{m}):S(\phi)\leq s\} \label{Def:LevelSets}$$ is a compact subset of $\mathcal{C}([0,T];\mathbb{R}^{m})$. 2. For every bounded and continuous function $h$ mapping $\mathcal{C}([0,T];\mathbb{R}^{m})$ into $\mathbb{R}$ $$\lim_{\epsilon\downarrow0}-\epsilon\ln\mathbb{E}_{x_{0},y_{0}}\left[ \exp\left\{ -\frac{h(X^{\epsilon,\gamma})}{\epsilon}\right\} \right] = \inf_{\phi\in \mathcal{C}([0,T];\mathbb{R}^{m}), \phi_{0}=x_{0}}\left[ S(\phi)+h(\phi)\right] . $$ almost surely with respect to $\gamma\in\Gamma$. In other words, under the imposed assumptions, $\{X^{\epsilon,\gamma},\epsilon>0\}$ satisfies the quenched large deviations principle with action functional $S$. Actually, it turns out that in this case we can compute the quenched action functional in closed form. \[T:MainTheorem3\] Let $\{\left(X^{\epsilon,\gamma},Y^{\epsilon,\gamma}\right),\epsilon>0\}$ be, for fixed $\gamma\in\Gamma$, the unique strong solution to (\[Eq:Main\]). Under Conditions \[A:Assumption1\] and \[A:Assumption2\], $\{X^{\epsilon,\gamma},\epsilon>0\}$ satisfies, almost surely with respect to $\gamma\in\Gamma$, the large deviations principle with rate function $$S(\phi)=\begin{cases} \frac{1}{2}\int_{0}^{T}(\dot{\phi}(s)-r(\phi(s)))^{T}q^{-1}(\phi(s))(\dot{\phi}(s)-r(\phi(s)))ds & \text{if }\phi\in\mathcal{AC}([0,T];\mathbb{R}^{m}) \text{ and } \phi(0)=x_{0}\\ +\infty & \text{otherwise.}\end{cases} \label{Eq:ActionFunctional1}$$ where $$r(x)=\lim_{\rho\downarrow0}\mathrm{E}^{\pi}\left[ \tilde{c}(x,\cdot) +D\tilde{\chi}_{\rho }(\cdot) \tilde{g}(x,\cdot)\right] =\mathrm{E}^{\pi}[\tilde{c}(x,\cdot)+\tilde{\xi}(\cdot)\tilde{g}(x,\cdot)]$$$$\begin{aligned} q(x)&=\lim_{\rho\downarrow0}\mathrm{E}^{\pi}\left[ (\tilde{\sigma}(x,\cdot)+D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{1}(\cdot))(\tilde{\sigma}(x,\cdot)+D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{1}(\cdot))^{T}+\left(D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{2}(\cdot)\right)\left(D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{2}(\cdot)\right)^{T}\right]\nonumber\\ &=\mathrm{E}^{\pi}\left[ (\tilde{\sigma}(x,\cdot)+\tilde{\xi}(\cdot)\tilde{\tau}_{1}(\cdot))(\tilde{\sigma}(x,\cdot)+\tilde{\xi} (\cdot)\tilde{\tau}_{1}(\cdot))^{T}+\left(\tilde{\xi} (\cdot)\tilde{\tau}_{2}(\cdot)\right)\left(\tilde{\xi} (\cdot)\tilde{\tau}_{2}(\cdot)\right)^{T}\right]\nonumber\end{aligned}$$ Notice that the coefficients $r(x)$ and $q(x)$ that enter into the action functional are those obtained if we had first taken to (\[Eq:Main\]) $\delta\downarrow 0$ with $\epsilon$ fixed and then consider the large deviations for the homogenized system. This is in accordance to intuition since in the case $\epsilon/\delta\uparrow\infty$, $\delta$ goes to zero faster than $\epsilon$. This implies that homogenization should occur first as it indeed does and then large deviations start playing a role. Proof of Theorem \[T:MainTheorem1\] {#S:ProofLLN} =================================== In this section we prove Theorem \[T:MainTheorem1\]. Tightness is established in Subsection \[S:Tightness\], whereas the identification of the limit point is done in Subsection \[S:WeakConvergence\]. Tightness of the controlled pair $\left\{ (\bar{X}^{\epsilon,\gamma },\mathrm{P}^{\epsilon,\Delta,\gamma}), \epsilon,\Delta>0 \right\} $. {#S:Tightness} ---------------------------------------------------------------------- In this section we prove that the family $\{(\bar{X}^{\epsilon,\gamma },\mathrm{P}^{\epsilon,\Delta,\gamma}),\epsilon>0\}$, is almost surely tight with respect to $\gamma\in\Gamma$ where $\Delta =\Delta(\epsilon)\downarrow0$. The following proposition takes care of tightness and uniform integrability of $\{\mathrm{P}^{\epsilon,\Delta,\gamma },\epsilon>0\}$. \[L:TightnessOccupationalMeasures\] Assume Conditions \[A:Assumption1\] and \[A:Assumption2\]. Let $\{u^{\epsilon,\gamma},\epsilon>0,\gamma\in\Gamma\}$ be a family of controls in $\mathcal{A}$ such that Conditions \[Eq:UniformlySquareIntegrableControlsAdditional\] and \[Eq:UniformlySquareIntegrableControlsAdditionalb\] of Lemma \[L:ErgodicTheorem4\] hold. The following hold 1. [For every $\eta>0$, there is a set $N_{\eta}$ (the same $N_{\eta}$ identified in Lemma \[L:ErgodicTheorem4\]) with $\pi(N_{\eta})\geq 1-\eta$ such that for every $\gamma\in N_{\eta}$ and for every bounded sequence $\Delta\in \mathcal{H}^{N_{\eta}}_{1}$ (i.e. a sequence that satisfies Condition \[A:h\_functionUniform\]), the family $\{\mathrm{P}^{\epsilon,\Delta,\gamma},\epsilon>0\}$ is tight as $\epsilon\downarrow 0$.]{} 2. [The family $\{\mathrm{P}^{\epsilon,\Delta,\gamma},\epsilon>0\}$ is uniformly integrable, in the sense that $$\lim_{M\rightarrow\infty}\sup_{\epsilon>0,\gamma\in\Gamma}\mathbb{E}\int_{\{(z_{1},z_{1})\in\mathcal{Z}^{2}:\left[\left\Vert z_{1}\right\Vert+\left\Vert z_{2}\right\Vert\right]\geq M\}\times\Gamma\times\lbrack 0,T]}\left[\left\Vert z_{1}\right\Vert+\left\Vert z_{2}\right\Vert\right]\mathrm{P}^{\epsilon,\Delta,\gamma}(dz_{1}dz_{1}d\tilde{\gamma}dt)=0$$ ]{} (i). Let us first prove the first part of the Lemma. It is clear that we can write $$\mathrm{P}^{\epsilon,\Delta,\gamma}(A_{1}\times A_{2}\times B\times\Theta)=\int_{\Theta}\mathrm{P}^{\epsilon,\Delta,\gamma}_{t}(A_{1}\times A_{2}\times B)dt$$ where $$\mathrm{P}^{\epsilon,\Delta,\gamma}_{t}(A_{1}\times A_{2}\times B)=\left[ \frac{1}{\Delta}\int_{t}^{t+\Delta}1_{A_{1}}(u_{1}^{\epsilon,\gamma }(s))1_{A_{2}}(u_{2}^{\epsilon,\gamma}(s))1_{B}\left( \tau_{\bar{Y}_{s}^{\epsilon}}\gamma\right) ds\right] dt,$$ Let us denote by $\mathrm{P}^{\epsilon,\Delta,\gamma}_{1,t}(A_{1}\times A_{2})$ and by $\mathrm{P}^{\epsilon,\Delta,\gamma}_{2,t}( B)$ the first and second marginals of $\mathrm{P}^{\epsilon,\Delta,\gamma}_{t}(A_{1}\times A_{2}\times B)$ respectively. Namely, $$\mathrm{P}^{\epsilon,\Delta,\gamma}_{1,t}(A_{1}\times A_{2})=\mathrm{P}^{\epsilon,\Delta,\gamma}_{t}(A_{1}\times A_{2}\times \Gamma),\text{ and }\mathrm{P}^{\epsilon,\Delta,\gamma}_{2,t}(B)=\mathrm{P}^{\epsilon,\Delta,\gamma}_{t}(\mathcal{Z}\times \mathcal{Z}\times B)$$ It is clear that tightness of $\{\mathrm{P}^{\epsilon,\Delta,\gamma},\epsilon>0\}$ is a consequence of tightness of $\{\mathrm{P}^{\epsilon,\Delta,\gamma}_{1,t},\epsilon>0\}$ and of $\{\mathrm{P}^{\epsilon,\Delta,\gamma}_{2,t},\epsilon>0\}$. Let us first consider tightness of $\{\mathrm{P}^{\epsilon,\Delta,\gamma}_{1,t},\epsilon>0\}$. For this purpose, we claim that the function $$g(r)=\int_{\mathcal{Z}\times\mathcal{Z}\times\lbrack0,T]}\left[\left\Vert z_{1}\right\Vert ^{2}+\left\Vert z_{2}\right\Vert ^{2}\right]r(dz_{1}dz_{2}dt),\hspace{0.2cm}r\in\mathcal{P}(\mathcal{Z}\times\mathcal{Z}\times\lbrack0,T])$$ is a tightness function, i.e., it is bounded from below and its level sets $R_{k}=\{r\in\mathcal{P}(\mathbb{R}^{2k}\times\lbrack 0,T]):g(r)\leq k\}$ are relatively compact for each $k<\infty$. Notice that the second marginal of every $r\in\mathcal{P}(\mathcal{Z} \times\mathcal{Z}\times\lbrack0,T])$ is the Lebesgue measure. Chebyshev’s inequality implies $$\sup_{r\in R_{k}}r\left( \{(z_{1},z_{2})\in\mathcal{Z}\times\mathcal{Z}:\left[\left\Vert z_{1}\right\Vert+\left\Vert z_{2}\right\Vert\right] >M\}\times\lbrack0,T]\right) \leq\sup_{r\in R_{k}}\frac {g(r)}{M^{2}}\leq\frac{k}{M^{2}}.$$ Hence, $R_{k}$ is tight and thus relatively compact as a subset of $\mathcal{P}$. Since $g$ is a tightness function, by Theorem A.3.17 of [@DupuisEllis] tightness of $\{\mathrm{P}^{\epsilon,\Delta,\gamma}_{1,t},\epsilon>0\}$ will follow if we prove that $$\sup_{\epsilon\in(0,1]}\mathbb{E}\left[ g(\mathrm{P}^{\epsilon ,\Delta,\gamma}_{1,t}\otimes \text{Leb}_{[0,T]})\right] <\infty,$$ where $\text{Leb}_{[0,T]}$ denotes Lebesgue measure in $[0,T]$. However, by (\[Eq:Ubound\]) $$\begin{aligned} \sup_{\epsilon\in(0,1]}\mathbb{E}\left[ g(\mathrm{P}^{\epsilon ,\Delta,\gamma}_{1,t}\otimes \text{Leb}_{[0,T]})\right] & =\sup_{\epsilon\in(0,1]}\mathbb{E}\left[ \int_{0}^{T}\int_{\mathcal{Z}\times\mathcal{Z}}\left[\left\Vert z_{1}\right\Vert^{2}+\left\Vert z_{2}\right\Vert^{2}\right]\mathrm{P}^{\epsilon,\Delta,\gamma}_{1,t}(dz_{1}dz_{2})dt\right] \nonumber\\ & =\sup_{\epsilon\in(0,1]}\mathbb{E}\int_{0}^{T}\frac{1}{\Delta}\int_{t}^{t+\Delta}\left[\left\Vert u_{1}^{\epsilon}(s)\right\Vert ^{2}+\left\Vert u_{2}^{\epsilon}(s)\right\Vert ^{2}\right]dsdt\nonumber\\ & <\infty,\nonumber\end{aligned}$$ uniformly in $\gamma\in\Gamma$, which concludes the tightness proof for $\{\mathrm{P}^{\epsilon,\Delta,\gamma}_{1,t},\epsilon>0\}$. Let us now consider tightness of $\{\mathrm{P}^{\epsilon,\Delta,\gamma}_{2,t},\epsilon>0\}$. For this purpose we notice that for every $\gamma\in\Gamma$ and every $\tilde{\phi}\in L^{2}(\Gamma)\cap L^{1}(\pi)$ we have $$\int_{\Gamma}\tilde{\phi}(\tilde{\gamma})\mathrm{P}^{\epsilon,\Delta,\gamma}_{2,t}(d\tilde{\gamma})=\frac{1}{\Delta}\int_{t}^{t+\Delta}\tilde{\phi} \left(\tau_{\bar{Y}^{\epsilon}_{s}}\gamma\right)ds= \frac{1}{\Delta}\int_{t}^{t+\Delta}\phi\left(\bar{Y}^{\epsilon}_{s},\gamma\right)ds.$$ Let us fix $\eta>0$. Then, by Lemma \[L:ErgodicTheorem4\] we know that there exists $N_{\eta}\subset \Gamma$ with $\pi(N_{\eta})\geq 1-\eta$ such that for every bounded sequence $\Delta\in \mathcal{H}^{N_{\eta}}_{1}$ we have $$\lim_{\epsilon\downarrow0}\sup_{\gamma\in N_{\eta}}\sup_{0\leq t\leq T}\mathbb{E}\left\vert \frac{1}{\Delta}\int_{t}^{t+\Delta}\phi\left(\bar{Y}_{s}^{\epsilon},\gamma\right)ds-\bar{\phi}\right\vert =0$$ or equivalently $$\lim_{\epsilon\downarrow0}\sup_{\gamma\in N_{\eta}}\sup_{0\leq t\leq T}\mathbb{E}\left\vert \int_{\Gamma}\tilde{\phi}(\tilde{\gamma})\mathrm{P}^{\epsilon,\Delta,\gamma}_{2,t}(d\tilde{\gamma})-\bar{\phi}\right\vert =0\label{Eq:UsingErgodicThmForTightness}$$ Now, as a probability measure in a Polish space $\pi$ is itself tight. So, there exists a compact subset of $\Gamma$, say $K_{\eta}$, such that $$\pi(K_{\eta})\geq 1-\eta/2.$$ Therefore, using (\[Eq:UsingErgodicThmForTightness\]) and the latter bound, we get that for $\epsilon$ sufficiently small, say $\epsilon<\epsilon_{0}(\eta)$ and for every $\gamma\in N_{\eta}$ and $t\in[0,T]$, we have $$\inf_{\epsilon\in(0,\epsilon_{0}(\eta))}\mathbb{E}\left[\mathrm{P}^{\epsilon,\Delta,\gamma}_{2,t}(K_{\eta})\right]\geq 1-\eta$$ which implies that, uniformly in $\gamma\in N_{\eta}$, the measure valued random variables $\{\mathrm{P}^{\epsilon,\Delta,\gamma}_{2,t}(\cdot),\epsilon\in(0,\epsilon_{0}(\eta))\}$ are tight. (ii). Uniform integrability of the family $\{\mathrm{P}^{\epsilon,\Delta,\gamma},\epsilon>0\}$ follows by $$\begin{aligned} &\mathbb{E}\left[ \int_{\{(z_{1},z_{2})\in\mathcal{Z}\times\mathcal{Z}:\left[\left\Vert z_{1}\right\Vert+\left\Vert z_{2}\right\Vert\right] >M\}\times\Gamma\times\lbrack0,T]}\left[\left\Vert z_{1}\right\Vert +\left\Vert z_{2}\right\Vert\right]\mathrm{P}^{\epsilon,\Delta}(dz_{1}dz_{2}d\tilde{\gamma}dt)\right] \nonumber\\ &\quad\leq\frac{2}{M}\mathbb{E}\left[ \int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma\times\lbrack0,T]}\left[\left\Vert z_{1}\right\Vert ^{2}+\left\Vert z_{2}\right\Vert ^{2}\right]\mathrm{P}^{\epsilon,\Delta}(dz_{1}dz_{2}d\tilde{\gamma}dt)\right]\nonumber\\ & \quad = \frac{2}{M} \mathbb{E}\int_{0}^{T}\frac{1}{\Delta}\int_{t}^{t+\Delta}\left[\left\Vert u_{1}^{\epsilon}(s)\right\Vert ^{2}+\left\Vert u_{2}^{\epsilon}(s)\right\Vert ^{2}\right]dsdt\nonumber\end{aligned}$$ and the fact that $$\sup_{\epsilon>0,\gamma\in\Gamma}\mathbb{E}\int_{0}^{T}\frac{1}{\Delta}\int_{t}^{t+\Delta}\left[\left\Vert u_{1}^{\epsilon}(s)\right\Vert ^{2}+\left\Vert u_{2}^{\epsilon}(s)\right\Vert ^{2}\right]dsdt <\infty.$$ This concludes the proof of the lemma. \[L:TightnessControlledProcess\] Assume Conditions \[A:Assumption1\] and \[A:Assumption2\]. Let $\{u^{\epsilon,\gamma},\epsilon>0,\gamma\in\Gamma\}$ be a family of controls in $\mathcal{A}$ as in Lemma \[L:TightnessOccupationalMeasures\]. Moreover, fix $\eta>0$, and consider the set $N_{\eta}$ with $\pi(N_{\eta})\geq 1-\eta$ from Lemma \[L:ErgodicTheorem4\]. Then, for every $\gamma\in N_{\eta}$, the family $\{\bar{X}^{\epsilon,\gamma},\epsilon>0\}$ is relatively compact as $\epsilon\downarrow 0$. It suffices to prove that for every $\eta>0$ $$\lim_{\theta\downarrow0}\limsup_{\epsilon\downarrow0}\mathbb{P}\left[ \sup_{t_{1},t_{2}<T, |t_{1}-t_{2}|<\theta}\left\Vert \bar{X}^{\epsilon,\gamma}_{t_{1}}-\bar{X}^{\epsilon,\gamma}_{t_{2}}\right\Vert >\eta\right] =0$$ Recalling the auxiliary problem (\[Eq:RandomCellProblem\]) and the discussion succeeding it, we apply Itô formula (see also [@Osada1983]), to rewrite $\bar{X}^{\epsilon,\gamma}_{t_{1}}-\bar {X}^{\epsilon,\gamma}_{t_{2}}$ as $$\begin{aligned} \bar{X}^{\epsilon,\gamma}_{t_{1}}-\bar{X}^{\epsilon,\gamma}_{t_{2}} &=& \int_{t_{1}}^{t_{2}} \lambda\left( \bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{s}, u_{1}(s),u_{2}(s)\right) ds\nonumber\\ & &\quad -\delta\left[ \chi_{0}\left( \bar{Y}^{\epsilon,\gamma}_{t_{2}}\right) -\chi_{0}\left( \bar{Y}^{\epsilon,\gamma}_{t_{1}} \right)\right] \nonumber\\ & &\quad+ \sqrt{\epsilon}\int_{t_{1}}^{t_{2}} \left(\sigma+\xi\tau_{1}\right)\left( \bar{X}^{\epsilon,\gamma}_{s}, \bar{Y}^{\epsilon,\gamma}_{s}\right) dW_{s}+\sqrt{\epsilon}\int_{t_{1}}^{t_{2}}\xi\tau_{2}\left( \bar{Y}^{\epsilon,\gamma}_{s}\right)dB_{s}\nonumber\\ & &=B^{\epsilon,\gamma}_{1}+ B^{\epsilon,\gamma}_{2}+ B^{\epsilon,\gamma}_{3}\nonumber\end{aligned}$$ where $B^{\epsilon,\gamma}_{i}$ is the $i^{th}$ line of the right hand side of the last display. First we treat the term $B^{\epsilon,\gamma}_{3}$. It suffices to discuss one of the two stochastic integrals, let’s say the first one. In particular, by Itô isometry, Lemma \[L:ErgodicTheorem4\], we have, that there is a set $N_{\eta}$ with $\pi(N_{\eta})\geq 1-\eta$ such that for every $\gamma\in N_{\eta}$, $$\lim_{\epsilon\downarrow0}\left| \mathbb{E}\left\Vert \int_{t_{1}}^{t_{2}}\left(\tilde{\sigma}\left( \bar{X}^{\epsilon,\gamma}_{s}, \cdot\right)+\tilde{\xi}\tilde{\tau}_{1}\left(\cdot\right)\right)dW_{s} \right\Vert ^{2}- \int_{t_{1}}^{t_{2}}\mathrm{E}^{\pi}\left[ \left\Vert \left( \tilde{\sigma}\left( \bar{X}^{\epsilon,\gamma}_{s}, \cdot\right)+\tilde{\xi}\tilde{\tau}_{1}\left(\cdot\right) \right) \right\Vert ^{2}\right]ds \right| \rightarrow0$$ as $\epsilon\downarrow0$. In a similar fashion we can also treat the stochastic integral with respect to the Brownian motion $B$. Hence, for every $\gamma\in N_{\eta}$ $$\lim_{\epsilon\downarrow0}\mathbb{E}\left\Vert B^{\epsilon,\gamma}_{3}\right\Vert^{2}=0$$ Next, we treat $B^{\epsilon,\gamma}_{1}$. Lemma \[L:ErgodicTheorem4\] and the uniform bound (\[Eq:UniformlySquareIntegrableControlsAdditional\]), implies that for every $\gamma\in N_{\eta}$ $$\lim_{|t_{2}-t_{1}|\rightarrow0 }\lim_{\epsilon\downarrow0} \mathbb{E}\left\Vert B^{\epsilon,\gamma,t_{2}-t_{1}}_{1}\right\Vert^{2}=0$$ Similarly, one can show that $\lim_{\epsilon\downarrow0}\mathbb{E}\left\Vert B^{\epsilon,\gamma}_{2} \right\Vert =0$. Therefore, tightness of $\{\bar{X}^{\epsilon,\gamma},\epsilon>0\}$ follows for $\gamma\in N_{\eta}$. Identification of the limit points. {#S:WeakConvergence} ----------------------------------- In this section we prove that any weak limit point of the tight sequence $\left\{ (\bar{X}^{\epsilon,\gamma},\mathrm{P}^{\epsilon,\Delta,\gamma}), \epsilon>0 \right\} $ is a viable pair, i.e., it satisfies Definition \[D:ViablePair\]. Let $(\bar{X},\mathrm{P})$ be an accumulation point (in distribution) of $(\bar{X}^{\epsilon,\gamma},\mathrm{P}^{\epsilon ,\Delta,\gamma})$ as $\epsilon,\Delta\downarrow0$. Due to the Skorokhod representation, we may assume that there is a probability space, where this convergence holds with probability $1$. The constraint (\[Eq:Ubound\]) and Fatou’s lemma guarantee that with probability $1$, $$\int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma\times[0,T]}\left[\left\Vert z_{1}\right\Vert^{2}+\left\Vert z_{2}\right\Vert^{2}\right]\bar{\mathrm{P}}(dz_{1}dz_{2} d\gamma dt)<\infty.$$ Moreover, since $\mathrm{P}^{\epsilon,\Delta,\gamma}\left( \mathcal{Z}\times\mathcal{Z}\times\Gamma\times[0,t]\right) =t$ for every $t\in[0,T]$ and using the fact that $\bar{\mathrm{P}}\left( \mathcal{Z}\times\mathcal{Z}\times\Gamma\times[0,t]\right) $ is continuous as a function of $t\in[0,T]$ and that $\bar{\mathrm{P}}\left( \mathcal{Z}\times\mathcal{Z}\times\Gamma\times\left\{ t\right\} \right) =0$ we obtain $\bar{\mathrm{P}}\left( \mathcal{Z}\times\mathcal{Z}\times\Gamma\times[0,t]\right) =t$ and that $\bar{\mathrm{P}}$ can be decomposed as $\mathrm{P}(dz_{1}dz_{2} d\gamma dt)=\mathrm{P}_{t}(dz_{1}dz_{2} d\gamma)dt$ with $\mathrm{P}_{t}(\mathcal{Z}\times\mathcal{Z}\times \Gamma)=1$. Let us next prove that $(\bar{X},\bar{\mathrm{P}})$ satisfy (\[Eq:ViablePair1\]). We will use the martingale problem. In particular, let $\zeta$ be a smooth bounded function, $\phi\in\mathcal{C}^{2}(\mathbb{R}^{m})$ compactly supported, $\left\{ \tilde{z}_{j}\right\} _{j=1}^{q}$ be a family of bounded, smooth and compactly supported functions and for $r\in\mathcal{P}\left( \mathcal{Z}\times\mathcal{Z}\times\Gamma\times[0,T]\right) $, $t\in[0,T]$ define $$\left( r,\tilde{z}_{j}\right) _{t}=\int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma \times[0,t]} \tilde{z}_{j}(z_{1},z_{2},\gamma,s)r(dz_{1}dz_{2}d\gamma ds)$$ Then, in order to show (\[Eq:ViablePair1\]), it is enough to show that for any $0<t_{1}<t_{2}<\cdots<t_{m}<t<t+r\leq T$, the following limit holds almost surely with respect to $\gamma\in\Gamma$ as $\epsilon\downarrow0$ $$\begin{aligned} \mathbb{E} & \left\{ \zeta\left( \bar{X}^{\epsilon,\gamma}_{t_{i}},(\mathrm{P}^{\epsilon,\Delta,\gamma},z_{j})_{t_{i}}, i\leq m, j\leq q\right) \left[ \phi(\bar{X}^{\epsilon,\gamma}_{t+r})-\phi(\bar{X}^{\epsilon,\gamma}_{t})\right. \right. \nonumber\\ & \left. \left. \hspace{2cm}-\int_{t}^{t+r}\left[ \lim_{\rho\rightarrow 0}\int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma}\tilde{\lambda}_{\rho}(\bar{X}^{\epsilon ,\gamma}_{s},\gamma,z_{1},z_{2})\mathrm{P}_{s}(dz_{1}dz_{2}d\gamma) \right] \nabla\bar{\phi}(\bar {X}^{\epsilon,\gamma}_{s})ds\right] \right\} \rightarrow0 \label{Eq:TargetForConvergenceControlled1}$$ Let us define $$\mathcal{L}^{\epsilon,\Delta,\rho}_{s}\phi(x)=\int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma }\tilde{\lambda}_{\rho}(x,\gamma,z_{1},z_{2})\mathrm{P}^{\epsilon,\Delta,\gamma}_{s}(dz_{1}dz_{2} d\gamma) \nabla \phi(x)$$ where $$\mathrm{P}^{\epsilon,\Delta,\gamma}_{s}(dz_{1}dz_{2} d\gamma) =\frac{1}{\Delta}\int _{s}^{s+\Delta}1_{z_{1}}(u^{\epsilon}_{1}(\theta))1_{z_{2}}(u^{\epsilon}_{2}(\theta))1_{B}\left( \tau_{\bar {Y}^{\epsilon,\gamma}_{\theta}}\gamma\right) d\theta$$ Then, weak convergence of the pair $(\bar{X}^{\epsilon,\gamma},\mathrm{P}^{\epsilon,\Delta,\gamma})$ and uniform integrability of $\mathrm{P}^{\epsilon,\Delta,\gamma}$ as indicated by Lemma \[L:TightnessOccupationalMeasures\], shows that almost surely with respect to $\gamma\in\Gamma$ $$\begin{aligned} \mathbb{E} & \left[ \int_{t}^{t+r}\mathcal{L}^{\epsilon ,\Delta,\rho}_{s}\phi(\bar{X}^{\epsilon,\gamma}_{s})ds-\int_{t}^{t+r}\left[ \lim_{\rho\rightarrow0}\int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma}\tilde{\lambda}_{\rho }(\bar{X}^{\epsilon,\gamma}_{s},\gamma,z_{1},z_{2})\mathrm{P}_{s}(dz_{1}dz_{2}d\gamma) \right] \nabla \phi(\bar{X}^{\epsilon,\gamma}_{s})ds\right] \rightarrow0\nonumber\end{aligned}$$ as $\epsilon\downarrow0$ and $\rho=\rho(\epsilon)\downarrow 0$. Hence, in order to prove (\[Eq:TargetForConvergenceControlled1\]), it is sufficient to prove that almost surely with respect to $\gamma\in\Gamma$ $$\mathbb{E} \left\{ \zeta\left( \bar{X}^{\epsilon,\gamma}_{t_{i}},(\mathrm{P}^{\epsilon,\Delta,\gamma},z_{j})_{t_{i}}, i\leq m, j\leq q\right) \left[ \phi(\bar{X}^{\epsilon,\gamma}_{t+r})-\phi(\bar{X}^{\epsilon ,\gamma}_{t})-\int_{t}^{t+r}\mathcal{L}^{\epsilon,\Delta,\rho}_{s}\phi(\bar {X}^{\epsilon,\gamma}_{s})ds\right] \right\} \rightarrow0$$ Recall the auxiliary problem (\[Eq:RandomCellProblem\]) and consider a function $\phi\in\mathcal{C}^{2}(\mathbb{R}^{m})$ with compact support. Let us write $\chi_{\rho}=\left( \chi_{1,\rho},\ldots,\chi_{m,\rho}\right) $ for the components of the vector solution to (\[Eq:RandomCellProblem\]), and consider $\psi_{\ell,\rho}(x,y,\gamma)=\chi_{\ell,\rho}(y,\gamma )\partial_{x_{\ell}}\phi(x)$ for $\ell\in\{1,\ldots,m\}$. Set $\psi_{\rho }(x,y,\gamma)=\left( \psi_{1,\rho},\ldots,\psi_{m,\rho}\right) $. It is easy to see that $\tilde{\psi}_{\rho}(x,\gamma)$ satisfies the resolvent equation $$\rho\tilde{\psi}_{\ell,\rho}(x,\cdot)-\tilde{L}\tilde{\psi}_{\ell,\rho }(x,\cdot)=\tilde{h}_{\ell}(x,\cdot) \label{Eq:RandomCellProblem1}$$ where we have defined $\tilde{h}_{\ell}(x,\cdot)=\tilde{b}_{\ell}(\cdot)\partial_{x_{\ell}}\phi(x)$. By Itô formula and making use of (\[Eq:RandomCellProblem1\]), we obtain $$\begin{aligned} & \mathbb{E}\left\{ \zeta\left( \bar{X}^{\epsilon,\gamma}_{t_{i}},(\mathrm{P}^{\epsilon,\Delta,\gamma},z_{j})_{t_{i}}, i\leq m, j\leq q\right) \left[ \phi(\bar{X}^{\epsilon,\gamma}_{t+r})-\phi(\bar{X}^{\epsilon ,\gamma}_{t})-\int_{t}^{t+r}\mathcal{L}^{\epsilon,\Delta,\rho}_{s}\phi(\bar {X}^{\epsilon,\gamma}_{s})ds\right] \right\} \nonumber\\ & = \mathbb{E}\left\{ \zeta\left( \cdots\right) \left[ \int _{t}^{t+r}\lambda_{\rho}\left( \bar{X}^{\epsilon,\gamma}_{s},\bar {Y}^{\epsilon,\gamma}_{s},\gamma,u^{\epsilon}_{1}(s),u^{\epsilon}_{2}(s)\right) \nabla \phi(\bar{X}^{\epsilon,\gamma}_{s})ds- \int_{t}^{t+r}\mathcal{L}^{\epsilon ,\Delta,\rho}_{s}\phi(\bar{X}^{\epsilon,\gamma}_{s})ds\right] \right\} \nonumber\\ & \hspace{0.2cm}+\delta\mathbb{E}\left\{ \zeta\left( \cdots\right) \int_{t}^{t+r}\sum_{\ell=1}^{m}\left( (c+\sigma u^{\epsilon}_{1}(s)) \partial_{x} \psi_{\ell,\rho} +\epsilon\frac{1}{2} \text{tr}\left[ \partial^{2}_{x}\psi_{\ell,\rho}\right] \right) \left( \bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{s}\right) \right\} ds\nonumber\\ & \hspace{0.2cm}+\epsilon\mathbb{E}\left\{ \zeta\left( \cdots\right) \int_{t}^{t+r}\sum_{\ell=1}^{m}\text{tr}\left[\sigma\tau_{1}^{T} D\partial _{x}\psi_{\ell,\rho}\right] \left( \bar{X}^{\epsilon,\gamma}_{s},\bar {Y}^{\epsilon,\gamma}_{s}\right) \right\} ds\nonumber\\ & \hspace{0.2cm}+\epsilon2\mathbb{E}\left\{ \zeta\left( \cdots\right) \int_{t}^{t+r} \text{tr}\left[ \sigma\sigma^{T}\left( \bar{Y}^{\epsilon,\gamma}_{s}\right) \nabla^{2}\phi\left( \bar{X}^{\epsilon,\gamma}_{s}\right) \right] \right\} ds\nonumber\\ & \hspace{0.2cm} +\frac{\epsilon}{\delta}\rho\mathbb{E}\left\{ \zeta\left( \cdots\right) \int_{t}^{t+r}\chi_{\rho}\left( \bar {Y}^{\epsilon,\gamma}_{s}\right) \nabla \phi(\bar{X}^{\epsilon,\gamma }_{s})ds\right\} \nonumber\\ & \hspace{0.2cm}-\delta\sum_{\ell=1}^{m}\mathbb{E}\left\{ \zeta\left( \cdots\right) \left( \psi_{\ell,\rho}\left( \bar{X}^{\epsilon,\gamma}_{t+r},\bar {Y}^{\epsilon,\gamma}_{t+r}\right) -\psi_{\ell,\rho}\left( \bar{X}^{\epsilon,\gamma}_{t},\bar {Y}^{\epsilon,\gamma}_{t}\right) \right) \right\} \nonumber\\ & = \sum_{i=1}^{6}\mathbb{E}B_{i}^{\epsilon,\gamma} \label{Eq:TargetForConvergenceControlled2}$$ where $\mathbb{E}B_{i}^{\epsilon,\gamma}$ is the $i^{\text{th}}$ line on the right hand side of (\[Eq:TargetForConvergenceControlled2\]). We want to show that each of those terms goes to zero almost surely with respect to $\gamma\in\Gamma$. Condition \[A:Assumption1\] and the bound (\[Eq:Ubound\]) give us that $$\mathbb{E}\left| B_{2}^{\epsilon,\gamma}\right| +\mathbb{E}\left| B_{3}^{\epsilon,\gamma}\right| \rightarrow0, \quad\text{ as }\epsilon\downarrow0$$ Due to the boundedness and compact support of functions $\zeta$ and $\phi$, we also get that almost surely in $\gamma\in\Gamma$ $$\mathbb{E}\left| B_{4}^{\epsilon,\gamma}\right| \rightarrow0, \quad\text{ as }\epsilon\downarrow0$$ By choosing $\rho=\rho(\epsilon)=\frac{\delta^{2}}{\epsilon}$, we also have that almost surely in $\gamma\in\Gamma$ $$\mathbb{E}\left| B_{5}^{\epsilon,\gamma}\right| +\mathbb{E}\left| B_{6}^{\epsilon,\gamma}\right| \rightarrow0, \quad\text{ as }\epsilon\downarrow0$$ Let us next consider $B_{1}^{\epsilon,\gamma}$. We have $$\begin{aligned} \mathbb{E}B_{1}^{\epsilon,\gamma} & =\mathbb{E}\left\{ \zeta\left( \cdots\right) \left[ \int_{t}^{t+r}\lambda_{\rho}\left( \bar{X}^{\epsilon ,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{s},\gamma,u^{\epsilon}_{1}(s),u^{\epsilon}_{2}(s)\right) \nabla \phi(\bar{X}^{\epsilon,\gamma}_{s}) ds- \int_{t}^{t+r}\mathcal{L}^{\epsilon,\Delta,\rho}_{s}\phi(X^{\epsilon,\gamma}_{s})ds\right] \right\} \nonumber\\ & = \mathbb{E}\left\{ \zeta\left( \cdots\right) \left[ \int _{t}^{t+r}\lambda_{\rho}\left( \bar{X}^{\epsilon,\gamma}_{s},\bar {Y}^{\epsilon,\gamma}_{s},\gamma,u^{\epsilon}_{1}(s),u^{\epsilon}_{2}(s)\right) \nabla \phi(\bar{X}^{\epsilon,\gamma}_{s}) ds-\right.\right.\nonumber\\ &\hspace{3cm}\left.\left.-\int_{t}^{t+r}\frac{1}{\Delta}\int_{s}^{s+\Delta} \lambda_{\rho}\left( \bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{\theta},\gamma,u^{\epsilon}_{1}(\theta), u^{\epsilon}_{2}(\theta)\right) \nabla \phi(\bar{X}^{\epsilon,\gamma}_{s}) d\theta ds\right] \right\} \nonumber\\ & = \mathbb{E}\left\{ \zeta\left( \cdots\right) \left[ \int _{t}^{t+r}\frac{1}{\Delta}\int_{s}^{s+\Delta}\lambda_{\rho}\left( \bar {X}^{\epsilon,\gamma}_{\theta},\bar{Y}^{\epsilon,\gamma}_{\theta},\gamma,u^{\epsilon}_{1}(\theta),u^{\epsilon}_{2}(\theta)\right) \nabla \phi(\bar{X}^{\epsilon,\gamma}_{\theta}) d\theta ds-\right. \right. \nonumber\\ & \hspace{3cm}\left. \left. -\int_{t}^{t+r}\frac{1}{\Delta}\int _{s}^{s+\Delta} \lambda_{\rho}\left( \bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{\theta},\gamma,u^{\epsilon}_{1}(\theta),u^{\epsilon}_{2}(\theta) \right) \nabla \phi(\bar{X}^{\epsilon,\gamma}_{s}) d\theta ds\right] \right\} \nonumber\\ & + \mathbb{E}\left\{ \zeta\left( \cdots\right) \left[ \int _{t}^{t+r}\lambda_{\rho}\left( \bar{X}^{\epsilon,\gamma}_{s},\bar {Y}^{\epsilon,\gamma}_{s},\gamma,u^{\epsilon}_{1}(s),u^{\epsilon}_{2}(s)\right) \nabla \phi(\bar{X}^{\epsilon,\gamma}_{s}) ds-\right.\right.\nonumber\\ &\hspace{3cm}\left.\left.-\int_{t}^{t+r}\frac{1}{\Delta}\int_{s}^{s+\Delta} \lambda_{\rho}\left( \bar{X}^{\epsilon,\gamma}_{\theta},\bar{Y}^{\epsilon,\gamma}_{\theta},\gamma,u^{\epsilon}_{1}(\theta),u^{\epsilon}_{2}(\theta)\right) \nabla \phi(\bar{X}^{\epsilon,\gamma}_{\theta}) d\theta ds\right] \right\} \nonumber\\ & = \mathbb{E}B_{1,1}^{\epsilon,\gamma}+\mathbb{E}B_{1,2}^{\epsilon,\gamma}\nonumber\end{aligned}$$ Let us first treat $\mathbb{E}B_{1,1}^{\epsilon,\gamma}$. $$\begin{aligned} & \mathbb{E}B_{1,1}^{\epsilon,\gamma}=\nonumber\\ & = \mathbb{E}\left\{ \zeta\left( \cdots\right) \left[ \int _{t}^{t+r}\frac{1}{\Delta}\int_{s}^{s+\Delta}\lambda_{\rho}\left( \bar {X}^{\epsilon,\gamma}_{\theta},\bar{Y}^{\epsilon,\gamma}_{\theta},\gamma,u^{\epsilon}_{1}(\theta),u^{\epsilon}_{2}(\theta)\right) \nabla \phi(\bar{X}^{\epsilon,\gamma}_{\theta}) d\theta ds-\right. \right. \nonumber\\ & \hspace{6cm}\left. \left. - \int_{t}^{t+r}\frac{1}{\Delta}\int _{s}^{s+\Delta} \lambda_{\rho}\left( \bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{\theta},\gamma,u^{\epsilon}_{1}(\theta),u^{\epsilon}_{2}(\theta) \right) \nabla \phi(\bar{X}^{\epsilon,\gamma}_{s}) d\theta ds\right] \right\} \nonumber\\ & = \mathbb{E}\left\{ \zeta\left( \cdots\right) \left[ \int _{t}^{t+r}\frac{1}{\Delta}\int_{s}^{s+\Delta}\tilde{\lambda}_{\rho}\left( \bar{X}^{\epsilon,\gamma}_{\theta},\tau_{\bar{Y}^{\epsilon,\gamma }_{\theta}}\gamma,u^{\epsilon}_{1}(\theta),u^{\epsilon}_{2}(\theta)\right) \nabla \phi(\bar{X}^{\epsilon,\gamma}_{\theta}) d\theta ds-\right. \right. \nonumber\\ & \hspace{6cm}\left. \left. -\int_{t}^{t+r}\frac{1}{\Delta}\int _{s}^{s+\Delta} \tilde{\lambda}_{\rho}\left( \bar{X}^{\epsilon,\gamma}_{s},\tau_{\bar{Y}^{\epsilon,\gamma}_{\theta}}\gamma ,u^{\epsilon}_{1}(\theta),u^{\epsilon}_{2}(\theta)\right)\nabla \phi(\bar{X}^{\epsilon,\gamma}_{s}) d\theta ds\right] \right\} \nonumber\\ & \rightarrow0, \quad\text{ as }\epsilon\downarrow0,\nonumber\end{aligned}$$ by continuity of $\tilde{\lambda}_{\rho}$ on the first argument, stationarity and the uniform integrability obtained in Lemma \[L:TightnessOccupationalMeasures\]. Next we treat $\mathbb{E}B_{1,2}^{\epsilon,\gamma}$. We have $$\begin{aligned} \mathbb{E} \left| B_{1,2}^{\epsilon,\gamma}\right| & \leq C_{0}\left\{ \mathbb{E} \int_{0}^{\Delta}\left| \lambda_{\rho }\left( \bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{s},\gamma,u^{\epsilon}_{1}(s),u^{\epsilon}_{2}(s)\right) \nabla \phi(\bar{X}^{\epsilon,\gamma}_{s}) \right| ds\right.\nonumber\\ &\hspace{3cm}\left.+ \mathbb{E} \int_{t}^{t+\Delta}\left| \lambda_{\rho}\left( \bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{s},\gamma,u^{\epsilon}_{1}(s),u^{\epsilon}_{2}(s)\right) \nabla \phi(\bar{X}^{\epsilon,\gamma}_{s}) \right| ds\right\} \nonumber\end{aligned}$$ where $C_{0}$ is a finite constant. Choose $\Delta\downarrow0$ such that $\Delta /\frac{\delta^{2}}{\varepsilon}\uparrow\infty$. Then, we have [$$\begin{aligned} & \mathbb{E} \int_{0}^{\Delta}\left| \lambda_{\rho}\left( \bar {X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{s} ,\gamma,u^{\epsilon}_{1}(s),u^{\epsilon}_{2}(s)\right) \nabla \phi(\bar{X}^{\epsilon,\gamma}_{s}) \right| ds \nonumber\\ &\leq\mathbb{E} \int _{0}^{\Delta}\left| \left( c\left( \bar {X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{s} ,\gamma\right)+D \chi_{\rho}\left( \bar{Y}^{\epsilon,\gamma}_{s},\gamma\right)g\left( \bar {X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{s} ,\gamma\right) \right)\nabla \phi(\bar{X}^{\epsilon,\gamma}_{s}) \right| ds\nonumber\\ & + \mathbb{E} \int_{0}^{\Delta}\left|\left( \sigma\left( \bar{X}^{\epsilon,\gamma}_{s}, \bar{Y}^{\epsilon,\gamma}_{s},\gamma\right) u^{\epsilon}_{1}(s)+D \chi_{\rho}\left( \bar{Y}^{\epsilon,\gamma}_{s},\gamma\right) \left[\tau_{1}\left( \bar{Y}^{\epsilon,\gamma}_{s},\gamma\right) u^{\epsilon}_{1}(s)+ \tau_{2}\left( \bar{Y}^{\epsilon,\gamma}_{s},\gamma\right) u^{\epsilon}_{2}(s)\right]\right)\nabla \phi(\bar{X}^{\epsilon,\gamma}_{s})\right| ds\nonumber\\ & \leq\Delta\frac{\frac{\delta^{2}}{\epsilon}}{\Delta} \mathbb{E} \int_{0}^{\Delta/\frac{\delta^{2}}{\epsilon}}\left|\left( c\left( \bar{X}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s},\bar{Y}^{\epsilon,\gamma}_{(\delta ^{2}/\epsilon)s},\gamma\right)+D \chi_{\rho}\left( \bar{Y}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s },\gamma\right)g\left( \bar{X}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s},\bar{Y}^{\epsilon,\gamma}_{(\delta ^{2}/\epsilon)s},\gamma\right) \right)\nabla \phi(\bar{X}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s})\right| ds\nonumber\\ & + \sqrt{\Delta}\sqrt{\frac{\frac{\delta^{2}}{\epsilon}}{\Delta }\mathbb{E} \int_{0}^{\Delta/\frac{\delta^{2}}{\epsilon}}\left\Vert \left(\sigma\left( \bar{X}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s}, \bar{Y}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s},\gamma\right) +D \chi_{\rho}\left( \bar{Y}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s},\gamma\right) \tau_{1}\left( \bar{Y}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s},\gamma\right) \right)\nabla \phi \right\Vert ^{2}ds \mathbb{E} \int_{0}^{\Delta}\left\Vert u^{\epsilon}_{1}(s)\right\Vert ^{2} ds }\nonumber\\ & + \sqrt{\Delta}\sqrt{\frac{\frac{\delta^{2}}{\epsilon}}{\Delta }\mathbb{E} \int_{0}^{\Delta/\frac{\delta^{2}}{\epsilon}}\left\Vert\left( D \chi_{\rho}\left( \bar{Y}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s},\gamma\right)\tau_{2}\left( \bar{Y}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s},\gamma\right)\right)\nabla \phi \right\Vert ^{2}ds \mathbb{E} \int_{0}^{\Delta}\left\Vert u^{\epsilon}_{2}(s)\right\Vert ^{2} ds }\nonumber\\ & \leq\Delta\frac{\frac{\delta^{2}}{\epsilon}}{\Delta} \mathbb{E} \int_{0}^{\Delta/\frac{\delta^{2}}{\epsilon}}\left\Vert\left( \tilde{c}\left( \bar{X}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s},\tau_{\bar{Y}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s}}\gamma\right)+D \tilde{\chi}_{\rho}\left( \tau_{\bar{Y}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s}}\gamma\right)\tilde{g}\left( \bar{X}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s},\tau_{\bar{Y}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s}}\gamma\right) \right)\nabla \phi \right\Vert ds\nonumber\\ & + \sqrt{\Delta}\sqrt{\frac{\frac{\delta^{2}}{\epsilon}}{\Delta }\mathbb{E} \int_{0}^{\Delta/\frac{\delta^{2}}{\epsilon}}\left\Vert\left( \tilde{\sigma}\left( \bar{X}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s}, \tau_{\bar{Y}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s}}\gamma\right) +D \tilde{\chi}_{\rho}\left( \tau_{\bar{Y}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s}}\gamma\right) \tilde{\tau}_{1}\left( \tau_{\bar{Y}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s}}\gamma\right)\right)\nabla \phi \right\Vert ^{2}ds \mathbb{E} \int_{0}^{\Delta}\left\Vert u^{\epsilon}_{1}(s)\right\Vert^{2} ds }\nonumber\\ & + \sqrt{\Delta}\sqrt{\frac{\frac{\delta^{2}}{\epsilon}}{\Delta }\mathbb{E} \int_{0}^{\Delta/\frac{\delta^{2}}{\epsilon}}\left\Vert\left( D \tilde{\chi}_{\rho}\left( \tau_{\bar{Y}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s}}\gamma\right) \tilde{\tau}_{2}\left(\tau_{\bar{Y}^{\epsilon,\gamma}_{(\delta^{2}/\epsilon)s}}\gamma\right) \right)\nabla \phi \right\Vert ^{2}ds \mathbb{E} \int_{0}^{\Delta}\left\Vert u^{\epsilon}_{2}(s)\right\Vert ^{2} ds }\nonumber\\ & \rightarrow0, \quad\text{as } \epsilon,\Delta\downarrow 0, \Delta /\frac{\delta^{2}}{\varepsilon} \uparrow\infty,\nonumber\end{aligned}$$]{} by Lemma \[L:ErgodicTheorem4\], Condition \[A:Assumption1\] and the uniform bound (\[Eq:Ubound\]). Hence, we obtain that almost surely with respect to $\gamma\in\Gamma$, $$\mathbb{E}\left| B_{1,2}^{\epsilon,\gamma}\right| \rightarrow0.$$ This concludes the proof of (\[Eq:ViablePair1\]). Next, we treat (\[Eq:ViablePair2\]). Consider $\tilde{\phi}\in L^{2}(\Gamma)$ stationary, ergodic random field on $\mathbb{R}^{d-m}$. Let $\phi(y,\gamma)=\tilde{\phi}(\tau_{y}\gamma)$ and assume that $\phi(\cdot,\gamma)\in C^{2}_{b}(\mathbb{R}^{d-m})$. Define the formal operators $$\mathcal{G}^{0,\gamma}_{x,y,\gamma,z_{1},z_{2}}\phi(y,\gamma)=\left[ g(x,y,\gamma )+\tau_{1}(y,\gamma) z_{1}+\tau_{2}(y,\gamma) z_{2}\right] D\phi(y,\gamma)$$ and $$\mathcal{G}^{1,\epsilon,\gamma}_{x,y,\gamma, z_{1},z_{2}}\phi(y,\gamma)=\frac{\epsilon }{\delta^{2}}\mathcal{L}^{\gamma}\phi(y,\gamma)+\frac{1}{\delta}\mathcal{G}^{0,\gamma}_{x,y,z_{1},z_{2}}\phi(y,\gamma)$$ Following the customary notation we write $\tilde{\mathcal{G}}^{0,\gamma }_{x,\gamma,z_{1},z_{2}}\tilde{\phi}(\gamma)=\left[ \tilde{g}(x,\gamma)+ \tilde{\tau}_{1}(\gamma) z_{1}+\tilde{\tau}_{2}(\gamma) z_{2}\right] D\tilde{\phi}(\gamma)$ and analogously for $\tilde{\mathcal{G}}^{1,\epsilon,\gamma}_{x,\gamma, z_{1},z_{2}}\tilde{\phi}(\gamma)$. For each fixed $\gamma\in\Gamma$, the process $$\begin{aligned} M^{\epsilon,\gamma}_{t} & =\phi(\bar{Y}^{\epsilon,\gamma}_{t})- \phi(\bar {Y}^{\epsilon,\gamma}_{0})-\int_{0}^{t}\mathcal{G}^{1,\epsilon,\gamma}_{\bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{s},u^{\epsilon}_{1}(s),u^{\epsilon}_{2}(s)}\phi(\bar{Y}^{\epsilon,\gamma}_{s})ds\nonumber\\ & =\frac{\sqrt{\epsilon}}{\delta}\int_{0}^{t}\left< D\phi(\bar {Y}^{\epsilon,\gamma}_{s}),\tau_{1}(\bar{Y}^{\epsilon,\gamma}_{s})dW_{s}\right>+\frac{\sqrt{\epsilon}}{\delta}\int_{0}^{t}\left< D\phi(\bar {Y}^{\epsilon,\gamma}_{s}),\tau_{2}(\bar{Y}^{\epsilon,\gamma}_{s})dB_{s}\right>\nonumber\end{aligned}$$ is an $\mathfrak{F}_{t}-$martingale. Set $h(\epsilon)=\frac{\delta^{2}}{\epsilon}$ and write $$\begin{aligned} h(\epsilon)M^{\epsilon,\gamma}_{t} & -h(\epsilon)\left[ \phi(\bar{Y}^{\epsilon,\gamma}_{t})- \phi(\bar{Y}^{\epsilon,\gamma}_{0})\right] \nonumber\\ & +h(\epsilon)\left[ \int_{0}^{t}\frac{1}{\Delta}\left( \int_{s}^{s+\Delta }\mathcal{G}^{1,\epsilon,\gamma}_{\bar{X}^{\epsilon,\gamma}_{\theta},\bar {Y}^{\epsilon,\gamma}_{\theta},u^{\epsilon}_{1}(\theta),u^{\epsilon}_{2}(\theta)}\phi(\bar{Y}^{\epsilon ,\gamma}_{\theta})d\theta\right) ds -\int_{0}^{t}\mathcal{G}^{1,\epsilon ,\gamma}_{\bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{s},u^{\epsilon}_{1}(s),u^{\epsilon}_{1}(s)}\phi(\bar{Y}^{\epsilon,\gamma}_{s})ds\right] \nonumber\\ & =-\frac{\delta}{\epsilon}\int_{0}^{t}\frac{1}{\Delta}\left[ \int _{s}^{s+\Delta}\left( \mathcal{G}^{0,\gamma}_{\bar{X}^{\epsilon,\gamma }_{\theta},\bar{Y}^{\epsilon,\gamma}_{\theta},u^{\epsilon}_{1}(\theta),u^{\epsilon}_{2}(\theta)}\phi(\bar {Y}^{\epsilon,\gamma}_{\theta}) -\mathcal{G}^{0,\gamma}_{\bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{\theta},u^{\epsilon}_{1}(\theta), u^{\epsilon}_{2}(\theta)}\phi(\bar{Y}^{\epsilon,\gamma}_{\theta})\right) d\theta\right] ds\nonumber\\ & \hspace{0.5cm}-\frac{\delta}{\epsilon}\int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma \times[0,t]}\tilde{\mathcal{G}}^{0,\gamma}_{\bar{X}^{\epsilon,\gamma}_{s},\gamma,z_{1},z_{2}}\tilde{\phi}(\gamma)\bar{\mathrm{P}}^{\epsilon,\Delta,\gamma }(dz_{1}dz_{2}d\gamma ds)\nonumber\\ & \hspace{0.5cm}-\int_{0}^{t}\frac{1}{\Delta}\int_{s}^{s+\Delta}\mathcal{L}^{\gamma}\phi(\bar{Y}^{\epsilon,\gamma}_{\theta},\gamma)d\theta ds \label{Eq:Part2ofViability}$$ The boundedness of $\phi$ and of its derivatives imply that almost surely in $\gamma\in\Gamma$ $$\mathbb{E}\left[ \left| h(\epsilon)M^{\epsilon,\gamma}_{t}\right| ^{2}+\left| h(\epsilon)\left[ \phi(\bar{Y}^{\epsilon,\gamma}_{t})- \phi(\bar {Y}^{\epsilon,\gamma}_{0})\right] \right| \right] \rightarrow 0,\quad\text{as }\epsilon\downarrow0$$ Moreover, we have almost surely in $\gamma\in\Gamma$ $$\begin{aligned} & h(\epsilon) \mathbb{E}\left| \int_{0}^{t}\frac{1}{\Delta}\left( \int_{s}^{s+\Delta}\mathcal{G}^{1,\epsilon,\gamma}_{\bar{X}^{\epsilon,\gamma }_{\theta},\bar{Y}^{\epsilon,\gamma}_{\theta},u^{\epsilon}_{1}(\theta),u^{\epsilon}_{2}(\theta)}\phi(\bar {Y}^{\epsilon,\gamma}_{\theta})d\theta\right) ds -\int_{0}^{t}\mathcal{G}^{1,\epsilon,\gamma}_{\bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma }_{s},u^{\epsilon}_{1}(s),u^{\epsilon}_{2}(s)}\phi(\bar{Y}^{\epsilon,\gamma}_{s})ds\right| \nonumber\\ & \leq h(\epsilon) \mathbb{E}\int_{0}^{\Delta}\left| \mathcal{G}^{1,\epsilon,\gamma}_{\bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma }_{s},u^{\epsilon}_{1}(s),u^{\epsilon}_{2}(s)}\phi(\bar{Y}^{\epsilon,\gamma}_{s})\right| ds +h(\epsilon) \mathbb{E}\int_{t}^{t+\Delta}\left| \mathcal{G}^{1,\epsilon,\gamma}_{\bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma }_{s},u^{\epsilon}_{1}(s),u^{\epsilon}_{2}(s)}\phi(\bar{Y}^{\epsilon,\gamma}_{s})\right| ds\nonumber\\ & \leq\mathbb{E}\int_{0}^{\Delta}\left| \mathcal{L}\phi(\bar {Y}^{\epsilon,\gamma}_{s})\right| ds+\frac{\delta}{\epsilon}\mathbb{E}^{\gamma}\int_{0}^{\Delta}\left| \mathcal{G}^{0,\gamma}_{\bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{s},u^{\epsilon}_{1}(s),u^{\epsilon}_{2}(s)}\phi(\bar{Y}^{\epsilon,\gamma}_{s})\right| ds\nonumber\\ & +\mathbb{E}\int_{t}^{t+\Delta}\left| \mathcal{L}\phi(\bar {Y}^{\epsilon,\gamma}_{s})\right| ds+\frac{\delta}{\epsilon}\mathbb{E}^{\gamma}\int_{t}^{t+\Delta}\left| \mathcal{G}^{0,\gamma}_{\bar{X}^{\epsilon,\gamma}_{s},\bar{Y}^{\epsilon,\gamma}_{s},u^{\epsilon}_{1}(s),u^{\epsilon}_{2}(s)}\phi(\bar{Y}^{\epsilon,\gamma}_{s})\right| ds\nonumber\\ & \leq\Delta C_{0}\left[ 1+\frac{\delta}{\epsilon}\mathbb{E}\int_{0}^{T}\left\Vert u^{\epsilon}(s)\right\Vert ^{2}ds\right] \nonumber\\ & \rightarrow0, \quad\text{as } \epsilon\downarrow0,\nonumber\end{aligned}$$ due to (\[Eq:Ubound\]) and $\Delta=\Delta(\epsilon )\downarrow0$. The constant $C_{0}$ depends on the upper bound of the coefficients and on $\beta,T$. The first term on the right hand side of (\[Eq:Part2ofViability\]) goes to zero in probability, almost surely with respect to $\gamma\in\Gamma$, due to continuous dependence of $\mathcal{G}^{0,\gamma}_{x,y,z_{1},z_{2}}\phi(y,\gamma)$ on $x\in\mathbb{R}^{m}$, tightness of $\bar{X}^{\epsilon,\gamma}$, stationarity and $\delta/\epsilon\downarrow0$. The second term on the right hand side of (\[Eq:Part2ofViability\]) also goes to zero in probability, almost surely with respect to $\gamma\in\Gamma$, due to continuous dependence of $\mathcal{G}^{0,\gamma}_{x,y,z_{1},z_{2}}\phi(y,\gamma)$ on $x\in\mathbb{R}^{m}$, tightness of $(\bar{X}^{\epsilon,\gamma},\mathrm{P}^{\epsilon,\Delta,\gamma})$, uniform integrability of $\mathrm{P}^{\epsilon,\Delta,\gamma}$ (Lemma \[L:TightnessOccupationalMeasures\]) and the fact that $\delta/\epsilon\downarrow0$. Lastly, we consider the third term on the right hand side of (\[Eq:Part2ofViability\]). We have $$\begin{aligned} \int_{0}^{t}\frac{1}{\Delta}\int_{s}^{s+\Delta}\mathcal{L}^{\gamma}\phi(\bar {Y}^{\epsilon,\gamma}_{\theta},\gamma)d\theta ds & = \int_{0}^{t}\frac {1}{\Delta}\int_{s}^{s+\Delta}\tilde{\mathcal{L}}\tilde{\phi}(\tau_{\bar {Y}^{\epsilon,\gamma}_{\theta}}\gamma)d\theta ds\nonumber\\ & =\int_{0}^{t}\int_{\mathcal{Z}\times\Gamma}\tilde{\mathcal{L}}\tilde {\phi}(\gamma)\mathrm{P}^{\epsilon,\Delta,\gamma}(dzd\gamma ds)\nonumber\end{aligned}$$ Due to weak convergence of $(\bar{X}^{\epsilon,\gamma},\mathrm{P}^{\epsilon,\Delta,\gamma})$, the last term converges, almost surely with respect to $\phi\in\Gamma$ to $\int_{0}^{t}\int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma }\tilde{\mathcal{L}}\tilde{\phi}(\gamma)\bar{\mathrm{P}}(dz_{1}dz_{2}d\gamma ds)$. Hence, since the rest of the terms converge to $0$, as $\epsilon\downarrow0$, we obtain in probability, almost surely in $\gamma\in\Gamma$ $$\int_{0}^{t}\int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma}\tilde{\mathcal{L}}\tilde{\phi}(\gamma)\bar{\mathrm{P}}(dz_{1}dz_{2}d\gamma ds) =0$$ for almost all $t\in[0,T]$, which together with continuity in $t\in[0,T]$ conclude the proof of (\[Eq:ViablePair2\]). Compactness of level sets and quenched lower and upper bounds {#S:LDPrelaxedForm} ============================================================= Compactness of level sets of the rate function is standard and will not be repeated here (e.g., Subsection 4.2. of [@DupuisSpiliopoulos] or [@FW1]). Let us now prove the quenched lower bound. First we remark that we can restrict attention to controls that satisfy Conditions \[Eq:UniformlySquareIntegrableControlsAdditional\] and \[Eq:UniformlySquareIntegrableControlsAdditionalb\], which are required in order for Lemma \[L:ErgodicTheorem4\] to be true. For this purpose we have the following lemma, whose proof is deferred to the end of this section. \[L:RestrictingTheControl\] Let $(\bar{X}_{s}^{\epsilon,\gamma}, \bar{Y}_{s}^{\epsilon,\gamma})$ be the strong solution to (\[Eq:Main2\]) and assume Conditions \[A:Assumption1\] and \[A:Assumption2\]. Then, the infimum of the representation in (\[Eq:VariationalRepresentation\]) can be taken over all controls that satisfy Conditions \[Eq:UniformlySquareIntegrableControlsAdditional\] and \[Eq:UniformlySquareIntegrableControlsAdditionalb\]. Based on Lemma \[L:RestrictingTheControl\], we can restrict attention to controls satisfying Conditions \[Eq:UniformlySquareIntegrableControlsAdditional\] and \[Eq:UniformlySquareIntegrableControlsAdditionalb\]. Given such controls, we construct the controlled pair $(\bar {X}^{\epsilon,\gamma},\mathrm{P}^{\epsilon,\Delta,\gamma})$ based on such a family of controls. Then, Theorem \[T:MainTheorem1\] implies tightness of the pair $\left\{ (\bar{X}^{\epsilon,\gamma},\mathrm{P}^{\epsilon ,\Delta,\gamma}), \epsilon,\Delta>0 \right\} $. Let us denote by $(\bar {X},\bar{\mathrm{P}})\in\mathcal{V}$ an accumulation point of the controlled pair in distribution, almost surely with respect to $\gamma\in\Gamma$. Then, by Fatou’s lemma we conclude the proof of the lower bound. Indeed $$\begin{aligned} \liminf_{\epsilon\downarrow0}-\epsilon\log\mathbb{E}\left[ e^{-\frac{1}{\epsilon}h(X^{\epsilon})}\right] & \geq\liminf_{\epsilon \downarrow0}\mathbb{E}\left[ \frac{1}{2}\int_{0}^{T}\left[\left\Vert u^{\epsilon,\gamma}_{1}(s)\right\Vert^{2}+\left\Vert u^{\epsilon,\gamma}_{2}(s)\right\Vert^{2}\right]ds+h(\bar{X}^{\epsilon,\gamma})\right] \nonumber\\ & \geq\liminf_{\epsilon\downarrow0}\mathbb{E}\left[ \frac{1}{2}\int_{0}^{T}\int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma}\left[\left\Vert z_{1}\right\Vert^{2}+\left\Vert z_{2}\right\Vert^{2}\right]\mathrm{P}^{\epsilon,\Delta,\gamma}(dz_{1}dz_{2} d\gamma ds)+h(\bar{X}^{\epsilon,\gamma})\right] \nonumber\\ & \geq\inf_{(\phi,\mathrm{P})\in\mathcal{V} }\left[ \frac{1}{2}\int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma\times[0,T]}\left[\left\Vert z_{1}\right\Vert^{2}+\left\Vert z_{2}\right\Vert^{2}\right]\mathrm{P}(dz_{1}dz_{2} d\gamma ds)+h(\phi)\right] \nonumber\end{aligned}$$ which concludes the proof of the Laplace principle lower bound. It remains to prove the quenched upper bound for the Laplace principle. To do so, we fix a bounded and continuous function $h:\mathcal{C}\left( [0,T];\mathbb{R}^{m}\right) \mapsto\mathbb{R}$, and we show that $$\limsup_{\epsilon\downarrow0}-\epsilon\log\mathbb{E}\left[ e^{-\frac{1}{\epsilon}h(X^{\epsilon})}\right] \leq\inf_{\phi\in \mathcal{C}\left( [0,T];\mathbb{R}^{m}\right) }\left\{ S(\phi )+h(\phi)\right\}$$ The idea is to fix a nearly optimizer of the right hand side of the last display and construct the control which attains the given upper bound. Fix $\eta>0$ and consider $\psi\in\mathcal{C}\left( [0,T];\mathbb{R}^{m}\right) $ with $\psi_{0}=x_{0}$ such that $$S(\psi)+h(\psi)\leq\inf_{\phi\in\mathcal{C}\left( [0,T];\mathbb{R}^{m}\right) }\left\{ S(\phi)+h(\phi)\right\} +\eta<\infty$$ Boundedness of $h$ implies that $S(\psi)<\infty$ which means that $\psi$ is absolutely continuous. Since the local rate function $L^{o}(x,v)$, defined in (\[Eq:LocalRateFunction\]), is continuous and bounded as a function of $(x,v)\in\mathbb{R}^{m}$, standard mollification arguments (Lemmas 6.5.3 and 6.5.5 in [@DupuisEllis]) allow to assume that $\dot{\psi}$ is piecewise constant. Next, we define the elements of $L^{2}(\Gamma)$ $$\tilde{u}_{1,\rho}(t,x,\gamma)=\left(\tilde{\sigma}(x,\gamma)+ D\tilde {\chi}_{\rho}(\gamma)\tilde{\tau}_{1}(\gamma)\right)^{T}q^{-1}(x)(\dot{\psi}_{t}-r(x))$$ and $$\tilde{u}_{2,\rho}(t,x,\gamma)=\left(D\tilde {\chi}_{\rho}(\gamma)\tilde{\tau}_{2}(\gamma)\right)^{T}q^{-1}(x)(\dot{\psi}_{t}-r(x))$$ and the associated stationary fields $u_{1,\rho}(t,x,y,\gamma)=\tilde{u}_{1,\rho }(t,x,\tau_{y}\gamma)$ and $u_{2,\rho}(t,x,y,\gamma)=\tilde{u}_{2,\rho }(t,x,\tau_{y}\gamma)$. We recall that $\tilde{\chi}_{\rho}$ satisfies the auxiliary problem in (\[Eq:RandomCellProblem\]). Let us consider now the solution $\left(\bar{X}^{\epsilon}_{t},\bar{Y}^{\epsilon}_{t}\right)$ of (\[Eq:Main2\]) with the control $u(t)=\left(u_{1}(t),u_{2}(t)\right)$ being $$u^{\epsilon,\rho,\gamma}_{t}=\left(u_{1,\rho}\left( t,\bar{X}^{\epsilon}_{t},\bar{Y}^{\epsilon}_{t},\gamma\right), u_{2,\rho}\left( t,\bar{X}^{\epsilon}_{t},\bar{Y}^{\epsilon}_{t},\gamma\right)\right).$$ Then, replacing $c(x,y,\gamma)$ by $c(t,x,y,\gamma)=c(x,y,\gamma)+\sigma(y,\gamma)u_{1,\rho}(t,x,y,\gamma)$, and $g(x,y,\gamma)$ by $g(t,x,y,\gamma)=g(x,y,\gamma)+\tau_{1}(y,\gamma)u_{1,\rho}(t,x,y,\gamma)+\tau_{2}(y,\gamma)u_{2,\rho}(t,x,y,\gamma)$ Theorem \[L:ErgodicTheorem4\] implies that $$\bar{X}^{\epsilon}\rightarrow\bar{X} \quad\text{in law, almost surely with respect to }\gamma\in\Gamma,$$ as $\epsilon\downarrow0$ where we have that w.p. 1 the limit is $$\begin{aligned} \bar{X}_{t} & =x_{0}+\int_{0}^{t}\lim_{\rho\downarrow0}\mathrm{E}^{\pi }\left[ \tilde{c}(\bar{X}_{s},\cdot)+D\tilde{\chi}_{\rho}(\cdot)\tilde{g}(\bar{X}_{s},\cdot) + \left(\tilde{\sigma}(\bar{X}_{s},\cdot)+D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{1}(\cdot)\right) \tilde{u}_{1,\rho}(s,\bar{X}_{s},\cdot)\right.\nonumber\\ &\qquad\qquad\left.+\left(D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{2}(\cdot)\right) \tilde{u}_{2,\rho}(s,\bar{X}_{s},\cdot)\right] ds\nonumber\\ & =x_{0}+\int_{0}^{t}\lim_{\rho\downarrow0}\mathrm{E}^{\pi}\left[ \tilde{c}(\bar{X}_{s},\cdot)+D\tilde{\chi}_{\rho}(\cdot)\tilde{g}(\bar{X}_{s},\cdot)\right] ds\nonumber\\ &\qquad\qquad+\int_{0}^{t}\lim_{\rho\downarrow0}\mathrm{E}^{\pi}\left[ \left(\tilde{\sigma}(\bar{X}_{s},\cdot)+D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{1}(\cdot)\right) \tilde{u}_{1,\rho}(s,\bar{X}_{s},\cdot)+\left(D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{2}(\cdot)\right) \tilde{u}_{2,\rho}(s,\bar{X}_{s},\cdot)\right] ds\nonumber\\ & =x_{0}+\int_{0}^{t}r(\bar{X}_{s})ds+\int_{0}^{t}\lim_{\rho\downarrow 0}\mathrm{E}^{\pi}\left\{ \left[ \left(\tilde{\sigma}(\bar{X}_{s},\cdot) +D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{1}(\cdot)\right) \left(\tilde{\sigma}(\bar{X}_{s},\cdot)+D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{1}(\cdot)\right)^{T} \right.\right.\nonumber\\ &\hspace{6cm}\left.\left.+\left(D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{2}(\cdot)\right) \left(D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{2}(\cdot)\right)^{T}\right]q^{-1}(\bar{X}_{s}) (\dot{\psi}_{s}-r(\bar{X}_{s}))\right\} ds\nonumber\\ & =x_{0}+\int_{0}^{t}r(\bar{X}_{s})ds+\int_{0}^{t}\mathrm{E}^{\pi}\left\{ \left[ \left(\tilde{\sigma}(\bar{X}_{s},\cdot)+D\tilde{\xi}(\cdot)\tilde{\tau}_{1}(\cdot)\right) \left(\tilde{\sigma}(\bar{X}_{s},\cdot)+D\tilde{\xi}(\cdot)\tilde{\tau}_{1}(\cdot)\right)^{T} \right.\right.\nonumber\\ &\hspace{6cm}\left.\left.+\left(D\tilde{\xi}(\cdot)\tilde{\tau}_{2}(\cdot)\right) \left(D\tilde{\xi}(\cdot)\tilde{\tau}_{2}(\cdot)\right)^{T}\right]q^{-1}(\bar{X}_{s}) (\dot{\psi}_{s}-r(\bar{X}_{s}))\right\}ds \nonumber\\ & =x_{0}+\int_{0}^{t}r(\bar{X}_{s})ds+\int_{0}^{t}q(\bar{X}_{s}) q^{-1}(\bar{X}_{s})(\dot{\psi}_{s}-r(\bar{X}_{s}))ds\nonumber\\ & =x_{0}+\psi_{t}-\psi_{0}\nonumber\\ & =\psi_{t}.\nonumber\end{aligned}$$ Moreover, by Theorem \[L:ErgodicTheorem4\] we have that for any $\eta>0$, there exists a $N_{\eta}$ with $\nu\left[ N_{\eta}\right] >1-\eta$ such that $$\begin{aligned} &\lim_{\epsilon\downarrow0}\sup_{\gamma\in N_{\eta}}\sup_{0\leq t\leq T}\mathbb{E}\left[ \frac{1}{2}\int_{0}^{T}\left[\left\Vert u^{\epsilon ,\gamma}_{1,\rho}(s)\right\Vert^{2}+\left\Vert u^{\epsilon ,\gamma}_{2,\rho}(s)\right\Vert^{2}\right]ds\right.\nonumber\\ &\hspace{4cm}\left.- \frac{1}{2}\int_{0}^{T}\lim_{\rho\downarrow 0}\mathrm{E}^{\pi}\left[\left\Vert u^{\epsilon ,\gamma}_{1,\rho}(s,X^{\epsilon}_{s},\cdot)\right\Vert^{2}+\left\Vert u^{\epsilon ,\gamma}_{2,\rho}(s,X^{\epsilon}_{s},\cdot)\right\Vert^{2}\right] ds\right| =0\nonumber\end{aligned}$$ Therefore, noticing that for each fixed $x\in\mathbb{R}^{m}$ and almost every $t\in[0,T]$ $$\begin{aligned} \lim_{\rho\downarrow0}\mathrm{E}^{\pi}\left[\left\Vert u^{\epsilon ,\gamma}_{1,\rho}(s,x,\cdot)\right\Vert^{2}+\left\Vert u^{\epsilon ,\gamma}_{2,\rho}(s,x,\cdot)\right\Vert^{2}\right] ds&= (\dot{\psi}_{s}-r(x))^{T}q^{-1}(x)q(x) q^{-1}(x)(\dot{\psi}_{s}-r(x))\nonumber\\ & =L^{0}(x,\dot{\psi}_{s}),\nonumber\end{aligned}$$ we finally obtain $$\begin{aligned} \limsup_{\epsilon\downarrow0}-\epsilon\log\mathbb{E}\left[ e^{-\frac{1}{\epsilon}h(X^{\epsilon})}\right] & =\limsup_{\epsilon \downarrow0}\inf_{u\in\mathcal{A}}\mathbb{E}\left[ \frac {1}{2}\int_{0}^{T}\left[\left\Vert u_{1}(s)\right\Vert^{2}+\left\Vert u_{2}(s)\right\Vert^{2}\right]ds+h(\bar{X}^{\epsilon})\right] \nonumber\\ & \leq\limsup_{\epsilon\downarrow0}\mathbb{E}\left[ \frac{1}{2}\int_{0}^{T}\left[\left\Vert u^{\epsilon,\gamma}_{1,\rho}(s)\right\Vert^{2}+\left\Vert u^{\epsilon,\gamma}_{2,\rho}(s)\right\Vert^{2}\right]ds+h(\bar{X}^{\epsilon })\right] \nonumber\\ & \leq\left[ S(\psi)+h(\psi)\right] \nonumber\\ & \leq\inf_{\phi\in\mathcal{C}\left( [0,T];\mathbb{R}^{m}\right) }\left\{ S(\phi)+h(\phi)\right\} +\eta.\nonumber\end{aligned}$$ The first line follows from the representation (\[Eq:VariationalRepresentation\]) and the second line from the choice of the particular control. The third line follows from he convergence of the $X^{\epsilon}$ and of the cost functional using the continuity of $h$. Then, the fourth line follows from the fact $\bar{X}_{t}=\psi_{t}$. Since the last statement is true for every $\eta>0$ the proof of the upper bound is done. We conclude this section with the proof of Lemma \[L:RestrictingTheControl\]. First, we explain why Condition \[Eq:UniformlySquareIntegrableControlsAdditional\] can be assumed without loss of generality. Without loss of generality, we can consider a function $h(x)$ that is bounded and uniformly Lipschitz continuous in $\mathbb{R}^{m}$. Namely, there exists a constant $L_{h}$ such that $$|h(x)-h(y)|\leq L_{h}\left\Vert x-y\right\Vert$$ and $\left\Vert h\right\Vert_{\infty}=\sup_{x\in\mathbb{R}^{m}}|h(x)|<\infty$. We recall that the representation $$-\epsilon\log\mathbb{E}\left[ e^{-\frac{1}{\epsilon }h(X^{\epsilon}_{T})}\right] =\inf_{u\in\mathcal{A}}\mathbb{E}\left[ \frac{1}{2}\int_{0}^{T}\left\Vert u(s)\right\Vert^{2}ds+h(\bar{X}^{\epsilon }_{T})\right] \label{Eq:RepresentationTheorem1}$$ is valid in a $\gamma$ by $\gamma$ basis. Fix $a>0$. Then for every $\epsilon>0$, there exists a control $u^{\epsilon}\in\mathcal{A}$ such that $$-\epsilon\log\mathbb{E}\left[ e^{-\frac{1}{\epsilon }h(X^{\epsilon}_{T})}\right] \geq \mathbb{E}\left[ \frac{1}{2}\int_{0}^{T}\left\Vert u^{\epsilon}(s)\right\Vert^{2}ds+h(\bar{X}^{\epsilon }_{T})\right]-a . \label{Eq:RepresentationTheorem1a}$$ So, letting $M_{0}=\left\Vert h\right\Vert_{\infty}=\sup_{x\in\mathbb{R}^{m}}|h(x)|$ we easily see that such a control $u^{\epsilon}$ should satisfy $$\sup_{\epsilon>0,\gamma\in\Gamma}\mathbb{E}\left[ \frac{1}{2}\int_{0}^{T}\left\Vert u^{\epsilon}(s)\right\Vert^{2} ds\right]\leq M_{1}=2M_{0}+a.$$ Given that the latter bound has been established, the claim that in proving the Laplace principle lower bound one can assume Condition \[Eq:UniformlySquareIntegrableControlsAdditional\] without loss of generality, follows by the last display and the representation (\[Eq:RepresentationTheorem1\]) as in the proof of Theorem 4.4 of [@BudhirajaDupuis2000]. In particular, it follows by the arguments in [@BudhirajaDupuis2000] that if the last display holds, then it is enough to assume that for given $a>0$ the controls satisfy the bound $$\int_{0}^{T}\left\Vert u^{\epsilon}(s)\right\Vert^{2}ds<N,$$ with $$N\geq \frac{4M_{0}(4M_{0}+a)}{a}$$ which proves that in proving the Laplace principle lower bound one can assume Condition \[Eq:UniformlySquareIntegrableControlsAdditional\] without loss of generality. Second, we explain why Condition \[Eq:UniformlySquareIntegrableControlsAdditionalb\] can be assumed without loss of generality. It is clear by the representation (\[Eq:RepresentationTheorem1\]) that the trivial bound holds $$-\epsilon\log\mathbb{E}\left[ e^{-\frac{1}{\epsilon }h(X^{\epsilon}_{T})}\right] \leq\mathbb{E} h(X^{\epsilon }_{T}),$$ where the control $u^{\epsilon}(\cdot)=0$ is used to evaluate the right hand side. Thus, we only need to consider controls that satisfy $$\begin{aligned} \mathbb{E}\left[ \frac{1}{2}\int_{0}^{T} \left\Vert u^{\epsilon,\gamma}(s)\right\Vert^{2}ds+h(\bar{X}^{\epsilon}_{T})\right] & \leq \mathbb{E} h(X^{\epsilon }_{T})\nonumber\end{aligned}$$ which by the Lipschitz assumption on $h$, implies that $$\begin{aligned} \mathbb{E}\left[ \int_{0}^{T}\left\Vert u^{\epsilon,\gamma}(s)\right\Vert^{2}ds\right] & \leq \mathbb{E} \left|h(X^{\epsilon}_{T})-h(\bar{X}^{\epsilon}_{T})\right|\nonumber\\ & \leq L_{h} \mathbb{E} \left\Vert X^{\epsilon}_{T}-\bar{X}^{\epsilon}_{T}\right\Vert.\nonumber\end{aligned}$$ Let us next define the processes $\left(\hat{\bar{X}}^{\epsilon}_{t},\hat{\bar{Y}}^{\epsilon}_{t}\right)=\left(\bar{X}^{\epsilon}_{\frac{\delta^{2}t}{\epsilon}},\bar{Y}^{\epsilon}_{\frac{\delta^{2}t}{\epsilon}}\right)$ and $\left(\hat{X}^{\epsilon}_{t},\hat{Y}^{\epsilon}_{t}\right)=\left(X^{\epsilon}_{\frac{\delta^{2}t}{\epsilon}},Y^{\epsilon}_{\frac{\delta^{2}t}{\epsilon}}\right)$. It is easy to see that $\left(\hat{\bar{X}}^{\epsilon}_{t},\hat{\bar{Y}}^{\epsilon}_{t}\right)$ satisfies the SDE $$\begin{aligned} d\hat{\bar{X}}^{\epsilon}_{t}&=& \delta b\left(\hat{\bar{Y}}^{\epsilon}_{t},\gamma\right)dt+\frac{\delta^{2}}{\epsilon}\left[c\left( \hat{\bar{X}}^{\epsilon}_{t},\hat{\bar{Y}}^{\epsilon}_{t},\gamma\right)+\sigma\left( \hat{\bar{X}}_{t}^{\epsilon},\hat{\bar{Y}}_{t}^{\epsilon},\gamma\right) u_{1}(\delta^{2}t/\epsilon)\right] dt+\delta\sigma\left( \hat{\bar{X}}^{\epsilon}_{t},\hat{\bar{Y}}^{\epsilon}_{t},\gamma\right) dW_{t}, \nonumber\\ d\hat{\bar{Y}}^{\epsilon}_{t}&=& f\left(\hat{\bar{Y}}^{\epsilon}_{t},\gamma\right)dt +\frac{\delta}{\epsilon}\left[ g\left( \hat{\bar{X}}^{\epsilon}_{t} ,\hat{\bar{Y}}^{\epsilon}_{t},\gamma\right)+\tau_{1}\left(\hat{\bar{Y}}^{\epsilon}_{t},\gamma\right)u_{1}(\delta^{2}t/\epsilon)+\tau_{2}\left(\hat{\bar{Y}}^{\epsilon}_{t},\gamma\right)u_{2}(\delta^{2}t/\epsilon)\right] dt\nonumber\\ & &\hspace{5cm}+\left[ \tau_{1}\left(\hat{\bar{Y}}^{\epsilon}_{t},\gamma\right) dW_{t}+\tau_{2}\left(\hat{\bar{Y}}^{\epsilon}_{t},\gamma\right)dB_{t}\right],\nonumber\\ \hat{\bar{X}}^{\epsilon}_{0}&=&x_{0},\hspace{0.2cm}\hat{\bar{Y}}^{\epsilon}_{0}=y_{0},\nonumber\end{aligned}$$ and $\left(\hat{X}^{\epsilon}_{t},\hat{Y}^{\epsilon}_{t}\right)$ satisfies the same SDE with the control $u^{\epsilon}_{1}(\cdot)=u^{\epsilon}_{2}(\cdot)=0$. So, we basically have that $$\begin{aligned} \frac{1}{\epsilon}\mathbb{E}\left[ \int_{0}^{\frac{\delta^{2}T}{\epsilon}}\left\Vert u^{\epsilon,\gamma}(s)\right\Vert^{2}ds\right] & \leq L_{h} \frac{1}{\epsilon} \mathbb{E} \left\Vert X^{\epsilon}_{\frac{\delta^{2}T}{\epsilon}}-\bar{X}^{\epsilon}_{\frac{\delta^{2}T}{\epsilon}}\right\Vert\nonumber\\ &= L_{h} \frac{\delta}{\epsilon} \mathbb{E} \left\Vert \frac{1}{\delta} \hat{X}^{\epsilon}_{T}-\frac{1}{\delta}\hat{\bar{X}}^{\epsilon}_{T}\right\Vert.\nonumber\end{aligned}$$ For notational convenience, we define $$\nu^{\epsilon}_{T}\doteq \frac{1}{\epsilon}\mathbb{E}\left[ \int_{0}^{\frac{\delta^{2}T}{\epsilon}}\left\Vert u^{\epsilon,\gamma}(s)\right\Vert^{2}ds\right]$$ and $$m^{\epsilon}_{T}\doteq \mathbb{E} \left\Vert \frac{1}{\delta}\hat{X}^{\epsilon}_{T}-\frac{1}{\delta}\hat{\bar{X}}^{\epsilon}_{T}\right\Vert^{2}+\mathbb{E} \left\Vert \hat{Y}^{\epsilon}_{T}-\hat{\bar{Y}}^{\epsilon}_{T}\right\Vert^{2}.$$ Since for $x>0$, the function $x^{2}$ is increasing, the latter inequality, followed by Jensen’s inequality give us $$\begin{aligned} \left|\nu^{\epsilon}_{T}\right|^{2}\leq \left|L_{h} \frac{\delta}{\epsilon}\right|^{2}m^{\epsilon}_{T}.\nonumber\end{aligned}$$ The next step is to derive an upper bound of $m^{\epsilon}_{T}$ in terms of $|\nu^{\epsilon}_{T}|^{2}$. Writing down the differences of $\hat{X}^{\epsilon}_{T}-\hat{\bar{X}}^{\epsilon}_{T}$ and $\hat{Y}^{\epsilon}_{T}-\hat{\bar{Y}}^{\epsilon}_{T}$, squaring, taking expectation and using Lipschitz continuity of the functions $b,c,f,g,\sigma,\tau_{1},\tau_{2}$ and boundedness of $\sigma,\tau_{1},\tau_{2}$ we obtain the inequality $$\begin{aligned} m^{\epsilon}_{T}&\leq C_{0}\int_{0}^{T} m^{\epsilon}_{s}ds+ C_{1}\left\{\left|\frac{\delta}{\epsilon}\mathbb{E} \left[\int_{0}^{T}\left\Vert u^{\epsilon,\gamma}_{1}\left(\frac{\delta^{2}s}{\epsilon}\right)\right\Vert ds\right]\right|^{2} +\left|\frac{\delta}{\epsilon}\mathbb{E} \left[\int_{0}^{T}\left\Vert u^{\epsilon,\gamma}_{2}\left(\frac{\delta^{2}s}{\epsilon}\right)\right\Vert ds\right]\right|^{2}\right\},\nonumber\end{aligned}$$ where the constants $C_{0}, C_{1}$ depends only on the Lipschitz constants of $b,c,f,g,\sigma,\tau_{1},\tau_{2}$ and on the sup norm of $\sigma,\tau_{1},\tau_{2}$. Defining for notational convenience $$\left|a^{\epsilon}_{T}\right|^{2}\doteq \frac{\delta}{\epsilon}\mathbb{E}\left[ \int_{0}^{T}\left\Vert u^{\epsilon,\gamma}_{1}\left(\frac{\delta^{2}s}{\epsilon}\right)\right\Vert ds\right]+\frac{\delta}{\epsilon}\mathbb{E} \left[\int_{0}^{T}\left\Vert u^{\epsilon,\gamma}_{2}\left(\frac{\delta^{2}s}{\epsilon}\right)\right\Vert ds\right].$$ Gronwall lemma, gives us $$\begin{aligned} m^{\epsilon}_{T}&\leq C_{1}\left|a^{\epsilon}_{T}\right|^{2}+ C_{0}C_{1}\int_{0}^{T}\left|a^{\epsilon}_{s}\right|^{2}e^{C_{0}(T-s)}ds.\nonumber\end{aligned}$$ Let us now rewrite and upper bound $\left|a^{\epsilon}_{T}\right|^{2}$. We notice that, Hölder inequality followed by Young’s inequality give us $$\begin{aligned} \left|a^{\epsilon}_{T}\right|^{2}&=\left|\frac{\epsilon}{\delta}\frac{1}{\epsilon}\mathbb{E}\left[ \int_{0}^{\frac{\delta^{2}T}{\epsilon}}\left\Vert u^{\epsilon,\gamma}_{1}\left(s\right)\right\Vert ds\right]\right|^{2}+\left|\frac{\epsilon}{\delta}\frac{1}{\epsilon}\mathbb{E}\left[ \int_{0}^{\frac{\delta^{2}T}{\epsilon}}\left\Vert u^{\epsilon,\gamma}_{2}\left(s\right)\right\Vert ds\right]\right|^{2}\nonumber\\ &\leq \frac{1}{\delta^{2}}\frac{\delta^{2}T}{\epsilon}\mathbb{E}\left[ \int_{0}^{\frac{\delta^{2}T}{\epsilon}}\left\Vert u^{\epsilon,\gamma}\left(s\right)\right\Vert^{2}ds\right]\nonumber\\ &= T \frac{1}{\epsilon}\mathbb{E}\left[ \int_{0}^{\frac{\delta^{2}T}{\epsilon}}\left\Vert u^{\epsilon,\gamma}\left(s\right)\right\Vert^{2}ds\right]\nonumber\\ &= T\nu^{\epsilon}_{T}\nonumber\\ &\leq \frac{T^{2}}{2}+\frac{\left|\nu^{\epsilon}_{T}\right|^{2}}{2}.\nonumber\end{aligned}$$ Putting these estimates together, we obtain $$\begin{aligned} \left|\nu^{\epsilon}_{T}\right|^{2}& \leq L_{h}^{2} \left|\frac{\delta}{\epsilon}\right|^{2} m^{\epsilon}_{T}\nonumber\\ &\leq L_{h}^{2}C_{1} \left|\frac{\delta}{\epsilon}\right|^{2} \left[ \left|a^{\epsilon}_{T}\right|^{2}+ C_{0}\int_{0}^{T}\left|a^{\epsilon}_{s}\right|^{2}e^{C_{0}(T-s)}ds\right]\nonumber\\ &\leq L_{h}^{2}C_{1} \left|\frac{\delta}{\epsilon}\right|^{2} \left[ \left(\frac{T^{2}}{2}+\frac{\left|\nu^{\epsilon}_{T}\right|^{2}}{2}\right)+ C_{0}\int_{0}^{T}\left(\frac{s^{2}}{2}+\frac{\left|\nu^{\epsilon}_{s}\right|^{2}}{2}\right)e^{C_{0}(T-s)}ds\right].\nonumber\end{aligned}$$ Therefore, by choosing $\delta/\epsilon$ sufficiently small such that $L_{h}^{2}C_{1} \left|\frac{\delta}{\epsilon}\right|^{2}\leq 1$, we have $$\begin{aligned} \frac{\left|\nu^{\epsilon}_{T}\right|^{2}}{2} &\leq L_{h}^{2}C_{1} \left|\frac{\delta}{\epsilon}\right|^{2} \left[ \frac{T^{2}}{2} + C_{0}\int_{0}^{T}\left(\frac{s^{2}}{2}+\frac{\left|\nu^{\epsilon}_{s}\right|^{2}}{2}\right)e^{C_{0}(T-s)}ds\right]\nonumber\\ &\leq L_{h}^{2}C_{1} \left|\frac{\delta}{\epsilon}\right|^{2} \left[ \frac{T^{2}}{2} + \frac{T^{2}}{2}(e^{C_{0}T}-1) +C_{0} \int_{0}^{T}\frac{\left|\nu^{\epsilon}_{s}\right|^{2}}{2}e^{C_{0}(T-s)}ds\right]\nonumber\\ &= L_{h}^{2}C_{1} \left|\frac{\delta}{\epsilon}\right|^{2} \left[ \frac{T^{2}}{2}e^{C_{0}T} +C_{0} \int_{0}^{T}\frac{\left|\nu^{\epsilon}_{s}\right|^{2}}{2}e^{C_{0}(T-s)}ds\right].\end{aligned}$$ Thus, we have $$\begin{aligned} e^{-C_{0}T}\frac{\left|\nu^{\epsilon}_{T}\right|^{2}}{2} &\leq L_{h}^{2}C_{1} \left|\frac{\delta}{\epsilon}\right|^{2} \left[ \frac{T^{2}}{2} +C_{0} \int_{0}^{T}e^{-C_{0}s}\frac{\left|\nu^{\epsilon}_{s}\right|^{2}}{2}ds\right].\end{aligned}$$ So, letting $\beta^{\epsilon}_{T}=L_{h}^{2}C_{1} \frac{T^{2}}{2}\left|\frac{\delta}{\epsilon}\right|^{2} $ and $\theta^{\epsilon}=L_{h}^{2}C_{1} C_{0}\left|\frac{\delta}{\epsilon}\right|^{2}$, Gronwall lemma guarantees that $$\begin{aligned} e^{-C_{0}T}\frac{\left|\nu^{\epsilon}_{T}\right|^{2}}{2}&\leq \beta^{\epsilon}_{T}+\theta^{\epsilon} \int_{0}^{T}\beta^{\epsilon}_{s}e^{\theta^{\epsilon}(T-s)}ds.\end{aligned}$$ Since $\beta^{\epsilon}_{T}$ and $\theta^{\epsilon}$ go uniformly in $\gamma\in\Gamma$ to zero at the speed $O(\left(\frac{\delta}{\epsilon}\right)^{2})$ as $\epsilon\downarrow 0$, we get that $$\left|\nu^{\epsilon}_{T}\right|^{2}\leq C (\delta/\epsilon)^{2},$$ where the constant $C$, depends on $T$, but not on $\epsilon,\delta$ or $\gamma$. This concludes the argument of why Condition \[Eq:UniformlySquareIntegrableControlsAdditionalb\] can be assumed without loss of generality. Proof of Theorem \[T:MainTheorem3\] {#S:ProofTheoremExplicitLDP} ==================================== In this section we prove that the explicit expression of the large deviation’s action functional is given by Theorem \[T:MainTheorem3\]. Due to Theorem \[T:MainTheorem2\], we only need to prove that the rate function given in (\[Eq:GeneralRateFunction\]) can be written in the form of Theorem \[T:MainTheorem3\]. First, we notice that one can write (\[Eq:GeneralRateFunction\]) in terms of a local rate function, in the form $$S(\phi)=\int_{0}^{T}L^{r}(\phi_{s},\dot{\phi}_{s})ds$$ where we have defined $$L^{r}(x,v)=\inf_{\mathrm{P}\in\mathcal{A}_{x,v}^{r}}\frac{1}{2}\int _{\mathcal{Z}\times\mathcal{Z}\times\Gamma}\left[\left\Vert z_{1}\right\Vert^{2}+\left\Vert z_{2}\right\Vert^{2}\right]\mathrm{P}(dz_{1}dz_{2}d\gamma)$$ and $$\begin{aligned} \mathcal{A}_{x,v}^{r} & =\left\{ \mathrm{P}\in\mathcal{P}\left( \mathcal{Z}\times\mathcal{Z}\times\Gamma\right) :\int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma}\tilde{L}\tilde{f}(\gamma)\mathrm{P}(dz_{1}dz_{2}d\gamma)=0,\quad\forall\quad\tilde{f}\in\mathcal{D}(\tilde{L})\right. \\ & \hspace{0.2cm} \left. \int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma}\left[\left\Vert z_{1}\right\Vert^{2}+\left\Vert z_{2}\right\Vert^{2}\right]\mathrm{P}(dz_{1}dz_{2}d\gamma)<\infty,\text{ and }v=\lim_{\rho\rightarrow0}\int_{\mathcal{Z}\times\mathcal{Z}\times\Gamma}\tilde{\lambda}_{\rho}(x,\gamma,z_{1},z_{2})\mathrm{P}(dz_{1}dz_{2}d\gamma)\right\}\end{aligned}$$ This follows directly by the definition of a viable pair (Definition \[D:ViablePair\]). We call this representation the relaxed formulation since the control is characterized as a distribution on $\mathcal{Z}\times\mathcal{Z}$ rather than an element of $\mathcal{Z}\times\mathcal{Z}$. However, as we shall demonstrate below, the structure of the problem allows us to rewrite the relaxed formulation of the local rate function in terms of an ordinary formulation of an equivalent local rate function, where the control is indeed given as an element of $\mathcal{Z}\times\mathcal{Z}$. In preparation for this representation, we notice that any element $\mathrm{P}\in\mathcal{P}\left( \mathcal{Z}\times\mathcal{Z}\times \Gamma\right) $ can be written of a stochastic kernel on $\mathcal{Z}\times\mathcal{Z}$ given $\Gamma$ and a probability measure on $\Gamma$, namely $$\mathrm{P}(dz_{1}dz_{2}d\gamma)=\eta(dz_{1}dz_{2}|\gamma)\pi(d\gamma).$$ Hence, by the definition of viability, we obtain for every $\tilde{f}\in\mathcal{D}(\tilde{L})$ that $$\int_{\Gamma}\tilde{L}\tilde{f}(\gamma)\pi(d\gamma)=0$$ where we used the independence of $\tilde{L}$ on $z$ to eliminate the stochastic kernel $\eta$. Then Proposition \[P:NewMeasureRandomCase\] guarantees that $\pi$ takes the form $$\pi(d\gamma)\doteq\frac{\tilde{m}(\gamma)}{\mathrm{E}^{\nu}\tilde{m}(\cdot )}\nu(d\gamma)$$ and is actually an invariant, ergodic and reversible probability measure for the process associated with the operator $\tilde{L}$, or equivalently for the environment process $\gamma_{t}$ as given by (\[Eq:EnvironmentProcess\]). Next, since the cost $\left\Vert z\right\Vert^{2}$ is convex in $z=(z_{1},z_{2})$ and $\tilde{\lambda}_{\rho}$ is affine in $z$, the relaxed control formulation can be easily written in terms of the ordinary control formulation $$L^{o}(x,v)=\inf_{\tilde{u}\in\mathcal{A}_{x,v}^{o}}\frac{1}{2}\mathrm{E}^{\pi }\left[\left\Vert \tilde{u}_{1}(\cdot)\right\Vert^{2}+\left\Vert \tilde{u}_{2}(\cdot)\right\Vert^{2}\right] \label{Eq:LocalRateFunction_OrdinaryFormulation}$$ and $$\mathcal{A}_{x,v}^{o}=\left\{ \tilde{u}=(\tilde{u}_{1},\tilde{u}_{2}):\Gamma\mapsto\mathbb{R}^{d}:\mathrm{E}^{\pi}\left[\left\Vert \tilde{u}_{1}(\cdot)\right\Vert^{2}+\left\Vert \tilde{u}_{2}(\cdot)\right\Vert^{2}\right] <\infty,\text{ and }v=\lim _{\rho\rightarrow0}\mathrm{E}^{\pi}\left[ \tilde{\lambda}_{\rho}(x,\cdot,\tilde{u}_{1}(\cdot),\tilde{u}_{2}(\cdot))\right] \right\} .$$ Jensen’s inequality and the fact that $\tilde{\lambda}_{\rho}(x,\gamma,z_{1},z_{2})$ is affine in $z$ imply $L^{r}(x,v)\geq L^{o}(x,v)$. For the reverse inequality, for given $\tilde{u}=(\tilde{u}_{1},\tilde{u}_{2})$ one can define a corresponding relaxed control by $\mathrm{P}(dz_{1}dz_{2}d\gamma)=\delta_{\left(\tilde{u}_{1}(\gamma),\tilde{u}_{2}(\gamma)\right)}(dz_{1}dz_{2})\pi(d\gamma)$. The next step is to prove that the infimization problem in (\[Eq:LocalRateFunction\_OrdinaryFormulation\]) can be solved explicitly and in particular that $$L^{o}(x,v)=\frac{1}{2}(v-r(x))^{T}q^{-1}(x)(v-r(x)) \label{Eq:LocalRateFunction}$$ where$$r(x)=\lim_{\rho\downarrow0}\mathrm{E}^{\pi}\left[ \tilde{c}(x,\cdot) +D\tilde{\chi}_{\rho }(\cdot) \tilde{g}(x,\cdot)\right] =\mathrm{E}^{\pi}[\tilde{c}(x,\cdot)+\tilde{\xi}(\cdot)\tilde{g}(x,\cdot)]$$$$\begin{aligned} q(x)&=\lim_{\rho\downarrow0}\mathrm{E}^{\pi}\left[ (\tilde{\sigma}(x,\cdot)+D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{1}(\cdot))(\tilde{\sigma}(x,\cdot)+D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{1}(\cdot))^{T}+\left(D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{2}(\cdot)\right)\left(D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{2}(\cdot)\right)^{T}\right]\nonumber\\ &=\mathrm{E}^{\pi}\left[ (\tilde{\sigma}(x,\cdot)+\tilde{\xi}(\cdot)\tilde{\tau}_{1}(\cdot))(\tilde{\sigma}(x,\cdot)+\tilde{\xi} (\cdot)\tilde{\tau}_{1}(\cdot))^{T}+\left(\tilde{\xi} (\cdot)\tilde{\tau}_{2}(\cdot)\right)\left(\tilde{\xi} (\cdot)\tilde{\tau}_{2}(\cdot)\right)^{T}\right]\end{aligned}$$ Let us first prove that for every $\tilde{u}=\left(\tilde{u}_{1},\tilde{u}_{2}\right)\in\mathcal{A}_{x,v}^{o}$ $$\mathrm{E}^{\pi}\left\Vert \tilde{u}(x,\cdot)\right\Vert^{2}\geq(v-r(x))^{T}q^{-1}(x)(v-r(x)). \label{Eq:cost_bound}$$ By definition, any $\tilde{u}=\left(\tilde{u}_{1},\tilde{u}_{2}\right)\in\mathcal{A}_{x,v}^{o}$ satisfies $$\begin{aligned} v&=\lim_{\rho\rightarrow0}\mathrm{E}^{\pi}\left[ \tilde{\lambda}_{\rho }(x,\cdot,\tilde{u}_{1}(\cdot),\tilde{u}_{1}(\cdot))\right]\nonumber\\ &=r(x)+\lim_{\rho\rightarrow 0}\mathrm{E}^{\pi}\left[ \left( \tilde{\sigma}(x,\cdot)+D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{1}(\cdot)\right)\tilde{u}_{1}(\cdot)+ D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{2}(\cdot)\tilde{u}_{2}(\cdot)\right] .\nonumber\end{aligned}$$ Treating $x$ as a parameter, define $$\hat{v}=v-r(x)=\lim_{\rho\rightarrow 0}\mathrm{E}^{\pi}\left[ \left( \tilde{\sigma}(x,\cdot)+D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{1}(\cdot)\right)\tilde{u}_{1}(\cdot)+ D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{2}(\cdot)\tilde{u}_{2}(\cdot)\right],$$ and for notational convenience set $$\tilde{\kappa}_{1,\rho}(x,\gamma)=\tilde{\sigma}(x,\gamma)+D\tilde{\chi}_{\rho}(\gamma)\tilde{\tau}_{1}(\gamma)\quad\textrm{ and } \tilde{\kappa}_{2,\rho}(x,\gamma)=D\tilde{\chi}_{\rho}(\gamma)\tilde{\tau}_{2}(\gamma)$$ Next, we drop writing explicitly the dependence on the parameter $x$ and we write $q^{-1}=W^{T}W$, where $W$ is an invertible matrix, so that $\hat{v}^{T}q^{-1}\hat{v}=\left\Vert W\hat{v}\right\Vert^{2}$. Without loss of generality, we assume that $\tilde{u}\in L^{2}(\Gamma)$ is such that $\mathrm{E}^{\pi}\left\Vert \tilde{u}(\cdot)\right\Vert^{2}=1$. By Cauchy-Schwartz inequality in $\mathbb{R}^{m}$ we have $$\begin{aligned} \left\Vert W\hat{v}\right\Vert^{2} & =\left\langle W\hat{v},W\lim_{\rho\downarrow0}\mathrm{E}^{\pi}\left[ \tilde{\kappa}_{1,\rho}(\cdot)\tilde{u}_{1}(\cdot)+ \tilde{\kappa}_{2,\rho}(\cdot)\tilde{u}_{2}(\cdot)\right] \right\rangle \nonumber\\ & =\lim_{\rho\downarrow0}\mathrm{E}^{\pi}\left[ \left\langle \tilde{u}_{1} (\cdot),\tilde{\kappa}_{1,\rho}^{T}(\cdot)W^{T}W\hat{v}\right\rangle + \left\langle \tilde{u}_{2} (\cdot),\tilde{\kappa}_{2,\rho}^{T}(\cdot)W^{T}W\hat{v}\right\rangle\right] \nonumber\\ & \leq\lim_{\rho\downarrow0}\left( \mathrm{E}^{\pi}\left[\left\Vert \tilde{\kappa }_{1,\rho}^{T}(\cdot)W^{T}W\hat{v}\right\Vert^{2}+\left\Vert \tilde{\kappa }_{2,\rho}^{T}(\cdot)W^{T}W\hat{v}\right\Vert^{2}\right] \right) ^{1/2}\nonumber\\ & =\lim_{\rho\downarrow0}\left( \hat{v}^{T}W^{T}W\mathrm{E}^{\pi}\left[ \tilde{\kappa}_{1,\rho}(\cdot)\tilde{\kappa}_{1,\rho}^{T}(\cdot)+ \tilde{\kappa}_{2,\rho}(\cdot)\tilde{\kappa}_{2,\rho}^{T}(\cdot)\right] W^{T}W\hat{v}\right) ^{1/2}\nonumber\\ & =\left( \hat{v}^{T}W^{T}WqW^{T}W\hat{v}\right) ^{1/2}\nonumber\\ & =\left\Vert W\hat{v}\right\Vert.\nonumber\end{aligned}$$ If $\left\Vert W\hat{v}\right\Vert=0$, then (\[Eq:cost\_bound\]) holds automatically. If $\left\Vert W\hat{v}\right\Vert\neq0$, then the last display implies $\left\Vert W\hat{v}\right\Vert\leq1$, which directly proves that $$\mathrm{E}^{\pi}\left\Vert\tilde{u}(x,\cdot)\right\Vert^{2}=1\geq\left\Vert W\hat{v}\right\Vert^{2}=(v-r(x))^{T}q^{-1}(x)(v-r(x)).$$ To prove that the inequality becomes an equality when taking the infimum over all $\tilde{u}\in\mathcal{A}_{x,v}^{o}$, we need to find a $\tilde{u}\in L^{2}(\Gamma)$ which attains the infimum. Define the elements of $L^{2}(\Gamma)$ $$\tilde{u}_{1,\rho}(x,\gamma;v)=\left(\tilde{\sigma}(\gamma)+ D\tilde {\chi}_{\rho}(\gamma)\tilde{\tau}_{1}(\gamma)\right)^{T}q^{-1}(x)(v-r(x))$$ and $$\tilde{u}_{2,\rho}(x,\gamma;v)=\left(D\tilde {\chi}_{\rho}(\gamma)\tilde{\tau}_{2}(\gamma)\right)^{T}q^{-1}(x)(v-r(x))$$ and set $\tilde{u}_{\rho}(x,\cdot;v)=\left(\tilde{u}_{1,\rho}(x,\cdot;v),\tilde{u}_{2,\rho}(x,\cdot;v)\right)$. A straightforward computation yields $$\begin{aligned} \mathrm{E}^{\pi}\left\Vert \tilde{u}_{\rho}(x,\cdot;v)\right\Vert^{2}&=(v-r(x))^{T}q^{-1}(x) \mathrm{E}^{\pi}\left[ (\tilde{\sigma}(x,\cdot)+D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{1}(\cdot))(\tilde{\sigma}(x,\cdot)+D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{1}(\cdot))^{T}+\right.\nonumber\\ &\hspace{6cm}\left.+\left(D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{2}(\cdot)\right)\left(D\tilde{\chi}_{\rho}(\cdot)\tilde{\tau}_{2}(\cdot)\right)^{T}\right] q(x)(v-r(x))\end{aligned}$$ Thus, letting $\rho\downarrow0$, we obtain $$\lim_{\rho\downarrow0}\mathrm{E}^{\pi}\left\Vert\tilde{u}_{\rho}(x,\cdot;v)\right\Vert^{2}=(v-r(x))^{T}q^{-1}(x)(v-r(x))$$ Hence, the element $\tilde{u}\in L^{2}(\Gamma)$ that we are looking for is the $L^{2}(\pi)$ limit of $\tilde{u}_{\rho}$ as defined above. This is well defined, since by Proposition 2.6 in [@Olla1994] $D\tilde{\chi}_{\rho}$ has a well defined $L^{2}(\pi)$ strong limit. Therefore, we have proven that $$L^{o}(x,v)=\frac{1}{2}(v-r(x))^{T}q^{-1}(x)(v-r(x))$$ which concludes the proof of Theorem \[T:MainTheorem3\]. Quenched ergodic theorems {#S:QuenchedErgodicTheorems} ========================== In this appendix we prove quenched ergodic theorems that are required for the proof of Theorem \[T:MainTheorem1\]. For notational convenience and without loss of generality, we mostly consider a process $Y$ driven by a single Brownian motion with diffusion coefficient $\kappa(y,\gamma)$ such that $\kappa\kappa^{T}=\tau_{1}\tau_{1}^{T}+\tau_{2}\tau_{2}^{T}$. We prove the required ergodic result, Lemma \[L:ErgodicTheorem4\] in a progressive way. First, in Lemma \[L:ErgodicTheorem0\] we recall the classical ergodic theorem. This is strengthened in Lemma \[L:ErgodicTheorem1u\] to cover cases of time shifts, uniformly with respect to $t\in[0,T]$. Then, in Lemmas \[L:ErgodicTheorem2\]-\[L:ErgodicTheorem3\] we consider the case of perturbing the drift of the process by small perturbations (uncontrolled and controlled case). The latter result together with the standard technique of freezing the slow component yield the proof of the ergodic statement in Lemma \[L:ErgodicTheorem4\]. No time shifts, i.e. $t=0$ -------------------------- \[L:ErgodicTheorem0\] Consider the process $Y_{t}^{\epsilon,y_{0},\gamma}$ satisfying the SDE $$Y_{t}^{\epsilon,y_{0},\gamma}=y_{0}+\frac{\epsilon}{\delta^{2}}\int_{0}^{t}f(Y_{s}^{\epsilon,y_{0},\gamma},\gamma)ds+\frac{\sqrt{\epsilon}}{\delta }\int_{0}^{t}\kappa(Y_{s}^{\epsilon,y_{0},\gamma},\gamma)dW_{s}. \label{Eq:BasicSDE}$$ Consider also a function $\tilde{\Psi}\in L^{2}(\Gamma)\cap L^{1}(\pi )$ and define $\Psi (y,\gamma)=\tilde{\Psi}(\tau_{y}\gamma)$. Assume that $\Psi:\mathbb{R}^{d-m}\times\Gamma\mapsto\mathbb{R}$ is measurable. Denote $\bar{\Psi}\doteq\int_{\Gamma}\tilde{\Psi}(\gamma)\pi(d\gamma)$. Then for any sequence $h(\epsilon)$ that is bounded from above and such that $\delta^{2}/[\epsilon h(\epsilon)]\downarrow0$ (note that in particular $h(\epsilon)$ could be a constant), there is a set $N$ of full $\pi-$measure such that for every $\gamma\in N$ $$\lim_{\epsilon\downarrow0}\mathbb{E}\left\vert \frac{1}{h(\epsilon )}\int_{0}^{h(\epsilon)}\Psi(Y_{s}^{\epsilon,y_{0},\gamma},\gamma)ds-\bar{\Psi}\right\vert =0.$$ \[Proof of Lemma \[L:ErgodicTheorem0\]\]Let $\hat{Y}_{t}^{y_{0},\gamma }=Y_{\delta^{2}t/\epsilon}^{\epsilon,y_{0},\gamma}$. Note that $\hat{Y}_{t}^{y_{0},\gamma}$ satisfies $$\hat{Y}_{t}^{y_{0},\gamma}=y_{0}+\int_{0}^{t}f(\hat{Y}_{s}^{y_{0},\gamma },\gamma)ds+\int_{0}^{t}\kappa(\hat{Y}_{s}^{y_{0},\gamma},\gamma)dW_{s}, \label{Eq:Yhat}$$ and also that $\pi(d\gamma)$ is the invariant ergodic probability measure for the environment process $\gamma_{t}=\tau_{\hat{Y}_{t}^{y_{0},\gamma}}\gamma$ (Proposition \[P:NewMeasureRandomCase\]). Suppose that $\delta^{2}/[\epsilon h(\epsilon)]\downarrow0$. By the ergodic theorem, there is a set $N$ of full $\pi-$measure such that for any $\gamma\in N$ $$\begin{aligned} \lim_{\epsilon\downarrow0}\mathbb{E}\left[ \frac{1}{h(\epsilon)}\int_{0}^{h(\epsilon)}\Psi(Y_{s}^{\epsilon,y_{0},\gamma},\gamma)ds\right] & =\lim_{\epsilon\downarrow0}\mathbb{E}\left[ \frac{\delta^{2}}{\epsilon h(\epsilon)}\int_{0}^{\frac{\epsilon h(\epsilon)}{\delta^{2}}}\Psi(\hat{Y}_{s}^{y_{0},\gamma},\gamma)ds\right] \\ & \qquad=\lim_{\epsilon\downarrow0}\mathbb{E}\left[ \frac {\delta^{2}}{\epsilon h(\epsilon)}\int_{0}^{\frac{\epsilon h(\epsilon)}{\delta^{2}}}\tilde{\Psi}(\tau_{\hat{Y}_{s}^{y_{0},\gamma}}\gamma)ds\right] \\ & \qquad=\lim_{\epsilon\downarrow0}\mathbb{E}\left[ \frac {\delta^{2}}{\epsilon h(\epsilon)}\int_{0}^{\frac{\epsilon h(\epsilon)}{\delta^{2}}}\tilde{\Psi}(\gamma_{s})ds\right] \\ & \qquad=\bar{\Psi}.\end{aligned}$$ It follows from Egoroff’s theorem that for every $\eta>0$ there is a set $N_{\eta}$ with $\pi\left[ N_{\eta}\right] >1-\eta$, such that $$\lim_{\epsilon\downarrow0}\sup_{\gamma\in N_{\eta}}\mathbb{E}\left\vert \frac{1}{h(\epsilon)}\int_{0}^{h(\epsilon)}\Psi(Y_{s}^{\epsilon,y_{0},\gamma})ds-\bar{\Psi}\right\vert =0.$$ Time shifts and uniformity -------------------------- For notational purposes we will write that $h(\epsilon)\in \mathcal{H}_{1}^{N_{\eta}}$, if the pair $(h(\epsilon),N_{\eta})$ satisfies Condition \[A:h\_functionUniform\]. \[A:h\_functionUniform\] Let $\tilde{\Psi}\in L^{2}(\Gamma)\cap L^{1}(\pi )$ and define the measurable function $\Psi (y,\gamma)=\tilde{\Psi}(\tau_{y}\gamma)$. For $\gamma\in\Gamma$ define $$\theta^{\gamma}(u)=\sup_{r>u}\mathbb{E}\left| \frac{1}{r}\int_{0}^{r}\Psi(\hat{Y}_{s}^{y_{0},\gamma},\gamma)ds-\bar{\Psi}\right| .$$ For any $\eta\in(0,1)$, there exists a set $N_{\eta}$ with $\pi(N_{\eta})\geq1-\eta$ and a sequence $\{h(\epsilon),\epsilon>0\}$ such that the following are satisfied: 1. [$\frac{\delta^{2}/\epsilon}{h(\epsilon)}\rightarrow0$ as $\epsilon \downarrow0$, ]{} 2. [there exists $\beta\in(0,1)$ such that $\frac{\sup_{\gamma\in N_{\eta}} \theta^{\gamma}\left( \frac{1}{\left( \delta^{2}/\epsilon\right) ^{\beta}}\right) }{h(\epsilon )}\rightarrow0$, as $\epsilon\downarrow0$, and]{} 3. [$\frac{1}{h(\epsilon)}\sup_{\gamma\in N_{\eta}}\sup_{t\in[0,T]}\mathbb{E}\left| \left( \delta^{2}/\epsilon\right) \int^{\frac{t}{\delta^{2}/\epsilon}}_{0}\Psi(\hat{Y}_{s}^{y_{0},\gamma},\gamma)ds-t \bar{\Psi}\right| \rightarrow0$ as $\epsilon\downarrow0$]{} Lemma \[L:ErgodicTheorem1u\] shows that one in fact can find a pair $(h(\epsilon),N_{\eta})$ satisfies Condition \[A:h\_functionUniform\] in order to prove a uniform in time $t\in[0,T]$, ergodic theorem. \[L:ErgodicTheorem1u\] Consider the setup and notations of Lemma \[L:ErgodicTheorem0\]. Fix $\eta>0$. Then there exists a set $N_{\eta}$ such that $\pi(N_{\eta})\geq 1-\eta$ and $h(\epsilon)\in \mathcal{H}_{1}^{N_{\eta}} $ such that $$\lim_{\epsilon\downarrow0}\sup_{\gamma\in N_{\eta}}\sup_{t\in[0,T]}\mathbb{E}\left\vert \frac{1}{h(\epsilon)}\int_{t}^{t+h(\epsilon)}\Psi(Y_{s}^{\epsilon,y_{0} ,\gamma},\gamma)ds-\bar{\Psi}\right\vert =0.\label{Eq:ErgodicTheoremUniform}$$ \[Proof of Lemma \[L:ErgodicTheorem1u\]\] We start with the following decomposition $$\begin{aligned} & \mathbb{E}\left| \frac{1}{h(\epsilon)}\int_{t}^{t+h(\epsilon )}\Psi(Y_{s}^{\epsilon,y_{0},\gamma},\gamma)ds-\bar{\Psi}\right| = \mathbb{E}\left| \frac{\delta^{2}/\epsilon}{h(\epsilon)}\int _{\frac{t}{\delta^{2}/\epsilon}}^{\frac{t+h(\epsilon)}{\delta^{2}/\epsilon}}\Psi(\hat{Y}_{s}^{y_{0},\gamma},\gamma)ds-\bar{\Psi}\right| \nonumber\\ & \quad=\mathbb{E}\left| \frac{\delta^{2}/\epsilon}{h(\epsilon )}\int_{0}^{\frac{t+h(\epsilon)}{\delta^{2}/\epsilon}}\Psi(\hat{Y}_{s}^{y_{0},\gamma},\gamma)ds-\frac{\delta^{2}/\epsilon}{h(\epsilon)}\int^{\frac{t}{\delta^{2}/\epsilon}}_{0}\Psi(\hat{Y}_{s}^{y_{0},\gamma },\gamma)ds-\bar{\Psi}\right| \nonumber\\ & \quad=\mathbb{E}\left| \frac{t+h(\epsilon)}{h(\epsilon)} \left( \frac{\delta^{2}/\epsilon}{t+h(\epsilon)}\int_{0}^{\frac{t+h(\epsilon)}{\delta^{2}/\epsilon}}\Psi(\hat{Y}_{s}^{y_{0},\gamma},\gamma )ds-\bar{\Psi}\right) -\frac{t}{h(\epsilon)}\left( \frac{\delta^{2}/\epsilon}{t}\int^{\frac{t}{\delta^{2}/\epsilon}}_{0}\Psi(\hat{Y}_{s}^{y_{0},\gamma},\gamma)ds-\bar{\Psi}\right) \right| \nonumber\\ & \quad\leq\frac{T+h(\epsilon)}{h(\epsilon)} \mathbb{E}\left| \frac{\delta^{2}/\epsilon}{t+h(\epsilon)}\int_{0}^{\frac{t+h(\epsilon)}{\delta^{2}/\epsilon}}\Psi(\hat{Y}_{s}^{y_{0},\gamma},\gamma )ds-\bar{\Psi}\right| + \frac{1}{h(\epsilon)}\mathbb{E}\left| \delta^{2}/\epsilon\int^{\frac{t}{\delta^{2}/\epsilon}}_{0}\Psi(\hat{Y}_{s}^{y_{0},\gamma},\gamma)ds-t \bar{\Psi}\right| \nonumber\\ & \quad\leq\frac{T+h(\epsilon)}{h(\epsilon)} \sup_{r>\frac{t+h(\epsilon )}{\delta^{2}/\epsilon}}\mathbb{E}\left| \frac{1}{r}\int_{0}^{r}\Psi(\hat{Y}_{s}^{y_{0},\gamma},\gamma)ds-\bar{\Psi}\right| + \frac {1}{h(\epsilon)}\mathbb{E}\left| \delta^{2}/\epsilon\int^{\frac {t}{\delta^{2}/\epsilon}}_{0}\Psi(\hat{Y}_{s}^{y_{0},\gamma},\gamma)ds-t \bar{\Psi}\right| \nonumber\\ & \quad\leq\frac{T+1}{h(\epsilon)} \theta^{\gamma}\left( \frac{t+h(\epsilon )}{\delta^{2}/\epsilon}\right) + \frac{1}{h(\epsilon)}\mathbb{E}^{\gamma }\left| \left( \delta^{2}/\epsilon\right) \int^{\frac{t}{\delta ^{2}/\epsilon}}_{0}\Psi(\hat{Y}_{s}^{y_{0},\gamma},\gamma)ds-t \bar{\Psi}\right|\nonumber\end{aligned}$$ by choosing $h(\epsilon)<1$ and defining $$\theta^{\gamma}(u)=\sup_{r>u}\mathbb{E}\left| \frac{1}{r}\int_{0}^{r}\Psi(\hat{Y}_{s}^{y_{0},\gamma},\gamma)ds-\bar{\Psi}\right| .$$ Thus, we have proven that $$\begin{aligned} & \sup_{t\in[0,T]}\mathbb{E}\left| \frac{1}{h(\epsilon)}\int _{t}^{t+h(\epsilon)}\Psi(Y_{s}^{\epsilon,y_{0},\gamma},\gamma )ds-\bar{\Psi}\right| \leq\nonumber\\ & \qquad\leq\frac{T+1}{h(\epsilon)} \sup_{t\in[0,T]}\theta^{\gamma}\left( \frac{t+h(\epsilon)}{\delta^{2}/\epsilon}\right) +\frac{1}{h(\epsilon)}\sup_{t\in[0,T]}\mathbb{E}\left| \left( \delta^{2}/\epsilon\right) \int^{\frac{t}{\delta^{2}/\epsilon}}_{0}\Psi(\hat{Y}_{s}^{y_{0},\gamma },\gamma)ds-t \bar{\Psi}\right| \label{Eq:BoundForUniformStatement}$$ Let us first treat the second term on the right hand side of (\[Eq:BoundForUniformStatement\]). By the ergodic theorem, Lemma \[L:ErgodicTheorem0\], and Egoroff’s theorem we know that there exists a set $N_{\eta}$ with $\pi(N_{\eta})\geq 1-\eta$ such that $$\lim_{\epsilon\downarrow0}\sup_{\gamma\in N_{\eta}}\sup_{t\in[0,T]}\mathbb{E}\left| \left( \delta^{2}/\epsilon\right) \int^{\frac{t}{\delta^{2}/\epsilon}}_{0}\Psi (\hat{Y}_{s}^{y_{0},\gamma},\gamma)ds-t \bar{\Psi}\right| =0$$ So, if we choose $h(\epsilon)\downarrow0$ such that $$\lim_{\epsilon\downarrow0}\frac{1}{h(\epsilon)}\sup_{\gamma\in N_{\eta}}\sup_{t\in[0,T]}\mathbb{E}^{\gamma}\left| \left( \delta^{2}/\epsilon\right) \int^{\frac{t}{\delta ^{2}/\epsilon}}_{0}\Psi(\hat{Y}_{s}^{y_{0},\gamma},\gamma)ds-t \bar{\Psi}\right| =0$$ we have that the second term on the right hand side of (\[Eq:BoundForUniformStatement\]) goes to zero. Next, we treat the first term on the right hand side of (\[Eq:BoundForUniformStatement\]). Since, the function $\theta^{\gamma}(u)$ is decreasing, we get that $$\theta^{\gamma}\left( \frac{t+h(\epsilon)}{\delta^{2}/\epsilon}\right) \leq \theta^{\gamma}\left( \frac{h(\epsilon)}{\delta^{2}/\epsilon}\right)$$ Thus, we have obtained that for every $\gamma\in \Gamma$ $$\sup_{t\in[0,T]}\theta^{\gamma}\left( \frac{t+h(\epsilon)}{\delta^{2}/\epsilon }\right) \leq\sup_{t\in[0,T]}\theta^{\gamma}\left( \frac{h(\epsilon)}{\delta ^{2}/\epsilon}\right) =\theta^{\gamma}\left( \frac{h(\epsilon)}{\delta ^{2}/\epsilon}\right)$$ Notice that because $h(\epsilon)$ is chosen such that $\frac{\delta ^{2}/\epsilon}{h(\epsilon)}\downarrow0$, Lemma \[L:ErgodicTheorem0\] and Egoroff’s theorem, imply that $$\lim_{\epsilon\downarrow0}\sup_{\gamma\in N_{\eta}}\theta^{\gamma}\left( \frac{h(\epsilon)}{\delta ^{2}/\epsilon}\right) =0$$ Therefore, the first term on the right hand side of (\[Eq:BoundForUniformStatement\]) goes to zero, if we can choose $h(\epsilon)$, such that $\sup_{\gamma\in N_{\eta}}\theta^{\gamma}\left( \frac{h(\epsilon)}{\delta ^{2}/\epsilon}\right) /h(\epsilon)\downarrow0$. This is a little bit tricky here because the argument of $\theta$ depends on $h(\epsilon)$. However, this can be done as follows. Fix $\beta\in(0,1)$ (e.g., $\beta=1/2$) and choose $h(\epsilon )\geq\left( \delta^{2}/\epsilon\right) ^{1-\beta}$. Then, the monotonicity of $f$, implies that $$\theta^{\gamma}\left( \frac{h(\epsilon)}{\delta^{2}/\epsilon}\right) \leq \theta^{\gamma}\left( \frac{1}{\left( \delta^{2}/\epsilon\right) ^{\beta}}\right) \downarrow0$$ This proves that we can choose $h(\epsilon)$ such that the first term of the right hand of (\[Eq:BoundForUniformStatement\]) goes to zero. The claim follows, by noticing that the previous computations imply that we can choose $h(\epsilon)$ that may go to zero, but slowly enough, such that both the first and the second term on the right hand side of (\[Eq:BoundForUniformStatement\]) go to zero. Ergodic theorems with perturbation by small drift-Uncontrolled case ------------------------------------------------------------------- \[L:ErgodicTheorem2\] Consider the process $Y_{t}^{\epsilon,y_{0},\gamma}$ satisfying the SDE $$Y_{t}^{\epsilon,x,y_{0},\gamma}=y_{0}+\frac{\epsilon}{\delta^{2}}\int_{0}^{t}f(Y_{s}^{\epsilon,x,y_{0},\gamma},\gamma)ds+\frac{1}{\delta}\int_{0}^{t}g(s,x,Y_{s}^{\epsilon,x,y_{0},\gamma},\gamma)ds+\frac{\sqrt{\epsilon}}{\delta}\int_{0}^{t}\kappa(Y_{s}^{\epsilon,x,y_{0},\gamma},\gamma)dW_{s}$$ Let us consider a function $\tilde{\Psi}:[0,T]\times\mathbb{R}^{m}\times \Gamma$ such that $\tilde{\Psi}(t,x,\cdot)\in L^{2}(\Gamma)\cap L^{1}(\pi )$ and define $\Psi(t,x,y,\gamma)=\tilde{\Psi }(t,x,\tau_{y}\gamma)$. We assume that the function $\Psi:[0,T]\times \mathbb{R}^{m}\times\mathbb{R}^{d-m}\times\Gamma\mapsto\mathbb{R}$ is measurable, piecewise constant in $t$ and uniformly continuous in $x$ with respect to $(t,y)$. Denote $\bar{\Psi}(t,x)\doteq\int_{\Gamma}\tilde{\Psi}(t,x,\gamma)\pi (d\gamma)$ for all $(t,x)\in[0,T]\times\mathbb{R}^{m}$. Fix $\eta>0$. Then there exists a set $N_{\eta}$ such that $\pi(N_{\eta})\geq 1-\eta$ and $h(\epsilon)\in \mathcal{H}_{1}^{N_{\eta}} $ such that $$\lim_{\epsilon\downarrow0}\sup_{\gamma\in N_{\eta}}\sup_{0\leq t\leq T}\mathbb{E}\left\vert \frac{1}{h(\epsilon)}\int_{t}^{t+h(\epsilon)}\Psi(s,x,Y_{s}^{\epsilon,x,y_{0},\gamma},\gamma)ds-\bar{\Psi}(t,x)\right\vert =0$$ locally uniformly with respect to the parameter $x\in\mathbb{R}^{m}$. \[Proof of Lemma \[L:ErgodicTheorem2\]\]Let us set $\hat{Y}_{t}^{\epsilon, x,y_{0},\gamma}=Y_{\delta^{2}t/\epsilon}^{\epsilon,x,y_{0},\gamma}$. Notice that $\hat{Y}_{t}^{\epsilon,x,y_{0},\gamma}$ satisfies $$\hat{Y}_{t}^{\epsilon,x,y_{0},\gamma}=y_{0}+\int_{0}^{t}f(\hat{Y}_{s}^{\epsilon,x,y_{0},\gamma},\gamma)ds+\frac{\delta}{\epsilon}\int_{0}^{t}g\left( \frac{\delta^{2}}{\epsilon}s,x,\hat{Y}_{s}^{\epsilon, x,y_{0},\gamma},\gamma\right) ds+\int_{0}^{t}\kappa(\hat{Y}_{s}^{\epsilon,x,y_{0},\gamma},\gamma)dW_{s}$$ Slightly abusing notation, we denote by $Y_{t}^{\epsilon,y_{0},\gamma}$ and $\hat{Y}^{y_{0},\gamma}_{t}$ the processes corresponding to $Y_{t}^{\epsilon,x,y_{0},\gamma}$ and $\hat{Y}^{\epsilon,x,y_{0},\gamma}_{t}$ with $c(t,x,y)=0$. Lemma \[L:ErgodicTheorem1u\] guarantees that the statement of the Lemma is true for $Y_{t}^{\epsilon,y_{0},\gamma}$, namely that there exists a set $N_{\eta}$ such that $\pi(N_{\eta})\geq 1-\eta$ and $h(\epsilon)\in \mathcal{H}_{1}^{N_{\eta}} $ such that $$\lim_{\epsilon\downarrow0}\sup_{\gamma\in N_{\eta}}\sup_{t\in[0,T]}\mathbb{E}\left\vert \frac{1}{h(\epsilon)}\int_{t}^{t+h(\epsilon)}\Psi(s,x,Y_{s}^{\epsilon,y_{0},\gamma},\gamma)ds-\bar{\Psi}(t,x)\right\vert =0. \label{Eq:ErgodicTheorem1}$$ The fact that the convergence is also locally uniform with respect to the parameter $x\in\mathbb{R}^{m}$ follows by the uniform continuity of $\Psi$ in $x$. This implies that in Lemma \[L:ErgodicTheorem1u\], we can choose the sequence $h(\epsilon)$ so that the convergence holds uniformly with respect to $x$ in each bounded region, see for example Theorem II.3.11 in [@Skorokhod1987]. To translate this statement to what we need we use Girsanov’s theorem on the absolutely continuous change of measures on the space of trajectories in $C([0,T];\mathbb{R}^{d-m})$. Let $$\phi(s,x,y,\gamma)=-\kappa^{-1}(y,\gamma)g(s,x,y,\gamma)$$ and define the quantity $$M^{\epsilon,\gamma}_{T}=e^{\frac{\delta}{\epsilon}\frac{1}{\sqrt{2}} \int _{0}^{T}\phi(\delta^{2}s/\epsilon,x,\hat{Y}^{y_{0},\gamma}_{s},\gamma)dW_{s}-\frac{1}{2}\left( \frac{\delta}{\epsilon}\right) ^{2}\frac {1}{2}\int_{0}^{T}\left\Vert \phi(\delta^{2}s/\epsilon,x,\hat{Y}^{y_{0},\gamma}_{s},\gamma)\right\Vert ^{2}ds}$$ Then, by the aforementioned Girsanov’s theorem, for each$\gamma\in\Gamma$, $M^{\epsilon,\gamma}_{T}$ is a $\mathbb{P}^{\gamma}$ martingale. Therefore, we obtain $$\begin{aligned} \mathbb{E}\frac{1}{h(\epsilon)}\int_{t}^{t+h(\epsilon)}\Psi(s,x,Y^{\epsilon,x,y_{0},\gamma}_{s},\gamma)ds & = \mathbb{E}\frac{\delta^{2}/\epsilon}{h(\epsilon)}\int_{\frac{t}{\delta ^{2}/\epsilon}}^{\frac{t+h(\epsilon)}{\delta^{2}/\epsilon}}\Psi(s,x,\hat {Y}^{\epsilon,x,y_{0},\gamma}_{s},\gamma)ds\nonumber\\ & =\mathbb{E}\left[ \left( \frac{\delta^{2}/\epsilon}{h(\epsilon )}\int_{\frac{t}{\delta^{2}/\epsilon}}^{\frac{t+h(\epsilon)}{\delta ^{2}/\epsilon}}\Psi(s,x,\hat{Y}^{y_{0},\gamma}_{s},\gamma)ds\right) M^{\epsilon,\gamma}_{T}\right]\nonumber\end{aligned}$$ Next, we prove that, for every $\gamma\in\Gamma$, $M^{\epsilon,\gamma}_{T}$ converges to $1$ in probability as $\epsilon\downarrow 0$. For this purpose, let us write $M^{\epsilon,\gamma}_{T}=e^{\mathcal{E}^{\epsilon,\gamma}_{T}}$, where $$\mathcal{E}^{\epsilon,\gamma}_{T}=\frac{\delta}{\epsilon}\frac{1}{\sqrt{2}} \int_{0}^{T}\phi(\delta^{2}s/\epsilon,x,\hat{Y}^{y_{0},\gamma}_{s},\gamma)dW_{s}-\frac{1}{2}\left( \frac{\delta}{\epsilon}\right) ^{2}\frac{1}{2}\int_{0}^{T}\left\Vert \phi(\delta^{2}s/\epsilon,x,\hat{Y}^{y_{0},\gamma}_{s},\gamma)\right\Vert ^{2}ds$$ Notice that $$\mathcal{E}^{\epsilon,\gamma}_{T}=N^{\epsilon,\gamma}_{T}-\frac{1}{2}\left< N^{\epsilon,\gamma}\right> _{T}$$ where $$\begin{aligned} N^{\epsilon,\gamma}_{T}=\frac{\delta}{\epsilon}\frac{1}{\sqrt{2}} \int_{0}^{T}\phi(\delta^{2}s/\epsilon,x,\hat{Y}^{y_{0},\gamma}_{s},\gamma)dW_{s}\nonumber\end{aligned}$$ Since, $\phi$ is by assumption bounded, we obtain that $N^{\epsilon,\gamma }_{T}$ is a continuous martingale and $\left< N^{\epsilon,\gamma}\right> _{T}$ is its quadratic variation. Boundedness of $\phi$ and the assumption $\delta/\epsilon\downarrow0$ as $\epsilon\downarrow0$, implies that $$\begin{aligned} \lim_{\epsilon\downarrow0}\sup_{\gamma\in\Gamma}\mathbb{E}\left< N^{\epsilon,\gamma }\right> _{T} & =\lim_{\epsilon\downarrow0}\sup_{\gamma\in\Gamma}\frac{1}{2}\left( \frac{\delta }{\epsilon}\right) ^{2}\mathbb{E}\int_{0}^{T}\left\Vert \phi(\delta ^{2}s/\epsilon,x,\hat{Y}^{y_{0},\gamma}_{s},\gamma)\right\Vert ^{2}ds\nonumber\\ & =0. \label{Eq:QuadraticVariationOfExponent}$$ Hence, uniformly in $\gamma\in\Gamma$, $\left< N^{\epsilon,\gamma}\right> _{T}$ converges to $0$ in probability and by Problem 1.9.2 in [@LipsterShirayev1989], the same convergence holds for the martingale $N^{\epsilon,\gamma}_{T}$ as well. Thus, we have obtained that uniformly in $\gamma\in\Gamma$ $$M^{\epsilon,\gamma}_{T}=e^{\mathcal{E}^{\epsilon,\gamma}_{t}}\text{ converges to }1\text{ in probability, as }\epsilon \downarrow0. \label{Eq:ExponentialmartingaleConvergence}$$ Moreover, (\[Eq:ExponentialmartingaleConvergence\]) together with Scheffé’s theorem (Theorem 16.12 in [@Billingsley]) imply that $$\sup_{\gamma\in\Gamma}\mathbb{E}\left| M^{\epsilon,\gamma}_{T}-1\right| \rightarrow 0\text{, as }\epsilon\downarrow0. \label{Eq:ExponentialmartingaleConvergenceL1}$$ In fact, boundedness of $\phi$ implies that for every $\epsilon\in(0,1)$ and $\gamma\in\Gamma$, $M^{\epsilon,\gamma}_{T}$ is a square integrable martingale. The latter statement and convergence in probability (\[Eq:ExponentialmartingaleConvergence\]), imply that $$\sup_{\gamma\in\Gamma}\mathbb{E}\left| M^{\epsilon,\gamma}_{T}-1\right| ^{2}\rightarrow0\text{, as }\epsilon\downarrow0. \label{Eq:ExponentialmartingaleConvergenceL2}$$ Now that (\[Eq:ExponentialmartingaleConvergenceL2\]) has been established, we continue with the proof of the lemma. Choose $h(\epsilon)$, such that (\[Eq:ErgodicTheorem1\]) holds, we obtain $$\begin{aligned} & \mathbb{E}\left| \frac{1}{h(\epsilon)}\int_{t}^{t+h(\epsilon )}\Psi(s,x,Y_{s}^{\epsilon,x,y_{0},\gamma},\gamma)ds-\bar{\Psi }(t,x)\right| \nonumber\\ & \qquad= \mathbb{E}\left| \left( \frac{\delta^{2}/\epsilon }{h(\epsilon)}\int_{\frac{t}{\delta^{2}/\epsilon}}^{\frac{t+h(\epsilon )}{\delta^{2}/\epsilon}}\Psi(s,x,\hat{Y}^{y_{0},\gamma}_{s},\gamma)ds\right) M^{\epsilon,\gamma}_{T}-\bar{\Psi}(t,x)\right| \nonumber\\ & \qquad\leq\mathbb{E}\left| \frac{\delta^{2}/\epsilon}{h(\epsilon)}\int_{\frac{t}{\delta^{2}/\epsilon}}^{\frac{t+h(\epsilon)}{\delta^{2}/\epsilon}}\Psi(s,x,\hat{Y}^{y_{0},\gamma}_{s},\gamma)ds-\bar{\Psi}(t,x)\right| \nonumber\\ & \qquad\quad+\mathbb{E}\left| \left( \frac{\delta^{2}/\epsilon }{h(\epsilon)}\int_{\frac{t}{\delta^{2}/\epsilon}}^{\frac{t+h(\epsilon )}{\delta^{2}/\epsilon}}\Psi(s,x,\hat{Y}^{y_{0},\gamma}_{s},\gamma)ds\right) (M^{\epsilon,\gamma}_{T}-1)\right|\nonumber\end{aligned}$$ Clearly, the first term converges to zero by (\[Eq:ErgodicTheorem1\]). The second term also converges to zero by Hölder’s inequality, Lemma \[L:ErgodicTheorem1u\] applied to $\Psi^{2}$ and (\[Eq:ExponentialmartingaleConvergenceL2\]). The claim that the convergence is locally uniformly with respect to the parameter $x\in\mathbb{R}^{m}$ follows by the fact that this is true for (\[Eq:ErgodicTheorem1\]). Ergodic theorems with perturbation by small drift-Controlled case ----------------------------------------------------------------- \[L:ErgodicTheorem3\] Fix $T<\infty$ and consider $\mathcal{A}$ to be the set of progressively measurable controls such that $$\int_{0}^{T}\left\Vert u^{\epsilon}(s)\right\Vert^{2}ds< N , \label{Eq:UniformlySquareIntegrableControlsAdditional}$$ where the constant $N$ does not depend on $\epsilon,\delta, T$ or $\gamma$ and additionally such that for $\delta/\epsilon\ll 1$ $$\frac{1}{\epsilon}\mathbb{E}\int_{0}^{\frac{\delta^{2}T}{\epsilon}}\left\Vert u^{\epsilon}(s)\right\Vert^{2}ds\leq C \delta/\epsilon, \label{Eq:UniformlySquareIntegrableControlsAdditionalb}$$ where the constant $C$ depends on $T$, but not on $\epsilon,\delta$ or $\gamma$. Consider the process $\bar{Y}_{t}^{\epsilon,x,y_{0},\gamma}$ satisfying the SDE $$\begin{aligned} \bar{Y}_{t}^{\epsilon,x,y_{0},\gamma} & =y_{0}+\frac{\epsilon}{\delta^{2}}\int_{0}^{t}f(\bar{Y}_{s}^{\epsilon,x,y_{0},\gamma},\gamma)ds+\frac{1}{\delta}\int_{0}^{t}\left[ g(s,x,\bar{Y}_{s}^{\epsilon,x,y_{0},\gamma},\gamma)+\kappa(\bar{Y}_{s}^{\epsilon,x,y_{0},\gamma},\gamma) u^{\epsilon}(s)\right] ds\nonumber\\ & \qquad+\frac{\sqrt{\epsilon}}{\delta}\int_{0}^{t}\kappa(\bar{Y}_{s}^{\epsilon,x,y_{0},\gamma},\gamma)dW_{s}\nonumber\end{aligned}$$ Let us consider a function $\tilde{\Psi}:[0,T]\times\mathbb{R}^{m}\times \Gamma$ such that $\tilde{\Psi}(t,x,\cdot)\in L^{2}(\Gamma)\cap L^{1}(\pi)$ and define $\Psi(t,x,y,\gamma)=\tilde{\Psi }(t,x,\tau_{y}\gamma)$. We assume that the function $\Psi:[0,T]\times \mathbb{R}^{m}\times\mathbb{R}^{d-m}\times\Gamma\mapsto\mathbb{R}$ is measurable, piecewise constant in $t$ and uniformly continuous in $x$ with respect to $(t,y)$. Denote $\bar{\Psi}(t,x)\doteq\int_{\Gamma}\tilde{\Psi}(t,x,\gamma)\pi (d\gamma)$ for all $(t,x)\in[0,T]\times\mathbb{R}^{m}$. Fix $\eta>0$. Then there exists a set $N_{\eta}$ such that $\pi(N_{\eta})\geq 1-\eta$ and $h(\epsilon)\in \mathcal{H}_{1}^{N_{\eta}} $ such that $$\lim_{\epsilon\downarrow0}\sup_{\gamma\in N_{\eta}}\sup_{0\leq t\leq T}\mathbb{E}\left\vert \frac{1}{h(\epsilon)}\int_{t}^{t+h(\epsilon)}\Psi(s,x,\bar{Y}_{s}^{\epsilon,x,y_{0},\gamma},\gamma)ds-\bar{\Psi}(t,x)\right\vert =0$$ locally uniformly with respect to the parameter $x\in\mathbb{R}^{m}$. \[Proof of Lemma \[L:ErgodicTheorem3\]\]Let us set $\hat{\bar{Y}}_{t}^{\epsilon,x,y_{0},\gamma}=\bar{Y}_{\delta^{2}t/\epsilon}^{\epsilon,x,y_{0},\gamma}$. Notice that $\hat{\bar{Y}}_{t}^{\epsilon,x,y_{0},\gamma}$ satisfies $$\begin{aligned} \hat{\bar{Y}}_{t}^{\epsilon,x,y_{0},\gamma} & =y_{0}+\int_{0}^{t}f(\hat {\bar{Y}}_{s}^{\epsilon,x,y_{0},\gamma},\gamma)ds+\frac{\delta}{\epsilon}\int_{0}^{t}\left[ g\left( \frac{\delta^{2}}{\epsilon}s,x,\hat{\bar{Y}}_{s}^{\epsilon,x,y_{0},\gamma},\gamma\right) +\kappa(\hat{\bar{Y}}_{s}^{\epsilon,x,y_{0},\gamma},\gamma) u^{\epsilon}\left(\delta^{2}s/\epsilon \right)\right] ds\nonumber\\ & \qquad+\int_{0}^{t}\kappa(\hat{\bar{Y}}_{s}^{x,y_{0},\gamma ,\epsilon},\gamma)dW_{s}\nonumber\end{aligned}$$ Essentially, based on the condition of the allowable controls (\[Eq:UniformlySquareIntegrableControlsAdditional\]), the arguments of the uncontrolled case, Lemma \[L:ErgodicTheorem2\], go through verbatim. The only place that needs some discussion is in regards to the proof of the statement corresponding to (\[Eq:QuadraticVariationOfExponent\]). Let us show now how this term can be treated. In the controlled case we have that $$\phi(s,x,y,\gamma)=-\kappa^{-1}(y,\gamma)g(s,x,y,\gamma)-u^{\epsilon}(s)$$ and we want to prove that for every $\gamma\in \Gamma$ $$\begin{aligned} \lim_{\epsilon\downarrow0}\mathbb{E}\left< N^{\epsilon,\gamma }\right> _{T} & =\lim_{\epsilon\downarrow0}\frac{1}{2}\left( \frac{\delta }{\epsilon}\right) ^{2}\mathbb{E}\int_{0}^{T}\left\Vert \phi(\delta ^{2}s/\epsilon,x,\hat{Y}^{y_{0},\gamma}_{s},\gamma)\right\Vert ^{2}ds=0. \label{Eq:QuadraticVariationOfExponentControlled}$$ It is clear that $$\begin{aligned} \mathbb{E}\left< N^{\epsilon,\gamma}\right> _{T} & =\frac{1}{2}\left( \frac{\delta}{\epsilon}\right) ^{2}\mathbb{E}\int_{0}^{T}\left\Vert \phi(\delta^{2}s/\epsilon,x,\hat{Y}^{y_{0}}_{s},\gamma)\right\Vert ^{2}ds\nonumber\\ & \leq \left( \frac{\delta}{\epsilon}\right) ^{2}\mathbb{E}\int _{0}^{T}\left\Vert \kappa^{-1}(\hat{Y}^{y_{0}}_{s},\gamma)g(\delta ^{2}s/\epsilon,x,\hat{Y}^{y_{0}}_{s},\gamma)\right\Vert ^{2}ds+ \left( \frac{\delta}{\epsilon}\right) ^{2}\mathbb{E}\int_{0}^{T}\left\Vert u^{\epsilon}\left(\delta^{2}s/\epsilon\right)\right\Vert ^{2}ds\nonumber\end{aligned}$$ The first term of the right hand side of the last display goes to zero by the boundedness of $\left\Vert \kappa^{-1}g\right\Vert ^{2}$ (as in Lemma \[L:ErgodicTheorem2\]). So we only need to consider the second term. Here we use Condition \[Eq:UniformlySquareIntegrableControlsAdditionalb\]. In particular, we notice that Condition \[Eq:UniformlySquareIntegrableControlsAdditionalb\] gives $$\begin{aligned} \lim_{\epsilon\downarrow 0}\left( \frac{\delta}{\epsilon}\right) ^{2}\mathbb{E}\int_{0}^{T}\left\Vert u^{\epsilon }\left(\delta^{2}s/\epsilon\right)\right\Vert ^{2}ds & =\lim_{\epsilon\downarrow 0}\frac{1}{\epsilon}\mathbb{E}\int _{0}^{\delta^{2}T/\epsilon}\left\Vert u^{\epsilon}(s)\right\Vert ^{2}ds=0\nonumber\end{aligned}$$ uniformly in $\gamma\in\Gamma$. Thus we have completed the proof of (\[Eq:QuadraticVariationOfExponentControlled\]). This concludes the proof of the lemma. Ergodic theorem with explicit dependence on the slow process. -------------------------------------------------------------- In this subsection we consider the pair $(\bar{X}_{s}^{\epsilon,\gamma}, \bar{Y}_{s}^{\epsilon,\gamma})$ satisfying (\[Eq:Main2\]) and the purpose is to prove Lemma \[L:ErgodicTheorem4\]. \[L:ErgodicTheorem4\] Consider the set-up, assumptions and notations of Lemma \[L:ErgodicTheorem3\]. Fix $\eta>0$. Then there exists a set $N_{\eta}$ such that $\pi(N_{\eta})\geq 1-\eta$ and $h(\epsilon)\in \mathcal{H}_{1}^{N_{\eta}} $ such that $$\lim_{\epsilon\downarrow0}\sup_{\gamma\in N_{\eta}}\sup_{0\leq t\leq T}\mathbb{E}\left\vert \frac{1}{h(\epsilon)}\int_{t}^{t+h(\epsilon)}\Psi\left(s,\bar{X}_{s}^{\epsilon,\gamma}, \bar{Y}_{s}^{\epsilon,\gamma},\gamma\right)ds-\bar{\Psi}(t,\bar{X}_{t}^{\epsilon,\gamma})\right\vert =0.$$ \[Sketch of proof of Lemma \[L:ErgodicTheorem4\]\] Due to Lemma \[L:ErgodicTheorem3\], the statement follows by using the standard argument of freezing the slow component, see for example Chapter 7.9 of [@FW1] or [@PardouxVeretennikov1]. Details are omitted. Acknowledgements ================ The author would like to thank Paul Dupuis for discussions on aspects of this work. The author was partially supported by the National Science Foundation (DMS 1312124). 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--- abstract: | The spectrum of $^9$He was studied by means of the $^8$He($d$,$p$)$^9$He reaction at a lab energy of 25 MeV/n and small center of mass (c.m.) angles. Energy and angular correlations were obtained for the $^9$He decay products by complete kinematical reconstruction. The data do not show narrow states at $\sim $1.3 and $\sim $2.4 MeV reported before for $^9$He. The lowest resonant state of $^9$He is found at about 2 MeV with a width of $\sim $2 MeV and is identified as $1/2^-$. The observed angular correlation pattern is uniquely explained by the interference of the $1/2^-$ resonance with a virtual state $1/2^+$ (limit on the scattering length is obtained as $a > -20$ fm), and with the $5/2^+$ resonance at energy $\geq 4.2$ MeV. author: - 'M.S. Golovkov' - 'L.V. Grigorenko' - 'A.S. Fomichev' - 'A.V. Gorshkov' - 'V.A. Gorshkov' - 'S.A. Krupko' - 'Yu.Ts. Oganessian' - 'A.M. Rodin' - 'S.I. Sidorchuk' - 'R.S. Slepnev' - 'S.V. Stepantsov' - 'G.M. Ter-Akopian' - 'R. Wolski' - 'A.A. Korsheninnikov' - 'E.Yu. Nikolskii' - 'V.A. Kuzmin' - 'B.G. Novatskii' - 'D.N. Stepanov' - 'S. Fortier' - 'P. Roussel-Chomaz' - 'W. Mittig' date: '. [File: he9-11.tex ]{}' title: 'New insight into the low-energy $^9$He spectrum' --- *Introduction.* — Since the first observation of $^9$He in the experiment [@set87] it was studied in relatively small number of works compared to the neighbouring exotic neutron dripline nuclei. This can be connected, on one hand, to the facts that technical difficulty of the precision measurements rapidly grows with move away from the stability line. On the other hand, already in the first experiment (pion double charge exchange on the $^9$Be nucleus [@set87]) several narrow resonances were observed above the $^8$He+$n$ threshold. This observation was confirmed in Ref. [@boh99], where the $^9$Be($^{14}$C,$^{14}$O)$^9$He reaction was used, and now the experimental situation with the low-energy spectrum of $^9$He is considered to be well established. A new rise of interest to $^9$He was connected with the question of a possible $2s$ state location in the framework of the shell inversion problem in nuclei with large neutron excess. The recent experiment [@che01] was focused on the search for the virtual state in $^9$He. An upper limit on the scattering length $a<-10$ fm was established in this work. The properties of states in $^{9}$He were inferred in [@gol03] basing on studies of isobaric partners in $^{9}$Li. The available results are summarized in Table \[tab:exp\]. Interpretation of the $^9$He spectrum as provided in [@set87; @boh99] faces certain difficulties which were not unnoticed (e.g. Ref. [@bar04]). Indeed, the ground $1/2^-$ state is expected to be single particle state with width estimated as $0.8-1.3$ MeV at $E = 1.27$ MeV for typical channel radii $3 - 6$ fm. This requires spectroscopic factor $S\sim 0.1$ which contradicts single particle character of the state. F. Barker in Ref. [@bar04] concludes on this point that “some configuration mixing in either the $^{9}$He($1/2^-$) or $^{8}$He($0^+$) state or both is possible, but is unlikely to be large enough to reduce the calculated width to the experimental value”. The next, presumably $3/2^-$ state, should be a complicated particle-hole excitation as $p_{3/2}$ subshell is occupied. However, a much larger spectroscopic factor $S\sim 0.3-0.4$ is required for its widths found in a range $2.0-2.6$ MeV. ![Experimental setup, angles, and momenta.[]{data-label="fig:setup"}](setup){width="44.00000%"} Having in mind the mentioned problematic issues we decided to study the $^9$He in the “classical” one-neutron transfer ($d$,$p$) reaction well populating single particle states. In contrast with the previous works complete kinematics studies were foreseen to reveal the low-energy $s$-wave mode. Following the experimental concept of [@gol04a; @gol05b], where correlation studies of $^5$H continuum were accomplished by means of the $^3$H($t$,$p$) transfer reaction, this work was performed in the so called “zero geometry”. ---------- ------------ ---------- ---------- ---------- ---------- ------------ ---------- 1/2$^+$ Ref. $a$ (fm) $E$ $\Gamma$ $E$ $\Gamma$ $E$ $\Gamma$ [@set87] 1.13(10) small 2.3 small 4.9 [@boh99] 1.27(10) 0.1(0.6) 2.42(10) 0.7(2) 4.3 small [@che01] $< \! -10$ [@gol03] 1.1 2.2 4.0 Our $> \! -20$ 2.0(0.2) 2 $\geq 4.2$ $> 1$ ---------- ------------ ---------- ---------- ---------- ---------- ------------ ---------- : Experimental positions of states in $^9$He relative to the $^8$He+$n$ threshold (energies and widths are given in MeV). \[tab:exp\] {width="86.00000%"} *Experiment.* — The experiment was done at the U-400M cyclotron of the Flerov Laboratory of Nuclear Reactions, JINR (Dubna, Russia). A 34 MeV/nucleon $^{11}$B primary beam delivered by the cyclotron hitted a 370 mg/cm$^2$ Be production target. The modified ACCULINNA fragment separator [@rod97] was used to produce a $^8$He secondary beam with a typical intensity of $2\times 10^4$ s$^{-1}$. The beam was focused on a cryogenic target [@yuk03] filled with deuterium at 1020 mPa pressure and cooled down to 25 K. The 4 mm thick target cell was supplied with 6 $\mu$m stainless steel windows, 30 mm in diameter. Experimental setup and kinematical diagram for the $^{2}$H($^8$He,$p$)$^9$He reaction are shown in Fig.\[fig:setup\]. Slow protons escaping from the target in the backward direction hitted on an annular 300 $\mu$m silicon detector with an active area of the outer and inner diameters of 82 mm and 32 mm, respectively, and a 28 mm central hole. The detector was installed 100 mm upstream of the target. It was segmented in 16 rings on one side and 16 sectors on the other side providing a good position resolution. The detection threshold for the protons ($\sim $1.2 MeV) corresponded to a $\sim$5.5 MeV cutoff in the missing mass of $^9$He. We did not use here particle identification because, due to the kinematical constraints of the $^8$He+$^2$H collisions, only protons can be emitted in the backward direction. The main cause of the background was due to evaporation protons originating from the interaction of $^8$He beam with the material of target windows. This background was almost completely suppressed by the coincidence with $^8$He. The detection of such coincidences fixed the complete kinematics for the experiment. Energy-momentum conservation was used for cleaning the spectra. Finally the comparison with an empty target run has shown that only $\sim 2 \%$ events can be treated as a background. The $^8$He nuclei resulting from the $^9$He decay, focused in narrow angular cone relative to the beam direction, were detected by a Si-CsI telescope mounted in air just behind the exit window of the scattering chamber. The 82 mm diameter exit window was closed by a 125 $\mu$m capton foil. The Si-CsI telescope consisted of two 1 mm thick silicon detectors and 16 CsI crystals with photodiode readouts. The $6 \times 6$ cm Si detectors were segmented in 32 strips both in horizontal and vertical directions, providing position resolution and particle identification by the $\Delta E$-$E$ method (together with following thick CsI detector). Sixteen $1.5 \times 2 \times 2$ cm CsI crystals were arranged as a $4 \times 4$ wall just behind the Si detectors. A $ 2 \%$ energy resolution of the CsI detectors allowed particle identification even for $Z=1$ nuclei. The distance between the target and the telescope (50 cm) was sufficient to provide a good efficiency for the detection of the $^8$He nuclei in coincidence with protons in the whole range of accessible $^9$He energies. To eliminate signals in the proton telescope coming from the beam halo a veto detector was installed upstream the proton telescope. The energy of the $^8$He beam in the middle of the target was $\sim $25 MeV/nucleon. Energy spread of the beam, angular divergence, and position spread on the target were $8.5 \%$, $0.23^{\circ}$, and about 5.4 mm respectively. A set of beam detectors was installed upstream of the veto (not shown in the Fig. \[fig:setup\]). The beam energy was measured by two time-of-flight plastic scintillators with a 785 cm base. The overall time resolution was 0.8 ns. Beam tracking was made by two multiwire proportional chambers installed 26 and 80 cm upstream of the target. Each chamber had two perpendicular planes of wires with a 1.25 mm pitch. Energy resolution was estimated by Monte-Carlo (MC) code taking into account all experimental details. It was found to be 0.8 MeV (FWHM) for the $^9$He missing mass. {width="90.00000%"}\ {width="85.00000%"}\ {width="85.00000%"}\ {width="85.00000%"} *Qualitative considerations.* — Data obtained in the experiment are shown in Figs. \[fig:expth\] and \[fig:distrib\]. The total number of counts corresponds to cross section of $\sim$7 mb/sr. This value is consistent with the direct one-neutron transfer reaction mechanism at forward angles. The narrow states known from the literature do not show up in the data. Instead, we get two broad peaks at about 2 and 4.5 MeV. Near the threshold the $^9$He spectrum exhibits behavior (although distorted by the finite energy resolution of the experiment) which is more consistent with $s$-wave ($\sigma \sim \sqrt{E}$) rather than $p$-wave ($\sigma \sim E^{3/2}$). This is an indication for a possible virtual state in $^9$He. An important feature of the data is a prominent forward-backward asymmetry with $^8$He flying preferably in the backward direction in the $^{9}$He c.m. system. This is not feasible if the narrow (means long-living) states of the $^9$He are formed. To describe such an asymmetry the interference of opposite parity states is unavoidable. As far as asymmetry is observed even at very low energy, the $s$-$p$ interference is compulsory. Such an interference can provide only a very smooth distribution described by the first order polynomial \[Eq. (\[eq:sigma-full\]) and Fig. \[fig:distrib\], $E<2.2$ MeV\]. Since above 3 MeV the character of the distribution changes to a higher polynomial, but asymmetry does not disappear, the $p$-$d$ interference is also needed. This defines the minimal set of states as $s$, $p$, and $d$. *Theoretical model.* — In the zero geometry approach the resonant states of interest are identified by the observation of recoil particle (here proton) at zero (in reality small, $3^{\circ} \leq \theta_{p}(\text{c.m.})\leq 7^{\circ}$) angle. This means that the angular momentum transferred to the studied system should have zero projection on the axis of the momentum transfer. As a result we get a complete (strong) alignment for states with $J>1/2$ in the produced system. In the $^9$He case only the magnetic substates with $M= \pm 1/2$ should be populated for $J^{\pi}=5/2^+$, $3/2^-$, and $3/2^+$ states. This strongly reduces possible ambiguity in the analysis of correlation patterns. E.g. in the case of zero spin particles the zero geometry experiments give very clean pictures with angular distributions described by pure Legendre polynomial $|P^0_l|^2$. The situation in the case of nonzero spin particles involved is more complicated (see detailed discussion in Ref. [@gol05b]), and diverse correlation patterns are possible. We have found that the observed experimental picture (Figs. \[fig:expth\], \[fig:distrib\]) can be well explained in a simple model involving only three low-lying states: $1/2^+$, $1/2^-$, and $5/2^+$. The inelastic cross in the DWBA ansatz is written as $$\begin{aligned} \frac{d\sigma(\Omega_{^9\text{He}})}{dE \,d \Omega_{^8\text{He}}} \sim \frac{v_{f}}{v_{i}}\,\sqrt{E} \, \sum \nolimits_{MM_S} \left| \sum \nolimits_J\langle \Psi^{JMM_S}_f \left| V \right| \Psi_i \rangle \right|^2 \label{eq:sigma}\\ % = \frac{v_f}{v_i} \sum _{MM_S} \sum _{JJ'} \sum _{M'_lM_l}\! \rho_{JM}^{J'M} C^{J'M'}_{l'M'_lSM_S} C^{JM}_{lM_lSM_S} \, Y^{\ast}_{lM'_l} Y_{lM_l}\,. \nonumber\end{aligned}$$ For the density matrix the generic symmetries are $$\rho_{JM}^{J'M'}= \left(\rho^{JM}_{J'M'}\right)^* \quad ; \qquad \rho_{JM}^{JM'}= (-)^{M+M'}\rho^{J-M}_{J-M'} \; ,$$ and properties specific to coordinate choice (spirality representation) and setup (zero geometry) are $$\rho_{JM}^{J'M'} \sim \; \delta_{M,M'}(\delta_{M,1/2}+\delta_{M,-1/2}) \; .$$ For the density matrix parametrization we use the following model for the transition matrix. The wave function (WF) $\Psi_f$ is calculated in the $l$-dependent square well (with depth parameters $V_l$). The well radius is taken $r_0=3$ fm, which is consistent with typical R-matrix phenomenology $1.4 A^{1/3}$. The energy dependence of the velocities $v_i$, $v_f$ (in the incoming $^8$He-$d$ and outgoing $^9$He-$p$ channels) and WF $\Psi_i$ is neglected for our range of $^9$He energies. The term $ V \left| \Psi_i \right. \rangle $, describing the reaction mechanism, is approximated by radial $\theta$-function: $$V \left| \Psi_i \right. \rangle \rightarrow C_l \;r^{-1}\,\theta(r_0-r) \; [Y_{l}(\hat{r}) \otimes \chi_{S}]_{JM} \; ,$$ where $C_l$ is (complex) coefficient defined by the reaction mechanism. For $\bigl|\rho_{J\pm 1/2}^{J'\pm 1/2}\bigr|$ denoted as $A_{l'l}$ the cross section as a function of energy $E$ and $x=\cos(\theta_{^8\text{He}})$ is $$\begin{aligned} \frac{d\sigma(\Omega_{^9\text{He}})}{dE \;d x} \sim \frac{1}{\sqrt{E}} \left[ \rule{0pt}{12pt} 4 A_{00} + 4 A_{11} + 3(1-2x^2+5x^4) A_{22} \right. \nonumber \\ % + \left. 8 \, x \cos(\phi_{10})A_{10} + 4 \sqrt{3} \, x(5x^2 - 3) \cos(\phi_{12}) A_{12} \right] \, . \, \label{eq:sigma-full} \\ % A_{l'l}=|A_{l'}| |A_{l}|\; , \;\;A_l= C_l N_l(E)\,e^{i\delta_l(E)}\int_0^{r_0}dr j_l(q_lr)\; , \nonumber \\ % q_l=\sqrt{2M(E-V_l)}\; , \;\; \phi_{l'l}(E) = \phi^{(0)}_{l'l} + \delta_{l'}(E) - \delta_{l}(E)\, , \nonumber %\end{aligned}$$ where $N_l$ is defined by matching condition on the well boundary for internal function $j_l(q_lr)$. The three coefficients $C_l$ give rise to the two phases $\phi^{(0)}_{10}$ and $\phi^{(0)}_{12}$. Positions and widths of the states are fixed by the three parameters $V_l$. Their relative contributions to the inclusive energy spectrum (Fig. \[fig:distrib\]a–c) are fixed by the three parameters $|C_l|$. The phase $\phi^{(0)}_{10}$ is fixed by the angular distributions at low energy, where the contribution of the $d$-wave resonance is small. After that the phase $\phi^{(0)}_{12}$ was varied to fit the angular distributions at higher energies (Fig. \[fig:distrib\], $E>2.2$ MeV). So, the model does not have redundant parameters and the ambiguity of the theoretical interpretation is defined by the quality of the data. Set $|C_0|^2$ $|C_1|^2$ $V_0$ $V_1$ $V_2$ $\phi^{(0)}_{10}$ $\phi^{(0)}_{12}$ ----- ----------- ----------- ---------- --------- --------- ------------------- ------------------- 1 0.26 0.35 $-4.0$ $-20.7$ $-43.4$ $0.80 \pi$ $-0.03 \pi$ 2 0.03 0.52 $-4.0$ $-20.7$ $-43.4$ $0.85 \pi$ $-0.02 \pi$ 3 0.12 0.43 $-5.817$ $-20.7$ $-43.4$ $1.00 \pi$ $-0.03 \pi$ : Parameters of theoretical model used in the work. $V_i$ values are in MeV, weight coefficients for different states are normalized to unity $\sum |C_i|^2=1$. \[tab:th\] We have found that the weight and interaction strength for the $1/2^+$ state can be varied in a relatively broad range, still providing a good description of data. The results of MC simulations of the experiment with different $s$-wave contributions are shown in Fig. \[fig:distrib\]a–c. The parameter sets of the model are given in Table \[tab:th\]; sets 1 and 2 correspond to small scattering length ($a=-4$ fm) and different weights of $s$-wave (largest and lowest possible), set 3 has $a=-25$ fm and largest possible weight of $s$-wave. It can be seen that the agreement with the data deteriorates when the population of the $s$-wave continuum falls, say, below $15-25 \%$ of the $p$-wave. On the other hand the large negative scattering length has a drastic effect below 0.5 MeV. The energy resolution and the quality of the measured angular distributions are not sufficient to draw solid conclusions about the exact properties of the $s$-wave contribution. The situations with the large contribution of the $s$-wave cross section but with moderate scattering length (say $a > -20$ fm) seem to be more plausible. Measurements with better resolution are required to refine the properties of the $1/2^+$ continuum. Position of the $d$-wave resonance is not well defined in our analysis of data due to the efficiency fall in the high-energy side of the spectrum. This can be well seen from the comparison of theoretical inputs and MC results in Fig.\[fig:distrib\]a–c. The lower limit for the resonance energy of 4.2 MeV is in a good agreement with the value 4.0 MeV found in [@gol03]. A broader energy range measured for $^9$He is needed to resolve the $5/2^+$ state completely and to make the angular distribution analysis more restrictive. *Discussion.* — It should be noted that the interference of any other combination of $s$- $p$- $d$-wave states [*can not*]{} lead to the required forward-backward asymmetry in the whole energy range. The correlation terms \[square brackets in Eq. (\[eq:sigma-full\])\] are $$\begin{aligned} % \left[\rule{0pt}{9pt}\ldots \right] &= &2 A_{00} + 2 A_{11} + (1+3x^2) A_{22} + 4 x \cos(\phi_{10}) A_{10} \nonumber \\ % & +& 2 \sqrt{2} (3x^2 - 1) \cos(\phi_{20}) A_{20}\, , \nonumber \\ % \left[ \rule{0pt}{9pt}\ldots \right] &=& 4 A_{00} + 2 (1+3x^2) A_{11} + 3(1-2x^2+5x^4) A_{22} \,, \nonumber \\ % \left[ \rule{0pt}{9pt} \ldots \right] &= &2 A_{00} + (1+3x^2) (A_{11} + A_{22}) + 2 \, x (9x^2-5) \nonumber \\ % & \times & \cos(\phi_{12})A_{12} + 2 \sqrt{2} (3x^2 - 1) \cos(\phi_{20}) A_{20} \, , \nonumber %\end{aligned}$$ for $\{s_{1/2},p_{1/2},d_{3/2} \}$, $\{s_{1/2},p_{3/2},d_{5/2} \}$, and $\{s_{1/2},p_{3/2},$ $d_{3/2} \}$ sets of states respectively. The asymmetric term ($\sim x)$ is present here either for $s$-$p$ interference only or for $p$-$d$ only or for neither. The angular distributions in energy bins (Fig. \[fig:distrib\]) provide a good indication that a [*narrow*]{} $p_{1/2}$ state is not populated in the reaction. Fig. \[fig:qualit\] shows qualitatively what happens with angular distribution if there is a narrow resonance. The phase shift changes across the narrow resonance to a value close to $\pi$ and the character of angular distribution should change drastically within this energy range. No trend of this kind is observed in Fig. \[fig:distrib\]. Phase shift for the [*broad*]{} $1/2^-$ state changes slowly and hardly achieves $\pi /2$ in our calculations. This allows to explain the smooth behaviour of asymmetry up to 3 MeV. ![Schematic illustration of possible behavior of angular distributions (shown by inserts for angles $\theta_{^8\text{He}}$ from $0^{\circ}$ to $180^{\circ}$) due to $s_{1/2}$-$p_{12}$ interference around a narrow resonance for different phases $\phi_{10}^{(0)}$.[]{data-label="fig:qualit"}](qualit){width="45.00000%"} The existing experimental data can be regarded as not contradicting our results. In Refs. [@set87; @boh99] the narrow states were observed with low statistics (15–40 events/state). If the narrow states really exist they should be observable even at such a low counting rates. However, simulations show that if the cross section behavior is smooth, “statistically driven” narrow structures are quite probable in such a situation. A look at the data of Ref. [@gol03] also shows that states (which should have analogues in $^9$He) at 2.2 MeV and 4.0 MeV are absolutely evident in the data. However, the presence of a narrow 1.1 MeV state is more likely not to contradict the data, rather than necessarily follow from these. The idea that only the $1/2^-$ resonance state can be found in the low energy region not only looks natural, but also finds support in the recent theoretical studies. In Ref. [@vol05], which deals with the whole chain of helium isotopes in continuum shell model, the $1/2^-$ state is located at 1.6 MeV above the $^8$He+$n$ threshold and the width is $\sim 0.6$ MeV indicating the dominant single particle component in the WF. The $3/2^-$ state is predicted to be at 6.6 MeV and relatively narrow ($\sim 2.5$ MeV), what is natural for complicated particle-hole excitations. *Conclusions.* — We would like to emphasize the following results of our study. \(i) Our data show two broad overlapping peaks (at 2 and at 4.5 MeV) in the $^9$He spectrum. Statistics obtained in our experiment is about factor of 10 higher than in the previous works [@set87; @boh99]. Our resolution is sufficient to resolve narrow low-lying states. Even if a narrow $p_{1/2}$ is not resolved, the rapid change of phase around resonance energy should produce the change in the forward-backward asymmetry, which is also not seen in the data. \(ii) An essential contribution of the $s$-wave $1/2^+$ state is evident from the data. It is manifested in two ways: (a) Large forward-backward asymmetry at $E \leq 3$ MeV and (b) accumulation of counts around the threshold, which should not take place for typical cross section behavior for higher $l$-values. A limit $a > -20$ fm is obtained for the scattering length of this state. \(iii) The proposed spin assignment $\{s_{1/2},p_{1/2},d_{5/2} \}$ is unique, as no other reasonable set of low-lying states can provide the observed correlation pattern. \(iv) The experimental data are well described in a simple single-particle potential model, involving only basic theoretical assumptions about the reaction mechanism and the low-energy spectrum of $^9$He. This supports the idea that $^8$He (having closed $p_{3/2}$ subshell) presents a “good” core in the $^{9}$He structure. *Acknowledgments.* — This work was supported by the Russian Foundation for Basic Research grants 02-02-16550, 02-02-16174, 05-02-16404, and 05-02-17535 by the INTAS grantS 03-51-4496 and 03-54-6545. LVG acknowledge the financial support from the Royal Swedish Academy of Science and Russian Ministry of Industry and Science grant NS-1885.2003.2. [99]{} K. Seth, *et al.*, Phys. Rev. Lett.  **58**, 1930 (1987). H.G. Bohlen, *et al.*, Prog. Part. Nucl. Phys. **42**, 17 (1999). L. Chen, *et al.*, Phys. Lett.  **B505**, 21 (2001). G.V. Rogachev, *et al.*, Phys. Rev. C **67**, 041603R (2003). F.C. Barker, Nucl. Phys. **A741**, 42 (2004). M. Golovkov, *et al.*, Phys. Rev. Lett. **93**, 262501 (2004). M. Golovkov, *et al.*, Phys. Rev. C **72**, 064612 (2005). A.M. Rodin, *et al.*, Nucl. Instr. Meth. **A** 391, 228 (1997). A.A. Yukhimchuk, *et al.*, Nucl. Instrum. Meth.  **A** 513, 439 (2003). A. Volya and V. Zelevinsky, Phys. Rev. Lett. **94**, 052501 (2005).

--- abstract: '[ The dependence of the critical current of spin transfer torque-driven magnetization dynamics on the free-layer thickness was studied by taking into account both the finite penetration depth of the transverse spin current and spin pumping. We showed that the critical current remains finite in the zero-thickness limit of the free layer for both parallel and anti-parallel alignments. We also showed that the remaining value of the critical current of parallel to anti-parallel switching is larger than that of anti-parallel to parallel switching. ]{}' author: - 'Tomohiro Taniguchi${}^{1,2}$' - 'Hiroshi Imamura${}^{1}$[^1]' title: 'Dependence of critical current of spin transfer torque-driven magnetization dynamics on free layer thickness' --- Spin transfer torque (STT)-driven magnetization dynamics is a promising technique to operate spin-electronics devices such as a non-volatile magnetic random access memory (MRAM) and a microwave generator [@slonczewski96; @berger96]. STT is the torque due to the transfer of the transverse (perpendicular to magnetization) spin angular momentum from the conducting electrons to the magnetization of the ferromagnetic metal. One of the most important quantities of STT-driven magnetization dynamics is the critical current over which the dynamics of the magnetization is induced. The typical value of the critical current density is on the order of $10^{6}-10^{8}$ \[A/cm${}^{2}$\] [@kiselev03; @seki06; @chen06]. Control of the value of the critical current is required to reduce the energy consumption of spin-electronics devices. In Slonczewski’s theory of STT [@slonczewski96], the critical current of P-to-AP (AP-to-P) switching is expressed as [@sun00; @grollier03] $$I_{\rm c}^{{\rm P}\to{\rm AP}({\rm AP}\to{\rm P})} = \frac{2eMSd}{\hbar\gamma\eta_{\rm P(AP)}} \alpha_{0}\omega_{\rm P(AP)}\ , \label{eq:critical_current}$$ where $e$ is the absolute value of the electron charge, $\hbar$ is the Dirac constant, and $M$, $\gamma$, $S$, $d$ and $\alpha_{0}$ are the magnetization, gyromagnetic ratio, cross section area, thickness and the intrinsic Gilbert damping constant of the free layer, respectively [@chen06]. $\omega_{\rm P(AP)}$ is the angular frequency of the magnetization around the equilibrium point. The coefficient $\eta_{\rm P,AP}$ characterizes the strength of STT, and depends only on the relative angle of the magnetizations of the fixed and free layer [@slonczewski96; @sun00; @grollier03]. According to Eq. (\[eq:critical\_current\]), the critical current vanishes in the zero-thickness limit of the free layer, $d\!\to\!0$. {width="0.95\columnwidth"} \[fig:fig1\] However, recently, Chen [[ *et al.* ]{}]{}[@chen06] reported that the critical current of STT-driven magnetization dynamics of a CPP-GMR spin valve remains finite even in the zero-thickness limit of the free layer. What are missed in the above naive considerations based on Slonczewski’s theory are the effects of the finite penetration depth of the transverse spin current, $\lambda_{\rm t}$, [@zhang02; @zhang04; @taniguchi08a] and of spin pumping [@mizukami02b; @tserkovnyak02; @tserkovnyak03; @taniguchi07]. We investigated the critical current of STT-driven magnetization switching from AP to P alignment by taking into account both the finite penetration depth of the transverse spin current and the spin pumping, and showed that the critical current remains finite in the zero-thickness limit of the free layer [@taniguchi08b]. We also showed that the remaining value of the critical current is mainly determined by spin pumping. Although our results [@taniguchi08b] agree well with the experimental results of Chen [[ *et al.* ]{}]{}[@chen06], we investigated only the critical current of AP-to-P switching, $I_{\rm c}^{{\rm AP}\to{\rm P}}$. For the manipulation of spin-electronics devices, the thickness dependence of the critical current of P-to-AP switching, $I_{\rm c}^{{\rm P}\to{\rm AP}}$, should also be investigated. In this paper, we study the critical current of STT-driven magnetization switching both from P to AP alignment and from AP to P alignment by taking into account both the finite penetration depth of the transverse spin current and the spin pumping. We show that both critical currents, $I_{\rm c}^{{\rm P}\to{\rm AP}}$ and $I_{\rm c}^{{\rm AP}\to{\rm P}}$, remain finite in the zero-thickness of the free layer. We also show that $I_{\rm c}^{{\rm P}\to{\rm AP}}$ is larger than $I_{\rm c}^{{\rm AP}\to{\rm P}}$ over the whole range of the free layer thickness, and thus, the remaining value of $I_{\rm c}^{{\rm P}\to{\rm AP}}$ is larger than that of $I_{\rm c}^{{\rm AP}\to{\rm P}}$. The difference between the remaining values of the critical currents, $I_{\rm c}^{{\rm P}\to{\rm AP}}$ and $I_{\rm c}^{{\rm AP}\to{\rm P}}$, can be explained by considering how the strength of STT, $\eta$, depends on the magnetic alignment. A schematic view of the system we consider is shown in Fig. \[fig:fig1\]. Two ferromagnetic layers (F${}_{1}$ and F${}_{2}$) are sandwiched by the nonmagnetic layers N${}_{i}$ $(i=1-7)$. The F${}_{1}$ and F${}_{2}$ layers correspond to the free and fixed layers, respectively. $\mathbf{m}_{k}$ $(k=1,2)$ is the unit vector pointing in the direction of the magnetization of the F${}_{k}$ layer. $I$ is the electric current flowing perpendicular to the film plane. The electric current and pumped spin current at the F${}_{k}$/N${}_{i}$ interface (into N${}_{i}$) is obtained by using the circuit theory [@tserkovnyak02; @brataas01]: $$\begin{aligned} & I^{{\rm F}_{k}/{\rm N}_{i}} \!=\! \frac{eg}{2h} \left[ 2(\mu_{{\rm F}_{k}}-\mu_{{\rm N}_{i}}) \!+\! p\mathbf{m}_{k}\!\cdot\!(\bm{\mu}_{{\rm F}_{k}}-\bm{\mu}_{{\rm N}_{i}}) \right]\ , \label{eq:electric_current} \\ & \mathbf{I}_{s}^{\rm pump} \!=\! \frac{\hbar}{4\pi} \left( g_{\rm r}^{\uparrow\downarrow} \mathbf{m}_{1}\!\times\! \frac{{{\rm d}}\mathbf{m}_{1}}{{{\rm d}}t} \!+\! g_{\rm i}^{\uparrow\downarrow} \frac{{{\rm d}}\mathbf{m}_{1}}{{{\rm d}}t} \right)\ , \label{eq:pump_current}\end{aligned}$$ where $h\!=\!2\pi\hbar$ is the Planck constant, $g\!=\!g^{\uparrow\uparrow}\!+\!g^{\downarrow\downarrow}$ is the sum of the spin-up and spin-down conductances, $p\!=\!(g^{\uparrow\uparrow}\!-\!g^{\downarrow\downarrow})/(g^{\uparrow\uparrow}\!+\!g^{\downarrow\downarrow})$ is the spin polarization of the conductances, and $g_{\rm r(i)}$ is the real (imaginary) part of the mixing conductance. $\mu_{{\rm N}_{i},{\rm F}_{k}}$ and $\bm{\mu}_{{\rm N}_{i},{\rm F}_{k}}$ are the charge and spin accumulation, respectively. The spin current at each F${}_{k}$/N${}_{i}$ and N${}_{i}$/N${}_{j}$ interface (into N${}_{i}$) is given by [@taniguchi08a; @brataas01] $$\begin{aligned} \!\!\!\! &\mathbf{I}_{s}^{{\rm F}_{k}/{\rm N}_{i}} \!= \frac{1}{4\pi}\! \left[ g \left\{\! p(\mu_{{\rm F}_{k}}\!-\!\mu_{{\rm N}_{i}}) \!+\! \frac{1}{2} \mathbf{m}_{k}\!\cdot\!(\bm{\mu}_{{\rm F}_{k}}\!-\!\bm{\mu}_{{\rm N}_{i}})\! \right\} \mathbf{m}_{k} \right.\nonumber \\ &\hspace{3em}- g_{\rm r}^{\uparrow\downarrow} \mathbf{m}_{k}\!\times\!(\bm{\mu}_{{\rm N}_{i}}\!\times\!\mathbf{m}_{k}) \!-\! g_{\rm i}^{\uparrow\downarrow} \bm{\mu}_{{\rm N}_{i}}\!\times\!\mathbf{m}_{k} \nonumber\\ &\hspace{3em} + \left. t_{\rm r}^{\uparrow\downarrow} \mathbf{m}_{k}\!\times\!(\bm{\mu}_{{\rm F}_{k}}\!\times\!\mathbf{m}_{k}) \!+\! t_{\rm i}^{\uparrow\downarrow} \bm{\mu}_{{\rm F}_{k}}\!\times\!\mathbf{m}_{k} \right]\ , \label{eq:spin_current_FN}\\ &\mathbf{I}_{s}^{{\rm N}_{i}/{\rm N}_{j}} \!=\! -\frac{g_{{\rm N}_{i}/{\rm N}_{j}}}{4\pi} (\bm{\mu}_{{\rm N}_{i}}\!-\!\bm{\mu}_{{\rm N}_{j}})\ , \label{eq:spin_current_NN}\end{aligned}$$ where $t_{\rm r(i)}^{\uparrow\downarrow}$ is the real (imaginary) part of the transmission mixing conductance at the F${}_{k}$/N${}_{i}$ interface and $g_{{\rm N}_{i}/{\rm N}_{j}}$ is the conductance of the one spin channel at the N${}_{i}$/N${}_{j}$ interface. The spin accumulations in the N and F layer obey the diffusion equation [@zhang02; @taniguchi08a; @valet93]. The spin accumulation in the N layer, $\bm{\mu}_{\rm N}$, decays exponentially with the spin diffusion length $\lambda_{\rm sd(N)}$. The longitudinal and transverse spin accumulations in the F layer are defined as $(\mathbf{m}\cdot\bm{\mu}_{\rm F})\mathbf{m}$ and $\mathbf{m}\times(\bm{\mu}_{\rm F}\times\mathbf{m})$, respectively. The longitudinal and transverse spin accumulations decay exponentially with the spin diffusion length $\lambda_{\rm sd(F_{L})}$ and with the penetration depth of the transverse spin current $\lambda_{\rm t}$, respectively. The total spin currents across the N${}_{3}$/F${}_{1}$ and F${}_{1}$/N${}_{4}$ interfaces, i.e., $\mathbf{I}_{s}^{(1)}\!=\!\mathbf{I}_{s}^{\rm pump}\!+\!\mathbf{I}_{s}^{\rm F_{1}/N_{3}}$ and $\mathbf{I}_{s}^{(2)}\!=\!\mathbf{I}_{s}^{\rm pump}\!+\!\mathbf{I}_{s}^{\rm F_{1}/N_{4}}$, exert the torque $\bm{\tau}\!=\mathbf{m}_{1}\!\times\![(\mathbf{I}_{s}^{(1)}\!+\!\mathbf{I}_{s}^{(2)})\!\times\!\mathbf{m}_{1}]$ on the magnetization $\mathbf{m}_{1}$. In order to obtain the spin current $\mathbf{I}_{s}^{(1,2)}$, we solve the diffusion equation of spin accumulation in each layer. The boundary conditions are as follows. We assume that the thicknesses of the N${}_{1}$ and N${}_{7}$ layer are much larger than their spin diffusion length, and that the spin current is zero at the outer boundary of the N${}_{1}$ and N${}_{7}$ layer. We also assume that the spin current is continuous at all interfaces and that the electric current is constant through the entire structure. The torque $\bm{\tau}$ modifies the Landau-Lifshitz-Gilbert (LLG) equation of magnetization $\mathbf{m}_{1}$ as [@tserkovnyak02] $$\begin{split} \frac{{{\rm d}}\mathbf{m}_{1}}{{{\rm d}}t} \!=\! & -\!\gamma \mathbf{m}_{1}\!\times\!\mathbf{B}_{\rm eff} \!+\! \frac{\gamma}{MSd} \bm{\tau} \!+\! \alpha_{0} \mathbf{m}_{1} \!\times\! \frac{{{\rm d}}\mathbf{m}_{1}}{{{\rm d}}t} \\ &=\! -\!\gamma_{\rm eff} \mathbf{m}_{1}\!\times\!\mathbf{B}_{\rm eff} \!+\! \frac{\gamma_{\rm eff}}{\gamma} (\alpha_{0}+\alpha^{'}) \mathbf{m}_{1}\!\times\!\frac{{{\rm d}}\mathbf{m}_{1}}{{{\rm d}}t}\ , \end{split}$$ where $\mathbf{B}_{\rm eff}$ is the effective magnetic field, and $\alpha^{'}\!=\!\alpha_{c}\!+\!\alpha_{\rm pump}$ is the enhancement of the Gilbert damping constant. The enhancement $\alpha_{c}$ is proportional to the electric current and independent of the pumped spin current. The enhancement $\alpha_{\rm pump}$ represents the contribution from the pumped spin current and is independent of the electric current. The enhancement of the gyromagnetic ratio, $\gamma_{\rm eff}/\gamma$, is a function of both the electric current and the pumped spin current. The critical current of the STT-driven magnetization dynamics is defined by the electric current that satisfies the condition, $\alpha_{0}\!+\!\alpha_{c}\!+\!\alpha_{\rm pump}\!=\!0$, and given by $$I_{\rm c}^{{\rm P}\to{\rm AP}({\rm AP}\to{\rm P})} \!=\! \frac{2eMSd}{\hbar\gamma\tilde{\eta}_{\rm P(AP)}} (\alpha_{0}\!+\!\alpha_{\rm pump}) \omega_{\rm P(AP)}\ , \label{eq:critical_current_TT}$$ where the coefficient $\tilde{\eta}_{\rm P,AP}$ characterizes the strength of STT due to the electric current, and is determined by the diffusion equations of the spin accumulations. Thus, $\tilde{\eta}_{\rm P,AP}$ is the function of $d/\lambda_{\rm sd(F_{L})}$, $d/\lambda_{\rm t}$ and the relative angle of the magnetizations of the F${}_{1}$ and F${}_{2}$ layers. {width="0.8\columnwidth"} \[fig:fig2\] We performed numerical calculation to obtain the critical currents $I_{\rm c}^{{\rm P}\to{\rm AP}}$ and $I_{\rm c}^{{\rm AP}\to{\rm P}}$. The system consists of nine layers as shown in Fig. \[fig:fig1\], where F${}_{1}$ and F${}_{2}$ are Co, N${}_{1}$, N${}_{3}$, N${}_{4}$, N${}_{5}$ and N${}_{7}$ are Cu, and N${}_{2}$ and N${}_{6}$ are Pt. The thicknesses of the N${}_{3}$, N${}_{4}$ and N${}_{5}$ layers are 10 nm, the thicknesses of the N${}_{2}$ and N${}_{6}$ layers are 3 nm and the thickness of the F${}_{2}$ layer is 12 nm [@chen06]. The thickness of the N${}_{1}$ and N${}_{7}$ layers are taken to be 10 $\mu$m. The spin diffusion length of Cu and Pt are 1000 and 14 nm, respectively [@bass07]. The conductance at the Cu/Pt interface is 35 nm${}^{-2}$ [@bass07]. The magnetization, the intrinsic Gilbert damping constant and the gyromagnetic ratio of Co are 0.14 T, 0.008 and $1.89 \! \times \! 10^{11}$ Hz/T, respectively [@chen06; @beaujour06]. The polarization $p$ is taken to be $0.46$ for Co [@bass07]. The spin diffusion length of Co is 40 nm [@bass07]. The penetration depth of the transverse spin current of Co is 4.2 nm [@zhang04; @taniguchi08b]. The conductances at the Co/Cu interface, $g/S$, $g_{\rm r}^{\uparrow\downarrow}/S$ and $g_{\rm i}^{\uparrow\downarrow}/S$, are 50, 27 and 0.4 nm${}^{-2}$, respectively [@tserkovnyak02; @tserkovnyak03; @brataas01]. We assume that $t_{\rm r} \! = \! t_{\rm i}$ where $t_{\rm r,i}/S$ at the Co/Cu interface is taken to be 6.0 nm${}^{-2}$. The angular frequency is $\omega_{\rm P(AP)} \! = \! \gamma[B_{\rm appl} \!-\!(+)4\pi M]$ where the strength of the applied magnetic field $B_{\rm appl}$ is 7 T [@chen06]. {width="0.8\columnwidth"} \[fig:fig3\] Figure \[fig:fig2\] shows the dependence of the critical current density of Eq. for P-to-AP switching, $I_{\rm c}^{{\rm P}\to{\rm AP}}/S$, and AP-to-P switching, $I_{\rm c}^{{\rm AP}\to{\rm P}}/S$, on the free layer thickness, $d$. As shown in Fig. \[fig:fig2\], both $I_{\rm c}^{{\rm P}\to{\rm AP}}$ and $I_{\rm c}^{{\rm AP}\to{\rm P}}$ remain finite in the zero-thickness limit of the free layer. We show that the critical current $I_{\rm c}^{{\rm P}\to{\rm AP}}$ is larger than $I_{\rm c}^{{\rm AP}\to{\rm P}}$ over the whole range of the free layer thickness, and thus, the remaining value of $I_{\rm c}^{{\rm P}\to{\rm AP}}$ is larger than that of $I_{\rm c}^{{\rm AP}\to{\rm P}}$. As shown in Ref. [@taniguchi08b], the remaining value of the critical current is mainly determined by spin pumping. It should be noted that the magnitude of the enhancement of the Gilbert damping constant due to spin pumping, $\alpha_{\rm pump}$, is the same for both P-to-AP switching and AP-to-P switching [@tserkovnyak03; @taniguchi07]. Thus, the fact that the remaining values $I_{\rm c}^{{\rm P}\to{\rm AP}}$ and $I_{\rm c}^{{\rm AP}\to{\rm P}}$ are different from each other implies that the strength of STT, $\tilde{\eta}_{\rm P,AP}$, depends on the alignment of the magnetizations. As shown in Fig. \[fig:fig3\], $\tilde{\eta}_{\rm P,AP}$ decreases with a decreasing free layer thickness. On the other hand, the number of localized magnetic moments in the free layer, and therefore the STT per magnetic moment, is inversely proportional to the free layer thickness $d$. According to Eq. (\[eq:critical\_current\_TT\]), the remaining value of the critical current is proportional to $(\tilde{\eta}_{\rm P,AP}/d)^{-1}$ with $d\!\to\!0$, where $\tilde{\eta}_{\rm P}/d\simeq 0.44$ nm${}^{-1}$ and $\tilde{\eta}_{\rm AP}/d\simeq 1.47$ nm${}^{-1}$ in the limit of $d\!\to\!0$ are estimated by Fig. \[fig:fig3\]. Thus, the remaining value of $I_{\rm c}^{{\rm P}\to{\rm AP}}$ is larger than that of $I_{\rm c}^{{\rm AP}\to{\rm P}}$. In summary, we studied the critical current of STT-driven magnetization dynamics by taking into account the finite penetration depth of the transverse spin current and spin pumping for both P and AP magnetic alignments. We showed that the critical current remains finite in the zero thickness limit of the free layer for both P-to-AP and AP-to-P switching. We also showed that the critical current for P-to-AP switching is larger than that for AP-to-P switching over the whole range of the free layer thickness. The authors would like to acknowledge the valuable discussions they had with K. Matsushita, J. Sato and N. Yokoshi. This work was supported by JSPS and NEDO. [19]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , ****, (). , , , , , , , ****, (). , , , , ****, (). , , , , , ****, (). , ****, (). , , , , , , , , , ****, (). , , , ****, (). , , , , ****, (). , , , , ****, (). , , , ****, (). , , , ****, (). , , , ****, (). , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, (). , , , , ****, (). [^1]: Corresponding author. Email address: h-imamura@aist.go.jp

--- abstract: 'The super-inflationary phase is predicted by the Loop Quantum Cosmology. In this paper we study the creation of gravitational waves during this phase. We consider the inverse volume corrections to the equation for the tensor modes and calculate the spectrum of the produced gravitons. The amplitude of the obtained spectrum as well as maximal energy of gravitons strongly depend on the evolution of the Universe after the super-inflation. We show that a further standard inflationary phase is necessary to lower the amount of gravitons below the present bound. In case of the lack of the standard inflationary phase, the present intensity of gravitons would be extremely large. These considerations give us another motivation to introduce the standard phase of inflation.' author: - Jakub Mielczarek - 'Marek Szyd[ł]{}owski' title: 'Relic gravitons from super-inflation' --- Introduction {#sec:intro} ============ The cosmological creation of the gravitational waves was proposed by Grishchuk [@Grishchuk:1974ny] in the mid-seventies. Since that time this phenomenon has been studied extensively, especially in the context of the inflation. The accelerating expansion phase gives the conditions for the abundant creation of the gravitational waves. Gravitons produced during the inflation fill the entire space in the form of a stochastic background. Together with the scalar modes, produced during the inflation, they form primordial perturbations leading to the structure formation. The analysis of the cosmic microwave background (CMB) and large scale structures gives therefore the possibility of testing inflationary models. In the case of the CMB the impact of the gravitational waves comes from the primordial spectrum and from tensor Sachs-Wolfe effects. The Sachs-Wolfe effect is somehow secondary and leads to the CBM anisotropies as the result of the scattering of CMB photons on the relic gravitons. The form of this anisotropies is given by $$\left( \frac{\Delta \text{T}}{\text{T}} \right)_{\text{t}} = -\frac{1}{2} \int_{\tau_1}^{\tau_{2}} d \tau \ h'_{ij} n^i n^j$$ where $h_{ij}$ describes tensor modes and $n^i$ is the vector parallel to the unperturbed geodesics. The influence of the gravitational waves for the CMB is however to weak to be observed directly with the present observational abilities. Another possible method to detect gravitational waves is to use of the antennas like LIGO, VIRGO, TAMA or GEO600 [@Abbott:2003vs; @Cella:2007jh] . Although these detectors are now very sensitive this is still not enough to detect directly the gravitational waves background [@Abbott:2007wd]. It may look pessimistic, we hope however that some further improvement of the observational skills bring us the observational evidence, so needful for the further theoretical improvements. In this paper we consider a new type of the inflation which naturally occurs in the Loop Quantum Cosmology [@Bojowald:2006da]. This is so called the super-inflationary scenario [@Bojowald:2002nz; @Copeland:2007qt] and is a result of the quantum nature of spacetime in the Planck scales. The spacetime is namely discrete in the quantum regime and its evolution is governed by discrete equations. However for the scales greater than $a_i=\sqrt{\gamma}l_{\text{Pl}}$ ($\gamma $ is so called Barbero-Immirzi parameter) the evolution of the spacetime can be described by the Einstein equations with quantum corrections. For typical values of the quantum numbers, super-inflationary phase takes place in this semi-classical region. Our goal is to describe the production of the gravitational waves during the super-inflation. This problem was preliminary analysed in Ref. [@Mielczarek:2007zy], but quantum corrections to the equation for tensor modes was not included to calculate the spectrum of the gravitons. In this paper we include so called inverse volume corrections to the equation for evolution of the tensor modes and then calculate the spectrum of produced gravitational waves. The equations for the tensor modes was recently derived by Bojowald and Hossain [@Bojowald:2007cd]. They had analysed the inverse volume corrections and corrections from holonomies. In this paper we concentrate on these first ones. The quantum corrections are generally complicated functions but they have simple asymptotic behaviours. To calculate the productions of gravitons during some process we need somehow to know only initial and final states, where asymptotic solutions are good approximation. In these regimes calculations can be done analytically. We use numerical solutions to match them. The organization of the text is the following. In section II we fix the semi-classical dynamics. Then in section III we consider creation of the gravitons on the defined background. In section IV we summarize the results. Background dynamics =================== The formulation of Loop Quantum Gravity bases on the Ashtekar variables [@Ashtekar:1987gu] and holonomies. The Ashtekar variables replace the spatial metric field $q_{ab}$ in the canonical formulation as follow $$\begin{aligned} A^i_a &=& \Gamma^i_a+\gamma K_a^i , \\ E^a_i &=& \sqrt{|\det q|} e^{a}_i\end{aligned}$$ where $\Gamma^i_a$ is the spin connection defined as $$\Gamma^i_a = -\epsilon^{ijk}e^b_j(\partial_{[a}e^k_{b]}+\frac{1}{2}e^c_k e^l_a \partial_{[c}e^l_{b]} )$$ and the $K_a^i$ is the intrinsic curvature. The $e^{a}_i$ is the inverse of the co-triad $e^i_a$ defined as $q_{ab}=e_a^ie_b^j$. In terms of the Ashtekar variables the full Hamiltonian for general relativity is a sum of constraints $$H_{\text{G}}^{\text{tot}}= \int d^3 {\bf x} \, (N^i G_i + N^a C_a + N h_{\text{sc}}),$$ where $$\begin{aligned} C_a &= E^b_i F^i_{ab} - (1-\gamma^2)K^i_a G_i ,\nonumber \\ G_i &= D_a E^a_i\end{aligned}$$ and the scalar constraint has a form $$\begin{aligned} \label{ham} &H_{\text{G}}:=\int d^3{\bf x} \, N(x) h_{\rm sc}= \nonumber \\ &\frac{1}{16 \pi G} \int d^3{\bf x} \, N(x)\left( \frac{E^a_i E^b_j}{\sqrt{|\det E|}} {\varepsilon^{ij}}_k F_{ab}^k - 2(1+\gamma^2) \frac{E^a_i E^b_j}{\sqrt{|\det E|}} K^i_a K^j_b \right) \end{aligned}$$ with $F=dA + \frac{1}{2}[A,A]$. The full Hamiltonian of theory is a sum of the gravitational and matter part. With convenience as a matter part we choose the scalar field with the Hamiltonian $$H_{\phi}=\int d^3{\bf x} \, N(x)\left( \frac{1}{2}\frac{\pi^2_{\phi}}{\sqrt{|\det E|}} + \frac{1}{2} \frac{E^a_i E^b_i \partial_a \phi \partial_b \phi }{\sqrt{|\det E|}} + \sqrt{|\det E|} V(\phi) \right).$$ We assume here that field $\phi$ is homogeneous and start his evolution from the minimum of potential $ V(\phi)$. The second assumption states that contribution from potential term is initially negligible. So the density of Hamiltonian $H_{\phi}$ is simplified to the form $\mathcal{H}_{\phi}=(1/2)\pi^2_{\phi}/\sqrt{|\det E|}$. The term $1/\sqrt{|\det E|}$ for the classical FRW universe corresponds to $1/a^3$ where $a$ is the scale factor. On the quantum level term $1/\sqrt{|\det E|}$ is quantised and have discrete spectrum. In the regime $a \gg a_i$ we can however use the approximation $1/\sqrt{|\det E|}=D/a^3$ where $$D(q)=q^{3/2} \left\{ \frac{3}{2l} \left( \frac{1}{l+2}\left[(q+1)^{l+2}-|q-1|^{l+2} \right]- \frac{q}{1+l}\left[(q+1)^{l+1}-\mbox{sgn}(q-1)|q-1|^{l+1} \right] \right) \right\}^{3/(2-2l)} \label{correction}$$ and $q=(a/a_*)^2$ with $a_*=\sqrt{\gamma j / 3}l_{\text{Pl}} $. Function (\[correction\]) depends on the ambiguity parameter $l$. As it was shown by Bojowald [@Bojowald:2002ny] the value of this parameter is quantised according to $l_k=1-(2k)^{-1} \geq 1/2 , \ k \in \mathbb{N} $. For the further investigations we choose the representative value $l=3/4$. In the semi-classical region $ a_* \gg a \gg a_i$ expression (\[correction\]) simplify to the form $$D=D_*a^n$$ where $$D_* = \left( \frac{3}{1+l} \right)^{3/(2-2l)} a_*^{-3(2-l)/(1-l)} \ \ \text{and} \ \ n = 3(2-l)/(1-l) \ .$$ Now, due to the Hamilton equations we can derive the Friedmann and Raychaudhuri equations for the flat FRW universe filled with a homogeneous scalar field $$\begin{aligned} H^2 &=& \frac{8\pi G}{3} \left[ \frac{\dot{\phi}^2}{2D} +V(\phi) \right] \ , \label{Fried1} \\ \frac{\ddot{a}}{a} &=& -\frac{8\pi G}{3} \left[ \frac{\dot{\phi}^2}{D} \left( 1-\frac{\dot{D}}{4HD} \right) -V(\phi) \right]. \label{Raych1}\end{aligned}$$ The equation of motion for the scalar field with quantum corrections has the form $$\ddot{\phi}+\left(3H - \frac{\dot{D}}{D} \right)\dot{\phi} + D\frac{dV}{d\phi} = 0. \label{eom}$$ As we mentioned before, for the further investigations we simplify equations (\[Fried1\]), (\[Raych1\]) and (\[eom\]) assuming $V(\phi) = 0$. The expression for the quantum correction $D$ is complicated and it is impossible to find an analytical solution for the equations of motion. In fact we even do not need it for the future investigations. To calculate the spectrum of gravitons we need to know analytical solutions only for the inner and outer states. We choose the $| \text{in} \rangle$ and $| \text{out} \rangle$ states respectively in the quantum and classical regimes. The expression for the quantum correction (\[correction\]) simplifies to the form $D=D_*a^n$ for the $a_i < a \ll a_*$ and $D=1$ for $a \gg a_* $. In these limits we can find the analytical solutions for the equations of motion (\[Fried1\]), (\[Raych1\]) and (\[eom\]). It is useful to introduce the conformal time $d\tau = dt/a $ to solve equations and for the further investigations. In the next step we must to fit obtained asymptotic solutions using a global numerical solution. The solution for the evolution of the scale factor in the quantum limit has the form $$a=\xi(-\tau+\beta)^p \label{solution1}$$ where $p=2/(4-n)$. The solution in the classical limit we obtain putting simply $l=2$ what gives $D=1$ and $p=1/2$. The constants of integration $\xi$ and $\beta$ we fix with the use of a numerical solution applying formula $$\begin{aligned} \xi = a|_{\tau} \left[-\frac{a'|_{\tau}}{p \ a|_{\tau} } \right]^p \ \ \text{and} \ \ \beta = -\tau-p\frac{a|_{\tau}}{a'|_{\tau}}. \label{fixing}\end{aligned}$$ The value of the conformal time $\tau$ must be chosen in a proper way for the given regions. We will discuss this question in more details later. Our point of reference is the numerical solution. To make this description complete we must choose the proper boundary conditions for the numerical solution. We use here the condition for the Hubble radius which must be larger than the limiting value $a_i$ [@Lidsey:2004ef], what gives us $$k \simeq |H|a < \frac{a}{a_i} \ \ \Rightarrow \ \ |H|a_i < 1. \label{condition1}$$ The next condition requires that the scale factor must be greater than $a_i$ at the bounce. It is fulfilled taking $a|_{\tau_0} = a_{*}$ for some value of the conformal time $\tau_{0}$. In fact the conformal time is unphysical variable and their value can be chosen arbitrary. The physical outcomes do not depend on coordinates because the theory is invariant under local diffeomorphisms. So as an example we can choose $$\begin{aligned} a|_{\tau_0=-4} &=& a_{*} \label{init1} \\ a'|_{\tau_0=-4} &=& l_{\text{Pl}} \label{init2}\end{aligned}$$ The chosen value of $a'|_{\tau_0=-4}$ holds the condition (\[condition1\]). Namely for $j=100$ we have $|H_*|a_i = 0.084 < 1$. The numerical solution is shown in Fig. \[fig:solution1\] as a black line. ![Numerically calculated evolution of the scale factor (top black line) and the Hubble parameter (bottom black line) for the model with $j=100$, $l=3/4$ and with initial conditions (\[init1\]) and (\[init2\]). The approximated solutions described by (\[solution1\]) and (\[solution2\]).[]{data-label="fig:solution1"}](plot1.eps){width="7cm"} We fix the boundary approximations in $\tau_1=-20$ and $\tau_2=-1$. The the numerical solution gives us for these points $$\begin{aligned} a|_{\tau_1=-20} &=& 0.536 l_{\text{Pl}} \ , \ a'|_{\tau_1=-20} = 0.007 l_{\text{Pl}} \\ a|_{\tau_2=-1} &=& 3.912 l_{\text{Pl}} \ , \ a'|_{\tau_2=-1} = 0.461 l_{\text{Pl}}.\end{aligned}$$ Now with the use of expressions (\[fixing\]) we can fix the approximated solutions. However when we use formula (\[fixing\]) directly to calculate parameters in solution (\[solution1\]) for the outer state we obtain complex $\xi$. It is due to the expression under square $(p=1/2)$ is negative. So to put away complex numbers we redefine the exit solution to the form $$a=\kappa\sqrt{\tau+\zeta} \label{solution2}$$ where $$\begin{aligned} \kappa &=& -i \xi \\ \zeta &=& - \beta.\end{aligned}$$ We show in Fig. \[fig:solution1\] how these approximated solutions match with solutions obtained numerically. As we can see these approximated solutions well describe the evolution in the neighbourhood of $a_*$. Gravitational waves =================== We have already mentioned in section \[sec:intro\] that gravitational waves can be abundantly produced during the accelerating phase. In this section we want to show in details how it works and calculate properties of produced gravitons. To describe the spectrum of gravitons it is common to use the parameter $$\Omega_{\text{gw}}(\nu) =\frac{\nu}{\rho_c}\frac{d \rho_{\text{gw}}}{d \nu} \label{omegaGW}$$ where $\rho_{\text{gw}} $ is the energy density of gravitational waves and $\rho_c$ is present critical energy density. Our goal in this section is to calculate the function $\Omega_{\text{gw}}(\nu)$ for the gravitons produced during the super-inflationary phase. The gravitational waves $h_{ij}$ are the perturbations of the background spacetime in the form $$ds^2=a^2(\tau) \left[ -d\tau^2 +(\delta_{ij}+h_{ij} )dx^i dx^j \right] \label{metric1}$$ where $|h_{ij}|\ll 1$. Using constraints $h^i_i=\nabla_ih^i_j=0 $ we can see that tensor $h_{ij}$ have only two independent components $h^1_1=-h^2_2=h_+$ and $h^2_1=h^1_2=h_{\times}$. These components correspond to two different polarisations of gravitational waves. Inserting the perturbed metric (\[metric1\]) to the Hilbert-Einstein action $S_{\text{H-E}}=(1/16 \pi G) \int d^4x \sqrt{-g}R$ gives the series $S_{\text{H-E}}=S^{(0)}+S^{(1)}+S^{(2)}+\dots$, where the second order term have a form $$S^{(2)}_t=\frac{1}{64\pi G} \int d^4x a^3 \left[ \partial_t h^i_j\partial_t h^j_i-\frac{1}{a^2}\nabla_k h^i_j\nabla_k h^j_i \right] = \frac{1}{32\pi G} \int d^4x a^3 \left[ \dot{h}_{\times}^2+\dot{h}_{+}^2- \frac{1}{a^2}\left(\vec{\nabla} h_{\times} \right)^2-\frac{1}{a^2}\left(\vec{\nabla} h_{+}\right)^2 \right] \label{action1}$$ and give us the action for the gravitational waves. The two kinds of polarisations are not coupled and can be treated separately. To normalise the action and simplify the notation it us useful to introduce the variable $$h=\frac{h_{+}}{\sqrt{16\pi G}}=\frac{h_{\times}}{\sqrt{16\pi G}},$$ what leads to the expression for the action in the form $$S_t = \frac{1}{2} \int d^4x a^3 \left[ \dot{h}^2-\frac{1}{a^2} \left( \vec{\nabla} h \right)^2 \right]. \label{actionh}$$ This action is the same like the action for an inhomogeneous scalar field without the potential. Inverse volume corrections can be therefore introduced in the same way like in the case of the scalar field. As we mentioned in Introduction there are also holonomy corrections to this action. Here we consider however the influence from the better examined inverse volume corrections. As it was shown by Mulryne and Nunes [@Mulryne:2006cz], in the context of scalar field perturbations, it is useful to introduce the variable $u=a D^{-1/2} h $ and rewrite the action (\[actionh\]) with quantum corrections to the form $$S_{\text{t}} = \frac{1}{2}\int d \tau d^3 {\bf x} [ u^{'2}-D \delta^{ij} \partial_i u \partial_j u -m^2_{\text{eff}}u^2 ]$$ where $$m^2_{\text{eff}} = - \frac{\sqrt{D}}{a} \left( \frac{a}{\sqrt{D}} \right)^{''} .$$ Till now the considerations of the gravitational waves has been purely classical. The next step is the quantisation of the classical gravitational waves what brings us the concept of gravitons. To quantise the field $u$ we need to firstly calculate conjugated momenta $$\pi(\tau,{\bf x})=\frac{\delta S_{\text{t}}}{\delta u'} = u '.$$ The procedure of quantisation is the simple change of fields $u$ and $\pi$ for the operators just adding hats and to introduce the relations of commutation. We decompose operators considered for the Fourier modes $$\begin{aligned} \hat{u}(\tau,{\bf x} ) &=& \frac{1}{2(2\pi)^{3/2}} \int d^3{\bf k } \left[ \hat{u}_{{\bf k}}(\tau) e^{i{\bf k}\cdot {\bf x}} + \hat{u}_{{\bf k}}^{\dagger}(\tau) e^{-i{\bf k}\cdot {\bf x}} \right], \label{decomp1} \\ \hat{\pi}(\tau,{\bf x} ) &=& \frac{1}{2(2\pi)^{3/2}} \int d^3{\bf k } \left[ \hat{\pi}_{{\bf k}}(\tau) e^{i{\bf k}\cdot {\bf x}} + \hat{\pi}_{{\bf k}}^{\dagger}(\tau) e^{-i{\bf k}\cdot {\bf x}} \right], \label{decomp2} \end{aligned}$$ where the Fourier components fulfil the relations of commutation $$\begin{aligned} \left[ \hat{u}_{{\bf k}}(\tau) ,\hat{\pi}_{{\bf p}}^{\dagger}(\tau) \right] &=& i \delta^{(3)}({\bf k} - {\bf p}),\label{com1}\\ \left[ \hat{u}_{{\bf k}}(\tau)^{\dagger} ,\hat{\pi}_{{\bf p}}(\tau) \right] &=& i \delta^{(3)}({\bf k} - {\bf p}),\label{com2} \\ \left[ \hat{u}_{{\bf k}}(\tau) ,\hat{\pi}_{{\bf p}}(\tau) \right] &=& i \delta^{(3)}({\bf k} + {\bf p}), \label{com3} \\ \left[ \hat{u}_{{\bf k}}(\tau)^{\dagger} ,\hat{\pi}_{{\bf p}}^{\dagger}(\tau) \right] &=& i \delta^{(3)}({\bf k} + {\bf p}). \label{com4}\end{aligned}$$ To express the Fourier modes in terms of the annihilation and creation operators we need to solve the quantum Hamilton equations $$\begin{aligned} \hat{u}^{'} &=& i [ \hat{H}_{\text{t}}, \hat{u} ], \label{Ham1} \\ \hat{\pi}^{'} &=& i [ \hat{H}_{\text{t}}, \hat{\pi} ]. \label{Ham2}\end{aligned}$$ The Hamilton operator have the form $$\begin{aligned} \hat{H}_{\text{t}} &=& \frac{1}{2}\int d^3 {\bf x} [ \hat{\pi}^{2}+D \delta^{ij} \partial_i \hat{u} \partial_j \hat{u} +m^2_{\text{eff}}\hat{u}^2 ] \nonumber \\ &=& \frac{1}{2} \frac{1}{4(2\pi)^{3}} \int d^3 {\bf x} d^3 {\bf k} d^3 {\bf q} \left[ \hat{\pi}_{{\bf k}} e^{i{\bf k}\cdot {\bf x}} + \hat{\pi}_{{\bf k}}^{\dagger} e^{-i{\bf k}\cdot {\bf x}} \right] \left[ \hat{\pi}_{{\bf q}} e^{i{\bf q}\cdot {\bf x}} + \hat{\pi}_{{\bf q}}^{\dagger} e^{-i{\bf q}\cdot {\bf x}} \right] \nonumber \\ &+& D \delta^{ij} i \left[ k_i\hat{u}_{{\bf k}} e^{i{\bf k}\cdot {\bf x}} - k_i\hat{u}_{{\bf k}}^{\dagger} e^{-i{\bf k}\cdot {\bf x}} \right] i\left[q_j \hat{u}_{{\bf q}} e^{i{\bf q}\cdot {\bf x}} - q_j\hat{u}_{{\bf q}}^{\dagger} e^{-i{\bf q}\cdot {\bf x}} \right] \nonumber \\ &+& m^2_{\text{eff}} \left[ \hat{u}_{{\bf k}} e^{i{\bf k}\cdot {\bf x}} + \hat{u}_{{\bf k}}^{\dagger} e^{-i{\bf k}\cdot {\bf x}} \right] \left[ \hat{u}_{{\bf q}} e^{i{\bf q}\cdot {\bf x}} + \hat{u}_{{\bf q}}^{\dagger} e^{-i{\bf q}\cdot {\bf x}} \right] \nonumber \\ &=& \frac{1}{4} \int d^3 {\bf k}\left[ \hat{\pi}_{{\bf k}} \hat{\pi}_{{\bf k}}^{\dagger}+ \hat{\pi}_{{\bf k}}^{\dagger} \hat{\pi}_{{\bf k}} +\left(Dk^2 + m^2_{\text{eff} }\right) \left( \hat{u}_{{\bf k}} \hat{u}_{{\bf k}}^{\dagger}+ \hat{u}_{{\bf k}}^{\dagger} \hat{u}_{{\bf k}} \right) \right] \label{Ham}\end{aligned}$$ where we inserted decompositions (\[decomp1\]) and (\[decomp2\]). When we apply the Hamiltonian (\[Ham\]) and the decompositions (\[decomp1\]) and (\[decomp2\]), the Hamilton equations (\[Ham1\]) and (\[Ham2\]) take the forms $$\begin{aligned} \hat{u}^{'}_{{\bf k}} &=& \hat{\pi} _{{\bf k}}, \label{Ham11} \\ \hat{\pi}^{'}_{{\bf k}} &=& -\left(Dk^2 + m^2_{\text{eff} }\right) \hat{u}_{{\bf k}}. \label{Ham22} \end{aligned}$$ The general solution of these equations has the form $$\begin{aligned} \hat{u}_{ {\bf k} }(\tau) &=& \hat{a}_{{\bf k} } f(k,\tau)+ \hat{a}_{-{\bf k} }^{\dagger} f^{*}(k,\tau), \label{sol11} \\ \hat{\pi}_{{\bf k} }(\tau) &=& \hat{a}_{{\bf k} } g(k,\tau)+ \hat{a}_{-{\bf k} }^{\dagger} g^{*}(k,\tau). \label{sol22}\end{aligned}$$ where $f(k,\tau)'=g(k,\tau)$. When we insert these solutions to the Fourier decompositions (\[decomp1\]) and (\[decomp2\]) we simply obtain $$\begin{aligned} \hat{u}(\tau,{\bf x} ) &=& \frac{1}{(2\pi)^{3/2}} \int d^3{\bf k } \left[ f(k,\tau) \hat{a}_{{\bf k}} e^{i{\bf k}\cdot {\bf x}} + f^*(k,\tau) \hat{a}_{{\bf k}}^{\dagger} e^{-i{\bf k}\cdot {\bf x}} \right] \label{decomp11} \ , \\ \hat{\pi}(\tau,{\bf x} ) &=& \frac{1}{(2\pi)^{3/2}} \int d^3{\bf k } \left[ g(k,\tau) \hat{a}_{{\bf k}} e^{i{\bf k}\cdot {\bf x}} + g^*(k,\tau) \hat{a}_{{\bf k}}^{\dagger} e^{-i{\bf k}\cdot {\bf x}} \right]. \label{decomp22} \end{aligned}$$ The mode functions fulfils the so called Wronskian condition $$f^*(k,\tau) g(k,\tau) - f(k,\tau) g^*(k,\tau)=-i \label{Wronskian}$$ as a result of relations of commutation (\[com1\]-\[com4\]). These relations is important to normalise properly the mode functions. The Hamilton equations (\[Ham11\]) and (\[Ham22\]) together with (\[sol11\]) give us the equation for the mode function $$\frac{d^2}{d\tau^2}f(k,\tau) + \left[ D k^2 +m^2_{\text{eff}} \right] f(k,\tau) = 0. \label{modeeq}$$ This equations has two regimes. The first one called adiabatic corresponds to the situation when $D k^2 +m^2_{\text{eff}} \equiv \Gamma \gg 0 $. The second one leads to the super-adiabatic amplification and corresponds to the situation when $\Gamma \ll 0$. The creation of the gravitational waves corresponds to the case of the super-adiabatic amplification. We can now investigate which modes are amplified. The condition $\Gamma \ll 0$ corresponds to $D k^2 \ll -m^2_{\text{eff}}$. In Fig. \[pumpfield\] we see the evolution of $-m^2_{\text{eff}}$. As we see the condition for the creation of gravitational waves is fulfilled in the region of the super-inflation $( \tau \in [ \sim -6 , \sim -4 ] )$ . In the right panel we can see the evolution of $\Gamma$ for three different values of $k$. $\begin{array}{cc} \includegraphics[width=6cm,angle=270]{plot2.eps} & \includegraphics[width=6cm,angle=270]{plot3.eps} \end{array} $ The value of $k$ is dimensionless in the undertaken scheme. To obtain a dimensional value we must multiply it simply by the corresponding scale factor which has a dimension of length. The wave number $k$, for example for the final state $a_{\text{f}}$, corresponds to the frequency $$\nu = \frac{k}{2\pi a_{\text{f}}} \left( \frac{a_{\text{f}}}{a_{\text{today}}} \right) \label{freq}$$ measured today. The important task is to calculate the factor $ a_{\text{today}}/a_{\text{f}} $. We can make the decomposition $$\frac{a_{\text{today}}}{a_{\text{f}}} = \frac{a_{\text{today}}}{a_{\text{dec}}} \frac{a_{\text{dec}}}{a_{\text{rh}}} \frac{a_{\text{rh}}}{a_{\text{i-end}}} \frac{a_{\text{i-end}}}{a_{\text{i-star}}} \frac{a_{\text{i-start}}}{a_{\text{f}}}$$ where $$\begin{aligned} a_{\text{dec}} &\rightarrow& \text{ photons decoupling} , z _{\text{dec}}\simeq 1070 , T_{\text{dec}} \simeq 3\cdot 10^3 \ \text{K} = 0.2 \ \text{eV} \nonumber \\ a_{\text{rh}} &\rightarrow& \text{ reheating phase } , T_{\text{rh}} = T_{\text{GUT}} \simeq 10^{14} \ \text{GeV}\nonumber \\ a_{\text{i-end}} &\rightarrow& \text{end of inflation } \nonumber \\ a_{\text{i-start}} &\rightarrow& \text{beginning of inflation } \nonumber\end{aligned}$$ We can assume the sudden reheating approximation $(a_{\text{rh}} \simeq a_{\text{i-end}} )$ and the standard value of e-folding number for inflation $N \equiv \ln ( {a_{\text{i-end}}}/{a_{\text{i-star}}} ) = 63 $. Then increase of the scale factor forms the final state till present value assumes $$\frac{a_{\text{today}}}{a_{\text{f}}} \simeq (1+z _{\text{dec}}) \cdot \frac{ T_{\text{GUT}} }{ T_{\text{dec}} } \cdot e^{N} \cdot \frac{a_{\text{i-start}}}{a_{\text{f}}} \simeq 10^{56}.$$ We had assumed here that ${a_{\text{i-start}}}/{a_{\text{f}}}\simeq 10^2 $, this value can be obtained from numerical simulations like these in Ref. [@Tsujikawa:2003vr]. Now we can return to equation (\[freq\]) and then calculate the maximal frequency of produced gravitons. From the right panel in Fig. \[pumpfield\] we can see that in the vicinity of $\tau \sim -6$ we have the transition from positive to negative values of $\Gamma$. So as we can see for some $10<k_{\text{k}}<1000$ we have the transition from the adiabatic to the super-adiabatic regime. From the numerical investigation we obtain $k_{\text{tr}} \simeq 500$. In fact the function $\Gamma$, for small values of $\tau$, is always slightly below zero even for greater values of $k$ than $k_{\text{tr}}$. But we assume that this effect is negligible. In fact higher values of $k$ easily reach the transplanckian scales and it is not clear that we should trust the standard physics in this regime. The wave number $k_{\text{tr}}$ corresponds roughly to the scales $a_*$. So the value of $k_{\text{tr}}$ corresponds to $ k_{\text{f}} = (a_*/a_{\text{f}}) k_{\text{tr}} \simeq 250 $ in the final state $a_{\text{f}}(\tau=-1)=3.912 l_{\text{Pl}}$. Applying equation (\[freq\]) we obtain the maximal frequency for the present epoch $\nu_{\text{max}} \simeq 2 \cdot 10^{-12} \ \text{Hz}$. This value corresponds to the scales $\lambda_{\text{min}} \simeq 5 \ \text{kpc}$. It is instructive to consider the model without inflation. In this situation the maximal frequency of the relic gravitons would be $\nu_{\text{max}} \simeq 2 \cdot 10^{15} \ \text{Hz}$ what is extremely huge number. In fact it is possible that GUT energy scale and inflation cover each other in some place. It was somehow one of the motivation to introduce inflation to solve the problem of topological defects. In the case when the GUT scale occurs after thee reheating the problem of topological defects must be solved in a different way. We mention this problem to show that it is possible that the value ${a_{\text{today}}}/{a_{\text{f}}}$ can be lower than calculated before. After this analysis we can return now to equation (\[modeeq\]). We need to use the definition of the quantum correction $D$ and effective mass $m^2_{\text{eff}}$ in the quantum regime. It is also useful to rescale the conformal time and introduce $-\eta=-\tau+\beta$. We will back later to the previous definition because an additional degree of freedom $\beta$ is necessary to fit properly the boundary solutions with numerical one. Equation (\[modeeq\]) takes the form $$\frac{d^2}{d\eta^2}f(k,\eta)+\left[ D_*\xi^n(-\eta)^{np}k^2-p(p-1) \right]f(k,\eta) = 0$$ and the general solution in terms of Bessel functions have the form $$f (k,\eta)= C_1\sqrt{-\eta}J_{|\nu|}(x)+ C_2\sqrt{-\eta}Y_{|\nu|}(x) \label{solmodes}$$ with $$\begin{aligned} x &=& k \frac{2\sqrt{D_*\xi^n}}{|2+np|} (-\eta)^{(2+np)/2}, \\ \nu &=& -\frac{\sqrt{1+4p(p-1)}}{2+np}.\end{aligned}$$ With the use of the Wronskian condition (\[Wronskian\]) we can rewrite the solution (\[solmodes\]) to the form $$f (k,\eta) = \sqrt{\frac{\pi}{2|2+np|}}\sqrt{-\eta}\left[ D_1 H^{(1)}_{|\nu|}(x) + D_2 H^{(2)}_{|\nu|}(x) \right] \label{solmodes2}$$ where we introduced Hankel functions defined as $$\begin{aligned} H^{(1)}_{|\nu|}(x) &=& J_{|\nu|}(x) +i Y_{|\nu|}(x) \\ H^{(2)}_{|\nu|}(x) &=& J_{|\nu|}(x) -i Y_{|\nu|}(x).\end{aligned}$$ and the constants $D_1$ and $D_2$ enjoy the relation $|D_1|^2-|D_2|^2=1$. To fix values of the constants $D_1$ and $D_2$ we must consider the high energy limit, namely $x \gg 1 $. In this limit the Bessel functions behave as follow $$\begin{aligned} J_{|\nu|}(x) \rightarrow \sqrt{\frac{2}{\pi x}} \sin \left(x- \frac{|\nu|\pi}{2} -\frac{\pi}{4} \right), \\ Y_{|\nu|}(x) \rightarrow \sqrt{\frac{2}{\pi x}} \cos \left(x- \frac{|\nu|\pi}{2} -\frac{\pi}{4} \right),\end{aligned}$$ what give us $H^{(1)}_{|\nu|}(x) \rightarrow \sqrt{{2}/{(\pi x)}} \exp{\left[ix - i{|\nu|\pi}/{2} -i{\pi}/{4} \right]}$ and $H^{(2)}_{|\nu|}(x) \rightarrow \sqrt{{2}/{(\pi x)}} \exp{\left[ -ix +i{|\nu|\pi}/{2} +i{\pi}/{4} \right]}$. Classically the limit $x \gg 1 $ corresponds to advanced solution called the Bunch-Davies vacuum $e^{-ik\tau}/\sqrt{2k}$. Generally the limit obtained here differs from the classical one but can be restored taking $l=2$. Then to obtain the proper high energy limit we must choose $D_2=0$ and $D_1=\exp{\left[ i{|\nu|\pi}/{2} +i{\pi}/{4} \right]}$. Applying evaluated values of $D_1$ and $D_2$ to the solution (\[solmodes2\]) we finally obtain mode functions for the initial state $$\begin{aligned} f_i(k,\tau) &=& \mathcal{N} \frac{1}{\sqrt{k}}\sqrt{-k\tau+k\beta} H^{(1)}_{|\nu|}(x) \\ g_i(k,\tau) &=& \mathcal{N} \sqrt{k}\sqrt{-k\tau+k\beta} \left[ -\frac{1}{2} \frac{ H^{(1)}_{|\nu|}(x) }{-k\tau+k\beta } -\frac{2+np}{|2+np|}\sqrt{D_*\xi^n}(-\tau+\beta)^{\frac{np}{2}} \left( \frac{|\nu|}{x}H^{(1)}_{|\nu|}(x)-H^{(1)}_{|\nu|+1}(x) \right) \right]\end{aligned}$$ where $$\mathcal{N}=e^{i\left( \frac{|\nu|\pi}{2} +\frac{\pi}{4} \right)} \sqrt{\frac{\pi}{2|2+np|}}.$$ The functions $g_i(k,\tau)$ were calculated from relation $f(k,\tau)'=g(k,\tau)$ showed earlier. We had used also the expression for derivative of the Hankel functions in the form $$\frac{dH^{(1)}_{|\nu|}(x)}{dx} =\frac{|\nu|}{x}H^{(1)}_{|\nu|}(x)-H^{(1)}_{|\nu|+1}(x).$$ Similar investigations lead to the expression for the mode functions for the final state. We must remember that it is however not only a simple set $l=2$ but also the change of the solution for the scale factor to this expressed by (\[solution2\]). In this case we obtain $$\begin{aligned} f_f(k,\tau) &=& e^{i\frac{\pi}{4}} \sqrt{\frac{\pi}{4k}} \ \sqrt{k\tau+k\zeta}H^{(1)}_0(k\tau+k\zeta), \\ g_f(k,\tau) &=& e^{i\frac{\pi}{4}} \sqrt{\frac{\pi k}{4}}\sqrt{k\tau+k\zeta} \left[ \frac{1}{2} \frac{H^{(1)}_0(k\tau+k\zeta)}{k\tau+k\zeta} - H^{(1)}_1(k\tau+k\zeta) \right].\end{aligned}$$ Now we are ready to consider the creation of the gravitons during the transition from some initial to final states. The initial vacuum state $| 0_{\text{in}}\rangle$ is determined by $\hat{a}_{\text{k}}| 0_{\text{in}}\rangle = 0$, where $\hat{a}_{\text{k}}$ is the initial annihilation operator for $\tau_i$. The relation between annihilation and creation operators for the initial and final states is given by the Bogoliubov transformation $$\begin{aligned} \hat{b}_{{\bf k}} &=& B_{+}(k) \hat{a}_{{\bf k}} + B_{-}(k)^{*} \hat{a}_{-{\bf k}}^{\dagger} \ , \label{Bog1} \\ \hat{b}_{{\bf k}}^{\dagger} &=& B_{+}(k)^{*}\hat{a}_{{\bf k}}^{\dagger} + B_{-}(k) \hat{a}_{-{\bf k}} \label{Bog2} \end{aligned}$$ where $|B_{+}|^2-|B_{-}|^2=1$. Because we are working in the Heisenberg description the vacuum state does not change during the evolution. It results that $\hat{b}_{{\bf k}}| 0_{\text{in}}\rangle=B_{-}(k)^{*} \hat{a}_{-{\bf k}}^{\dagger}| 0_{\text{in}}\rangle $ is differ from zero when $B_{-}(k)^{*}$ is a nonzero function. This means that in the final state graviton field considered is no more in the vacuum state without particles. The number of produced particles in the final state is given by $$\bar{n}_{{\bf k}} = \frac{1}{2} \langle 0_{\text{in}} |\left[ \hat{b}_{{\bf k}}^{\dagger}\hat{b}_{{\bf k}}+ \hat{b}_{-{\bf k}}^{\dagger}\hat{b}_{-{\bf k}} \right]| 0_{\text{in}} \rangle =|B_{-}(k)|^2. \label{particles}$$ Using relations (\[sol11\])and (\[sol22\]) and the Bogoliubov transformation (\[Bog1\]) and (\[Bog2\]) we obtain $$\begin{aligned} B_{-}(k)&=& \frac{f_i(k,\tau_i)g_f(k,\tau_f) -g_i(k,\tau_i) f_f(k,\tau_f)}{f_f^*(k,\tau_f)g_f(k,\tau_f)-g_f^*(k,\tau_f)f_f(k,\tau_f)} = i\left[ f_i(k,\tau_i)g_f(k,\tau_f) -g_i(k,\tau_i) f_f(k,\tau_f) \right] \ , \\ B_{+}(k)&=& \frac{f_i(k,\tau_i)g_f^*(k,\tau_f) -g_i(k,\tau_i) f_f^*(k,\tau_f)}{f_f(k,\tau_f)g_f^*(k,\tau_f)-g_f(k,\tau_f)f_f^*(k,\tau_f)} =-i \left[ f_i(k,\tau_i)g_f^*(k,\tau_f) -g_i(k,\tau_i) f_f^*(k,\tau_f) \right] \end{aligned}$$ where simplifications come from the Wronskian condition (\[Wronskian\]). In the calculations we set $\tau_{{i}}=\tau_1=-20$ and $\tau_{{f}}=\tau_2=-1$. These boundaries fully cover the region of gravitational waves creation. The energy density of gravitons is given by $$d\rho_{\text{gw}} = 2 \cdot \hslash \omega \cdot \frac{4 \pi \omega^2 d\omega}{(2\pi c)^3} \cdot |B_{-}(k)|^2.$$ where we used definition (\[particles\]). The expression for the parameter $\Omega_{\text{gw}}$ defined by (\[omegaGW\]) takes now the form $$\Omega_{\text{gw}}(\nu) =\Omega_0 \cdot \nu^4 \cdot \bar{n}\left[ k = \nu \cdot 2\pi a_{\text{f}} \cdot \left( \frac{a_\text{today}}{a_{\text{f}}} \right) \right] \label{omegaGW1}$$ where $$\Omega_0 = \frac{\hslash c}{c^4}\frac{16\pi^2}{\rho_c} = \frac{ 16 \pi^2 \cdot 197.3 \cdot 10^{-15}[\text{MeV} \cdot \text{m}]}{3^4 \cdot 10^{32}[\text{m}^4/\text{s}^4] 1.05 \cdot 10^{-5} \cdot h^2_0 [\text{GeV}/\text{cm}^3] } = 3.66 \cdot h^{-2}_0 \cdot 10^{-49} \ [\text{Hz}^{-4}].$$ In the calculations we set present value of the Hubble factor for $h_{0}=0.7$. In Fig. \[spect1\] we show spectrum calculated with formula (\[omegaGW1\]). The obtained spectrum is extremely weak in the present epoch. The reason of this tiny amount of the background gravitons is the presence of the standard inflationary phase. The super-inflationary phase is placed before the inflation so the energy of gravitons decreases about $10^{27}$ times during this further phase. ![Spectrum of relic gravitons for the model with $j=100$ and $l=3/4$. Frequency scale in Hertz.[]{data-label="spect1"}](plot4.eps){width="7cm"} To see better how presence on the inflation affect this spectrum we show in Fig. \[spect1\] the spectrum of relic gravitons in the model without the inflationary phase. ![Spectrum of relic gravitons for the model without inflation. Frequency scale in Hertz.[]{data-label="spect2"}](plot5.eps){width="7cm"} It is clear that in such a model amount of relic gravitons would be extremely large. As we mentioned before it is possible that GUT energy scales cover partially with the inflation. In this situation the present amount of the relic gravitons would be higher than this in Fig. \[spect1\] and reaches to higher energies. For the present state we do not know the duration of the inflationary phase exactly. To estimate this value it is necessary to measure the spectrum of both scalar and tensor parts of primordial fluctuations produced during the standard inflation. For the present day we know only the contribution from the scalar part. The further generation of CMB telescopes in needful to improve the knowledge the properties of inflation. From our calculation we can see however that the inflationary phase is necessary. Without the inflation after the super-inflation the present amount of gravitons would be easily in reach of present observational skills. From this point of view we have found the next motivation to support the inflationary model. Summary ======= In summary, we have calculated the spectrum of gravitons produced during the super-inflationary phase induced by Loop Quantum Gravity effects. In the calculations we considered inverse-volume corrections to the dynamics and to the equation for the tensor modes. We have solved analytically equation for the tensor modes in the quantum and classical regimes. Both solutions we had matched by numerical solution for background dynamics. We have obtained spectra of relic gravitons for the models considered. In the first model we assumed the presence of the inflationary phase after the super-inflation. In that case we obtained presently a negligible amount of relic gravitons. However in the second model without inflation the present amount of graviton background would be unnaturally high. This results state that the period of inflation after the super-inflation is necessary to avoid the problem of relic gravitons. Nowadays the main motivation to introduce the inflationary phase comes from ability to creation of fluctuations. Our investigations based on loop quantum cosmology support the inflationary model. Results obtained in this work differ from our previous investigations [@Mielczarek:2007zy]. 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--- abstract: '[We prove a conjecture by W. Bergweiler and A. Eremenko on the traces of elements of modular group in this paper. ]{}' address: - 'Department of Mathematics, Changshu Institute of Technology, Changshu 215500, China' - 'Department of Mathematics, University of Texas of the Permian Basin, Odessa, TX, 79762' author: - Bin Wang and Xinyun Zhu title: On the traces of elements of modular group --- Introduction ============ W. Bergweiler and A. Eremenko made a remarkable conjecture on the traces of elements of modular group in [@E]. The main result of this paper is to prove their conjecture. We expect this result to have future applications in some fields such as control theory. Let $A =\left( \begin{array}{cc} 1 & 2 \\ 0 & 1 \end{array} \right)$ and $B =\left( \begin{array}{cc} 1 & 0 \\ -2 & 1 \end{array} \right)$. These two matrices generate the free group which is called $\Gamma(2)$, the principal congruence subgroup of level 2. With arbitrary integers $m_{j} \neq 0, n_{j} \neq 0$, consider the trace of the product $$p_k(m_1, n_1,..., m_k, n_k) = tr(A^{m_1}B^{n_1} \cdots A^{m_k}B^{n_{k}}).$$ It is easy to see that $p_k$ is a polynomial in $2k$ variables with integer coefficients. This polynomial can be written explicitly though the formula is somewhat complicated. Choosing an arbitrary sequence $\sigma$ of $2k$ signs $\pm $, we make a substitution $$p_{k}^{\sigma}(x_1, y_1,...x_k, y_k)=p_k(\pm (1+x_1), \pm (1+y_1),\cdots , \pm (1+x_k), \pm (1+y_k)).$$ Our main theorem is the following one. \[main\] The polynomial $p_k$, for every $k>0$, has the property that for every $\sigma$, all the coefficients of the polynomial $p_{k}^{\sigma}$ are of the same sign, that is, the sequence of coefficients of $p_{k}^{\sigma}$ has no sign changes. which was conjectured by W. Bergweiler and A. Eremenko in [@E]. We prove the theorem by induction on $k$. However it is not easy to pass from level $k$to level $k+1$since that $p_k$ has the above property does not simply imply that $p_{k+1}$ has the same one. The idea here is to substitute $p_k$’s with a suitable set of polynomials containing the $p_k$’s so that the difficulty disappears. This idea is explained in section 2 (see Proposition \[p1\]) and the theorem is showed in section 3. [**Acknowledgment**]{} We would like to thank Alex Eremenko for his helpful comments on the earlier draft of this paper. The first author is grateful to Jianming Chang for introducing the topic to him and for many helpful talks. Traces ====== Good polynomials ---------------- Set $$F_k = \begin{pmatrix} f_k &h_k \\ t_k &g_k \end{pmatrix} = A^{x_1}B^{y_1}A^{x_2}B^{y_2}\cdots A^{x_k}B^{y_k}$$ where $A= \begin{pmatrix} 1 &2 \\ 0 &1 \end{pmatrix},\,B= \begin{pmatrix} 1 &0 \\ -2 &1 \end{pmatrix}.$ Then the trace $p_k = trF_k = f_k + g_k$ and all $f_k, h_k, t_k, g_k$ are the polynomials in $2k$ variables $x_1, y_1, \cdots x_k, y_k$ with integer coefficients whose explicit formula can be found in [@E]. A sequence $\sigma$ of $2k$ signs $\pm$ can be viewed as a function $\sigma : \{1, 2, \cdots 2k \}\rightarrow \{1, -1\}$. For any polynomial $f$ in variables $x_1, y_1, \cdots x_k, y_k$, set $$f^{\sigma}=f(\si (1)(1+x_1), \si (2)(1+y_1), \cdots , \si (2k-1)(1+x_k), \si (2k)(1+y_{k}))$$ [A polynomial f in $2k$ variables is said to be good if for arbitrary sequence $\sigma$ of $2k$ signs, all the coefficients of $f^{\sigma}$ have the same sign. ]{} Let $Mat(2,2)$ be the set of $2 \times 2$ matrices over $\R$, the set of real numbers. Denote by $F_k^{\sigma}$ the matrix $\begin{pmatrix} f_k^{\sigma} &h_k^{\sigma}\\ t_k^{\sigma} &g_k^{\sigma} \end{pmatrix}$. If $M= \begin{pmatrix} a &c \\ b &d \end{pmatrix} \in Mat(2,2)$, then $$\begin{array}{rl} tr(F_kM) &= af_k + bh_k + ct_k +dg_k\\ tr(F_k^{\sigma}M) &= af_k^{\sigma} + bh_k^{\sigma} + ct_k^{\sigma} +dg_k^{\sigma} \end{array}$$ Write $$\begin{aligned} A_1= \begin{pmatrix} 1 &0 \\ 0 &0 \end{pmatrix}&& A_2= \begin{pmatrix} 2 &1 \\ 0 &0 \end{pmatrix}&& A_3= \begin{pmatrix} 2 &-1 \\ 0 &0 \end{pmatrix}, \\ A_4= \begin{pmatrix} 3 &2 \\ -2 &-1 \end{pmatrix}&& A_5= \begin{pmatrix} 5 &2 \\ 2 &1 \end{pmatrix}&& A_6= \begin{pmatrix} 5 &-2 \\ -2 &1 \end{pmatrix}.\end{aligned}$$ Note that $$A_4 + A_5 = 4A_2, \, A_4 + A_6= 4A_3^t, \, A_4^t + A_5= 4A_2^t, \, A_4^t + A_6 = 4A_3, A_2 + A_3 = 4A_1,\label{semi}$$ $$A_4= -A^{-1}B^{-1},\, A^t_4=-AB, \, A_5= AB^{-1},\, A_6=A^{-1}B. \label{eq}$$ Let $S$ be a subset of $Mat(2,2)$, we have \[p1\] If $S$ satisfies that p1) : $ a > 0$, for all $M= \begin{pmatrix} a &c \\ b &d \end{pmatrix} \in S$, P2) : $tr(CM)\geq 0$, for each $C \in \{A_4, A_{4}^t, A_5, A_6 \},\, M \in S$,where $D^t$ stands for the transpose of the matrix $D$, P3) : $CS\subseteq S, \text{for each} \,\, C \in \{A_4, A_{4}^t, A_5, A_6 \},$ then $af_k + bh_k + ct_k +dg_k$ is good, for every $M= \begin{pmatrix} a &c \\ b &d \end{pmatrix}\in S, k\geq 1.$ \[r1\] $S$ satisfies the conditions P1), P2) P3) if and only if so does the cone $Cone(S)\triangleq \{ \sum a_iM_i \,|\, a_i\geq 0, M_i \in S \}$. Furthermore, any set $S$ satisfying P1) possesses the property that $af_k + bh_k + ct_k +dg_k$ is good, for every $M= \begin{pmatrix} a &c \\ b &d \end{pmatrix}\in S, k\geq 1$ if and only if $Cone(S)$ satisfying P1) possesses the same property. [The first assertion is obvious and the second one follows from the fact that the sign of the leading term of $F_k^{\sigma}M) = af_k^{\sigma} + bh_k^{\sigma} + ct_k^{\sigma} +dg_k^{\sigma}$ with $a>0$ is independent of $a$ (see the proof of Lemma \[pass\])]{} We shall prove several lemmas before proving this proposition. Definition of $M^{ij}$ ---------------------- Let $\sigma$ be a sequence of $2k$ signs and let $\sigma_i, i=0, 1, 2, 3$, be the sequence of $2k+2$ signs such that (a) $\sigma_i(j)= \sigma (j), \text{for each}\, 1\leq j\leq 2k$ and (b) $\sigma_0 (2k+1)=1, \sigma_0 (2k+2)=1, \sigma_1 (2k+1)=1, \sigma_1 (2k+2)=-1, \sigma_2 (2k+1)=-1, \sigma_2 (2k+2)=1, \sigma_3 (2k+1)=-1, \sigma_3 (2k+2)=-1.$ Obviously every sequence of $2k+2$ signs equals to $\sigma_i$, for some $\sigma$ and $i$. For any $M= \begin{pmatrix} a &c \\ b &d \end{pmatrix}$, set $$\begin{array}{ll} M^{00} &= 4A_1M, \;\; M^{01} = 2A_3M, \;\; M^{02} = 2A_2^tM, \;\; M^{03} = A_4^tM \\ M^{10} &= 4A_1M, \;\; M^{11} = 2A_2M, \;\; M^{12} = 2A_2^tM, \;\; M^{13} = A_5M \\ M^{20} &= 4A_1M, \;\; M^{21} = 2A_3M, \;\; M^{22} = 2A_3^tM, \;\; M^{23} = A_6M \\ M^{30} &= 4A_1M, \;\; M^{31} = 2A_2M, \;\; M^{32} = 2A_3^tM, \;\; M^{33} = A_4M. \end{array}$$ For any $M= \begin{pmatrix} a &c \\ b &d \end{pmatrix} \in Mat(2,2)$ and $k\geq 1$, $$tr(F_{k+1}^{\sigma_i}M) = (-1)^{\tau (i)}(x_{k+1}y_{k+1}tr(F_k^{\sigma}M^{i0}) + x_{k+1}tr(F_k^{\sigma}M^{i1}) + y_{k+1}tr(F_k^{\sigma}M^{i2}) + tr(F_k^{\sigma}M^{i3}) \label{sign}$$ where $i= 0, 1, 2, 3, \, \tau (0)=\tau (3)= 1, \, \tau (1)= \tau (2)= 0.$ For $i=0$, $$\begin{aligned} F_{k+1}^{\sigma_0} &= F_k^{\sigma}A^{1+x_{k+1}}B^{1+y_{k+1}}=F_k^{\sigma}\begin{pmatrix} 1 &2+2x_{k+1} \\ 0 &1 \end{pmatrix} \begin{pmatrix} 1 &0 \\ -2-2y_{k+1} &1 \end{pmatrix}\\ &= F_k^{\sigma}\begin{pmatrix} -3-4x_{k+1}-4y_{k+1}-4x_{k+1}y_{k+1} &2+2x_{k+1} \\ -2-2y_{k+1} &1 \end{pmatrix}\\ &= -(x_{k+1}y_{k+1}F_k^{\sigma}(4A_1) + x_{k+1}F_k^{\sigma}(2A_3) + y_{k+1}F_k^{\sigma}(2A_2^t) + F_k^{\sigma}A_4^t)\end{aligned}$$ So, (\[sign\]) holds for $i=0$. Similarly for $i=1, 2, 3$. \[pass\]Let $\sigma$ be a sequence of $2k$ signs and $M= \begin{pmatrix} a &c \\ b &d \end{pmatrix}$, with $a > 0$ and assume further that the $(1,1)$ entry of $M^{ij}$ is also positive for every $i, j$. Then for any $0\leq i \leq 3$, all the coefficients of $tr(F_{k+1}^{\sigma_i}M)$ are of the same sign if and only if all the coefficients of $tr(F_{k}^{\sigma}M^{ij})$ are of the same sign, for $j= 0, 1, 2, 3.$ Note that the explicit formula of $f_k, h_k, t_k, g_k$ in [@E] implies that $deg(h_k)=deg(t_k)=2k-1, deg(g_k)=2k-2, deg(f_k)=2k$ and the leading term of $f_k$ is $(-1)^k4^kx_1y_1\cdots x_ky_k$. Hence if all the coefficients of $tr(F_{k}^{\sigma}M) = af_{k}^{\sigma}+ bh_{k}^{\sigma}+ ct_{k}^{\sigma} +dg_{k}^{\sigma}$, with $a > 0$, are of the same sign, then all the coefficients have the same sign with $(-1)^{k + \sharp(\sigma)}$ where $\sharp(\sigma)$ is the number of negative signs that $\sigma$ takes. Now the lemma follows immediately from $(\ref{sign})$. We have $f_1= 1- 4x_1y_1, \, h_1= 2x_1,\, t_1= -2y_1,\, g_1= 1.$ If set $F_0 = E$, the identity matrix, then $(\ref{sign})$ also holds for $k=0$, i.e. $$tr(F_1^{\sigma_i}M) = (-1)^{\tau (i)}(x_{1}y_{1}tr(M^{i0}) + x_{1}tr(M^{i1}) + y_{1}tr(M^{i2}) + tr(M^{i3}) \label{sign1}$$ where the sequences $\sigma_0, \sigma_1, \sigma_2, \sigma_3$ of $2$ signs are respectively $\{+, +\}, \{+, -\}, \{-, +\}, \{-, -\}$. Let $M= \begin{pmatrix} a &c \\ b &d \end{pmatrix}$. \[leval1\] For $a>0$ and assume the $(1,1)$ entry of $M^{ij}$ is positive, then $af_1 + bh_1 + ct_1 + dg_1$ is good if and only if $tr(M^{i3}) \geq 0, i=0, 1, 2, 3.$ It is easy to see by $(\ref{sign1})$ that $af_1 + bh_1 + ct_1 + dg_1$ is good if and only if $tr(M^{ij}) \geq 0, \text{for all} \,i, j$. Now the lemma follows immediately from (\[semi\]). Proof of Proposition \[p1\] --------------------------- We prove it by induction on $k$. For $k=1$, $M= \begin{pmatrix} a &c \\ b &d \end{pmatrix} \in S$, we have $af_1 + bh_1 + ct_1 + dg_1$ is good by Lemma \[leval1\]. Now assume $af_k + bh_k + ct_k + dg_k$ is good, for all $M= \begin{pmatrix} a &c \\ b &d \end{pmatrix} \in S$. One deduces first that all $M^{ij}$ are contained $Cone(S)$ by (\[semi\]) and the condition [p3)]{} that $S$ satisfies, and then that $tr(F_kM^{ij})$ is good by the induction hypothesis and Remark \[r1\]. Therefore $af_{k+1} + bh_{k+1} + ct_{k+1} + dg_{k+1}$ is good as well, by Lemma \[pass\]. Proof of the Main Theorem ========================= Decreasing matrices ------------------- A matrix $X = \begin{pmatrix} a &c\\b &d \end{pmatrix}$ is said to be decreasing if $|a| > |b| > |d| \,\text{and}\, |a| > |c| > |d|$, according to [@E]. The following lemma is proved in [@E]. \[dec\]Let $X \in \Gamma (2)$ be decreasing and $m, n \in \Z\backslash \{0\}$, then $Y=A^mB^nX$ is decreasing. Proof of Theorem \[main\] ------------------------- Let $\Delta = \{ K_1K_2\cdots K_n \,| \, n\geq 0, K_i = A_4, A_{4}^t, A_5, A_6 \}$ (for $n=0$, we mean the identity matrix $E$). By (\[eq\]) every matrix except the identity matrix in $\Delta$ is decreasing by Lemma \[dec\]. Now assume every word of length $n, M=K_1K_2\cdots K_n = \begin{pmatrix} a &c\\b &d \end{pmatrix}$ in $\Delta$, has the property that $a > 0$. Then for any word in $\Delta$ of length $n+1, M'=K_1K_2\cdots K_{n+1}= \begin{pmatrix} a' &c'\\b' &d' \end{pmatrix}$, it is easy to show that $a' > 0$ since $M$ is decreasing and $M'=MK_{n+1}$ with $K_{n+1}\in \{A_4, A_{4}^t, A_5, A_6 \}$. Hence we have proved, by induction, that for every $C = \begin{pmatrix} a &c\\b &d \end{pmatrix}\in \Delta,\, a > 0$, that is, $\Delta$ satisfies the condition P1). In addition it is easy to see that the trace of a decreasing matrix whose $(1,1)$ entry is positive is always positive. Thus $\Delta$ satisfies the condition P2) as well. Meanwhile $\Delta$ obviously satisfies the conditions P3) by the definition of $\Delta$. Therefore $af_k + bh_k + ct_k + dg_k$ is good, for all $k, \forall \, M= \begin{pmatrix} a &c \\ b &d \end{pmatrix} \in \Delta$, by Proposition \[p1\]. Now Theorem \[main\] follows. [999]{}=2ex=-1pt=-1pt W. Bergweiler and A. Eremenko, Goldberg’s constant, preprint.

--- abstract: 'The concept of a flock of a quadratic cone is generalized to arbitrary cones. Flocks whose planes contain a common point are called star flocks. Star flocks can be described in terms of their coordinate functions. If the cone is “big enough”, the star flocks it admits can be classified by means of a connection with minimal blocking sets of Rédei type. This connection can also be used to obtain examples of bilinear flocks of non-quadratic cones.' address: 'Department of Mathematical and Statistical Sciences, University of Colorado Denver, Campus Box 170, P.O. Box 173364, Denver, CO 80217-3364, U.S.A.' author: - William Cherowitzo title: 'Flocks of Cones: Star Flocks' --- Introduction ============ This is the second (see [@WEC2]) in a series of articles devoted to providing a foundation for a theory of flocks of arbitrary cones in $PG(3,q)$. The desire to have such a theory stems from a need to better understand the very significant and applicable special case of flocks of quadratic cones in $PG(3,q)$. Flocks of quadratic cones have connections with several other geometrical objects, including certain types of generalized quadrangles, spreads, translation planes, hyperovals (in even characteristic), ovoids, inversive planes and quasi-fibrations of hyperbolic quadrics. This rich collection of interconnections is the basis for the strong interest in such flocks. Cones and Flocks ================ Let $\pi_0$ be a plane and $V$ a point not on $\pi_0$ in $PG(3,q)$. Let $\mathcal{S}$ be any set of points in $\pi_0$ (including the empty set). A *cone*, $\Sigma = \Sigma(V,\mathcal{S})$ is the union of all points of $PG(3,q)$ on the lines $VP$ where $P$ is a point of $\mathcal{S}$. $V$ is called the *vertex* and $\mathcal{S}$ is called the *carrier* of $\Sigma$. $\pi_0$ is the *carrier plane* and the lines $VP$ are the *generators* of $\Sigma$. In the event that $\mathcal{S} = \emptyset$ we call $\Sigma$ the *empty cone* and by convention consider it to consist of only the point $V$. A *flock of planes* in $PG(3,q)$ is any set of $q$ *distinct* planes of $PG(3,q)$. As $q$ planes can not cover all the points of $PG(3,q)$, there always are points of the space which do not lie in any of the planes in a flock of planes. If $\Sigma$ is a cone of $PG(3,q)$, then a flock of planes, $\mathcal{F}$, is said to be a *flock of* $\Sigma$ when the vertex of $\Sigma$ lies in no plane of $\mathcal{F}$ and no two planes of $\mathcal{F}$ intersect at a point of $\Sigma$. Any flock of planes is a flock of a cone, possibly only the empty cone. In general, however, a given flock of planes will be a flock of several cones. In the literature on flocks of quadratic cones, the approach is always to consider a fixed quadratic cone and study the flocks of that cone. We will change the viewpoint and consider, for a fixed flock of planes, the various cones of which it is a flock. In the sequel we shall refer to a flock of planes simply as a *flock* and it shall be understood that it is always a flock of a cone, even if the cone is not explicitly indicated. Furthermore, we shall always assume, unless explicitly stated otherwise, that a flock is a flock of a non-empty cone. Let $\mathcal{F}$ be a flock. We can introduce coordinates in $PG(3,q)$ so that the plane $x_3 = 0$ is one of the planes of the flock and the point $V = ( 0,0,0,1 )$ is not in any plane of the flock. Since $V$ is not in any plane of $\mathcal{F}$, each of the planes of this flock has an equation of the form $Ax_0 + Bx_1 + Cx_2 - x_3 = 0$. We parameterize the planes of $\mathcal{F}$ with the elements of $GF(q)$ in an arbitrary way except that we will require that $0$ is the parameter assigned to the plane $x_3 = 0$. We can now describe the flock as, $\mathcal{F} = \{\pi_t\colon f(t)x_0 + g(t)x_1 + h(t)x_2 - x_3 = 0 \mid t \in GF(q)\}$ with $\pi_0 \colon x_3 = 0$. The functions $f,g \text{ and } h$ are called the *coordinate functions* of the flock. Note that the requirement on the parameter $0$ means that $f(0) = g(0) = h(0) = 0$. If $f,g \text{ and } h$ are the coordinate functions of the flock $\mathcal{F}$ we shall write $\mathcal{F} = \mathcal{F}(f,g,h)$. We remark that the coordinate functions of a flock depend on the parameterization of the flock. As all cones under consideration have vertex $V = ( 0,0,0,1 )$ and the plane $\pi_0$ as the carrier plane, a cone is determined when its carrier $\mathcal{S}$, a point set in $\pi_0$, is specified. Given a flock $\mathcal{F}$, there is a largest set $\mathcal{S}_0$ of $\pi_0$ such that $\mathcal{F}$ is a flock of the cone with carrier $\mathcal{S}_0$. This cone is called the *critical cone* of $\mathcal{F}$. If $\mathcal{C}$ is any subset of the carrier of the critical cone of a flock $\mathcal{F}$, then clearly $\mathcal{F}$ is also a flock of the cone with carrier $\mathcal{C}$. Thus, determining the critical cone of a flock implicitly determines all cones for which this flock of planes is a flock. \[Th:herd\] A point $( a,b,c,0 )$ is in the carrier of the critical cone of the flock $\mathcal{F} = \mathcal{F}(f,g,h)$ in $PG(3,q)$ if and only if the function $t \mapsto af(t) + bg(t) + ch(t)$ is a permutation of $GF(q)$. A point on the line $\langle (0,0,0,1), (a,b,c,0) \rangle$ other than the vertex has coordinates $( a,b,c,\lambda )$ with $\lambda \in GF(q)$. Such a point is on the plane $\pi_t$ of $\mathcal{F}$ if and only if $\lambda = af(t) + bg(t) + ch(t)$. No two planes of $\mathcal{F}$ will meet at the same point of this line if and only if $t \mapsto af(t) + bg(t) + ch(t)$ is a permutation of $GF(q)$. Thus, under this condition, the point $( a,b,c,0 )$ will be in the carrier of the critical cone of $\mathcal{F}$. The critical cone of a flock may be fairly “small”. Besides the empty cone, we will consider cones whose carriers consist of collinear points as being “small”. Cones of this type are called *flat cones*. For the most part, we shall regard flocks whose critical cones are flat as being uninteresting. For any nonempty set $S$ and any point $P$ of a projective plane, define $w_S(P)$ to be the number of lines through $P$ which contain an element of $S$. In the projective plane $\Pi$ we can define the *width* of a set $S$ to be $W_S = min \{ w_S(P) \mid P \in \Pi \}$. Clearly, $W_S = 1$ if and only if $S$ consists of a set of collinear points. If $S$ is an oval in a projective plane of order $q$ ($q+1$ points, no three of which are collinear), then $W_S = \frac{q+1}{2}$ if $q$ is odd or $W_S = \frac{q+2}{2}$ if $q$ is even. This can be written as $W_S = \lfloor \frac{q+2}{2} \rfloor$, independent of the parity of $q$. Using this idea we can provide an admittedly crude classification of critical cones. In $PG(3,q)$, if $S$ is the set of points of a carrier of a cone, then if $W_S < \lfloor \frac{q+2}{2} \rfloor$, we call the cone a *thin cone*. Cones which are not thin are called *wide* and a wide cone with at least $q+1$ points in a carrier are called *thick cones*. The class of thick cones contains the quadratic cones as well as all cones whose carrier contains any oval. Definitions and Basic Properties ================================ A *linear flock* is a flock whose planes share a common line. Any cone whose carrier is not a proper blocking set (a set of points which contains no line and which every line intersects) in its carrier plane admits linear flocks. Linear flocks can easily be characterized by their coordinate functions. \[Th:6.1\] A flock is a linear flock if and only if the three coordinate functions are scalar multiples of each other. A *star flock* is a flock whose planes share a common point and a *proper* star flock is one for which this common point is unique. The common point of a proper star flock is called the *star point*. Linear flocks are clearly star flocks, but not proper star flocks. We may also characterize star flocks in terms of their coordinate functions. \[Th:7.1\] A flock is a star flock if and only if the coordinate functions are linearly dependent over GF(q). Since a flock is a set of planes of $PG(3,q)$ it is natural to define the equivalence of flocks without reference to the cones that they are flocks of. While there are several ways to do this, the following has proved to be most useful. Two flocks, $\mathcal{F}_1$ and $\mathcal{F}_2$ are *equivalent* if they are in the same orbit of the group $P{\Gamma}L(4,q)_{V,x_3=0}$. That is, there is a collineation of $PG(3,q)$ fixing the point $V = ( 0,0,0,1 )$ and stabilizing the plane $x_3 = 0$ which maps the planes of $\mathcal{F}_1$ to those of $\mathcal{F}_2$. \[Th:11.1.1\] A star flock is equivalent to a flock $\mathcal{F}(t,g(t),0)$, where $0$ denotes the function which is identically zero. Let $P$ be a point in the carrier of the critical cone and $Q$ the star point of a star flock. By using a collineation of $PG(3,q)$ which fixes $(0,0,0,1)$ and stabilizes $x_3 = 0$, we may assume that $P = (1,0,0,0)$ and $Q = (0,0,1,0)$. If the star flock is given by $\mathcal{F} = \mathcal{F}(f,g,h)$, then by Theorem \[Th:herd\], $t \mapsto f(t)$ is a permutation. We can re-parameterize the planes of the flock so that $f(t) = t$. Since all the planes of the flock pass through $Q$, we must have $h \equiv 0$. Star Flocks of Wide Cones ========================= Viewing flocks in a dual setting has been an effective technique in the classical quadratic cone situation. It is especially useful in studying star flocks. Let $\mathcal{F}$ be a flock with critical cone $\Sigma = \Sigma(V,\mathcal{S})$ in $PG(3,q)$. By passing to the dual, $\mathcal{F}$ becomes a set of $q$ points in $PG(3,q)$ and the cone is the set of all planes passing through a set of lines in the plane corresponding to $V$, with the property that no line determined by a pair of the $q$ points lies in any of these planes. If $\mathcal{F}$ is a proper star flock, then the corresponding $q$ points in the dual are coplanar. We fix some notation for this situation. Let $\mathcal{F} = \mathcal{F}(t,g(t),0)$ be a star flock of the cone $\Sigma = \Sigma(V,\mathcal{S})$ with $V = ( 0,0,0,1)$ and the planes of $\mathcal{F}$ given by $tx_0 + g(t)x_1 + - x_3 = 0$. For the sake of clarity, we employ the notational device of using $(X,Y,Z,W)$ to refer to the generic coordinates of a point when viewed in the dual setting. Using the duality $(x,y,z,w) \leftrightarrow [x,y,z,-w]$, the flock becomes a set of points $D_F = \{(t, g(t), 0, 1) \colon t \in GF(q)\}$. $V$ becomes the plane with equation $W = 0$. The points on a generator of $\Sigma(V,\mathcal{S})$ become the set of all planes passing through a line in $W = 0$. The set of lines in $W = 0$, corresponding to the generators of $\Sigma$ will be denoted by $D_G$, and the set of all planes passing through the lines of $D_G$ will be denoted by $D_{\Sigma}$. The condition that lines formed by pairs of points of $D_F$ do not lie in the planes of $D_{\Sigma}$ is equivalent to the condition that these lines do not intersect the lines of $D_G$. The $q$ points of $D_F$ lie in the plane $Z = 0$, and since $\mathcal{F}$ is proper, they are not all collinear. Let $m$ be the line of intersection of the planes $W = 0$ and $Z = 0$. The $q$ points of $D_F$ lie in the affine plane obtained by removing the line $m$ from $Z = 0$. Since $\mathcal{F}$ is a flock, the line joining any two of these points can not intersect $W=0$ at a point on any line of $D_G$. Thus, the set of points on $m$ at which these lines meet $m$ is disjoint from the set of points on $m$ at which the lines of $D_G$ meet $m$. Let $M$ denote the set of points of $m$ at which lines determined by pairs of points of $D_F$ meet $m$, and let $N = |M|$. $N$ can be thought of as the number of “slopes” determined by the set of $q$ points of $D_F$. A *blocking set* in a projective plane is a set of points in the plane which every line intersects. Let $n$ be the largest number of collinear points in a blocking set. In a plane of order $q$, a blocking set of size $q+n$ is called a *Rédei blocking set*. \[Pr:Redei\] In $Z = 0$ the points of $D_F \cup M$ form a Rédei blocking set. Consider a point of $m$ which does not lie in $M$. The $q$ lines through this point other than $m$ must each contain exactly one point of $D_F$. Therefore, $D_F \cup M$ is a blocking set. Since the points of $D_F$ are not collinear, no collinear set of points of $D_F$ can contain more than $N-1$ points. Thus $D_F \cup M$ is of Rédei type of size $q+N$. The problem of determining the number of slopes determined by $q$ points in an affine plane has been well studied ([@LR:70],[@LoSc:81],[@BlBrSz:95],[@BlBaBrStSz:99],[@SB:02]) due to the connection with blocking sets of Rédei type. This problem is completely settled in the cases we are interested in and the relevant theorem (paraphrased in our terminology) is due to Ball [@SB:02] who settled two open cases of the main theorem first given by Blokhuis, Ball, Brouwer, Storme and Szönyi [@BlBaBrStSz:99]. \[Th:11.1.2\] Let $D_F$ consist of $q$ points of $PG(2,q)$ whose coordinates are given by $(t,g(t),1)$, $t \in GF(q)$, where $g$ is a permutation polynomial with $g(0) = 0$. If $q = p^n$ for some prime $p$, let $e$ be the largest integer so that any line of the plane containing at least two points of $D_F$ contains a multiple of $p^e$ points of $D_F$. If $N$ is the number of “slopes” determined by the points of $D_F$ then we have one of the following: 1. $e = 0$ and $\frac{q+3}{2} \le N \le q+1$, 2. $e \mid n$ and $p^{n-e} + 1 \le N \le \frac{q-1}{p^e -1}$, 3. $e = n$ and $N = 1$. Moreover, if $p^e > 2$, then g is a $p^e$-linearized polynomial. A *$p^e$-linearized polynomial* is one of the form: $$g(t) = \sum_{i=0}^{k-1} \alpha_i t^{p^{ie}}, \text{ where } \alpha_i \in GF(p^n), n = ke.$$ In this theorem, all the bounds for $N$ are sharp. The $p^e$-linearized permutation polynomials form a group under composition mod($x^q-x$). This is the Betti-Mathieu group (see [@LiNi:97]) and it is known to be isomorphic to $GL(k,p^e)$ where $q = p^{ek}$. Thus, \[Pr:13\] Let $q = p^n$ and $s = p^e$ with $n = ek$. Then the number of monic $s$-linearized permutation polynomials over $GF(q)$ is: $$\label{Eq:13} s^{\frac{k(k-1)}{2}}\prod_{i=1}^{k-1} (s^i-1).$$ Theorem \[Th:11.1.2\] has the following implication for star flocks of wide cones: \[Th:11.1.3\] If $q = p^n$ with $p$ a prime, then $\mathcal{F}$ is a proper star flock with a wide critical cone if and only if there exists a coordinatization such that $\mathcal{F} = \mathcal{F}(t,g(t),0)$ where $g$ is a non-linear $p^e$-linearized permutation polynomial for some $e \mid n$ with $e < n$. Furthermore, the critical cone of this star flock contains the points $(x, f(x), 1, 0)$ for $x \ne 0$ if and only if $$\label{Eq:12} f(x)g(t) + xt = 0 \mbox{ has no solutions with } xt \ne 0 .$$ By Theorem \[Th:11.1.1\] we may assume that $\mathcal{F} = \mathcal{F}(t,g(t),0)$ is a proper star flock of a wide cone with star point $P = (0,0,1,0)$ and $g$ a non-linear permutation polynomial (non-linearity follows from Theorem \[Th:6.1\]). Since the cone is wide, there are at least $\lfloor \frac{q+2}{2} \rfloor$ lines in $x_3 = 0$ through $P$ which contain the points of the critical cone of this flock. Thus, there are at least $\lfloor \frac{q+2}{2} \rfloor$ planes determined by these lines and the vertex of the cone. We now consider the dual setting and use the notation introduced at the beginning of this section. These planes correspond to points in $W=0$ which are also in $Z=0$, i.e., points on $m$. Since these planes contain points of the carrier, the points on $m$ that they correspond with are points where lines of $D_G$ meet $m$. Thus, for a wide cone, we must have $N \le q+1 - \lfloor \frac{q+2}{2} \rfloor = \lfloor \frac{q+1}{2} \rfloor$. Since the star flock is proper, we must also have $N> 1$. We can identify $Z=0$ with the projective plane $\Pi$ obtained by suppressing the third coordinates. Thus, the points of $D_F$ have coordinates $(t,g(t),1)$ in $\Pi$. By Theorem \[Th:11.1.2\], if $p^e = 1$ or $2$, the number of slopes determined by $D_F$ does not satisfy $1 < N \le \lfloor \frac{q+1}{2} \rfloor$. Thus, $p^e > 2$, and we have that $g$ is a $p^e$-linearized polynomial. Now, suppose that g is a non-linear $p^e$-linearized polynomial for some $e \mid n$ with $e < n$. Such functions are additive (i.e., $g(t+s) = g(t) + g(s)$), so to calculate $N$ we need only calculate the number of distinct values of $g(t)/t$ for $t \ne 0$. Let $K$ be the proper subfield of $GF(q)$ of order $p^e$. For each $c \in K$ we have $g(ct) = cg(t)$. Thus, for $c \in K \setminus \{0\}, g(ct)/ct = g(t)/t$, and the number of distinct non-zero values of $g(t)/t$ is at most $(q-1)/(p^e-1)$. Therefore, $N \le (q-1)/(p^e-1) + 1 < \lfloor \frac{q+1}{2} \rfloor$ if $p^e > 2$. We obtain a star flock of a wide cone from such a $g$, and it is proper since $g$ is not linear. Finally, For each $x \ne 0$, $f(x)g(t) + xt$ is a $p^e$-linearized polynomial (in $t$). Such polynomials represent permutations if and only if they vanish only at $t=0$ (Dickson [@LED:01]). Thus, (\[Eq:12\]) insures that each of the points $(x,f(x),1,0)$ is in the critical cone of $\mathcal{F}$ by Theorem \[Th:herd\]. \[Ex:1\] Let $q = p^h, \quad p$ an odd prime, and $h > 1$. Let $\sigma = p^i, 1 \le i \le h-1$. Then $t^{\sigma}$ is a $p^{gcd(i,h)}$-linearized polynomial. In $x_3 = 0$ the points $(x, -m/x, 1, 0), (1,0,0,0)$ and $(0,1,0,0)$ form a conic ($xy=-m$). $\frac{-m}{x}t^{\sigma} + xt = xt(\frac{-mt^{\sigma -1}}{x^2} + 1)$ will have no solutions with $xt\ne 0$ if and only if $m$ is a non-square in $GF(q)$. Thus, $\mathcal{F}(t, t^{\sigma}, 0)$ will be a (proper) star flock of the quadratic cone $xy = -m$ when $m$ is a non-square. These star flocks are known as the *Kantor-Knuth* (or K1) flocks of a quadratic cone. These examples come from a special case ($k=0$) of the fact that the points $(x,f(x),1,0), x \ne 0$ where $f(x) = \frac{-m}{x^{2k+1}}$, $0 \le k \le \frac{q-1}{2}$ are in the critical cone of this star flock when $m$ is a non-square. \[Co:11.1.1\] If $q$ is a prime, then all star flocks of wide cones in $PG(3,q)$ are linear. Since a prime field has no proper subfields, there are no $p^e$-linearized polynomials other than the linear ones, so by Theorem \[Th:11.1.3\] there are no proper star flocks. \[Co:11.1.2\] If $q = 2^p$, with $p$ a prime, then all star flocks of wide cones in $PG(3,q)$ are linear. If a star flock were not linear in this situation, then we would have $p^e = 2$ which is ruled out since it gives too large an $N$ for a wide cone. We examine two classes of examples of proper star flocks of wide cones which correspond to extreme values of $N$. Let $E = GF(q) = GF( p^n)$ with a subfield $K = GF(p^e)$, then the permutation function $g(t) = k(t^{p^e} - ct)$, where $c \ne {\beta}^{p^e - 1}$ with $k, \beta \in GF(q)^*$, gives $N = (q-1)/(p^e -1)$ and for $p^e > 2$ we have $N < \lfloor \frac{q+1}{2} \rfloor $, and so, $\mathcal{F}(t,g(t),-(at + bg(t)))$ for $a,b \in GF(q)$ is a star flock of a wide cone. These examples include those of Example \[Ex:1\] in odd characteristic, so we shall call them (or any flocks equivalent to them, see [@WEC2]) *Kantor-Knuth star flocks*. Under the same assumptions about $q$ and $e$, the function $g(t) = k_1(Tr_{E/K}(\frac{t}{k_2}) - ct)$, where $c \ne \frac{1}{\beta}Tr_{E/K}(\frac{\beta}{k_2})$, $k_1, k_2, \beta \in GF(q)^*$ and $Tr_{E/K}$ is the relative trace function from $E$ onto $K$, gives $N = p^{n-e} + 1$, and again, if $p^e > 2, N < \lfloor \frac{q+1}{2} \rfloor$. We call these proper star flocks of wide cones, *Holder-Megyesi star flocks* (L. Megyesi gave the example in the Rédei blocking set context and L. Holder, independently, gave the example in the conic blocking set context (see [@LH:04]). \[Co:11.1.3\] If $q=p^2$ for any prime $p$, or $p = 4$, then all star flocks of wide cones in $PG(3,q)$ are either linear or Kantor-Knuth star flocks. All flocks of wide cones in $PG(3,4)$ are linear (see [@WEC2]) so we may assume that $p > 2$. Up to scalar multiples the only $p$-linearized permutation polynomials are of the form $t^p - ct$, where $c \ne {\beta}^{p-1}$ for any $\beta \in GF(q)^*$ or $t$. In the case of $PG(3,16)$, the 2-linearized polynomials do not give star flocks of wide cones, so the only $s$-linearized polynomials giving star flocks of wide cones have $s = 4$. \[Ex:2\] In PG(2,16) there are two projectively inequivalent hyperovals, the hyperconic (a.k.a. regular hyperoval) and the Lunelli-Sce hyperoval. The cone over a hyperconic (quadratic cone) admits only linear star flocks (Thas [@JAT:87]), but a cone over the Lunelli-Sce hyperoval admits both linear and Kantor-Knuth star flocks. Let $\lambda$ be a primitive element of $GF(16)$ satisfying $\lambda^4 = \lambda + 1$. The point sets of $x_3 = 0$ given by $\mathcal{H}_i = \{(x,f_i(x),1,0)\colon x \in GF(16)\} \cup \{(0,1,0,0), (1,0,0,0)\}$ with $i=1,2$, where $$\begin{gathered} f_1(x) = \lambda^{13}x^{14} + \lambda^3 x^{12} + \lambda^6 x^{10} + x^8 + \lambda^6 x^6 + \lambda^3 x^4 + \lambda^{13}x^2 \makebox{ and } \\ f_2(x) = \lambda^4 x^{14} + \lambda^{10} x^{12} + \lambda^{11} x^{10} + \lambda^{11} x^8 + x^6 + \lambda^2 x^4,\end{gathered}$$ represent Lunelli-Sce hyperovals which intersect in nine points, the maximum number of points in the intersection of two hyperovals in $PG(2,16)$. Among the points of intersection are the points $(0,0,1,0),$ $(1,1,1,0),$ $(\lambda^{10}, \lambda^{13}, 1,0),$ $(\lambda^8, \lambda^2,1,0),$ $(\lambda^{14}, \lambda^5, 1,0)$ and $(\lambda^{13}, \lambda^{10},1,0)$. The slopes of the lines through $(0,0,0,1)$ and each of the other five points form a set of values $\{\lambda^{3i} \colon 0 \le i \le 4 \}$, i.e., the non-zero cubes in $GF(16)$. The symmetric difference of these two hyperovals, $\mathcal{H}_1 \triangledown \mathcal{H}_2$ is another Lunelli-Sce hyperoval (see [@BrCh:00]) with these five lines as exterior lines through the point $(0,0,1,0)$. The proper Kantor-Knuth star flock, $\mathcal{F}(t,t^4,0)$, with star point $(0,0,1,0)$ contains $\mathcal{H}_1 \triangledown \mathcal{H}_2$ in the carrier of its critical cone. \[Co:11.1.4\] If $q = p^3$ for any prime $p$ or $p = 4$, then all star flocks of wide cones in $PG(3,q)$ are either linear, Kantor-Knuth or Holder-Megyesi star flocks. If $p = 2$, then a star flock of a wide cone must be linear by Corollary \[Co:11.1.2\], so we may assume that $p > 2$. By Sherman [@BFS:02], only two values of $N$ occur when $q = p^3$, namely $p^2 + 1$ and $p^2 + p + 1$. Furthermore, when $N = p^2 + 1$ then $g(t) = k_1(Tr_{E/K}(\frac{t}{k_2}) - ct)$, where $c \ne \frac{1}{\beta}Tr_{E/K}(\frac{\beta}{k_2})$, $k_1, k_2, \beta \in GF(q)^*$. There are $(p^2 + p + 1)(p^3 - p^2 - 1)$ projectively equivalent monic polynomials of this type (Holder-Megyesi). There are $(p^3 - p^2 - p -1)^2$ projectively equivalent monic polynomials of the form $t^{p^2} - at^p - ct$ with $a \ne {\beta}^{p^2-p}$ for any $\beta \in GF(q)^*$ which are permutation polynomials. These are projectively equivalent to the $p^3 - p^2 - p - 1$ monic permutations of the form $t^p - at$ with $a \ne {\beta}^{p-1}$ for any $\beta \in GF(q)^*$ (Kantor-Knuth). Together with the single monic linear function, we have accounted for all the monic $p$-linearized permutation polynomials (Proposition \[Pr:13\]). In principle we could continue in this vein and classify all star flocks of wide cones. Indeed, it would be easy enough to do the next case of $q = p^4$, since Sherman [@BFS:02] has already determined the spectrum of values of $N$ in this case. Bilinear Star Flocks ==================== A *bilinear flock* is a flock in which each plane passes through at least one of two distinct lines of $PG(3,q)$ (carrier lines). If the carrier lines of a bilinear flock meet, the flock is a proper star flock. Bilinear flocks in $PG(3,K)$ with $K$ infinite have been studied (Biliotti and Johnson [@BiJo:99]), but no bilinear flocks of quadratic cones are known in the finite case. In the dual setting for $PG(3,q)$, a proper bilinear star flock corresponds to a Rédei blocking set whose $q$ affine points lie on two distinct lines. In this situation, using the notation of the previous section, there will exist lines containing exactly two points of $D_F$ and so, $p^e \le 2$. It follows from the proof of Theorem \[Th:11.1.3\] that any cone of such a flock must be thin. \[Th:nobi\] In $PG(3,q)$ there are no proper bilinear star flocks of wide cones. This result is sharp as the following examples show. Let $q$ be odd. A well known Rédei blocking set in $PG(2,q)$ is the *projective triangle* of side $(q+3)/2$ (see [@JWPH:79]). This consists of $3(q+1)/2$ points with $(q+3)/2$ points on each side of a triangle (including the vertices) such that the line joining any two points of the set on different sides of the triangle meets the third side at a point of the set. A representation of a projective triangle which includes the point $(0,0,1)$ has affine points of the form $(x,x^{\frac{q+1}{2}},1)$ and infinite points $(1,\frac{1+z}{1-z},0) \cup (1,-1,0) \cup (1,1,0)$ for each non-square $z \in GF(q)$. Note that the affine points of the set lie on the two lines $y = x$ (for square $x$) and $y = -x$ (for non-square $x$). This representation gives rise to the proper star flock $\mathcal{F}(t,t^{\frac{q+1}{2}},0)$ which can only be a flock of a thin cone. In particular, since the star point is $(0,0,1,0)$ the critical cone consists only of points other than $(0,0,1,0)$ on the lines $y = cx$ where $\frac{1-c}{1+c}$ is a non-zero square. There are $\frac{q-1}{2}$ such values of $c$ if $-1$ is not a square in $GF(q)$ and $\frac{q-3}{2}$ otherwise. The point $(0,1,0,0)$ is in the critical cone, and hence the points of the line $x = 0$ other than $(0,0,1,0)$ if and only if $-1$ is a square in $GF(q)$, that is, $q \equiv 1 \pmod{4}$. Therefore, for any odd $q$, the width of the base of this cone is $\frac{q-1}{2} < \lfloor \frac{q+2}{2} \rfloor$ and so the cone is thin, but any larger cone would be wide. A special case of this example occurs in $PG(2,q^2)$ for $q$ an odd prime power. The map $x \mapsto x^q$ is an involutory automorphism of $GF(q^2)$. The flock $\mathcal{F}(t,At^{\frac{q^2 + 1}{2}},0)$, where $A^q = -A$ has a critical cone which contains the points of $x^qy = z^{q+1}$ in the carrier plane $w = 0$. In the dual setting, the points of $Z = 0$ on the lines of $D_G$ are (with Z coordinate suppressed) $(0,1,0)$ and $\{(1,\alpha,0) \mid \alpha \in GF(q)\}$. The points of $W=0 $ on the lines determined by pairs of points of $D_F$ are (again with Z coordinate suppressed): $(1,A,0)$,$(1,-A,0)$ and $\{(1,A(\frac{1+z}{1-z}),0) \mid z \text{ is a nonsquare in }GF(q^2) \}$. All of these points lie on the line $m = \{W=0\} \cap \{Z=0\}$ and we wish to determine $A$ so that the sets are disjoint. Clearly, $A$ can not lie in $GF(q)$. The condition that $A(\frac{1+z}{1-z}) \in GF(q)$ is (with $\lambda$ a primitive element of $GF(q^2)$), $(A^{q-1}-1)(\lambda^{(q+1)(2k+1)} - 1) = (A^{q-1}+1)(\lambda^{(q-1)(2k+1)}-1)$ for all integer $k$. Assuming $A^q = -A$, this reduces to $\lambda^{(q+1)(2k+1)} = 1$, which implies that $q-1 \mid 2k + 1$, a contradiction since $q$ is odd. The analogous example for even $q$ is the *projective triad* of side $(q+2)/2$ (see [@JWPH:79]). This set consists of $(3q+2)/2$ points with $(q+2)/2$ points on each of three concurrent lines (point of intersection included) such that the line joining any two points of the set on different lines of the triad meets the third line at a point of the set. A representation of a projective triad which includes the point $(0,0,1)$ has affine points $(x,\mathrm{tr}(x),1)$, where “$\mathrm{tr}$” is the absolute trace function from $GF(q^2)$ to $GF(2)$, and infinite points $(1,\frac{1}{a},0) \cup (1,0,0)$ where $\mathrm{tr}(a) = 1$. The critical cone of the proper start flock which arises from this example consists only of points other than $(0,0,1,0)$ on the lines $y = cx$ where $\mathrm{tr}(c) = 0$. Note that $(0,1,0,0)$ is never in the critical cone since the absolute trace function is not a permutation. The width of the base of the critical cone is therefore $\frac{q}{2}$ and the critical cone is thin, but again any larger cone would be wide. 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--- abstract: 'Given number fields $L \supset K$, smooth projective curves $C$ defined over $L$ and $B$ defined over $K$, and a non-constant $L$-morphism $h \colon C \to B_L$, we consider the curve $C_h$ defined over $K$ whose $K$-rational points parametrize the $L$-rational points on $C$ whose images under $h$ are defined over $K$. Our construction gives precise criteria for deciding the applicability of Faltings’ Theorem and the Chabauty method to find the points of the curve $C_h$. We provide a framework which includes as a special case that used in Elliptic Curve Chabauty techniques and their higher genus versions. The set $C_h(K)$ can be infinite only when $C$ has genus at most $1$; we analyze completely the case when $C$ has genus 1.' address: - 'Mathematical Institute, University of Oxford, 24–29 St. Giles, Oxford OX1 3LB, United Kingdom' - 'Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom' author: - 'E.V. Flynn' - 'D. Testa' date: '30 September, 2012' title: Finite Weil restriction of curves --- Introduction {#introduction .unnumbered} ============ Let $L$ be a number field of degree $d$ over $\mathbb{Q}$, let $f \in L[x]$ be a polynomial with coefficients in $L$, and define $$A_f := \bigl\{ x \in L ~ \mid ~ f(x) \in \mathbb{Q} \bigr\} .$$ Choosing a basis of $L$ over $\mathbb{Q}$ and writing explicitly the conditions for an element of $L$ to lie in $A_f$, it is easy to see that the set $A_f$ is the set of rational solutions of $d-1$ polynomials in $d$ variables. Thus we expect the set $A_f$ to be the set of rational points of a (possibly reducible) curve $C_f$; indeed, this is always true when $f$ is non-constant. A basic question that we would like to answer is to find conditions on $L$ and $f$ that guarantee that the set $A_f$ is finite, and ideally to decide when standard techniques can be applied to explicitly determine this set. We formalize and generalize the previous problem as follows. Let $L \supset K$ be a finite separable field extension, let $B \to {{\rm Spec}}(K)$ be a smooth projective curve defined over $K$ and let $C$ be a smooth projective curve defined over $L$. Denote by $B_L$ the base-change to $L$ of the curve $B$, and suppose that $h \colon C \to B_L$ is a non-constant morphism defined over $L$. Then there is a (possibly singular and reducible) curve $C_h$ defined over $K$ whose $K$-rational points parametrize the $L$-rational points $p \in C(L)$ such that $h(p) \in B(K) \subset B_L(L)$. In this context, Theorem \[cocha\] identifies an abelian subvariety $F$ of the Jacobian of the curve $C_h$ and provides a formula to compute the rank of the Mordell-Weil group of $F$: these are the ingredients needed to apply the Chabauty method to determine the rational points of the curve $C_h$. To see the relationship of this general problem with the initial motivating question, we let $C:=\mathbb{P}^1_L$ and $B:=\mathbb{P}^1_\mathbb{Q}$. The polynomial $f$ determines a morphism $h \colon \mathbb{P}^1_L \to (\mathbb{P}^1_{\mathbb{Q}})_L$ and the $L$-points of $\mathbb{P}^1_L$ (different from $\infty$) with image in $\mathbb{P}^1(\mathbb{Q})$ correspond to the set $A_f$. Over an algebraic closure of the field $L$, the curve $C_h$ is isomorphic to the fibered product of morphisms $h_i \colon C_i \to B_L$ obtained from the initial morphism $h$ by taking Galois conjugates (Lemma \[profi\]). Thus we generalize further our setup: we concentrate our attention on the fibered product of finitely many morphisms $h_1 \colon C_1 \to B$, …, $h_n \colon C_n \to B$, where $C_1 , \ldots , C_n$ and $B$ are smooth curves and the morphisms $h_1,\ldots,h_n$ are finite and separable. We determine the geometric genus of the normalization of $C_h$, as well as a natural abelian subvariety $J_h$ of the Jacobian of $C_h$. Due to the nature of the problem and of the arguments, it is immediate to convert results over the algebraic closure to statements over the initial field of definition. Suppose now that $K$ is a number field. We may use the Chabauty method to find the rational points on $C_h$, provided the abelian variety $J_h$ satisfies the condition that the rank of $J_h(K)$ is less than the genus of $C_h$; by Chabauty’s Theorem (see [@prolegom; @chab1; @chab2]) this guarantees that $C_h(K)$ is finite. Chabauty’s Theorem has been developed into a practical technique, which has been applied to a range of Diophantine problems, for example in [@colemanchab; @colemanpadic; @flynnchab; @lortuck]. Further, if the set $C_h(K)$ is infinite, then, by Faltings’ Theorem [@faltings], the curve $C_h$ contains a component of geometric genus at most one; thus the computation of the geometric genus of the normalization of $C_h$ is a first step towards answering the question of whether $C_h(K)$ is finite or not. Moreover, since all the irreducible components of $C_h$ dominate the curve $C$, it follows that the set $C_h(K)$ can be infinite only in the case in which the curve $C$ has geometric genus at most one. In the case in which $C$ has genus zero, results equivalent to special cases of this question have already been studied ([@az; @bt; @pa; @sch; @za]). We shall analyze completely the case in which the genus of $C$ is one and the curve $C_h$ has infinitely many rational points. This covers as a special case the method called Elliptic Curve Chabauty (which is commonly applied to an elliptic curve $E$ defined over a number field $L \supset K$, satisfying that the rank of $E(L)$ is less than $[L : K]$ and we wish to find all $(x,y) \in E(L)$ subject to an arithmetic condition such as $x \in K$); see, for example, [@bruinth; @bruinchab; @flywet1; @flywet2; @wethth] and a hyperelliptic version in [@siksekchab]). Let $K$ be a number field and let $E : y^2 = (a_2 x^2 + a_1 x + a_0)(x + b_1 + b_2 \sqrt{d})$, with $a_0,a_1,a_2,b_1,b_2,d \in K$, be an elliptic curve defined over $L = K(\sqrt{d})$; suppose also that $b_2 \not= 0$, $d \not= 0$, $d \not\in (K^*)^2$, so that $E$ is not defined over $K$. We are interested in $(x,y) \in E(L)$ with $x \in K$. Let $y = r + s\sqrt{d}$, with $r,s \in K$. Equating coefficients of $1,\sqrt{d}$ gives $$r^2 + d s^2 = (a_2 x^2 + a_1 x + a_0) (x + b_1),\ \ 2 r s = (a_2 x^2 + a_1 x + a_0) b_2.$$ Let $t = s^2/(a_2 x^2 + a_1 x + a_0)$. Eliminate $r$ to obtain $b_2^2/(4 t) + d t = x + b_1$, so that $x = x(t) := b_2^2/(4 t) + d t - b_1$. Hence $s,t$ satisfy the curve $C : (st)^2 = t^3(a_2 x(t)^2 + a_1 x(t) + a_0)$, for which the right hand side is a quintic in $t$. One can check directly that this quintic has discriminant whose factors are powers of: $a_0, b_2, d$, the discriminant of $E$ and its conjugate; all of these are guaranteed to be nonzero, from our assumptions, so that $C$ has genus $2$. By Faltings’ Theorem, $C(K)$ is finite, so there are finitely many such $s,t\in K$, and hence finitely many $x = x(t)$, and so finitely many $(x,y) \in E(L)$ with $x \in K$. Furthermore, one can check directly from the induced map from $C$ to $E$ that, if $J$ is the Jacobian of $C$ then $\hbox{rank}\bigl(J(K)\bigr) = \hbox{rank}\bigl(E(L)\bigr)$; hence if $\hbox{rank}\bigl(E(L)\bigr) < [L:K] = 2$ (the condition for Elliptic Curve Chabauty) then $C$ satisfies the conditions for Chabauty’s Theorem over $K$. So, for this special case, there is a perfect match between both conditions. Let $f \in \mathbb{Q}(\sqrt{2})[x]$ be the polynomial $f(x) = x^2 (x - \sqrt{2})$ and suppose that we are interested in finding the set $A_f$ of values of $x \in \mathbb{Q}(\sqrt{2})$ such that $f(x) \in \mathbb{Q}$. Writing $x=a+b \sqrt{2}$ with $a,b \in \mathbb{Q}$ and substituting in $f$ we find that $f(a + b \sqrt{2})$ is a rational number if and only if the equality $a^2+2b^2=3a^2b+2b^3$ holds. Apart from the solution $(a,b)=(0,0)$, all the remaining solutions of the resulting equation can be determined by the substitution $a=tb$ and the set $A_f$ is the set $$\left\{ \frac{t^2+2}{(3t^2+2)} (t+\sqrt{2})~ \Bigl| ~ t \in \mathbb{Q} \right\} \cup \bigl\{ 0 \bigr\} .$$ Let $f \in \mathbb{Q}(\sqrt{2})[x]$ be the polynomial $f(x) = \frac{x (x - \sqrt{2})}{x-1}$. Arguing similarly to the previous example, we find that the set values of $x \in \mathbb{Q}(\sqrt{2})$ such that $f(x) \in \mathbb{Q}$ consists of the solutions to the equation $a^2 b - a^2 - 2 a b + a - 2 b^3 + 2 b^2 = 0$. The projective closure of the previous equation defines an elliptic curve $E$ with Weierstrass form $y^2=x^3-x$; by a 2-descent it is easy to show that $E(\mathbb{Q}) = E(\mathbb{Q})[2]$ and we conclude that the set we are seeking is the set $\{ 0 , \sqrt{2} \}$. The last two of these examples exhibit the two qualitative behaviours that we analyze in what follows. Relative Weil Restriction {#rwr} ========================= We begin this section by recalling the definition of the Weil restriction functor. The setup is quite general, though we will only use it in a very specialized context. We prefer to adopt this formal point of view at the beginning, since it simplifies the arguments; for the cases mentioned in the introduction, it is straightforward to translate all our arguments into explicit computations that are also easy to verify. Let $s \colon S' \to S$ be a morphism of schemes and let $X'$ be an $S'$-scheme; the contravariant functor $$\begin{array}{rcl} {{\mathfrak{R}}}_{S'/S}(X') \colon ({\rm Sch}/S) ^o & \longrightarrow & {\rm Sets} \\[5pt] T & \longmapsto & {\rm Hom}_{S'} (T \times _S S' , X') \end{array}$$ is the [*Weil restriction functor*]{}. If the functor ${{\mathfrak{R}}}_{S'/S}(X')$ is representable, then we denote by ${{\mathfrak{R}}}_{S'/S}(X')$ also the scheme representing ${{\mathfrak{R}}}_{S'/S}(X')$; sometimes, to simplify the notation, we omit the reference to $S$ and $S'$ and write that ${{\mathfrak{R}}}(X')$ is the Weil restriction of $X'$. The scheme ${{\mathfrak{R}}}(X')$ is determined by the isomorphism $${\rm Hom}_S ( - , {{\mathfrak{R}}}(X')) \stackrel{\sim}{\longrightarrow} {\rm Hom}_{S'} (- \times_S S', X')$$ of functors $({\rm Sch}/S)^o \to {\rm Sets}$. Informally this means that the $S$-valued points of ${{\mathfrak{R}}}(X')$ are the same as the $S'$-valued points of $X'$. Suppose now that $Y \to S$ is another morphism of schemes, denote by $Y'$ the fibered product $Y \times _S S'$ with natural morphism $b \colon Y' \to Y$, and let $h \colon X \to Y'$ be any scheme. We have the diagram $$\label{diato} \begin{minipage}{150pt} \xymatrix{ X \ar[d] _h \\ Y' \ar[d] \ar[r]^b & Y \ar[d] \\ S' \ar[r] ^{s} & S } \end{minipage}$$ and $X$ is therefore both a $Y'$-scheme and an $S'$ scheme. Thus there are two possible Weil restrictions we can construct: - the Weil restriction ${{\mathfrak{R}}}_{S'/S} (X)$, using the $S'$-scheme structure of $X$, - the Weil restriction ${{\mathfrak{R}}}_{Y'/Y} (X)$, using the $Y'$-scheme structure of $X$. Below we shall use the notation ${{\rm Res}}_h(X)$ for the Weil restriction ${{\mathfrak{R}}}_{Y'/Y}(X)$ and call it the [*[relative Weil restriction]{}*]{}. In order to relate the Weil restriction ${{\mathfrak{R}}}_{Y'/Y}(X)$ to the discussion in the introduction, we give an alternative definition of this functor and then proceed to prove the equivalence of the two. For concreteness, suppose that in diagram  the morphism $S' \to S$ is induced by a (finite, separable) field extension $L \supset K$; then there is a subset of the $L$-points of $X$ whose image under $h$ is not simply an $L$-point of $Y'$, but it is actually a $K$-point of $Y$. We would like to say that this set of $L$-points of $X$ with $K$-rational image under $h$ are the $K$-rational points of a scheme defined over $K$, and that this scheme [*[is]{}*]{} the relative Weil restriction of $X$. This is what we discussed in the introduction and we now formalize it. Hence, let $T$ be any $S$-scheme and denote by $T'$ the $S'$-scheme $T \times _S S'$. Pull-back by the morphism $s$ defines a function ${\rm Hom}_S (T,Y) \to {\rm Hom}_{S'} (T',Y')$; there is also a function ${\rm Hom}_{S'} (T',X) \to {\rm Hom}_{S'} (T',Y')$ determined by composition with $h$. Summing up, for any $S$-scheme $T$, we obtain a diagram $$\label{sta} \begin{minipage}{150pt} \xymatrix{{\rm Hom}(T,X/Y) \ar@{-->}[r] \ar@{-->}[d] & {\rm Hom}_{S'} (T',X) \ar[d] ^{h \circ -}\\ {\rm Hom}_S (T,Y) \ar[r] ^{s^*} & {\rm Hom}_{S'} (T',Y') . } \end{minipage}$$ We denote by ${\rm Hom}(T,X/Y)$ the pull-back of diagram , and we define the [*relative Weil restriction functor*]{} to be the functor $$\begin{array}{rcl} {{\rm Res}}_h \colon ({\rm Sch}/S)^o & \longrightarrow & {\rm Sets} \\[5pt] T & \longmapsto & {\rm Hom}(T,X/Y). \end{array}$$ If the functor ${{\rm Res}}_h$ is representable, we denote a scheme representing it by ${{\rm Res}}_h(X)$. \[lem:rwr\] Let $s \colon S' \to S$ be a morphism of schemes. Let $X$ be an $S'$-scheme and let $Y$ be an $S$-scheme; denote by $Y_{S'}$ the $S'$-scheme $Y \times _S S'$. Let $h \colon X \to Y_{S'}$ be an $S'$-morphism and assume that both Weil restriction functors ${{\mathfrak{R}}}(X)$ and ${{\mathfrak{R}}}(Y_{S'})$ are representable (notation as in ). Then the relative Weil restriction functor ${{\rm Res}}_h \colon ({\rm Sch}/S)^o \to {\rm Sets}$ is representable and the schemes ${{\rm Res}}_h(X)$ and ${{\mathfrak{R}}}_{Y'/Y}(X)$ coincide. Moreover, there is a commutative diagram of $S$-schemes $$\label{dere} \begin{minipage}{150pt} \xymatrix{ {{\rm Res}}_h(X) \ar[r] \ar[d] & {{\mathfrak{R}}}(X) \ar[d]^{h'} \\ Y \ar[r]^{\iota~~~} & {{\mathfrak{R}}}(Y_{S'}) } \end{minipage}$$ exhibiting ${{\rm Res}}_h(X)$ as a fibered product of ${{\mathfrak{R}}}(X)$ and $Y$ over ${{\mathfrak{R}}}(Y_{S'})$. We begin by constructing an $S$-morphism $h' \colon {{\mathfrak{R}}}(X) \to {{\mathfrak{R}}}(Y_{S'})$. By representability of ${{\mathfrak{R}}}(Y_{S'})$ and of ${{\mathfrak{R}}}(X)$ there are natural bijections $$\begin{array}{rcll} {\rm Hom}_{S} ({{\mathfrak{R}}}(X) , {{\mathfrak{R}}}(Y_{S'})) & \stackrel{\sim}{\longrightarrow} & {\rm Hom}_{S'} ({{\mathfrak{R}}}(X)_{S'} , Y_{S'}) & \quad {\textrm{and}} \\[7pt] {\rm Hom}_{S} ({{\mathfrak{R}}}(X) , {{\mathfrak{R}}}(X)) & \stackrel{\sim}{\longrightarrow} & {\rm Hom}_{S'} ({{\mathfrak{R}}}(X)_{S'} , X) . \end{array}$$ Let $\gamma \in {\rm Hom}_{S'} ({{\mathfrak{R}}}(X)_{S'} , X)$ be the $S'$-morphism corresponding to the identity in ${\rm Hom}_{S} ({{\mathfrak{R}}}(X) , {{\mathfrak{R}}}(X))$ and let $h' \colon {{\mathfrak{R}}}(X) \to {{\mathfrak{R}}}(Y_{S'})$ be the $S$-morphism corresponding to $h \circ \gamma \in {\rm Hom}_{S'} ({{\mathfrak{R}}}(X)_{S'} , Y_{S'})$. Note also that there is an $S$-morphism $\iota \colon Y \to {{\mathfrak{R}}}(Y_{S'})$ corresponding to the identity in ${\rm Hom}_{S'} (Y_{S'}, Y_{S'})$. Define ${{\rm Res}}_h(X) := Y \times _{{{\mathfrak{R}}}(Y_{S'})} {{\mathfrak{R}}}(X)$ so that there is a commutative diagram as in . To check that ${{\rm Res}}_h(X)$ represents ${{\rm Res}}_h$, let $T$ be any $S$-scheme; we have $$\begin{aligned} {\rm Hom}_S (T,{{\rm Res}}_h(X)) & = & {\rm Hom}_S (T,Y) \times_{{\rm Hom}_S (T , {{\mathfrak{R}}}(Y_{S'}))} {\rm Hom}_S (T,{{\mathfrak{R}}}(X)) = \\[5pt] & = & {\rm Hom}_S (T,Y) \times_{{\rm Hom}_S (T,{{\mathfrak{R}}}(Y_{S'}))} {\rm Hom}_{S'} (T_{S'},X) = {\rm Hom} (T,X/Y)\end{aligned}$$ as required. Thus, the scheme ${{\rm Res}}_h(X)$ defined above represents the functor ${{\rm Res}}_h$ and is obtained by the fibered product . Finally, we check that the schemes ${{\rm Res}}_h(X)$ and ${{\mathfrak{R}}}_{Y'/Y}(X)$ coincide. Let $T$ be any $Y$-scheme; we have $T \times_Y Y' = T \times _Y (Y \times_S S') = (T \times _Y Y) \times_S S' = T \times_S S'$ and $$\begin{aligned} {\rm Hom}_Y (T,{{\mathfrak{R}}}(X)) & = & {\rm Hom}_{Y'} (T \times_Y Y',X) = {\rm Hom}_{Y'} (T \times _S S',X) = \\[5pt] & = & \Bigl\{ f \in {\rm Hom}_{S'} (T \times_S S',X) ~\Bigl|~ h \circ f = s^*(T_Y \to Y) \Bigr\} = \\[5pt] & = & {\rm Hom} (T,X/Y) = {\rm Hom}_Y (T, {{\rm Res}}_h(X)) .\end{aligned}$$ We conclude using Yoneda’s Lemma that ${{\rm Res}}_h(X)$ and ${{\mathfrak{R}}}_{Y'/Y}(X)$ coincide, and hence the lemma follows. Informally, we may describe the scheme ${{\rm Res}}_h(X)$ as the scheme whose $S$-valued points correspond to the points in $X(S')$ lying above the points in $Y(S)$. Observe that, by the construction of ${{\rm Res}}_h(X)$ and diagram , there is an $S'$-morphism ${{\rm Res}}_h(X)_{S'} \to X$. The morphism ${{\rm Res}}_h(X)_{S'} \to X$ will prove useful later. We now introduce some notation that is used in the following lemma. Let $L \supset K$ be a finite, separable field extension of degree $n$, and let ${K^{\rm s}}$ be a separable closure of $K$. Set $S' := {\rm Spec}(L)$, $S := {\rm Spec}(K)$, and ${S^{\rm s}}:= {\rm Spec}({K^{\rm s}})$, denote by $s \colon S' \to S$ the morphism of schemes corresponding to the extension $L \supset K$ and suppose that $\varphi_1 , \ldots , \varphi_n \colon {S^{\rm s}}\to S'$ are the morphisms corresponding to all the distinct embeddings of $L$ into ${K^{\rm s}}$ fixing $K$. Given an $S'$-scheme $X$, we denote by $X_1 , \ldots , X_n$ the ${S^{\rm s}}$-schemes obtained by pulling back $X \to S'$ under the morphisms $\varphi_1 , \ldots , \varphi_n$. \[profi\] Let $X$ be an $S'$-scheme and let $Y$ be an $S$-scheme. Let $h \colon X \to Y_{S'}$ be an $S'$-morphism and assume that both Weil restriction functors ${{\mathfrak{R}}}(X)$ and ${{\mathfrak{R}}}(Y_{S'})$ are representable. Then the isomorphism $$\label{diago} {{\rm Res}}_h (X)_{{{S^{\rm s}}}} \simeq X_1 \times_{Y_{{{S^{\rm s}}}}} X_2 \times_{Y_{{{S^{\rm s}}}}} \cdots \times_{Y_{{{S^{\rm s}}}}} X_n$$ holds. More precisely, the isomorphism  holds replacing ${K^{\rm s}}$ by any normal field extension of $K$ containing $L$. To prove the result, it suffices to consider the case in which both $X$ and $Y$ are affine; thus suppose that $Y = {\rm Spec}(A)$, $X = {\rm Spec}(B)$, and that $\underline{x}$ denotes a set of generators of $B$ as an $A$-algebra. Let $\alpha_1 , \ldots , \alpha_n$ denote a basis of $K$ over $L$ and let $\underline{x}_1 , \ldots , \underline{x}_n$ denote disjoint sets of variables each in bijection with $\underline{x}$; whenever we denote a variable in $\underline{x}$ by a symbol such as $x$, we denote by the symbols $x_1 , \ldots x_n$ the variables corresponding to $x$ in $\underline{x}_1 , \ldots , \underline{x}_n$. For every $x \in \underline{x}$ define linear forms $\tilde{x}_1 := \sum _i x_i \varphi_1(\alpha_i) , \ldots , \tilde{x}_n := \sum _i x_i \varphi_n(\alpha_i)$ in $A[\underline{x}_1 , \ldots , \underline{x}_n]$. First we show that the forms $\tilde{x}_1 , \ldots , \tilde{x}_n$ are linearly independent. Indeed, let $\Delta$ be the matrix whose $(i,j)$ entry is $\varphi_j(\alpha_i)$; the $(i,j)$ entry of $\Delta \Delta^t$ is $\sum _k \varphi_k(\alpha_i \alpha_j) = {\rm Tr}_{L/K}(\alpha_i \alpha_j)$. Thus the determinant of $\Delta \Delta ^t$ is equal to the discriminant of $L$ over $K$, and it is therefore non-zero. We deduce that the matrix $\Delta$ is invertible and that the forms $\tilde{x}_1 , \ldots , \tilde{x}_n$ are independent. Thus, defining $\tilde{\underline{x}}_j := \{ \tilde{x}_j \mid x \in \underline{x} \}$ for $j \in \{1,\ldots,n\}$, we proved that there is an isomorphism ${K^{\rm s}}\otimes_K A[\underline{x}_1 , \ldots , \underline{x}_n] \simeq {K^{\rm s}}\otimes_K A[\tilde{\underline{x}}_1 , \ldots , \tilde{\underline{x}}_n]$. The relative Weil restriction of $X$ may be defined as follows. Let $g(\underline{x})$ be an element of the polynomial ring $A[\underline{x}]$ contained in the ideal defining $X$; evaluate $g(\underline{x})$ substituting for each variable $x \in \underline{x}$ the sum $\sum x_i \alpha_i$ and write the resulting polynomial in $L \otimes_K A[\underline{x}_1 , \ldots , \underline{x}_n]$ as $\sum g_i \alpha_i$, where $g_1 , \ldots , g_n$ are elements of $A[\underline{x}_1 , \ldots , \underline{x}_n]$; denote the sequence $(g_1 , \ldots , g_n)$ by $\tilde{g}$. Then the scheme ${{\rm Res}}(X)$ is the scheme in ${\rm Spec}(A[\underline{x}_1 , \ldots , \underline{x}_n])$ whose ideal is the ideal generated by the elements of $\tilde{g}$, as $g$ varies among all the elements of the ideal defining $X$. It is therefore clear that the ideal $I$ defining ${{\rm Res}}(X)$ in ${K^{\rm s}}\otimes_K A[\underline{x}_1 , \ldots , \underline{x}_n]$ contains, for every embedding $\varphi \colon L \to {K^{\rm s}}$ fixing $K$, the elements $\sum_i g_i \varphi(\alpha_i) = \varphi \bigl( \sum g_i \alpha_i) = \varphi(g)$, and conversely that the ideal containing all such elements contains $g_1 , \ldots , g_n$ and hence it contains $I$. Let $\mathscr{F} \in A[\underline{x}]$ be a set of generators of the ideal of $X$; since, for $i \in \{1,\ldots , n\}$, the scheme $X_i$ is defined by $\{\varphi_i(f) \mid f \in \mathscr{F} \}$ in $A[\tilde{\underline{x}}_i]$ the result follows. The case of curves over an algebraically closed field ===================================================== In this section we compute the geometric genus of the relative Weil restriction $\mathscr{C}$ of a curve and we also determine a natural abelian variety isogenous to a subvariety of the Jacobian of $\mathscr{C}$. To calculate the geometric genus in Theorem \[thm:fibre\] we may clearly assume that the ground field is algebraically closed; the proof reduces the computation to the étale local case, settled in Lemma \[lem:fibre\]. \[lem:fibre\] Let $n$ be a positive integer. Suppose that $k$ is an algebraically closed field and that $r_1 , \ldots , r_n$ are positive integers relatively prime with the characteristic of $k$. Let $R$ be the least common multiple of $r_1 , \ldots , r_n$ and let $C \subset \mathbb{A}_{x,y_1,\ldots,y_n}^{n+1}$ be the affine scheme defined by $$C \colon \left\{ \begin{array}{rcl} y_1^{r_1} & = & x , \\ & \vdots \\ y_n^{r_n} & = & x . \end{array} \right.$$ The scheme $C$ has $\frac{r_1 \cdots r_n}{R}$ irreducible components and the morphism induced by the projection onto the $x$-axis from the normalization of each component ramifies at the origin to order $R$. Observe that the curve $C$ carries the action $\rho$ of $\mathbb{G}_m$ defined by $$t \cdot (x,y_1,\ldots,y_n) = (t^R x , t^{R/r_1} y_1 , \ldots , t^{R/r_n} y_n).$$ First we show that the action $\rho$ has trivial stabilizer at all points of $C$ different from the origin. Indeed, let $q=(x,y_1,\ldots,y_n)$ be a point of $C$ different from the origin; it follows that all the coordinates of $q$ are non-zero and the equality $(x,y_1,\ldots,y_n) = (t^R x , t^{R/r_1} y_1 , \ldots , t^{R/r_n} y_n)$ implies that $t$ is a root of unity of order dividing $\gcd (R/r_1 , \ldots , R/r_n) = 1$. Thus the complement of the origin in $C$ is a principal homogeneous space for $\mathbb{G}_m$. There are $r_1 \cdots r_n$ points on $C$ such that $x=1$ and these are stabilized precisely by the action of the $R$-th roots of unity; we deduce that $C$ consists of $\frac{r_1 \cdots r_n}{R}$ irreducible components, each isomorphic to closure of the orbit of a point $(1,\eta_1 , \ldots , \eta_n)$, where $\eta_i$ is an $r_i$-th root of unity, for $i \in \{1,\ldots,n\}$. Therefore the normalization of $C$ consists of $\frac{r_1 \cdots r_n}{R}$ components each mapping to the $x$-axis by $(t^R,\eta_1 t^{R/r_1}, \ldots , \eta_n t^{R/r_n}) \mapsto t^R$, as required. \[thm:fibre\] Let $k$ be an algebraically closed field and let $n$ be a positive integer. Suppose that $S,C_1,\ldots,C_n$ are smooth curves; for $i \in \{1,\ldots,n\}$ let $f_i \colon C_i \to S$ be a finite separable morphism whose ramification indices are coprime to the characteristic of $k$. Let $f$ denote the morphism of the curve $C := C_1 \times _S \cdots \times _S C_n$ to $S$, and let $C'$ be the normalization of $C$; denote by $g_S$ and $g_{C'}$ the arithmetic genera of $S$ and $C'$ respectively. For any point $p \in C$ and any $i \in \{1,\ldots,n\}$ denote by $r_i=r_i(p)$ the ramification index of $f_i$ at the point corresponding to $p$ and let $R=R(p) := {\rm lcm} \{r_1(p) , \ldots , r_n(p) \}$. Then the curve $C$ is a local complete intersection and the curve $C'$ has arithmetic genus $$\begin{aligned} g_{C'} & = & 1 + (g_S-1) \prod_{i=1}^n \deg(f_i) + \frac{1}{2} \sum _{p \in C} r_1(p) \cdots r_n(p) \left( 1 - \frac{1}{R(p)} \right) = \\[5pt] & = & 1 + \frac{1}{2} (r - 2g_S - 2) \prod_{i=1}^n \deg(f_i) - \frac{1}{2} \sum _{p \in f^{-1} (R_{f})} \frac{r_1(p) \cdots r_n(p)}{R(p)} \end{aligned}$$ where $R_f \subset S$ is the union of the sets of branch points of the morphisms $f_1 , \ldots , f_n$ and $r$ is the cardinality of $R_f$. Let $\pi \colon C' \to S$ be the morphism induced by the structure morphism $f \colon C \to S$. Let $p' \in C'$ be a closed point, let $p$ be the corresponding point of $C$ and let $p_1 , \ldots , p_n$ be the points of $C_1 , \ldots , C_n$ respectively corresponding to $p$. We prove the result by finding a local model of $C$ near $p$ which is a local complete intersection, and then applying the Hurwitz formula to the morphism induced by $f$ on the normalization of such a model. Choosing a local coordinate $x$ on $S$ near $\pi(p')$ we reduce to the case in which $S$ is $\mathbb{A}^1$ and $\pi(p') = 0$. Similarly choose local coordinates $z_1 , \ldots , z_n$ on $C_1 , \ldots , C_n$ near $p_1 , \ldots , p_n$ respectively. Thus, near $p$, the curve $C$ is defined by $$C \colon \left\{ \begin{array}{rcl} z_1^{r_1} \varphi_1(z_1) & = & x , \\ & \vdots \\ z_n^{r_n} \varphi_n(z_n) & = & x , \end{array} \right.$$ where $(\varphi_1 , \ldots , \varphi_n)$ is a rational function on $C$ defined and non-zero at $p$, and $r_1 , \ldots , r_n$ are the local ramification indices. In particular, the curve $C$ is a local complete intersection near $p$. Denote by $\mathcal{O}_{p}$ the local ring of $C$ near $p$; the base-change defined by the inclusion $$\mathcal{O}_{p} \longrightarrow \mathcal{O}_{p} [t_1 , \ldots , t_n] / \bigl( t_1^{r_1} - \varphi_1(z_1) , \ldots , t_n^{r_n} - \varphi_n(z_n) \bigr)$$ is finite étale (of degree $r_1 \cdots r_n$) by the assumption that the ramification indices are relatively prime to the characteristic of the field $k$. Hence, each component of the resulting curve is locally isomorphic to the curve $C_{p} \subset \mathbb{A}^{n+1}_{x,y_1,\ldots,y_n}$ defined by $$C_{p} \colon \left\{ \begin{array}{rcl} y_1^{r_1} & = & x , \\ & \vdots \\ y_n^{r_n} & = & x , \end{array} \right.$$ where $y_1 = z_1 t_1 , \ldots , y_n = z_n t_n$. The morphism induced by $\pi$ on $C_{p}$ is the morphism induced by the coordinate $x$. Using Lemma \[lem:fibre\], we conclude that the contribution of the point $p$ to the Hurwitz formula is $\frac{r_1 \cdots r_n}{R} (R - 1) = r_1 \cdots r_n (1 - \frac{1}{R})$, and hence we obtain $$2g_C-2 = (2g_S-2) \prod \deg(f_i) + \sum _{p \in C} r_1(p) \cdots r_n(p) \left( 1 - \frac{1}{R(p)} \right)$$ and the first formula follows. To prove the second one note that the quantity $r_1(p) \cdots r_n(p) ({1 - \frac{1}{R(p)}})$ vanishes for $p \notin R_f$, since in this case all the local ramification indices equal $1$, and that for all points $s$ of $S$ we have $$\sum _{p \in f^{-1}(s)} r_1(p) \cdots r_n(p) = \prod _{i=1}^n \sum _{p \in f_i^{-1}(s)} r_i(p) = \prod \deg(f_i)$$ and we conclude. In the next results, for a projective scheme $Y$, we denote by ${{\rm Jac}}(Y)$ the [*[Jacobian variety of $Y$]{}*]{}, that is the connected component of the identity of the group ${{\rm Pic}}(Y)$. \[prodo\] Suppose that $X_1 , \ldots , X_n$ are smooth projective varieties defined over an algebraically closed field $k$ and let $X=X_1 \times \cdots \times X_n$. For $i \in \{1,\ldots,n\}$ let $\rho_i \colon X \to X_i$ denote the canonical projection. The morphism $$\rho_\bullet^* = (\rho_1^* , \ldots , \rho_n^*) \colon {{\rm Jac}}(X_1) \times \cdots \times {{\rm Jac}}(X_n) \to {{\rm Jac}}(X)$$ is an isomorphism. Choosing a point in each variety $X_1 , \ldots , X_n$ allows us to define, for $i \in \{1,\ldots,n\}$, an inclusion $X_i \hookrightarrow X$. These inclusions in turn determine a section ${{\rm Jac}}(X) \to {{\rm Jac}}(X_1) \times \cdots \times {{\rm Jac}}(X_n)$ of the morphism $\rho_\bullet^*$. We deduce that $\rho_\bullet^*$ is indeed an isomorphism of Jacobian varieties ${{\rm Jac}}(X_1) \times \cdots \times {{\rm Jac}}(X_n) \simeq {{\rm Jac}}(X)$. Maintaining the notation of the previous lemma, the morphisms $\rho_1,\ldots,\rho_n$ also induce a homomorphism $\psi^* \colon {{\rm Pic}}(X_1) \times \cdots \times {{\rm Pic}}(X_n) \to {{\rm Pic}}(X)$. The morphism $\psi$ though need not be an isomorphism. Since $\psi$ induces an isomorphism on the connected component of the identity, it factors through the Néron-Severi group and in particular its cokernel is a finitely generated abelian group. For instance, if $E$ is an elliptic curve and $X_1=X_2=E$, then the three classes $\{0\} \times E$, $E \times \{0\}$ and the diagonal are independent in ${\rm NS}(E \times E)$ so that ${\rm NS}(E \times E) \supset \mathbb{Z}^3$. On the other hand, the group ${\rm NS}(E) \times {\rm NS}(E)$ is isomorphic to $\mathbb{Z}^2$. For the next theorem we need to introduce some notation. Suppose that $B,X_1 , \ldots , X_n$ are smooth projective varieties defined over an algebraically closed field $k$. For each $i \in \{1,\ldots,n\}$ let $f_i \colon X_i \to B$ be a finite morphism. Denote by $X$ the product $X_1 \times \cdots \times X_n$, by $X_B$ the fibered product $X_1 \times _B \cdots \times _B X_n$ and by $\iota \colon X_B \to X$ the natural inclusion. For each $i \in \{1,\ldots,n\}$ - let $d_i$ denote the degree of $f_i$, - let $\rho_i \colon X \to X_i$ and $\pi_i=\rho_i \circ \iota \colon X_B \to X_i$ be the canonical projections, - let $\pi \colon X \to B$ be the composition $f_1 \circ \pi_1 = \cdots = f_n \circ \pi_n$, - let $d = d_1 \cdots d_n$ denote the degree of $\pi$. We summarize the notation in the case $n=2$ in . $$\label{diauno} \begin{array}{c} X := X_1 \times X_2 \\[25pt] X_B := X_1 \times _B X_2 \\ \vphantom{\begin{array}{c}W\\W\\W\\W\\W\end{array}} \end{array} \hspace{20pt} \begin{minipage}{150pt} \xymatrix{ & X_1 \times X_2 \ar[dddl] _{\rho_1} \ar[dddr] ^{\rho_2} \\ \\ & {\hphantom{{}_B} X_1} \times_B X_2 \ar[uu] _\iota \ar[dl] ^{\pi_1} \ar[dd] ^{\pi} \ar[dr] _{\pi_2} \\ X_1 \ar[dr] ^{f_1} & & X_2 \ar[dl] _{f_2} \\ & B } \end{minipage} \hphantom{\hspace{20pt}C}$$ Even though the diagram may suggest it, the identities $f_1 \circ \rho_1 = \cdots = f_n \circ \rho_n$ [*[do not]{}*]{} hold necessarily. On the other hand, the identities $f_1 \circ \pi_1 = \cdots = f_n \circ \pi_n = \pi$ hold. Let $\phi \colon {{\rm Jac}}(X) \to {{\rm Jac}}(X)$ be the isogeny defined by $$\begin{array}{ccl} \phi \colon \quad {{\rm Jac}}(X) & \longrightarrow & {{\rm Jac}}(X) \\[5pt] \rho_1^*D_1 + \cdots + \rho_n^*D_n & \longmapsto & \frac{d}{d_1}D_1 + \cdots + \frac{d}{d_n}D_n , \end{array}$$ where we used the identification of ${{\rm Jac}}(X)$ with ${{\rm Jac}}(X_1) \times \cdots \times {{\rm Jac}}(X_n)$ of Lemma \[prodo\]. Let $M'$ denote the kernel of the multiplication map ${{\rm Jac}}(B)^n \to {{\rm Jac}}(B)$, so that $M' \simeq {{\rm Jac}}(B)^{n-1}$; define the group $M$ as the image of $M'$ under the morphism $$\begin{array}{rcl} {{\rm Jac}}(B)^n & \longrightarrow & {{\rm Jac}}(X) \\[7pt] (D_1,\ldots,D_{n-1}) & \longmapsto & \rho_1^*f_1^*D_1 + \cdots + \rho_n^*f_n^*D_n. \end{array}$$ By construction, the group $M$ is connected and contained in the kernel of $\iota^*$. Moreover, it follows from Lemma \[prodo\] and the fact that the morphisms $f_1,\ldots,f_n$ are finite that the morphism $M' \to M$ is finite and hence that the dimension of the group $M$ is $(n-1) \dim ({{\rm Jac}}(B))$. \[thm:jaco\] Maintaining the notation introduced above, the group $M$ has finite index in $\ker(\iota^*)$. More precisely, for each element $D$ of $\ker(\iota^*)$ there are elements $D_1,\ldots,D_n$ of ${{\rm Jac}}(B)$ such that the identities $$\begin{array}{rcll} \phi(D) & = & \sum_i \rho_i^*f_i^*D_i & \quad {\textrm{and}} \\[7pt] d \sum_i D_i & = & 0 \end{array}$$ hold. In particular, the dimension of the kernel of $\iota^*$ is $(n-1) \dim ({{\rm Jac}}(B))$. By Lemma \[prodo\], the morphism $$\begin{array}{ccl} {{\rm Jac}}(X_1) \times \cdots \times {{\rm Jac}}(X_n) & \longrightarrow & {{\rm Jac}}(X) \\[5pt] (D_1 , \ldots , D_n) & \longmapsto & \rho_1^* D_1 + \cdots + \rho_n^* D_n \end{array}$$ is an isomorphism. Thus we identify the divisor classes in ${{\rm Jac}}(X)$ with $n$-tuples of divisor classes, one in each Jacobian variety ${{\rm Jac}}(X_1), \ldots , {{\rm Jac}}(X_n)$. Let $i,j$ be distinct indices in $\{1,\ldots,n\}$ and let $P$ be a divisor on $X_i$; it is easy to check that the identities $$\begin{aligned} \pi_i{}_*\pi_i^*(P) & = & \frac{d}{d_i} P \, , \\[5pt] \pi_i{}_*\pi_j^*(P) & = & \frac{d}{d_i d_j} f_i^*f_j{}_*(P) \end{aligned}$$ hold, and hence the class of the divisor $\pi_i{}_*\pi_j^*(P)$ is contained in $f_i^* {{\rm Pic}}(B)$. Suppose that $D=\rho_1^*D_1+\cdots+\rho_n^*D_n$ is a divisor on $X$ representing an element of ${{\rm Jac}}(X)$. Let $i$ be an index in $\{1,\ldots,n\}$; we have $$\pi_i{}_* \iota^* (D) = \pi_i{}_* (\pi_1^*D_1+\cdots+\pi_n^*D_n) \in \frac{d}{d_i}D_i + f_i^* {{\rm Jac}}(B)$$ and summing over all indices $i$, we find the equivalence $$\begin{array}{rcl} \displaystyle \sum _i \pi_i{}_* \iota^* (D) & \equiv & \displaystyle \sum_i \frac{d}{d_i}D_i \\[15pt] & \equiv & \displaystyle \phi(D) \pmod{f_1^*{{\rm Jac}}(B) \times \cdots \times f_n^* {{\rm Jac}}(B)} . \end{array}$$ In particular, if $D$ is contained in the kernel of $\iota^*$, then $\phi(D)$ is contained in $f_1^*{{\rm Jac}}(B) \times \cdots \times f_n^* {{\rm Jac}}(B)$, establishing the first of the two identities. Finally, let $D_1 , \ldots , D_n \in {{\rm Jac}}(B)$ be divisor classes such that the element $\rho_1^*f_1^*D_1 + \cdots + \rho_n^*f_n^*D_n$ of ${{\rm Jac}}(X)$ lies in $\ker (\iota^*)$. Then, the element $D_1 + \cdots + D_n$ of ${{\rm Jac}}(B)$ lies in the kernel of $\pi^*$, so that $d(D_1 + \cdots + D_n) = \pi_* \pi^*(D_1 + \cdots + D_n) = 0$, proving the second identity. It follows from what we just proved that the equalities $$\dim (\ker(\iota^*)) = \dim M = ({n-1}) \dim ({{\rm Jac}}(B))$$ hold, proving the final assertion of the theorem. Mordell-Weil groups and relative Weil restriction ================================================= From now on, we shall be in the following set up (specializing the assumptions of Lemma \[lem:rwr\]): - $L$ is a number field and $S':={{\rm Spec}}(L)$, - $K \subset L$ is a subfield and $S:={{\rm Spec}}(K)$, - $C$ is a smooth projective curve over $L$, - $B$ is a smooth projective curve over $K$, and - $h \colon C \to B_{S'}=B_L$ is a finite morphism. To simplify the notation, for any variety $Z$ defined over a number field $k$ denote by $mw_k(Z)$ the rank of the Mordell-Weil group of the Jacobian of $Z$; we are only going to apply this notation with $k \in \{K,L\}$ to varieties $Z$ that are either reduced curves or products of smooth integral curves. \[cocha\] Suppose that $C$ is a smooth projective curve defined over a number field $L$. Suppose that $B$ is a smooth projective curve defined over a subfield $K$ of $L$, and let $h \colon C \to B_L := B \times_{{{\rm Spec}}(K)} {{\rm Spec}}(L)$ be a finite morphism. Denote by $n$ the dimension of $L$ as a vector space over $K$ and by $g(C)$ and $g(B)$ the genera of $C$ and $B$ respectively. The Jacobian of ${{\rm Res}}_h(C)$ contains an abelian subvariety $F$ of dimension $n g(C) - (n - 1) g(B)$ defined over $K$ and with Mordell-Weil group over $K$ of rank $mw_L(C) - \bigl( mw_L (B_L) - mw_K(B) \bigr)$. The $L$-morphism $h \colon C \to B_L$ induces a $K$-morphism ${{\mathfrak{R}}}(C) \to {{\mathfrak{R}}}(B_L)$ which in turn induces a pull-back $K$-morphism ${{\rm Jac}}({{\mathfrak{R}}}(B_L)) \to {{\rm Jac}}({{\mathfrak{R}}}(C))$. Furthermore, from the inclusion $\iota \colon {{\rm Res}}_h(C) \subset {{\mathfrak{R}}}(C)$ we obtain a sequence of $K$-morphisms $${{\rm Jac}}({{\mathfrak{R}}}(B_L)) \longrightarrow {{\rm Jac}}({{\mathfrak{R}}}(C)) \stackrel{\iota^*}{\longrightarrow} {{\rm Jac}}({{\rm Res}}_h(C)) .$$ From the representability of ${{\mathfrak{R}}}(B_L)$, there is a $K$-morphism $\kappa \colon B \to {{\mathfrak{R}}}(B_L)$ associated to the identity $B_L \to B_L$ using the bijection ${{\rm Hom}}_K(B,{{\mathfrak{R}}}(B_L)) = {{\rm Hom}}_L(B_L,B_L)$. The morphism $\kappa$ induces a pull-back $K$-morphism $\kappa^* \colon {{\rm Jac}}({{\mathfrak{R}}}(B_L)) \to {{\rm Jac}}(B)$; we denote by $M$ the kernel of the morphism of $\kappa^*$. Geometrically, the Jacobian of ${{\mathfrak{R}}}(B_L)$ is isomorphic to the product of $n$ copies of the Jacobian of $B$ and the morphism $\kappa^*$ corresponds to the addition of the line bundles in the various components using the isomorphisms between them (defined over an algebraic closure of $K$). We obtain that the group $M$ is the specialization to our setting of the group denoted also by $M$ in Theorem \[thm:jaco\]. Therefore $M$ is geometrically isomorphic to ${{\rm Jac}}(B)^{n-1}$, and it is connected of dimension $(n-1) g(B)$. Thus we obtain the diagram $$\xymatrix{ M \ar@{^(->}[d] \ar[r] & {{\rm Jac}}({{\mathfrak{R}}}(C)) \ar[r]^{\iota^*} & {{\rm Jac}}({{\rm Res}}_h(C)) \\ {{\rm Jac}}({{\mathfrak{R}}}(B_L)) \ar[d]^{\kappa^*} \ar[ur] \\ {{\rm Jac}}(B) }$$ of $K$-morphisms and it follows from Theorem \[thm:jaco\] that the group $M$ has finite index in $\ker(\iota^*)$. We let $F$ be the connected component of the identity of the image of $\iota^*$ and we show that it has the required properties. First of all, $F$ is an abelian variety over $K$, isogenous over $K$ to ${{\rm Jac}}({{\mathfrak{R}}}(C))/M$, and hence the dimension of $F$ is $$\dim (F) = \dim \left( \frac{{{\rm Jac}}({{\mathfrak{R}}}(C))}{M} \right) = n g(C) - (n-1) g(B)$$ as needed. Next, we prove the statement about the Mordell-Weil rank of $F$. For a curve $D$ defined over $L$ we have $$\begin{aligned} mw_K \bigl( {{\mathfrak{R}}}(D) \bigr) & = & {{\rm rk}}\Bigl( {{\rm Jac}}\bigl( {{\mathfrak{R}}}(D) \bigr) (K) \Bigr) \\ & = & {{\rm rk}}\biggl( {{\rm Hom}}_K \Bigl( {{\rm Spec}}(K) , {{\rm Jac}}\bigl( {{\mathfrak{R}}}(D) \bigr) \Bigr) \biggr) \\ & = & {{\rm rk}}\biggl( {{\rm Hom}}_K \Bigl( {{\rm Spec}}(K) , {{\mathfrak{R}}}\bigl( {{\rm Jac}}(D) \bigr) \Bigr) \biggr) \\ & = & {{\rm rk}}\Bigl( {{\rm Hom}}_L \bigl( {{\rm Spec}}(L) , {{\rm Jac}}(D) \bigr) \Bigr) \\ & = & mw_L(D) .\end{aligned}$$ Since the abelian varieties $F$ and ${{\rm Jac}}({{\mathfrak{R}}}(C))/M$ are $K$-isogenous, the ranks of their Mordell-Weil groups are the same. By the previous computation we conclude that $$\begin{aligned} {{\rm rk}}\bigl( F(K) \bigr) & = & {{\rm rk}}\left( \frac{{{\rm Jac}}\bigl( {{\mathfrak{R}}}(C) \bigr)}{M} (K) \right) \\ & = & mw_K \bigl( {{\mathfrak{R}}}(C) \bigr) - \Bigl( mw_K \bigl( {{\mathfrak{R}}}(B_L) \bigr) - mw_K(B) \Bigr) \\ & = & mw_L(C) - \bigl( mw_L (B_L) - mw_K(B) \bigr) ,\end{aligned}$$ as required, and the result follows. The theorem just proved opens the way to applications of the Chabauty method to find the $L$-rational points of the curve $C$ with $K$-rational image in $B$. We show how this method works on an example. Let $d$ be a squarefree integer; we let $K = \mathbb{Q}$ and $L = \mathbb{Q}(\sqrt{d})$. Denote by $g(x) \in \mathbb{Q}[x]$ the polynomial $$g(x) = x^3 + a x + b$$ and by $f(x) \in \mathbb{Q}(\sqrt{d})[x]$ the polynomial $$f(x) = g(x^2+\sqrt{d}) .$$ Let $E$ be the elliptic curve over $\mathbb{Q}$ with Weierstrass equation $y^2=g(x)$ and let $C$ be the smooth projective model over $\mathbb{Q}(\sqrt{d})$ of the genus two hyperelliptic curve with affine equation $y^2=f(x)$. By construction, there is a morphism $\phi \colon C \to E$ given by $$\begin{array}{rcl} \phi \colon C & \longrightarrow & E \\[5pt] (x,y) & \longmapsto & ( x^2 + \sqrt{d} , y ). \end{array}$$ Suppose we wish to find all points $P$ in $C\bigl( \mathbb{Q}(\sqrt{d}) \bigr) $ such that $\phi(P)$ is in $E(\mathbb{Q})$. Such points are the rational points of the curve $D = {{\rm Res}}_\phi(C)$ over $\mathbb{Q}$, for which we now determine an explicit model. Let $x = x_1 + x_2 \sqrt{d}$ and $y = y_1 + y_2 \sqrt{d}$, where $ x_1, x_2,y_1,y_2$ are $\mathbb{Q}$-rational variables. Substituting $x = x_1+x_2\sqrt{d}$ and $y = y_1+y_2\sqrt{d}$ in the polynomial defining $C$ we find the polynomial $$r(x_1,x_2,y_1,y_2) = (y_1 + y_2 \sqrt{d})^2 - f(x_1 + x_2 \sqrt{d})$$ in $\mathbb{Q}(\sqrt{d})$, whose vanishing represents the condition that the point $P = ( x_1 + x_2 \sqrt{d}, y_1 + y_2 \sqrt{d})$ lies on $C$. Define polynomials with rational coefficients $$r_1 := \frac{r + \overline{r}}{2} \quad {\textrm{and}} \quad r_2 := \frac{r - \overline{r}}{2 \sqrt{d}}$$ where $\overline{r}$ denotes the polynomial obtained from $r$ by applying the nontrivial element of the Galois group of $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$; we have the identity $r = r_1 + \sqrt{d} r_2$. Thus, the two equations $r_1=r_2=0$ in $ x_1,y_1, x_2,y_2$ correspond to $P$ lying on $C$; these are also the equations defining (an affine model of) the Weil restriction of $C$ from $\mathbb{Q}(\sqrt{d})$ to $\mathbb{Q}$. To determine $D = {{\rm Res}}_\phi(C)$, we also wish $\phi(P)$ to be in $E(\mathbb{Q})$. Note that the coordinates of $\phi(P)$ are $$\phi(P) = \bigl( x_1^2 + d x_2^2 + \sqrt{d} (2 x_1 x_2 + 1) , y_1+\sqrt{d} y_2 \bigr)$$ and the condition that the point $\phi(P)$ lies in $\mathbb{Q}$ translates to the equations $2x_1 x_2 + 1 = y_2 = 0$. We have therefore obtained the four equations $$\label{rima} \left\{ \begin{array}{rcl} y_1^2 & = & (x_1^6 + d^3 x_2^6) + 3(x_1^2 + d x_2^2) \bigl( 5 d x_1^2 x_2^2 + 4 d x_1 x_2 + d + \frac{a}{3} \bigr) + b \\[7pt] 0 & = & (2 x_1 x_2+1) (3 x_1^4+10 x_1^2 x_2^2 d+4 x_1 x_2 d +3 x_2^4 d^2+a+d) \\[7pt] 0 & = & 2 x_1 x_2+1 \\[5pt] y_2 & = & 0 \end{array} \right.$$ in the variables $ x_1,y_1, x_2,y_2$. But of course the second equation is divisible by the third equation, and we may ignore it (this is not a coincidence, but it is a consequence of the fact that the curve $E$ is defined over $\mathbb{Q}$). Multiplying the first equation in  by $2^{12}x_1^6$, we can use the relation $2 x_1 x_2=-1$ to eliminate the variable $x_2$, obtaining the single equation $(2^6x_1^3 y_1)^2 = \overline{\rho} (x_1)$ in $x_1, y_1$ for $D$. After the birational substitution $x=2x_1$ and $y=2^6x_1^3 y_1$, we obtain that the curve $D$ is birational to the genus 5 curve with equation $$D \colon y^2 = x^{12} + 4(3 d + 4 a) x^8 + 64 b x^6 + 16d (3 d + 4 a) x^4 + 64 d^3 .$$ The curve $D$ admits the non-constant map $(x,y) \mapsto (x^2,y)$ to the genus 2 curve $F$ with equation $$F \colon y^2 = x^6 + 4(3 d + 4 a) x^4 + 64 b x^3 + 16d (3 d + 4 a) x^2 + 64 d^3.$$ Summarizing, the curve $D$ is a genus 5 curve defined over $\mathbb{Q}$ and it admits two morphisms defined over $\mathbb{Q}(\sqrt{d})$ to the curves $$C \colon y^2=f(x) = g(x^2+\sqrt{d}) \quad {\textrm{ and }} \quad \overline{C} \colon y^2=\overline{f}(x) = g(x^2-\sqrt{d}) .$$ Each of the two curves $C$ and $\overline{C}$ admits a $\mathbb{Q}(\sqrt{d})$-morphism to the elliptic curve $E$, and the corresponding two compositions $D \to E$ coincide, and are defined over $\mathbb{Q}$. We deduce that the Jacobians of $C$ and of $\overline{C}$ are contained, up to isogeny, in the Jacobian of $D$, and they have an isogenous copy of $E$ in common. This implies that the five-dimensional Jacobian of $D$ contains a further two-dimensional abelian variety: up to isogeny this is the Jacobian of the curve $F$. It is now easy to provide examples where the inequality of the Chabauty method is not satisfied for the curve $C$, nor for the curve $E$, but it satisfied for the curve $F$, so that we can still apply the Chabauty method to find the points on $D$. For example, setting $a = 1$, $b = 3$, and $d = 13$, we find ${{\rm rk}}\bigl( E(\mathbb{Q}) \bigr) = 1$ and ${{\rm rk}}\bigl( {{\rm Jac}}(F)(\mathbb{Q}) \bigr) = 1$, and moreover $${{\rm rk}}\left( {{\rm Jac}}(C) \bigl( \mathbb{Q}(\sqrt{d}) \bigr) \right) \geq {{\rm rk}}\left( E \bigl( \mathbb{Q}(\sqrt{d}) \bigr) \right) \geq 2 .$$ Thus, in this example the Chabauty method is not applicable to $C$ or $E$, but it is only applicable to $F(\mathbb{Q})$. Cases where Faltings’ Theorem does not apply ============================================ In this section we analyze the cases where the relative Weil restriction of a morphism of curves contains a component of geometric genus at most one. In such cases, Faltings’ Theorem cannot be applied to deduce the finiteness of rational points of the relative Weil restriction and we find explicit non-tautological examples in which these sets of rational points are infinite. The following remark is an immediate consequence of Faltings’ Theorem and guides the choice of cases we handle in this section. \[gamu\] Let $L/K$ be an extension of number fields, and suppose that $C$ is a curve defined over $L$, $B$ is a curve defined over $K$ and $h \colon C \to B$ is a non-constant morphism. If the curve ${{\rm Res}}_h(C)$ has infinitely many $K$-rational points, then the genus of $C$ is at most one. In the case of the relative Weil restriction of a morphism from a curve of genus one, we completely characterize the cases where the set of rational points is not finite. \[p:quo\] Let $C , X_1 , \ldots , X_n , B$ be smooth projective curves (not necessarily connected), and let $$\xymatrix{ & C \ar[dl] _{r_1} \ar[dr] ^{r_n} \\ X_1 \ar[dr] & \cdots & X_n \ar[dl] \\ & B }$$ be a commutative diagram, where all morphisms are finite and flat. There is a smooth projective curve $X_C$ and a commutative diagram $$\xymatrix{ & C \ar[dl] _{r_1} \ar[dr] ^{r_n} \\ X_1 \ar[dr] & \cdots & X_n \ar[dl] \\ & X_C }$$ such that for every projective curve $D$ fitting in the commutative diagram of solid arrows $$\xymatrix{ & C \ar[dl] _{r_1} \ar[dr] ^{r_n} \\ X_1 \ar[dr] \ar[ddr] & \cdots & X_n \ar[dl] \ar[ddl] \\ & X_C \ar@{-->}[d] \\ & D }$$ the dashed arrow $\xymatrix{X_C \ar@{-->}[r] & D}$ exists uniquely. Let $X$ be the disjoint union $X := X_1 \sqcup \cdots \sqcup X_n$. The morphisms $r_1,\ldots,r_n$ determine an equivalence relation $\sim_C$ on $X$, where $x \sim_C y$ if there is a sequence $(i_1 , \ldots , i_t)$ of elements of $\{1,\ldots,n\}$, and points $x_1 = x \in X_{i_1}$, $x_2 \in X_{i_2}$, …, $x_{t-1} \in X_{i_{t-1}}$, $x_t = y \in X_{i_t}$ and a sequence $c_1 , \ldots , c_{t-1}$ of points of $C$ such that for all $j \in \{1,\ldots,t-1\}$ we have $r_{i_j} (c_i) = r_{i_{j+1}} (c_i)$. Thus, a pair $(x,y) \in X \times X$ is in the relation determined by the morphisms $r_1,\ldots,r_n$ if and only if there is a sequence $I := (i_1 , \ldots , i_t)$ of elements of $\{1,\ldots,n\}$ such that the pair $(x,y)$ is in the image of the composition $$C \times _{X_{i_2}} \times \cdots \times _{X_{i_{t-1}}} C \stackrel{\pi}{\longrightarrow} C \times C \stackrel{(r_{i_1},r_{i_t})}{\longrightarrow} X_{i_1} \times X_{i_t} \subset X \times X$$ where $\pi$ is the projection to the first and last factor; denote by $C_I \subset X \times X$ the image of this morphism. Observe that, for every finite sequence $I$ of elements of $\{1 , \ldots , n\}$, the scheme $C_I$ is a closed subscheme of $X \times X$ that is finite and flat over each factor $X$ and hence also over $B$; in particular, each scheme $C_I$ has non-zero degree over $B$. Moreover, if the pair $(x,y)$ is in the relation $\sim_C$, then $x$ and $y$ have the same image in $B$. From this it follows that pairs $(x,y)$ in the relation $\sim_C$ are covered by at most $\bigl( \sum _i \deg(f_i) \bigr)^2$ of the schemes $C_I$ defined above since they are contained in $X \times _B X$. It follows that the whole graph $R_C \subset X \times X$ of the relation $\sim_C$ is a subscheme of finite type of $X \times X$ that is flat and proper over each factor. The hypotheses of [@SGA3], Théorème V.7.1, are therefore satisfied and the result follows. \[coge1\] With the notation of the previous proposition, assume further that the curves $C,X_1 , \ldots , X_n$ all have genus one. Then the curve $X_C$ is a torsor under ${{\rm Jac}}(C)/K$, where $K$ is the subgroup of ${{\rm Jac}}(C)$ generated by the kernels of the morphisms ${{\rm Jac}}(C) \to {{\rm Jac}}(X_1)$, …, ${{\rm Jac}}(C) \to {{\rm Jac}}(X_n)$. We can clearly assume that the ground field is algebraically closed, and further reduce to the case in which the morphisms $r_1, \ldots , r_n$ are all homomorphisms of elliptic curves. Thus $K \subset C$ is identified with the subgroup generated by the kernels of all the morphisms $r_1, \ldots , r_n$ and let $C' := C/K$ denote the quotient of $C$ by the subgroup $K$. Clearly, the curves $X_1 , \ldots , X_n$ all admit a morphism to $C'$ making the diagram of Proposition \[p:quo\] commute. It follows from the previous proposition that $X_C$ admits a morphism to $C'$ that is necessarily non-constant, and we conclude that $X_C$ has genus one. Moreover, it is also clear that the curve $X_C$ is isomorphic to the curve $C'$: if $D$ is any curve making the diagram of Proposition \[p:quo\] commute, then the fiber in $C$ of the morphism $C \to D$ over the image of the origin in $C$ contains all the kernels of the morphisms $r_1 , \ldots , r_n$, and hence it contains the subgroup $K$, since the morphism factors through the elliptic curve $X_C$. Thus we see that if the curves $X_1,\ldots,X_n$ have genus one and if the fibered product $X_1 \times _B \cdots \times _B X_n$ contains a geometrically integral curve of geometric genus one defined over the ground field, then the morphism $X_1 \to B$ factors through a morphism $E \to B$ defined over the ground field, where $E$ is a smooth geometrically integral curve of geometric genus one and the morphism $X_1 \to E$ is an isogeny defined over the extension field. Genus one --------- We specialize what we just proved to the case of the relative Weil restriction from an elliptic curve. Let $L/K$ be an extension of number fields; let $E$ be an elliptic curve defined $L$ and let $B$ be a smooth projective integral curve defined over $K$. Suppose that $h \colon E \to B_L$ is a non-constant morphism. The relative Weil restriction ${{\rm Res}}_h(E)$ is a curve defined over $K$. Fix an algebraic closure $\overline{K}$ of $K$, denote by $\sigma_1 , \ldots , \sigma_n$ the distinct embeddings of $L/K$ in $\overline{K}$, and let $E_1 , \ldots , E_n$ be the corresponding Galois conjugates of $E$. Over the field $\overline{K}$, the curve ${{\rm Res}}_h(E)$ is isomorphic to $E_1 \times _B \cdots \times_B E_n$ (Lemma \[profi\]). Suppose that the curve ${{\rm Res}}_h(E)$ contains infinitely many $K$-rational points. It follows from Faltings’ Theorem that there is a component $C$ of ${{\rm Res}}_h(E)$ of geometric genus at most one, defined over $K$; if $C$ is not normal, we replace it by its normalization. Since the morphism ${{\rm Res}}_h(E) \to B$ is flat, all its fibers are finite and therefore the curve $C$ is also finite over $B$. In particular, the $L$-morphisms $C \to E_1 , \ldots , C \to E_n$ are all finite, since all the curves are smooth. Let $E_C$ denote the universal curve fitting in the diagram $$\xymatrix{ & C \ar[dl] _{r_1} \ar[dr] ^{r_n} \\ E_1 \ar[dr] & \cdots & E_n \ar[dl] \\ & E_C }$$ of Proposition \[p:quo\]. Since the curves $C,E_1,\ldots,E_n$ all have genus one, we may therefore apply Corollary \[coge1\] to deduce that the curve $E_C$ is also a torsor under an elliptic curve, and therefore also $E_C$ has genus one. In the case in which $L$ is a number field, we obtain the following corollary. Let $L/K$ be an extension of number fields and suppose that $E$ is an elliptic curve over $L$, $B$ is a curve over $K$, and $h \colon E \to B_L$ is a non-constant morphism. If the set of $K$-rational points of ${{\rm Res}}_h(E)$ is infinite, then the curve $E$ is $L$-isogenous to an elliptic curve defined over $K$ having positive rank over $K$. By Faltings’ Theorem we deduce that ${{\rm Res}}_h(E)$ contains a geometrically integral component $E'$ of genus at most one defined over $K$ and having infinitely many $K$-rational points. Since $E'$ admits a non-constant $L$-morphism to $E$, it follows that $E'$ has genus one and that it is $L$-isogenous to $E$, as required. In the case in which $E$ and $B$ have genus one, then all the geometric components of ${{\rm Res}}_h(E)$ have genus one. Hence, finding the $K$-rational points of ${{\rm Res}}_h(E)$ is equivalent to finding the $K$-rational points of finitely many elliptic curves that are $K$-isogenous to $B$. This completes our analysis in the case in which the curve $C$ of Remark \[gamu\] has genus one. We next discuss the case in which $C$ has genus zero. We are not able to give a treatment of this case that is as detailed as the case of genus one. Genus zero ---------- We specialize to the case in which the curve $C$ is isomorphic to $\mathbb{P}^1_L$ and hence $B$ is isomorphic to $\mathbb{P}^1_K$. The morphism $h \colon \mathbb{P}^1_L \to \mathbb{P}^1_L = (\mathbb{P}^1_K)_L$ is therefore determined by a rational function $F \in L(x)$. The set of $K$-rational points of ${{\rm Res}}_h(\mathbb{P}^1_L)$ is essentially the set $A_F \subset L$ of values of $x \in L$ such that $F(x)$ lies in $K$, mentioned in the introduction. Let $G \colon C \to D$ be a finite morphism between smooth curves, let $p \in D$ be a geometric point. The [*[type]{}*]{} of $p$ is the partition $\lambda_p$ of $\deg(G)$ determined by the fiber $G^{-1}(p)$. We extend this definition to the case in which the curve $C$ is reduced, but not necessarily smooth, by replacing $G$ with the morphism $G^\nu \colon C^\nu \to D$, where $C^\nu$ is the normalization of $C$ and $G^\nu$ is the morphism induced by $G$. In this section we restrict our attention to field extensions $L \supset K$ of degree at most three. ### Degree two {#degree-two .unnumbered} Let $L \supset K$ be a field extension of degree two, let $F \colon \mathbb{P}^1_L \to \mathbb{P}^1_L$ be a morphism of degree three. We are interested in the values $\ell \in L$ such that $f(\ell) \in K$. Assume that $L = K(\sqrt{d})$ where $d \in K \setminus K^2$; write $\alpha = a + b \sqrt{d}$, for $a,b \in K$. Parameterizing the rational curve ${{\rm Res}}_F(F)$ we find that, for every element $t$ in $K$, the evaluation $$p \left( \frac{d b t^2+2 a t+b}{t (d t^2 + 3)} (\sqrt{d} t + 1) \right)$$ is also in $K$. ### Degree three {#degree-three .unnumbered} Suppose that $F \colon \mathbb{P}^1_L \to \mathbb{P}^1_L$ is a morphism of degree three, and suppose that the characteristic of $K$ is neither two nor three. As usual, we are interested in the values $\ell \in L$ such that $f(\ell) \in K$. Denote by $\overline{F}$ the morphism conjugate to $F$ under the Galois involution of $L/K$. We construct examples of morphisms $F$ such that ${{\rm Res}}_F(\mathbb{P}^1_L)$ is a geometrically integral curve of geometric genus zero. Note that the curve ${{\rm Res}}_F(\mathbb{P}^1_L)$ has a line bundle of degree nine given by pull-back of $\mathcal{O}_{\mathbb{P}^1_K}(1)$; since this curve is geometrically rational and it has a line bundle of odd degree, it follows that it is rational over $K$. Denote by ${{\rm Res}}_F(F)^\nu$ the composition of the normalization map of ${{\rm Res}}_F(\mathbb{P}^1_L)$ and ${{\rm Res}}_F(F)$. Applying the Hurwitz formula to the morphisms $F$, $\overline{F}$ and ${{\rm Res}}_F(F)^\nu$, we find that the respective total degrees of the ramification divisors are $4$, $4$ and $16$. We begin by analyzing the ramification patterns. For a geometric point $p \in \mathbb{P}^1$, Table \[fihu\] shows the possibilities of the types of the fibers of the three morphisms $F$, $\overline{F}$ and ${{\rm Res}}_F(F)^\nu$ and the contributions of each to the Hurwitz formula. In our setup, the Galois involution of $L/K$ induces a bijection between fiber types of $F$ and of $\overline{F}$: this is recorded in the last column of Table \[fihu\]. \[fihu\] ------------------------------------------- --------------------------- --------------------------------------------- ------------------ Fiber type of Fiber type of Hurwitz contribution to Symmetrized the pair $F,\overline{F}$ ${{\rm Res}}_F(F)$ $[F,\overline{F}]$ and $[{{\rm Res}}_F(F)]$ contribution \[5pt\] $\bigl( (1,1,1) , (1,1,1) \bigr)$ $(1,1,1 , 1,1,1 , 1,1,1)$ \[0,0\] , \[0\] \[0,0\] , \[0\] \[5pt\] $\bigl( (2,1) , (1,1,1) \bigr)$ $(2,2,2 , 1,1,1)$ \[1,0\] , \[3\] \[1,1\] , \[6\] \[5pt\] $\bigl( (3) , (1,1,1) \bigr)$ $(3,3,3)$ \[2,0\] , \[6\] \[2,2\] , \[12\] \[5pt\] $\bigl( (2,1) , (2,1) \bigr)$ $(2,2,2,2,1)$ \[1,1\] , \[4\] \[1,1\] , \[4\] \[5pt\] $\bigl( (2,1) , (3) \bigr)$ $(6,3)$ \[1,2\] , \[7\] \[3,3\] , \[14\] \[5pt\] $\bigl( (3) , (3) \bigr)$ $(3,3,3)$ \[2,2\] , \[6\] \[2,2\] , \[6\] \[5pt\] ------------------------------------------- --------------------------- --------------------------------------------- ------------------ : Fiber types and contributions to the Hurwitz formula for fiber products of morphisms of degree three It is now easy to check that the only possibilities for the fiber types of the morphisms $F,\overline{F}$ are $$\bigl( (3) , (3) \bigr) + \bigl( (2,1) , (1,1,1) \bigr) + \bigl( (1,1,1) , (2,1) \bigr) + \bigl( (2,1) , (2,1) \bigr) \label{pollo}$$ and $$\bigl( (2,1) , (2,1) \bigr) + \bigl( (2,1) , (2,1) \bigr) + \bigl( (2,1) , (2,1) \bigr) + \bigl( (2,1) , (2,1) \bigr) . \label{simme}$$ We only analyze the case . ### Fiber types  {#fiber-types .unnumbered} The fiber type $((3),(3))$ in  implies that the coordinate on $\mathbb{P}^1$ can be chosen so that the morphism $F$ is a polynomial and the fiber type $((2,1) , (2,1))$ shows that one of the ramification points in defined over $K$. This case is realized by morphisms $F \colon \mathbb{P}^1_L \to \mathbb{P}^1_L$ of the form $p(x) = x^2(x-\alpha)$, for $\alpha \in L$. Since the derivative of $p$ vanishes at $0$ and at $\frac{2\alpha}{3}$, it follows that the ramification types of $F,\overline{F}$ are of the form  when $\alpha \notin K$.

--- abstract: 'The helicity of a vector field is a measure of the average linking of pairs of integral curves of the field. Computed by a six-dimensional integral, it is widely useful in the physics of fluids. For a divergence-free field tangent to the boundary of a domain in 3-space, helicity is known to be invariant under volume-preserving diffeomorphisms of the domain that are homotopic to the identity. We give a new construction of helicity for closed $(k+1)$-forms on a domain in $(2k+1)$-space that vanish when pulled back to the boundary of the domain. Our construction expresses helicity in terms of a cohomology class represented by the form when pulled back to the compactified configuration space of pairs of points in the domain. We show that our definition is equivalent to the standard one. We use our construction to give a new formula for computing helicity by a four-dimensional integral. We provide a Biot-Savart operator that computes a primitive for such forms; utilizing it, we obtain another formula for helicity. As a main result, we find a general formula for how much the value of helicity changes when the form is pushed forward by a diffeomorphism of the domain; it relies upon understanding the effect of the diffeomorphism on the homology of the domain and the de Rham cohomology class represented by the form. Our formula allows us to classify the helicity-preserving diffeomorphisms on a given domain, finding new helicity-preserving diffeomorphisms on the two-holed solid torus and proving that there are no new helicity-preserving diffeomorphisms on the standard solid torus. We conclude by defining helicities for forms on submanifolds of Euclidean space. In addition, we provide a detailed exposition of some standard ‘folk’ theorems about the cohomology of the boundary of domains in ${\mathbb{R}}^{2k+1}$.' address: - 'Department of Mathematics, University of Georgia, Athens, GA 30602' - 'Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109' author: - Jason Cantarella - Jason Parsley bibliography: - 'helicity-forms.bib' - 'cantarella.bib' title: 'A new cohomological formula for helicity in ${\mathbb{R}}^{2k+1}$ reveals the effect of a diffeomorphism on helicity' --- Introduction ============ The linking number of a pair of closed curves $a$ and $b$ in ${\mathbb{R}}^3$ is a topological measure of their entanglement. We can define the linking number as the degree of the Gauss map $g \co S^1 {\times}S^1 \rightarrow S^2$ given by $g(\theta,\phi) =\left( a(\theta) - b(\phi) \right) / {\left| a(\theta) - b(\phi) \right|}$. This degree can be written combinatorially, by counting signed crossings of $a$ and $b$, but we can also write this degree as an integral by pulling back the area form on $S^2$ via the Gauss map and integrating over the torus $S^1 {\times}S^1$. This “Gauss integral formula” for linking number yields $${\operatorname{Lk}}(a,b) = \frac{1}{{\operatorname{vol}}(S^2)} \int a'(\theta) {\times}b'(\phi) \cdot \frac{a(\theta) - b(\phi)}{{\left| a(\theta) - b(\phi) \right|}^3} d\theta \, d\phi.$$ The linking number is a knot invariant, so it is invariant under any ambient isotopy of ${\mathbb{R}}^3$ carrying the curves to new curves $\tilde{a}$ and $\tilde{b}$. Given a divergence-free vector field $V$ on a domain $\Omega \subset {\mathbb{R}}^3$ with smooth boundary, we can define an analogous integral invariant known as *helicity*. The six-dimensional helicity integral, which measures the average linking number of pairs of integral curves of the field [@MR891881], is given by: $$\label{mhelicity} {\operatorname{H}}(V) = \frac{1}{{\operatorname{vol}}(S^2)} \int_{\Omega \times \Omega} {V(x) \times V(y) \cdot \frac{x-y}{|x-y|^3} \; {\operatorname{dvol}}_x \, {\operatorname{dvol}}_y }$$ Just as the linking number of a pair of curves is a knot invariant, we might expect the helicity of a vector field to be a diffeomorphism invariant. This is not always true, as we will demonstrate below, but it is true in enough cases to make helicity an important quantity in fluid dynamics and plasma physics [@MR819398]. The helicity invariant for vector fields was used in plasma physics as early as 1958 by L. Woltjer [@MR0096542]. Woltjer showed that helicity was an invariant of the equations of ideal magnetohydrodynamics for an isolated system, and as such it was immediately useful in the study of astrophysical plasmas. J.J. Moreau in 1961 [@MR0128195] first used the invariant to study fluid dynamics. In an influential 1969 paper [@mof1], Keith Moffatt proved that helicity is an invariant of the equations of ideal fluid flow, even in the presence of an external force on the fluid. The invariance of helicity has been reproved in various physical contexts ever since. For instance, Peradzynski showed that helicity was invariant under the equations of motion for superfluid helium [@peradzynski]. The same invariant was associated to foliations by Godbillon and Vey in 1970 [@MR0283816], by defining the foliation as the kernel of a 1-form and measuring the helicity of the form. In 1973, V.I. Arnol’d defined helicity for 2-forms in a 3-manifold [@MR891881] (the paper was published in English translation in 1986). His may be the first proof of the invariance of helicity under arbitrary volume-preserving diffeomorphisms (on a simply-connected domain)[^1]. The most general invariance theorem for helicity known is: \[classicalinvariance\] The helicity of a divergence-free vector field $V$ on a domain $\Omega \subset {\mathbb{R}}^3$ is invariant under any volume-preserving diffeomorphism of $\Omega$ which is homotopic to the identity. If $\Omega$ is simply connected, then helicity is invariant under any volume-preserving diffeomorphism of $\Omega$. If $V$ is a null-homologous vector field (meaning that its dual 2-form is exact) on a compact manifold $M^3$ without boundary, then helicity is invariant under any volume-preserving diffeomorphism of $M$. Also, if $V$ on $\Omega$ is fluxless (cf. section \[sec:fluxless\]) on a domain in ${\mathbb{R}}^3$ with boundary, then its helicity is invariant under any volume-preserving diffeomorphism [@MR1770976]. These invariance results leave open some natural questions: are these all of the helicity-preserving diffeomorphisms? If not, can we classify the diffeomorphisms that do preserve helicity? What is the effect of an arbitrary diffeomorphism on helicity? Figure \[tori\] depicts a diffeomorphism which does not preserve helicity. Here, the domain is a solid torus, and the vector field following its longitudes is divergence-free and null-homologous[^2]. Applying a Dehn twist will preserve the volume form but changes the helicity of the field, which we will calculate by Theorem \[torus-h\]. [straight\_field]{} [twisted\_field]{} To answer the questions above, we notice that in the theory developed so far, there exists an asymmetry between linking number and helicity – while there are several useful ways to obtain linking number, including a “purely homological” expression as the degree of a map and a combinational expression as the sum of signed crossing numbers as well as an integral expression, so far the helicity has only been expressed as an integral. In this paper we try to restore the balance between linking number and helicity by providing a purely cohomological definition for the helicity of $(k+1)$-forms on domains $\Omega$ in ${\mathbb{R}}^{2k+1}$ (Definition \[fthelicity\]). We work with forms $\omega$ that are closed and satisfy the following definition: A smooth $p$-form $\alpha$ defined on the domain $\Omega$ is a *Dirichlet form* if $\alpha$ vanishes when restricted to the boundary, i.e., if $V_1, \ldots, V_p$ are all tangent to ${\partial}\Omega$, then $\alpha(V_1, \ldots, V_p)=0$. For domains in ${\mathbb{R}}^3$, closed Dirichlet forms are dual to vector fields that are divergence-free and tangent to the boundary. In Appendix \[hodge\], we examine decompositions of differential forms; in particular we characterize the set of closed Dirichlet forms. Proposition \[alphaisexact\] guarantees that every closed Dirichlet form is exact. Arnol’d defined helicity as the integral of the wedge product of an exact form with a primitive for that form [@MR1612569]. But there are many primitives for a given exact form on a general domain – so it is clear that the choice of primitive must be important to the definition. The helicity integral implicitly solves this problem by constructing a particular inverse curl for the given vector field by integration. Khesin and Chekanov generalized this approach to forms by defining an primitive for a form on $\Omega$ by integrating the wedge product of the pullback of the form to $\Omega {\times}\Omega$ and a singular “linking form” over the fiber in the bundle $\Omega {\times}\Omega \rightarrow \Omega$ [@MR1028280; @MR1612569]. Our paper is divided into two parts. In the first part of our paper, we redevelop this standard theory of helicity using a new idea: instead of defining a singular linking form on $\Omega {\times}\Omega$, we use a nonsingular form on the compactified configuration space $C_2[\Omega]$ of pairs of distinct points on $\Omega$. This approach allows us to give simpler and clearer proofs of the standard results on helicity for forms which better expose the underlying topology of the problem. Here is a summary of our construction for helicity in a simple special case. Suppose that on a ball $\Omega \subset {\mathbb{R}}^3$ we consider the helicity of a divergence-free vector field $V$ that is tangent to ${\partial}\Omega$. If we associate a 2-form $\alpha$ to $V$ by pairing $V$ with the volume form in ${\mathbb{R}}^3$, we will prove that the helicity can be expressed as an integral over the 6-dimensional compactified configuration space $C_2[\Omega]$ of disjoint pairs of points in $\Omega$. Let us understand the topology of this configuration space. We note that $C_2[\Omega]$ is homeomorphic to $\Omega {\times}\left( \Omega - B_r(x) \right)$, where $B_r(x)$ is a small neighborhood of a point in $\Omega$. Since $\Omega$ is a ball, $\Omega - B_r(x) \simeq S^2 {\times}I$ and this space is $D^3 {\times}S^2 {\times}I \simeq D^4 {\times}S^2$. The 2-form $\alpha$ can be pulled back to a pair of 2-forms $\alpha_x$ and $\alpha_y$ on $C_2[\Omega]$ under the projections of $\{x,y\}$ to $x$ and $y$. We will show that $\alpha_x \wedge \alpha_y$ is a closed Dirichlet 4-form on $C_2[\Omega]$. Hence $\alpha_x \wedge \alpha_y$ will represent a class $h[g]$ in the 1-dimensional relative de Rham cohomology group $H^4(D^4 {\times}S^2,{\partial}(D^4 {\times}S^2))$ where $g$ is a generator. We show that if $[g]$ is the Poincaré dual of the standard area form on $S^2$, then $h$ is the helicity of $V$ divided by the square of the volume of $\Omega$. This gives a cohomological definition of helicity. This definition has several attractive consequences. First, our configuration space approach extends to the standard setting for higher dimensional helicity: $(k+1)$-forms on $(2k+1)$-dimensional domains [@MR2078570 Section 5] . We recover the expected result that for $(4n + 1)$-dimensional domains (that is, for even values of $k$) helicity can extend only to a function that is identically zero, but for $(4n+3)$-dimensional domains, helicity is a nontrivial invariant for differential forms. We will see immediately that for forms, the question of whether a given diffeomorphism is volume-preserving has no bearing on whether the diffeomorphism preserves helicity. We are then able to give a quick proof of some of the standard invariance results for helicity in Proposition \[homotopic-invariance\]. Some new constructions are immediately suggested by our definition: a new, “combinatorial” integral for helicity appears in Proposition \[prop:combinatorial\]. We complete our redevelopment of the standard theory by proving that our helicity integral can be written as integration against an appropriately chosen primitive in Proposition \[prop:potential\]. We then begin the second part of paper, in which we solve the problem of computing the effect of an arbitrary diffeomorphism $f\co \Omega \rightarrow \Omega'$ on the helicity of a closed Dirichlet $(k+1)$-form $\alpha$. The theorems in this section are all new. It is a standard fact (see Appendix B, Theorems \[alexander basis existence\] and \[alexander basis transformations\]) that the $k$-th homology of ${\partial}\Omega$ splits into two subspaces generated by cycles $s_1, \dots, s_n$ which bound relative $(k+1)$-cycles outside $\Omega$ and Poincaré dual cycles $t_1, \dots, t_n$ which bound relative $(k+1)$-cycles $\tau_1, \dots, \tau_n$ inside $\Omega$. With respect to a corresponding basis $\langle s'_1, \dots, s'_n, t'_1, \dots, t'_n \rangle$ for the $k$-th homology of ${\partial}\Omega'$, we can write the linear map $\partial f_*\co H_k({\partial}\Omega) \rightarrow H_k({\partial}\Omega')$ as a block matrix $${\partial}f_* = \left[ \begin{array}{c|c} I & 0 \\ \hline (c_{ij}) & I \end{array} \right],$$ where the $c_{ij}$ form a symmetric matrix when $k$ is odd and a skew-symmetric matrix when $k$ is even. We then have Let $\Omega^{2k+1}$ be a subdomain of ${\mathbb{R}}^{2k+1}$, and $f\co\Omega \rightarrow \Omega'$ be an orientation-preserving diffeomorphism. Consider a closed Dirichlet $(k+1)$-form $\alpha$. The change in the helicity of $\alpha$ under $f$ is $${\operatorname{H}}\left(\alpha' \right) - {\operatorname{H}}(\alpha) = \sum_{i,j} c_{ij} \cdot {\operatorname{Flux}}(\alpha, \tau_i) {\operatorname{Flux}}(\alpha, \tau_j) \tag{\ref{eq:general-change}}$$ where the constants $c_{ij}$ arise from the homology isomorphism induced by $f$ on $H_k({\partial}\Omega)$ as above. The $(2m+2)$-form $\alpha'$ is the ‘push-forward’ of $\alpha$ under $f$; more precisely, $\alpha'={\left( f^{-1} \right)^*\alpha}$ is the pullback of $\alpha$ under the inverse diffeomorphism. Note that for $k$ even (i.e., subdomains of ${\mathbb{R}}^{4m+1}$) the matrix $\left( c_{ij} \right)$ is skew-symmetric. So Theorem \[general-change\] implies that helicity does not change under any diffeomorphism of $\Omega$. This confirms our previous calculation that helicity is always zero in these dimensions. The simplest example of this theorem is attractive and easy to understand: a diffeomorphism of a solid torus in ${\mathbb{R}}^3$ isotopic to $j$ Dehn twists changes the helicity of a $2$-form on the torus by $j$ times the square of the integral of the form over a spanning disk, as in Figure \[tori\]. In general, this allows us (for $k$ odd) to classify the helicity-preserving diffeomorphisms from $\Omega$ to $\Omega$ as those maps for which the $c_{ij}$ are all zero. If a diffeomorphism acts trivially on the homology of $\Omega$, it is in this class if and only if it acts trivially on the homology of ${\partial}\Omega$ (Corollary \[classification\]). We finish our paper with some discussion of directions for future research, including defining the $(k,n,m)$-helicity of $(k+1)$-forms on $n$-dimensional submanifolds of ${\mathbb{R}}^m$, an application of our results to computing “cross-helicities” of vector fields in two disjoint domains, and some thoughts on defining generalized helicities in a way inspired by the construction of the finite-type invariants for knots. Defining helicity in terms of cohomology ======================================== Helicity is motivated by the classical linking number between $k$ and $l$ cycles in ${\mathbb{R}}^{k+l+1}$. If $k=l$, observe that this linking number is only defined in odd-dimensional ambient spaces, which is why the classical linking number and helicity are defined in ${\mathbb{R}}^3$. In general, one could attempt to define the helicity of a tuple of $k$ vector fields in ${\mathbb{R}}^{2k+1}$, applying a version of the Gauss linking integral. We find it more natural to write such a tuple as dual to a single $(k+1)$-form, which can be envisioned as the form constructed by contracting the $k$-tuple of vector fields with the volume form of ${\mathbb{R}}^{2k+1}$. In the 3-dimensional case, we take a single vector field $V$ and construct a dual 2-form $\alpha$ by contracting the vector field with the volume form of ${\mathbb{R}}^3$ according to the rule $$\alpha(W_1,W_2) = {\operatorname{dvol}}(V,W_1,W_2).$$ Under this correspondence between vector fields and forms, a divergence-free vector field tangent to the boundary of $\Omega$ becomes a closed Dirichlet $2$-form. In general, we begin with a closed Dirichlet $(k+1)$-form $\alpha$, defined on a domain $\Omega$ in ${\mathbb{R}}^{2k+1}$. Our first goal is to define the helicity of $\alpha$ in terms of the cohomology class represented by a form constructed from $\alpha$ in the configuration space $C_2[\Omega]$ of two disjoint points in $\Omega$. We will start by recalling the definition of $C_2[\Omega]$, which will be a smooth closed manifold with boundary and with corners, in Section \[configspaces\]. We will then give our construction of helicity in Section \[redefining\]. Extending helicity to cases where $k$ is odd produces a nontrivial function; when $k$ is even, helicity extends to a function that is always zero. Finally, we will prove that our helicity is the standard helicity of a vector field in ${\mathbb{R}}^3$ in Section \[helicitysame\]. The Fulton–MacPherson compactification of configuration spaces {#configspaces} -------------------------------------------------------------- We start with a piece of technology: the Fulton-MacPherson compactification of a configuration space. There are many versions of this classical material (see for instance [@MR1259368; @MR1258919]). We follow Sinha [@MR2099074] as this gives a geometric viewpoint appropriate to our setting. \[cnmo\] Given an $m$-dimensional manifold $M$, define the *configuration space* ${C_n(M)}$ to be the subspace of $n$-tuples $(x_i) := (x_1, \dots, x_n) \in M^n$ such that $x_i \neq x_j$ if $i\neq j$. Let $\iota$ denote the inclusion of ${C_n(M)}$ in $M^{n}$. The configuration space ${C_n(M)}$ may be thought of as the space of ordered $n$-tuples in $M^n$, without the diagonals. Given $n$ distinct points in $M$, there are $n!$ points in ${C_n(M)}$ corresponding to the permutations of the $n$ points. The Fulton–Macpherson compactification of ${C_n(M)}$, defined below, keeps track of the directions and relative rates of approach when configuration points come together. To simplify the definition of ${C_n[M]}$, we introduce a bit of notation. Let ${\genfrac{[}{]}{0pt}{}{n}{k}}$ be the set of ordered $k$-tuples chosen from a set of $n$ elements and $\#{\genfrac{[}{]}{0pt}{}{n}{k}}$ to be the number of such tuples. For $(i,j)\in {\genfrac{[}{]}{0pt}{}{n}{2}}$, let ${\pi_{ij}}\co C_n({\mathbb{R}}^m)\rightarrow S^{m-1}$ be the map which sends $(x_\ell)$ to the unit vector in the direction of $x_i-x_j$. Let $[0,\infty]$ be the one-point compactification of $[0,\infty)$. For $(i,j,k)\in {\genfrac{[}{]}{0pt}{}{n}{3}}$, let ${s_{ijk}}\co C_n({\mathbb{R}}^m)\rightarrow [0,\infty]$ be the map which sends $(x_\ell)$ to $|x_i-x_j|/|x_i-x_k|$. We work with an arbitrary smooth manifold $M$ by first embedding $M$ in ${\mathbb{R}}^m$ (Whitney embedding), then defining the maps ${\pi_{ij}}$ and ${s_{ijk}}$ by restriction. Thus $C_n(M)$ is a submanifold of $C_n({\mathbb{R}}^m)$. If $M={\mathbb{R}}^m$, then it is a submanifold of itself through the identity map. \[Sinha Definition 1.3\] Let $A_n[M]$ be the product $$A_n[M] = (M)^{n}\times (S^{m-1})^{\#{\genfrac{[}{]}{0pt}{}{n}{2}}} \times [0,\infty]^{\#{\genfrac{[}{]}{0pt}{}{n}{3}}}.$$ Define the *Fulton–MacPherson compactification* ${C_n[M]}$ to be the closure of the image of ${C_n(M)}$ under the map $\iota_n= \iota \times ({\pi_{ij}}\vert_{{C_n(M)}}) \times (({s_{ijk}})\vert_{{C_n(M)}}) \co C_n(M)\rightarrow A_n[M]$. Let ${\partial}{C_n[M]}:={C_n[M]}- {C_n(M)}$ denote the boundary of $C_n[M]$, the points that are added in the closure. We list a few properties of this compactification, from [@MR2099074] and Theorem 2.3 of [@MR2102844]. \[thm:config\] The spaces ${C_n[M]}$ and ${C_n(M)}$ have the following properties: - ${C_n[M]}$ is a “manifold with corners” with interior ${C_n(M)}$. It has the same homotopy type as ${C_n(M)}$. It is independent of the embedding of $M$ in ${\mathbb{R}}^m$, and it is compact if $M$ is. - \[surjection\] The inclusion of ${C_n(M)}$ in $(M)^n$ extends to a surjective map $p$ from ${C_n[M]}$ to $(M)^n$ which is a homeomorphism over points in ${C_n(M)}$. - The boundary of ${C_n[M]}$ is stratified into a collection of faces of various dimensions. - An embedding $f \co M \rightarrow N$ induces an embedding of manifolds with corners called the evaluation map $ev_n[f] \co C_n[M]\rightarrow C_n[N]$ which respects the stratifications on the boundaries. The stratification of boundary faces of ${C_n[M]}$ has a beautiful combinatorial structure: in general, the set of faces of all codimensions is a Stasheff associahedron. While this structure is very interesting, we will use very little of it below, so we do not describe it in detail. We will need only the following: If $\Omega$ is a $k$-manifold with boundary embedded in ${\mathbb{R}}^n$ then the boundary of the manifold $C_2[\Omega]$ has three smooth faces, ${\partial}\Omega {\times}\Omega$, $\Omega {\times}{\partial}{\Omega}$ and an “interior” face diffeomorphic to the unit tangent bundle $UT(\Omega)$ of $\Omega$. Following the notation of Sinha, we will call this last face $(12)$, meaning that points 1 and 2 come together on that face. These codimension-1 faces of the boundary of $C_2[\Omega]$ meet at faces of higher codimension. The space $C_2[\Omega]$ is a closed subspace of the larger space $A_2[{\mathbb{R}}^n]$ created by closing the image of $\Omega {\times}\Omega$ under the map $\iota$. In this larger space, the boundary of the image consists of the image of ${\partial}(\Omega {\times}\Omega)$ together with a new boundary face created by removing the diagonal of $\Omega {\times}\Omega$. We are only taking configurations of pairs of points in $\Omega$, so there are no $s_{ijk}$ maps, and only two $\pi_{ij}$ maps: $\pi_{12}$ and $\pi_{21}$. Along the new boundary face, then, the map $\iota$ records the location $z$ and limiting direction $u$ of approach of pairs of points in $M$. This direction, recorded by $\pi_{12}$ and $\pi_{21}$, is a unit vector in the tangent space to $\Omega$ at $z$. The set of all such pairs is a copy of $UT(\Omega)$. These boundary faces meet at pairs of points where, for instance, an interior point approaches a boundary point, or where both points in the pair are on the boundary of $\Omega$. Sinha shows that these are faces of higher codimension. \[c2obdy\] If $\Omega$ is a domain in ${\mathbb{R}}^{2k+1}$ with smooth boundary, then the boundary of $C_2[\Omega]$ consists of three faces diffeomorphic to $\Omega {\times}{\partial}\Omega$, ${\partial}\Omega {\times}\Omega$ and $\Omega {\times}S^{2k}$. Sinha additionally defines configuration spaces where one or more points in the configuration are fixed. In this case, $C_{n,k}[M]$ is the space of $n$ points where $k$ points are fixed and the remaining $n-k$ points vary. Redefining helicity {#redefining} ------------------- Motivated by the Bott-Taubes approach [@MR1295465] to defining finite-type knot invariants, we now seek to define helicity for a $(k+1)$-form $\alpha$ on a domain in ${\mathbb{R}}^{2k+1}$ by integration over an appropriate configuration space. We will construct a “universal” $2k$-form on $C_2[{\mathbb{R}}^{2k+1}]$ by a Gauss map and then define helicity to be the integral of the wedge product of the universal form and a form derived from $\alpha$ over $C_2[\Omega]$. The corresponding approach for knots is explained beautifully by Volic [@MR2300426]. So let $\Omega \subset {\mathbb{R}}^{2k+1}$ be a compact subdomain with piecewise smooth boundary and let $\alpha$ be a closed Dirichlet $(k+1)$-form on $\Omega$. In ${\mathbb{R}}^3$, we may equivalently start with a smooth vector field $V$ on $\Omega$ that is divergence-free and tangent to the boundary, and take $\alpha$ to be the dual 2-form to $V$. The divergence-free condition implies that $\alpha$ is closed, while the boundary condition on $V$ implies that $\alpha\vert_{{\partial}\Omega}=0$. \[alpha\] The closed $(k+1)$-form $\alpha$ on $\Omega$ pulls back to a pair of closed $(k+1)$-forms $\alpha_x$ and $\alpha_y$ on $C_2[\Omega]$. Hence, their wedge product $\alpha_x \wedge \alpha_y$ is a closed $(2k+2)$-form on $C_2[\Omega]$. We take the surjective map $p: C_2[\Omega] {\rightarrow}\Omega {\times}\Omega$, guaranteed by Theorem \[thm:config\], and compose it with either of the two projections from $\Omega {\times}\Omega {\rightarrow}\Omega$ to obtain a map $C_2[\Omega] {\rightarrow}\Omega$. If we take $(x,y) \in C_2[\Omega]$, then these maps send $(x,y) \mapsto x$ or $(x,y) \mapsto y$. The pullback of $\alpha$ under the first map will be denoted $\alpha_x$ and the pullback under the second will be denoted $\alpha_y$. Since $\alpha$ is closed, $\alpha_x$, $\alpha_y$ and $\alpha_x \wedge \alpha_y$ are all closed forms. We now want to study the pullback of $\alpha_x \wedge \alpha_y$ to the boundary of $C_2[\Omega]$. To do so, we must first introduce coordinates on that boundary. As ${C_2[\Omega]}$ has codimension $4k$ in the ambient space ${A_2[\Omega]}$, the $8k+2$ natural coordinates on ${A_2[\Omega]}$ ($2k+1$ on $\Omega_x$, $2k+1$ on $\Omega_y$, $2k$ on each $S^{2k}$) overdetermine coordinates on ${C_2[\Omega]}$. In a neighborhood near the “interior” boundary face $(12)$, which we recall is diffeomorphic to $\Omega {\times}S^{2k} \subset {C_2[\Omega]}$ by Corollary \[c2obdy\], it will be convenient to work with three different coordinate systems: - [*configuration coordinates*]{}: $\{x_i, y_j\}$. These induce well-defined values on $S^{2k}$ except on the face $(12)$ where $x=y$. - [*midpoint-offset coordinates*]{}: $\{m_i, o_j\}$. Define $m:=(x+y)/2$ to be the midpoint of $xy$ and $o:=(x-y)/2$ to be the offset between $x$ and $y$. These variables are defined so that $x = m + o$ and $y = m - o$. These also induce well-defined values on $S^{2k}$, except on $\{o=0\}$, which describes the boundary face $(12)$. - [*boundary spherical coordinates*]{}: $\{z_i, r, u_j\}$. Define $\{r, u_j\}$ as spherical coordinates on the $o_j$ variables above so that $u_j$ is always a unit vector and $$r = {\left| o \right|}.$$ These have the advantage of naturally extending to the boundary face $(12)$, described by $\{r=0\}$. On $(12)$, the boundary spherical coordinates provide natural coordinates $\{z_i, u_j\}$. The $\{z_i\}$ describe the point $x=y$ while the $u_j$ measure the limiting direction by which $x$ and $y$ approached each other. \[alphavanishesonbdy\] If $\alpha$ is a Dirichlet form on $\Omega$, then $\alpha_x \wedge \alpha_y$ is a Dirichlet form on $C_2[\Omega]$. As we saw in Corollary \[c2obdy\], the boundary of $C_2[\Omega]$ consists of three codimension one faces: ${\partial}\Omega {\times}\Omega$, $\Omega {\times}{\partial}\Omega$ and $(12)$. On the first two boundary faces, either $x$ or $y$ is on ${\partial}\Omega$. But $\alpha$ vanishes when pulled back to ${\partial}\Omega$, so $\alpha_x$ vanishes on ${\partial}\Omega_x$ and $\alpha_y$ on ${\partial}\Omega_y$. Thus $\alpha_x \wedge \alpha_y$ vanishes on these faces. The third codimension one face, which we call face $(12)$, is all that remains. For convenience, let $I=(i_1, \ldots, i_k)$ denote a multi-index, so that $dx_I = dx_{i_1} \wedge \dots \wedge dx_{i_k}$. Using this notation, we observe that $\alpha_x$ can only consist of terms such as $h_{I}(x) dx_I$, with no $y$ dependence. Similarly, $\alpha_y$ consists of terms with the same coefficient functions $h_{I}(y) dy_I$. The functions $h_{I}(x)$ are smooth functions of $x$ since our original 2-form $\alpha$ on $\Omega$ was smooth. Consider these terms on the boundary face $(12)$, which is a copy of $\Omega {\times}S^{2k}$. We will write $\alpha_x \wedge \alpha_y$ in the boundary spherical coordinates $\{z_i, u_j \}$. In “midpoint-offset” coordinates, $x=m+o$ and $y=m-o$. Thus each $dx_i = dm_i + do_i$. If we now convert to boundary spherical coordinates using $o = r u$, then we see that $do_i = u_i dr + r du_i$. Now on the boundary face $(12)$, we have $r = 0$. The coefficient functions $h_{I}$ are smooth at $r=0$, so the term $h_{I} r du_I = 0$ on $(12)$ and the pullback of $\alpha_x$ to the boundary can have no $du_i$ terms. Further, $dr$ vanishes when pulled back to the boundary $S^{2k}$, so no $dr$ terms can be involved either. This means that the $(k+1)$-form $\alpha_x$ is expressed entirely in terms of the $2k+1$ midpoint coordinates $dm_i$. But the same is true for $\alpha_y$, so the $(2k+2)$-form $\alpha_x \wedge \alpha_y$ involves only the $2k+1$ elementary 1-forms $dm_i$. Thus some $dm_i$ is repeated, forcing this form to be zero. In the original definition of helicity in , we integrated over $\Omega {\times}\Omega$ even though the integrand was not defined on the diagonal. To justify the integration, it would be enough to show that the integrand converged on the diagonal. In fact, we can show that the integrand vanishes as we approach the diagonal. Lemma \[alphavanishesonbdy\] is the appropriate version of that familiar statement in our new setting. We now give a definition: \[gaussmap\] The *Gauss map* $g \co C_2(\Omega) \rightarrow S^{2k}$ is given by $(x,y) \mapsto (y-x)/{\left| y-x \right|}$. \[gausslem\] The Gauss map is a smooth map defined on all of $C_2[\Omega]$, including the boundary. The pullback of the unit volume form ${\operatorname{vol}}$ on $S^{2k}$ by $g$ defines a closed $2k$-form $g^*{\operatorname{dvol}}$ on $C_2[\Omega]$. The Gauss map extends naturally to $\Omega \times {\partial}\Omega$ and ${\partial}\Omega \times \Omega$, so we only have to worry about the boundary face $(12)$ of $C_2[\Omega]$. But by construction, $(12)$ is a blow-up of the diagonal of $\Omega \times \Omega$ so that the maps $\pi_{ij}$ extend smoothly to the boundary. In this case, $\pi_{21} = g$, so the lemma is proven. This lemma demonstrates why $C_2[\Omega]$ was better for our construction than $\Omega {\times}\Omega$. While the latter is simpler to work with, we could not have extended the Gauss map smoothly to the diagonal of $\Omega {\times}\Omega$. In fact, the form $g^*{\operatorname{dvol}}$ is the same as the “linking form” of [@MR1028280] (which is defined on $\Omega {\times}\Omega$) on the interior of $C_2[\Omega]$. The essential difference is that $g^*{\operatorname{dvol}}$ extends smoothly to the boundary of $C_2[\Omega]$ while the linking form has a singularity on the diagonal of $\Omega {\times}\Omega$. We can now combine the observations of Lemmas \[alpha\], \[alphavanishesonbdy\], and \[gausslem\] to redefine helicity. We have shown that if $\alpha$ is a closed Dirichlet form on $\Omega$, then $\alpha_x \wedge \alpha_y$ is a closed Dirichlet form on $C_2[\Omega]$. Hence $\alpha_x \wedge \alpha_y$ represents a relative de Rham cohomology class $[\alpha_x \wedge \alpha_y]$ in $H^{2k+2}(C_2[\Omega],{\partial}C_2[\Omega]; {\mathbb{R}})$. Similarly, $g^*{\operatorname{dvol}}$ is closed so it represents an absolute de Rham cohomology class $[g^*{\operatorname{dvol}}]$ in $H^{2k}(C_2[\Omega]; {\mathbb{R}})$. We will use de Rham cohomology (and thus coefficients in ${\mathbb{R}}$) for the rest of the paper. We now make an observation about the volume form on $C_2[\Omega]$: \[lem:volumeform\] If $M$ has a volume form ${\operatorname{dvol}}_M$, then there is a natural volume form ${\operatorname{dvol}}_{C_2[\Omega]}$ with total volume ${\operatorname{vol}}(C_2[\Omega]) = {\operatorname{vol}}(\Omega)^2$. Just as we pulled back the $(k+1)$-form $\alpha$ to forms $\alpha_x$ and $\alpha_y$ on $C_2[\Omega]$, we can pull back ${\operatorname{dvol}}_\Omega$ to $({\operatorname{dvol}}_\Omega)_x$ and $({\operatorname{dvol}}_\Omega)_y$. Then ${\operatorname{dvol}}(C_2[\Omega]) = ({\operatorname{dvol}}_{\Omega})_x \wedge ({\operatorname{dvol}}_{\Omega})_y$. This lemma enables us to define helicity. \[fthelicity\] If $\alpha$ is a closed Dirichlet $(k+1)$-form on $\Omega \subset {\mathbb{R}}^{2k+1}$, then we have seen that $\alpha_x \wedge \alpha_y$ defines a cohomology class in $H^{2k+2}(C_2[\Omega],{\partial}C_2[\Omega])$. We also know that $g^*{\operatorname{dvol}}_{S^{2k}}$ defines a cohomology class in $H^{2k}(C_2[\Omega])$. Let $[{\operatorname{dvol}}_{C_2[\Omega]}] \in H^{4k+2}(C_2[\Omega],{\partial}C_2[\Omega]) \simeq {\mathbb{R}}$ be the top class of $C_2[\Omega]$ defined by the standard volume form. The cup product $[\alpha_x \wedge \alpha_y] \cup [g^*{\operatorname{dvol}}_{S^{2k}}]$ is in $H^{4k+2}(C_2[\Omega])$ and is hence a multiple of $[{\operatorname{dvol}}_{C_2[\Omega]}]$. We define the *helicity* ${\operatorname{H}}(\alpha)$ of $\alpha$ by $$[\alpha_x \wedge \alpha_y] \cup [g^*{\operatorname{dvol}}] = \frac{{\operatorname{H}}(\alpha)}{{\operatorname{vol}}(\Omega)^2} [{\operatorname{dvol}}_{C_2[\Omega]}].$$ We can calculate ${\operatorname{H}}(\alpha)$ explicitly as the integral $$\label{hdefn} {\operatorname{H}}(\alpha) = \int_{C_2[\Omega]} \alpha_x \wedge \alpha_y \wedge g^*{\operatorname{dvol}}_{S^{2k}}.$$ Let $\Phi=\alpha_x \wedge \alpha_y \wedge g^*{\operatorname{dvol}}_{S^{2k}}$ denote the integrand above. In Theorem \[ftclassical\], we will show that our definition agrees with the classical integral on three-dimensional domains and the usual extension to the helicity of $(k+1)$-forms on $(2k+1)$-dimensional domains. As we expect from the theory of the Hopf invariant [@MR658304 Proposition 17.22], \[even-k\] For even $k$ values, the helicity of every $(k+1)$-form is zero. Let us consider the automorphism $a$ of $C_2(\Omega)$ that interchanges $x$ and $y$; it extends naturally to $C_2[\Omega]$. It reverses the orientation of $C_2[\Omega]$, since it exchanges the order of a product of odd-dimensional spaces. We take the pullback $a^*\Phi = \alpha_y \wedge \alpha_x \wedge a^*g^*{\operatorname{dvol}}_{S^{2k}}$. The map $a$ induces an antipodal map on $S^{2k}$; such a map has degree $-1$. Hence, $a^*g^*{\operatorname{dvol}}_{S^{2k}}= -g^*{\operatorname{dvol}}_{S^{2k}}$. Also, $\alpha_y \wedge \alpha_x = (-1)^{(k+1)^2} \alpha_x \wedge \alpha_y$. Combining these results, $a^*\Phi = (-1)^k \Phi$. We then compute $$-{\operatorname{H}}(\alpha) = \int_{-C_2[\Omega]} \Phi = \int_{a\left(C_2[\Omega]\right)} \Phi = \int_{C_2[\Omega]} a^*\Phi = \int_{C_2} (-1)^k \Phi = (-1)^k {\operatorname{H}}(\alpha).$$ If $k$ is even, this implies that ${\operatorname{H}}(\alpha) = -{\operatorname{H}}(\alpha)$, i.e., that helicity is zero, and proves our proposition. If $k$ is odd, the conclusion is a tautology: ${\operatorname{H}}(\alpha) = {\operatorname{H}}(\alpha)$. Comparison with the standard definition of helicity {#helicitysame} --------------------------------------------------- This description of helicity as a cohomology class may seem quite different from the definition of helicity that we gave earlier. So before we explore the consequences of our new definition, we will reassure ourselves that this approach is correct by showing explicitly that for 2-forms defined on domains in ${\mathbb{R}}^3$, our 6-form $\Phi$ on $C_2(\Omega)$ is exactly the classical helicity integrand. \[lemma-int\] Let $\Omega$ be a compact subdomain of ${\mathbb{R}}^3$ with smooth boundary, and let $\alpha$ be a closed Dirichlet 2-form on $\Omega$. Let $V$ be the vector field dual to $\alpha$. Recall from Definition \[cnmo\] that the map $\iota$ naturally embeds $C_2(\Omega)$ into $\Omega \times \Omega$. Then, the integrand $\Phi$ from (\[hdefn\]), namely the 6-form $\alpha_x \wedge \alpha_y \wedge g^*{\operatorname{dvol}}$, is equal to the pullback via $\iota$ of the classical helicity integrand $$\label{hintegrand} \frac{1}{4\pi} V(x) \times V(y) \cdot \frac{x-y}{|x-y|^3} \; {\operatorname{dvol}}_x \, {\operatorname{dvol}}_y.$$ With the lemma in place, we now conclude that our definition of helicity really is the same as the standard one. \[ftclassical\] For three-dimensional domains, the helicity of Definition \[fthelicity\], equals the classical helicity (of equation \[mhelicity\]). More explicitly, for a vector field $V$ dual to a 2-form $\alpha$, $$\label{eqhelicity} \int_{C_2[\Omega]} \alpha_x \wedge \alpha_y \wedge g^*{\operatorname{dvol}}_{S^2} = \frac{1}{4\pi} \int_{\Omega \times \Omega} {V(x) \times V(y) \cdot \frac{x-y}{|x-y|^3} \; {\operatorname{dvol}}_x \, {\operatorname{dvol}}_y}$$ The idea of the proof is to remove small neighborhoods of the boundary of ${C_2[\Omega]}$ and of the diagonal $\Delta$ in $\Omega \times \Omega$. On the removed neighborhoods, the integrals each tend to zero. On what remains, one integral is simply the pullback of the other. Denote the integrand (\[hintegrand\]) as $\mu$. Let $U_\epsilon$ be an $\epsilon$-neighborhood of $\partial {C_2[\Omega]}$. Then, $$\int_{C_2[\Omega]} \Phi = \int_{C_2[\Omega] - U_\epsilon } \Phi \; + \; \int_{U_\epsilon} \Phi$$ As $\epsilon \rightarrow 0$, so does $\int_{U_\epsilon} \Phi$. Then, the above lemma guarantees that $\Phi=\iota^*\mu$ on $C_2[\Omega] - U_\epsilon$. Hence, $$\int_{C_2[\Omega] - U_\epsilon} \Phi \; = \; \int_{C_2[\Omega] - U_\epsilon} \iota^*\mu \; = \; \int_{\iota(C_2[\Omega] - U_\epsilon )} \mu$$ But, the image $\iota(C_2[\Omega] - U_\epsilon )$ is $\Omega \times \Omega$ with some neighborhood $V_\epsilon$, dependent upon $\epsilon$, removed. As $\epsilon \rightarrow 0$, the set $V_{\epsilon}$ approximates $\Delta$. While the integral $\int_{\Omega \times \Omega} \mu$ is improper along the diagonal, it does in fact converge. The contribution of $\mu$ integrated over neighborhoods of the diagonal converges to 0. See [@MR1770976] for details. Hence, $\int_{\iota({C_n(M)}- U_\epsilon )} \mu$ limits to the classical helicity integral $\int_{\Omega \times \Omega} \mu$. But it also limits to $\int_{C_2[\Omega]} \Phi$, so the two are equal. We now prove the above lemma in local coordinates at an arbitrary point in $C_2(\Omega)$. A choice of coordinates on $\Omega$ induces a set of configuration coordinates on $C_2(\Omega)$. At the point $p=(x,y) \in C_2(\Omega)$, we choose right-handed orthonormal coordinates $\{u_i\}$ on $\Omega$ so that $u_3$ points along the vector $y-x$ at $p$. Via the map $\iota$ from Definition \[cnmo\], these induce coordinates $\{x_i, y_i\}$ on $C_2(\Omega)$. We now calculate $\Phi$ and the classical helicity integrand in these coordinates at $p$. Begin by writing $$V(x) = v_1 {\frac{{\partial}}{{\partial}u_{1}}} + v_2 {\frac{{\partial}}{{\partial}u_{2}}} + v_3 {\frac{{\partial}}{{\partial}u_{3}}} \qquad \mathrm{and} \qquad V(y) = w_1 {\frac{{\partial}}{{\partial}u_{1}}} + w_2 {\frac{{\partial}}{{\partial}u_{2}}} + w_3 {\frac{{\partial}}{{\partial}u_{3}}}$$ so that $$\begin{aligned} \alpha_x & = v_1 \; dx_2 \wedge dx_3 \; + \; v_2 \;dx_3 \wedge dx_1 \; + \; v_3 \; dx_1 \wedge dx_2, \\ \alpha_y &= w_1 \; dy_2 \wedge dy_3 \; + \; w_2\; dy_3 \wedge dy_1 \; + \; w_3 \; dy_1 \wedge dy_2. \end{aligned}$$ By the choice of coordinates, ${\displaystyle}{\frac{x-y}{|x-y|^3} = - \frac{1}{|x-y|^2} \; {\frac{{\partial}}{{\partial}u_{3}}}}$. Then, the classical helicity integrand is $$\label{int-coords} \frac{1}{4\pi} V(x) \times V(y) \cdot \frac{x-y}{|x-y|^3} {\operatorname{dvol}}_x \, {\operatorname{dvol}}_y = \frac{1}{4\pi} \frac{1}{|x-y|^2} (v_2 w_1 - v_1 w_2) {\operatorname{dvol}}_x \, {\operatorname{dvol}}_y.$$ Now we calculate $\Phi$ in these coordinates; we start with $g^*{\operatorname{dvol}}$, the pullback of the unit area form on $S^2$ via the Gauss map. Moving the configuration points in the $x_3$ (or $y_3$) direction, that is moving them closer or further apart, has no impact upon the Gauss map $g$, so $g^*{\operatorname{dvol}}$ contains no $dx_3$ or $dy_3$ terms. Writing it in terms of the other 2-forms, we get $$\begin{aligned} g^*{\operatorname{dvol}}= & c_1 dx_1 \wedge dx_2 \; + \; c_2 dy_1 \wedge dy_2 \; + \; c_3 dx_1 \wedge dy_1 \; + \; c_4 dx_2 \wedge dy_2 \\ & + c_5 dx_1 \wedge dy_2 \; + \; c_6 dy_1 \wedge dx_2.\end{aligned}$$ So which bi-vectors on $C_2(\Omega)$ span area on $S^2$ under $g$? Neither ${\frac{{\partial}}{{\partial}x_{1}}} \wedge {\frac{{\partial}}{{\partial}y_{1}}}$ nor ${\frac{{\partial}}{{\partial}x_{2}}} \wedge {\frac{{\partial}}{{\partial}y_{2}}}$ does, so $c_3 = c_4 = 0$. The other bi-vectors do; their effect must be normalized by the distance squared between $x$ and $y$ and also by the fact that the area of the sphere integrates to $1$ (we are using the unit area form ${\operatorname{dvol}}$). Considering orientations, $c_1= c_2 = 1/4\pi|x-y|^2 = -c_5 = -c_6$. So $$g^*{\operatorname{dvol}}= \frac{1}{ 4\pi|x-y|^2} \left(dx_1 \wedge dx_2 \; + \; dy_1 \wedge dy_2 \; - \; dx_1 \wedge dy_2 \; - \; dy_1 \wedge dx_2 \right).$$ We compute the 6-form $\Phi$ to be $c_5 v_1 w_2 - c_6 v_2 w_1$, which by substituting becomes $$\begin{aligned} \Phi & = \frac{1}{4\pi}\frac{1}{{\left| x-y \right|}^2} (v_2 w_1 - v_1 w_2) \; dx_1 \wedge dx_2 \wedge dx_3 \wedge dy_1 \wedge dy_2\wedge dy_3 \label{phi-coords}\end{aligned}$$ Pulling the classical helicity integrand (\[int-coords\]) back via $\iota$, we obtain $\Phi$ since $\iota^*({\operatorname{dvol}}_x) = dx_1 \wedge dx_2 \wedge dx_3$ (and similarly for ${\operatorname{dvol}}_y$). Understanding the properties of helicity via cohomology ======================================================= We have now defined helicity as a cup product of cohomology classes and have shown in the case of vector fields in ${\mathbb{R}}^3$ that our definition is the standard helicity integral. We now consider the consequences of our new definition and try to provide some motivation for the definition now that we have made it. Invariance of helicity under diffeomorphisms homotopic to the identity ---------------------------------------------------------------------- In the Introduction, we discussed the development of the Helicity Invariance theorem, from the earliest versions of helicity as an invariant of ideal MHD through Arnold’s picture of helicity as invariant under all diffeomorphisms on simply-connected domains to the modern picture of helicity as invariant under diffeomorphisms which are homotopic to the identity. Our redefinition of helicity allows us to give a quick proof of this invariance result. \[homotopic-invariance\] Let $\Omega$ be any domain in ${\mathbb{R}}^{2k+1}$ and let $\alpha$ be a closed Dirichlet $(k+1)$-form on $\Omega$. Let $f\co\Omega {\times}I \rightarrow {\mathbb{R}}^{2k+1}$ be a smooth map. For each fixed $t$, define $f_t\co \Omega \rightarrow \Omega_t \subset {\mathbb{R}}^{2k+1}$ by $f_t(p) = f(p,t)$ and assume that each $f_t$ is a diffeomorphism, with $f_0$ the identity map. Let $\alpha_t = (f_t^{-1})^* \alpha$ on each $\Omega_t$. Then ${\operatorname{H}}(\alpha)$ on $\Omega_0 = \Omega$ is equal to ${\operatorname{H}}(\alpha_1)$ on $\Omega_1$. There is a natural projection $\Omega {\times}I \rightarrow \Omega$ given by $(p,t) \mapsto p$. Pulling back under this map, we can extend $\alpha$ to a form on $\Omega {\times}I$. Similarly, there are two obvious projections $\pi_x, \pi_y \co C_2[\Omega] {\times}I \rightarrow \Omega {\times}I$ given by $(x,y,t) \mapsto (x,t)$ and $(x,y,t) \mapsto (y,t)$. Pulling back under these maps, we can define a closed form $\alpha_x \wedge \alpha_y$ on $C_2[\Omega] {\times}I$. We next define an extended Gauss map on $C_2[\Omega] {\times}I$ by $$G(x,y,t) = \frac{f_t(x) - f_t(y)}{{\left| f_t(x) - f_t(y) \right|}}.$$ This map allows us to construct a closed $2k$-form $G^*{\operatorname{dvol}}_{S^{2k}}$ on $C_2[\Omega] {\times}I$. We note that the $(4k+2)$ helicity form $\alpha_x \wedge \alpha_y \wedge G^*{\operatorname{dvol}}_{S^{2k}}$ is a closed Dirichlet form (by Lemma \[alphavanishesonbdy\]) on $C_2[\Omega]$. By Stokes’ theorem, the integral of this form over ${\partial}(C_2[\Omega] {\times}I)$ is zero. But this means that $$\label{topbottom} \int_{C_2[\Omega] {\times}\{0\}} \alpha_x \wedge \alpha_y \wedge G^*{\operatorname{dvol}}_{S^{2k}} = \int_{C_2[\Omega] {\times}\{ 1 \}} \alpha_x \wedge \alpha_y \wedge G^*{\operatorname{dvol}}_{S^{2k}}.$$ We now prove that the left hand side is ${\operatorname{H}}(\alpha_0)$ and the right hand side is ${\operatorname{H}}(\alpha_1)$. Since $f_0$ is the identity map, $G(x,y,0) = g(x,y)$ and the left hand side is clearly ${\operatorname{H}}(\alpha) = {\operatorname{H}}(\alpha_0)$. On the right-hand side, we observe that by definition $${\operatorname{H}}(\alpha_1) = \int_{C_2[\Omega_1]} (\alpha_1)_x \wedge (\alpha_1)_y \wedge g^*{\operatorname{dvol}}_{S^{2k}} = \int_{C_2[\Omega_1]} (F^{-1})^*\alpha_x \wedge (F^{-1})^*\alpha_y \wedge g^*{\operatorname{dvol}}_{S^{2k}}.$$ where $F \co C_2[\Omega] \rightarrow C_2[\Omega_1]$ is the map of configuration spaces induced by $f_1$ (c.f., Theorem \[thm:config\]). We note that $G(x,y,1)=g \circ F(x,y)$. If we pull back the integral above to $C_2[\Omega] {\times}\{1\}$ using $F^{-1}$, we get the right hand side of . $$\begin{aligned} {\operatorname{H}}(\alpha_1) & = \int_{F^{-1}(C_2[\Omega_1])=C_2[\Omega]} \alpha_x \wedge \alpha_y \wedge F^*g^*{\operatorname{dvol}}_{S^{2k}} \\ & = \int_{C_2[\Omega] {\times}\{ 1 \}} \alpha_x \wedge \alpha_y \wedge G^*{\operatorname{dvol}}_{S^{2k}}. \qedhere\end{aligned}$$ The invariance theorems for helicity and finite-type invariants --------------------------------------------------------------- We could have proved this theorem in a new way, parallel to the proof of invariance for the finite-type invariants for knots. Let ${\mathfrak{Embed}(\Omega \hookrightarrow {\mathbb{R}}^{2k+1})}$, henceforth denoted ${\mathfrak{E}}$, consist of all diffeomorphic embeddings of $\Omega$ into ${\mathbb{R}}^{2k+1}$. Maps in each connected component of ${\mathfrak{E}}$ are diffeotopic to one another. Define a Gauss map $g_f$ by $$((x,y),f) \in C_2[\Omega] \times \mathfrak{E} \mapsto \displaystyle{\frac{f(x)-f(y)}{|f(x)-f(y)|}}.$$ Consider the following diagram: $$\label{embedbundle} \begin{CD} C_2[ \Omega] \times {\mathfrak{Embed}(\Omega \hookrightarrow {\mathbb{R}}^{2k+1})}@>g_f>> S^{2k} \\ @VV{\pi}V \\ {\mathfrak{Embed}(\Omega \hookrightarrow {\mathbb{R}}^{2k+1})}\end{CD}$$ where $\pi$ is the natural projection in the trivial bundle $C_2[\Omega] {\times}{\mathfrak{Embed}(\Omega \hookrightarrow {\mathbb{R}}^{2k+1})}\rightarrow {\mathfrak{Embed}(\Omega \hookrightarrow {\mathbb{R}}^{2k+1})}$. This is analogous to the corresponding diagram for knots introduced by Bott and Taubes [@MR1295465]. Define the $(4k+2)$-form $$\label{defgamma} \Phi = \alpha_x \wedge \alpha_y \wedge g_f^{*}({\operatorname{dvol}})$$ on $C_2[\Omega] {\times}{\mathfrak{E}}$ by pulling back $\alpha_x \wedge \alpha_y$ from $C_2[\Omega]$ and the volume form ${\operatorname{dvol}}$ from $S^{2k}$; we note that $\Phi$ is Dirichlet, by Lemma \[alphavanishesonbdy\], and is closed. We now observe that integration of $\Phi$ over the fiber in the bundle $C_2[\Omega] {\times}{\mathfrak{E}}\rightarrow {\mathfrak{E}}$ produces a $0$-form ${\operatorname{H}}(f)$ on ${\mathfrak{E}}$. The value of this $0$-form on any embedding is the helicity ${\operatorname{H}}((f^{-1})^*\alpha)$. Using Stokes’ Theorem, we compute $$d {\operatorname{H}}(f) = d\int_{C_2[\Omega]}{\Phi} = \int_{C_2[\Omega]}{d\Phi} - \int_{{\partial}C_2[\Omega]}{\Phi} = 0 - 0,$$ since $\Phi$ is a closed Dirichlet form. Since $d{\operatorname{H}}(f) = 0$, we conclude that ${\operatorname{H}}(f)$ is constant on each connected component of ${\mathfrak{E}}$. This reproves that helicity is invariant under diffeomorphisms homotopic to the identity. Invariance of helicity for forms and vector fields -------------------------------------------------- The original invariance theorem for helicity of vector fields (Theorem \[classicalinvariance\]) required that the diffeomorphisms be volume-preserving. Our theorems about the invariance of the helicity of forms, by contrast, have no such requirement. If we fix our attention on the case $2k+1 = 3$, and consider the duality between 2-forms and vector fields, we immediately observe where the volume-preserving condition arises. Start with $V$ dual to $\alpha$ on $\Omega$ and a diffeomorphism $f$ that lies in the same component of ${\mathfrak{E}}$ as the identity. The helicity of $\alpha$ on $\Omega$ is the same as the helicity of the 2-form $\tilde{\alpha} = (f^{-1})^* (\alpha)$ on $f(\Omega)$. However if $f$ is not volume-preserving, $\tilde\alpha(\cdot, \cdot)$ may not be dual to the pushforward vector field $f_*V$ because the duality operation explicitly involves the volume form on $f(\Omega)$. Hence, differential forms produce a stronger invariance than vector fields do. Invariance of helicity defined with cohomologous forms ------------------------------------------------------ Another interesting feature of Definition \[fthelicity\] is that the helicity of $\alpha$ depends only on the cohomology classes of $[\alpha_x \wedge \alpha_y]$ and $[g^*{\operatorname{dvol}}_{S^{2k}}]$. In particular, this means that we may define the helicity integrand using any volume form on $S^{2k}$ which integrates to $1$ over the sphere and get an alternate integral formula for helicity. We are motivated here by the combinatorial formula for linking number, which is derived from the Gauss integral formula for linking number by concentrating the mass of the sphere at the north pole. This gives us a recipe for constructing new helicity integrals. Given a point $x = (x_1,x_2,x_3)$ in a domain $\Omega$ in ${\mathbb{R}}^3$, let $x^+(\Omega)$ be the set of points $y = (x_1,x_2,y_3) \in \Omega$ with $y_3 > x_3$. We then have \[prop:combinatorial\] The helicity of a divergence-free vector field in ${\mathbb{R}}^3$ which is tangent to the boundary of a domain $\Omega$ is given by the $4$-dimensional integral $${\operatorname{H}}(V) = \frac{1}{4\pi} \int_{x \in \Omega} \int_{y \in x^+(\Omega)} V(x) \cdot V(y) {\times}(0,0,1) \; {\operatorname{dvol}}_x dy_3.$$ Consider a sequence of $2$-forms on $S^2$ converging to the $\delta$-form which concentrates the area of the sphere at the north pole where each has integral $4\pi$ over the entire sphere. These forms are cohomologous as $2$-forms on $S^2$ to the standard area form, so their pullbacks generate cohomologous $2$-forms on $C_2[\Omega]$. This means that the helicities derived from the forms in the sequence are all equal to the standard helicity. But the limit of these integrals is the formula above. We now do an explicit helicity computation using the formula to check that it works. It is an old theorem of Moffatt [@mof1] and Berger and Field [@MR770136] that the helicity of a divergence-free field tangent to the boundary of a pair of linked tubes is equal to the helicity of the fields in each tube plus twice the linking number of the tubes multiplied by the square of the flux of the field in the tubes (see [@MR2002f:53002] for a more general version of this theorem). Imagine then, a pair of singly-linked tubes that have rectangular cross-section with width $w$ and height $h$ and one overcrossing and that contain unit length fields parallel to the walls. We will assume that at the overcrossing the tubes are rectangular boxes in parallel planes, as below in Figure \[helicityfig\]. [squarehopf]{} (22,23)[$A$]{} (87,6)[$B$]{} [helicityexample]{} (93.5,65)[$w$]{} (76,36)[$\theta$]{} (93.5,8.5)[$w$]{} We can arrange the tubes so that for any pair $\{x,y\}$ with $x$ and $y$ in the same tube and $y \in x^+(\Omega)$, the vectors $V(x)$ and $V(y)$ are collinear. Since the integrand above vanishes for collinear vectors, these pairs will not contribute to the integral. We may further arrange the tubes so that there are only two regions where $x$ and $y$ are in different tubes and $y \in x^+(\Omega)$. The overcrossing pictured is one region, with $x$ lying in the right side of ring $A$ and $y$ above it in the upper segment of ring $B$. The other region has $x$ in the lower section of $B$ and $y$ in $A$. We now need to integrate over these pairs. The triple product in the integrand can be rewritten $(0,0,1) \cdot V(x) {\times}V(y)$. Since $V(x)$ and $V(y)$ lie in horizontal planes in this region, the integrand always takes the constant value $\sin\theta$. On the other hand, the domain of integration (for $x$) is a prism of height $h$ whose base is a parallelogram of length $w/\sin\theta$ and width $w$; the domain of integration for $y$ is a line segment above each point in the $x$ prism of length $h$. Thus the total (4-dimensional) volume of integration is $w^2 h^2/\sin\theta$, and the value of the integral is $w^2h^2 = (wh)^2$. This is exactly the square of the flux of the vector field, and we note that the crossing is positively oriented. The other region with $y \in x^+(\Omega)$ has $x$ in the lower section of tube $B$ and $y$ in tube $A$. This configuration is similar to the first, and makes the same contribution to the integral. Helicity as a wedge product with a primitive ============================================ Arnol’d [@MR891881] defines helicity for 2-forms on simply connected 3-manifolds as the integral of the wedge product of a form $\alpha$ and a primitive form $\beta$ with $d\beta = \alpha$. In section \[redefining\], we provided an alternate definition in terms of cohomology classes on configuration spaces. In this section, we reconcile these two approaches. Our efforts culminate in the next section with a formula for the change in helicity under an arbitrary diffeomorphism of $\Omega$. Constructing a primitive form. ------------------------------ We start by observing that there is a natural fiber bundle $$\label{bsbundle} \begin{CD} C_{2,1}[ \Omega] @>i>> C_2[\Omega] \\ @. @VV{\pi_x}V \\ @. \Omega \end{CD}$$ where $\pi_x$ is the projection where $(x,y) \mapsto x$. Consider the $(k+1)$-form $\alpha_y = \pi_y^*\alpha$ from Lemma \[alpha\] generated by pulling back $\alpha$ from $\Omega$ in the corresponding projection $\pi_y$ where $(x,y) \mapsto y$, and the $2k$-form $g^*{\operatorname{dvol}}$ generated by pulling back the volume form on $S^{2k}$ under the Gauss map. We now develop some standard properties of this bundle. We define the *Biot-Savart operator for forms* to be the operation on $(k+1)$-forms $\alpha$ on $\Omega$ defined by the integration over the fiber in the bundle , $${\operatorname{BS}}(\alpha) = \frac{1}{{\operatorname{vol}}(S^{2k})} \int_{C_{2,1}[\Omega]} \alpha_y \wedge g^*{\operatorname{dvol}}.$$ Proposition \[alphaisexact\] guarantees that any closed Dirichlet $(k+1)$-form $\alpha$ is exact. We now show that the Biot-Savart operator constructs a primitive for $\alpha$. \[bsoperator\] If $\alpha$ is a closed Dirichlet $(k+1)$-form on $\Omega$, then $BS(\alpha)$ is a primitive for $\alpha$: $$d({\operatorname{BS}}(\alpha)) = \alpha.$$ We will use Stokes’ Theorem for fiber bundles $F^n \rightarrow E \rightarrow B$ (see Appendix \[gv\]). If $\beta$ is a $k$-form on $E$, then $\int_F \beta$ is a $(k-n)$-form on $B$, and $$\label{stokes} d \int_F \beta = \int_F d\beta - \int_{{\partial}F} \beta.$$ By definition, ${\operatorname{BS}}(\alpha)$ is the integration over the fiber $C_{2,1}[\Omega]$ of the form $\alpha_y \wedge g^*{\operatorname{dvol}}$. Since $\alpha_y \wedge g^*{\operatorname{dvol}}$ is closed on $C_{2,1}[\Omega]$, we see $d{\operatorname{BS}}(\alpha) = \int_{{\partial}C_{2,1}[\Omega]} \alpha_y \wedge g^*{\operatorname{dvol}}$. Now consider the structure of the boundary of $C_{2,1}[\Omega]$. We are assuming that $x$ is the fixed point, so there are two codimension-one faces of ${\partial}C_{2,1}[\Omega]$: one consists of a copy of ${\partial}\Omega$ in the form of pairs $(x,y)$ where $y$ is on the boundary; the other is a copy of $S^{2k}$ where $y$ approaches $x$ from some direction. We note that the outward normal to the fiber points into this $S^{2k}$. On the ${\partial}\Omega$ face, $\alpha_y$ vanishes so there is no contribution to the integral. We now consider the term $\int_{S^{2k}} \alpha_y \wedge g^*{\operatorname{dvol}}$. What is this form? In the definition of integration over the fiber (see Appendix \[gv\]), we see that to integrate a $(3k+1)$-form over a $2k$-dimensional fiber and get a resulting $(k+1)$-form, we must write each tangent space to the total space of the bundle as a product of the $2k$-dimensional tangent space to the fiber and the tangent space to the base and decompose our $(3k+1)$-form locally into a wedge of forms on each of these spaces. The fiber portion of the form is then integrated, while the base portion remains. On $C_{2}[\Omega]$ we now establish the coordinates $x_i$, $z_i = y_i - x_i$, and write $z = r u$, where $u$ is a unit vector. In the bundle , the base directions are the ${\partial}/ {\partial}x_i$ and the fiber directions are the ${\partial}/ {\partial}z_i$. How do these coordinates extend to the boundary of the fiber? There is no difficulty in defining these coordinates on the boundary face where $y \in {\partial}\Omega$. But on the boundary face (12) where the two configuration points coalesce, i.e., where $r=0$, the situation requires a bit more care. Unlike the standard polar coordinates, in which the $u_i$ will have no meaning when $r=0$, our compactification of the configuration space ensures that the $S^{2k}$ defined by the $u$ coordinates will still be present when $r=0$. We now consider the forms $\alpha_y$ and $g^*{\operatorname{dvol}}$ on the boundary face where $r=0$ with an eye toward integration over the fiber. The form $\alpha_y$ is written entirely in terms of elementary forms chosen from the $dy_i$. But $dy_i = dx_i + dz_i$. And $dz_i = u_i dr + r du_i$, so on this face $\alpha_y$ is written entirely in terms of $dr$ and the $dx_i$. In fact, $\alpha_y$ contains a precise copy of $\alpha_x$ together with a collection of other terms involving $dr$. When we pull this form back to the boundary, the $dr$ terms vanish, leaving only a copy of $\alpha_x$. On the other hand, the form $g^*{\operatorname{dvol}}$ is exactly the volume form on the boundary $S^{2k}$, as the Gauss map in these coordinates is just $g(x,r,u) = u$. Integrating over the fiber, we obtain $$- \int_{{\partial}C_{2,1}[\Omega]} \alpha_y \wedge g^*{\operatorname{dvol}}= \int_{S^{2k}} \alpha_y \wedge g^*{\operatorname{dvol}}= ({\operatorname{vol}}{S^{2k}}) \alpha_x .$$ Since the standard $2k$-sphere has the opposite orientation of ${\partial}C_{2,1}[\Omega]$, the leading minus sign (from ) does not appear after the first equality. Inspired by the theory of self-adjoint curl operators in dimension $3$ (a long story, stretching from [@MR1055988] to [@hiptmair-2008]), we observe that \[self adjoint\] ${\operatorname{BS}}$ is a self-adjoint operator on closed Dirichlet $(k+1)$-forms on $\Omega$, for odd $k$. For any two such forms $\alpha$ and $\beta$, $$\int_{\Omega} \alpha \wedge {\operatorname{BS}}(\beta) = (-1)^{k+1} \int_{\Omega} {\operatorname{BS}}(\alpha) \wedge \beta.$$ We observe that $$d({\operatorname{BS}}(\alpha) \wedge {\operatorname{BS}}(\beta)) = \alpha \wedge {\operatorname{BS}}(\beta) + (-1)^k {\operatorname{BS}}(\alpha) \wedge \beta.$$ Integrating both sides over $\Omega$, we see that we must prove that $\int_{{\partial}\Omega} {\operatorname{BS}}(\alpha) \wedge {\operatorname{BS}}(\beta) = 0$. Since ${\operatorname{BS}}(\alpha)$ and ${\operatorname{BS}}(\beta)$ are closed forms on the boundary, this integral depends only on the cup product of the cohomology classes represented by ${\operatorname{BS}}(\alpha)$ and ${\operatorname{BS}}(\beta)$ in $H^k({\partial}\Omega)$. Borrowing from Theorem \[alexander basis existence\] of the Appendix, we know that the de Rham cohomology group $H^k({\partial}\Omega)$ splits into two subspaces: forms with no circulation around $k$-cycles which bound outside $\Omega$ and forms with no circulation around $k$-cycles which bound inside $\Omega$. Since $\alpha$ and $\beta$ vanish outside $\Omega$, ${\operatorname{BS}}(\alpha)$ and ${\operatorname{BS}}(\beta)$ are in the first subspace. Further, Theorem \[alexander basis existence\] asserts that the cup product of any two forms in the same subspace vanishes. This shows that $[{\operatorname{BS}}(\alpha)] \cup [{\operatorname{BS}}(\beta)] = 0$, as desired. An equivalent definition of helicity as a potential. ---------------------------------------------------- Motivated by Arnold’s approach, can we express helicity as the integral of $\alpha \wedge \beta$ for an arbitrary primitive $\beta$ of $\alpha$? Unfortunately not, except in special circumstances (see [@MR1770976]), since helicity is not gauge-invariant; we must choose an appropriate primitive. Below, we show that ${\operatorname{BS}}(\alpha)$ is an appropriate primitive and that we recover the same helicity as in Definition \[fthelicity\]. \[arnoldhelicity\] Let $\alpha$ be a closed Dirichlet $(k+1)$-form on a compact domain $\Omega$ in ${\mathbb{R}}^{2k+1}$. The Hodge decomposition theorem for manifolds with boundary tells us that $\alpha$ is exact. Then the *“Arnol’d helicity”* of $\alpha$ is given by $${\operatorname{H}}(\alpha) = \int_\Omega \alpha \wedge {\operatorname{BS}}(\alpha).$$ The following proposition ensures that “Arnol’d helicity” is equivalent to our original definition of helicity; thus we will refer to both as *helicity*. The proof is almost immediate. \[prop:potential\] Given any closed Dirichlet $(k+1)$-form $\alpha$ on a domain $\Omega$ in ${\mathbb{R}}^{2k+1}$, the “Arnol’d helicity” (via integrating a specific primitive) $${\operatorname{H}}(\alpha) = \int_\Omega \alpha \wedge {\operatorname{BS}}(\alpha)$$ of Definition \[arnoldhelicity\] is equal to the helicity (via cohomology classes) $${\operatorname{H}}(\alpha) = \int_{C_2[\Omega]} \alpha_x \wedge \alpha_y \wedge g^*{\operatorname{dvol}}$$ of Definition \[fthelicity\]. Using the bundle and the properties of integration over the fiber, we see that $$\int_{C_2[\Omega]} \alpha_x \wedge \alpha_y \wedge g^*{\operatorname{dvol}}= \int_\Omega \alpha_x \wedge \left( \int_{C_{2,1}[\Omega]} \alpha_y \wedge g^*{\operatorname{dvol}}\right) = \int_\Omega \alpha_x \wedge {\operatorname{BS}}(\alpha_x).$$ When is helicity invariant under a diffeomorphism? {#sec:invariance} ================================================== We have now completed our revision of the standard theory of helicity. With this in hand, we may now fully and precisely answer the question: is helicity a diffeomorphism invariant? The answer is negative, except in certain special cases (for one such case, see Proposition \[homotopic-invariance\]). In the main result of this section, we explicitly calculate the change in helicity of $\alpha$ under an arbitrary diffeomorphism of $\Omega$. Specific cases of this formula reproduce the known invariance results about helicity for domains in ${\mathbb{R}}^3$ (Theorem \[classicalinvariance\] and Proposition \[homotopic-invariance\]). After describing the topology of domains in ${\mathbb{R}}^{2k+1}$, we first derive the formula for the case where $\Omega$ is a solid torus in ${\mathbb{R}}^3$ before describing the general result. Even though helicity is the zero function for subdomains $\Omega \subset {\mathbb{R}}^{4k+1}$ (i.e., $k$ even; see Proposition \[even-k\]), we carry out this computation in general and note the instances in which the parity of $k$ matters. As a check, we confirm that for the $k$ even case, helicity is invariant under all diffeomorphisms. We begin by fixing an orientation-preserving diffeomorphism $f\co\Omega \rightarrow \Omega'$ between domains in ${\mathbb{R}}^{2k+1}$. Let $\alpha$ be a closed Dirichlet $(k+1)$-form on $\Omega$. Then its pullback $\alpha' = {\left( f^{-1} \right)^*\alpha}$ is a closed Dirichlet $(k+1)$-form on $\Omega'$. By Proposition \[alphaisexact\], both $\alpha$ and ${\left( f^{-1} \right)^*\alpha}$ are exact. Did the helicity of $\alpha$ change under the map $f$? That is, does ${\operatorname{H}}(\alpha)$ equal ${\operatorname{H}}(\alpha')$? We compute $$\begin{aligned} {\operatorname{H}}(\alpha) & = & \int_{\Omega} \alpha \wedge {\operatorname{BS}}(\alpha), \\ {\operatorname{H}}(\alpha') & = & \int_{\Omega' = f(\Omega)}{\left( f^{-1} \right)^*\alpha}\wedge {\operatorname{BS}}\left({\left( f^{-1} \right)^*\alpha}\right) \\ & = & \int_{\Omega} f^*\left( {\left( f^{-1} \right)^*\alpha}\wedge {\operatorname{BS}}({\left( f^{-1} \right)^*\alpha}) \right) \\ & = & \int_{\Omega} \alpha \wedge f^*{\operatorname{BS}}\left({\left( f^{-1} \right)^*\alpha}\right).\end{aligned}$$ Both terms integrate $\alpha$ wedged with a $k$-form, either ${\operatorname{BS}}(\alpha)$ or $f^*{\operatorname{BS}}(\alpha')$. Both $k$-forms are both primitives for $\alpha$, since the exterior derivative commutes with pullbacks, i.e., $df^*{\operatorname{BS}}\left({\left( f^{-1} \right)^*\alpha}\right) = f^* \left( d{\operatorname{BS}}\left({\left( f^{-1} \right)^*\alpha}\right)\right) = f^*\left({\left( f^{-1} \right)^*\alpha}\right) = \alpha$. However, ${\operatorname{BS}}$ does not in general commute with pullbacks, and so these two $k$-forms are not necessarily equal. So we calculate the difference $$\label{helicity-change} {\operatorname{H}}(\alpha') - {\operatorname{H}}(\alpha) = \int_{\Omega} \alpha \wedge \left( f^*{\operatorname{BS}}(\alpha') - {\operatorname{BS}}(\alpha) \right)$$ In general terms, given two primitives $\beta$ and ${\tilde{\beta}}$ of $\alpha$, we wish to compute $\int_{\Omega} \alpha\wedge ({\tilde{\beta}}- \beta)$. We first observe that the integrand is an exact $(2k+1)$-form. In particular, $$d\left(({\tilde{\beta}}- \beta) \wedge ({\tilde{\beta}}+ \beta) \right) = 2 (-1)^{k^2 + 2k} \alpha \wedge ({\tilde{\beta}}- \beta) = 2 (-1)^k \alpha \wedge ({\tilde{\beta}}- \beta).$$ Upon simplifying this potential $2k$-form, we conclude that $$\left(({\tilde{\beta}}- \beta) \wedge ({\tilde{\beta}}+ \beta) \right) = \begin{cases} 2 {\tilde{\beta}}\wedge \beta & \text{if $k$ is odd},\\ {\tilde{\beta}}\wedge {\tilde{\beta}}- \beta \wedge \beta & \text{if $k$ is even}. \end{cases}$$ Applying Stokes’ theorem, we obtain $$\begin{aligned} \int_{\Omega} \alpha\wedge({\tilde{\beta}}- \beta) & = & \int_{\Omega} { \tfrac{1}{2} (-1)^k d\left(({\tilde{\beta}}- \beta) \wedge ({\tilde{\beta}}+ \beta) \right)} \\ & = & (-1)^k \int_{{\partial}\Omega} { \tfrac{1}{2} \left(({\tilde{\beta}}- \beta) \wedge ({\tilde{\beta}}+ \beta) \right)} \\ & = & \begin{cases} \int_{{\partial}\Omega} \beta \wedge {\tilde{\beta}}& \text{if $k$ is odd},\\ \tfrac{1}{2} \int_{{\partial}\Omega} {\tilde{\beta}}\wedge {\tilde{\beta}}- \beta \wedge \beta & \text{if $k$ is even}. \end{cases} \end{aligned}$$ Since both $\beta$ and ${\tilde{\beta}}$ are primitives of $\alpha$, and $\alpha$ is Dirichlet, they both are closed on the boundary. On the $2k$-manifold ${\partial}\Omega$, the Hodge decomposition theorem tells us that every closed $k$-form can be written as the sum of an exact $k$-form and a $k$-form which represents a de Rham cohomology class in $H^k({\partial}\Omega)$. So write $$\label{beta-gamma} \beta = d\phi + \gamma, \quad {\tilde{\beta}}= d{\tilde{\phi}}+ {\tilde{\gamma}}.$$ We now use this decomposition to analyze $\beta \wedge {\tilde{\beta}}$ on ${\partial}\Omega$. Since ${\tilde{\beta}}$ is closed, $$d\phi \wedge {\tilde{\beta}}= d(\phi \wedge {\tilde{\beta}}).$$ Stokes’ Theorem implies that the integral of an exact form on a boundary is zero; thus, $\int_{{\partial}\Omega} d\phi \wedge {\tilde{\beta}}= 0$. Continuing this argument, we see that $\int_{{\partial}\Omega} \beta \wedge {\tilde{\beta}}= \int_{{\partial}\Omega} \gamma \wedge {\tilde{\gamma}}$. This integral is the cup product of the de Rham cohomology classes represented by $\gamma$ and ${\tilde{\gamma}}$ in $H^k({\partial}\Omega)$ evaluated on the top class of ${\partial}\Omega$. Similarly, $\int_{{\partial}\Omega} \beta \wedge \beta = \int_{{\partial}\Omega} \gamma \wedge \gamma$ and $\int_{{\partial}\Omega} {\tilde{\beta}}\wedge {\tilde{\beta}}= \int_{{\partial}\Omega} {\tilde{\gamma}}\wedge {\tilde{\gamma}}$. Viewing $\beta= {\operatorname{BS}}(\alpha)$ and ${\tilde{\beta}}= f^*{\operatorname{BS}}(\alpha')$, we may now represent the change in helicity (\[helicity-change\]) in terms of primitives that represent cohomology classes: $$\label{helicity-change-gamma} {\operatorname{H}}(\alpha') - {\operatorname{H}}(\alpha) = \begin{cases} \int_{{\partial}\Omega} \gamma \wedge {\tilde{\gamma}}& \text{if $k$ is odd},\\ \tfrac{1}{2} \int_{{\partial}\Omega} {\tilde{\gamma}}\wedge {\tilde{\gamma}}- \gamma \wedge \gamma & \text{if $k$ is even}. \end{cases}$$ Background on the homology of domains in ${\mathbb{R}}^{2k+1}$ -------------------------------------------------------------- Before proceeding, we list a couple of “folk theorems” about the homology and cohomology of domains in ${\mathbb{R}}^{2k+1}$. To aid the non-expert reader, we also provide an example in Figure \[fig:alexanderbasis\]. We furnish proofs of these results in Appendix \[hom\]. In all of these theorems, we use de Rham cohomology and so take our coefficients in ${\mathbb{R}}$. In this case, the Universal Coefficient Theorem gives us a natural duality isomorphism betweeen homology and cohomology. For a homology class $s$, we denote the dual cohomology class by $s^*$. We start with an existence theorem for a special basis for the $k$-th homology of ${\partial}\Omega$: Let $\Omega$ be a compact domain with smooth boundary in ${\mathbb{R}}^{2k+1}$ or $S^{2k+1}$ (with $k > 0$) and ${\bar\Omega}$ be the complementary domain ${\mathbb{R}}^{2k+1} - \Omega$ or $S^{2k+1} - \Omega$. Then if we take coefficients in ${\mathbb{R}}$, $H_k({\partial}\Omega) = H_k(\Omega) \oplus H_k({\bar\Omega})$. Further, given any basis $\langle s_1, \dots, s_n \rangle$ for $H_k(\Omega)$ there is a corresponding basis $\langle s_1, \dots, s_n, t_1, \dots, t_n \rangle$ for $H_k({\partial}\Omega)$ which we call the *Alexander basis* corresponding to $\langle s_1, \dots, s_n \rangle$ so that: 1. The inclusion ${\partial}\Omega \hookrightarrow \Omega$ maps $\langle s_1, \dots, s_n \rangle \in H_k({\partial}\Omega)$ to the original basis $\langle s_1, \dots, s_n \rangle$ for $H_k(\Omega)$ and the inclusion ${\partial}\Omega \hookrightarrow {\bar\Omega}$ maps $\langle t_1, \dots, t_n \rangle$ to a basis for $H_k({\bar\Omega})$. 2. $s_i = {\partial}\sigma_i$ for $\sigma_i \in H_{k+1}({\bar\Omega},{\partial}{\bar\Omega})$, where the $\sigma_i$ form a basis for $H_{k+1}({\bar\Omega},{\partial}{\bar\Omega})$. Similarly, $t_i = {\partial}\tau_i$ for $\tau_i \in H_{k+1}(\Omega,{\partial}\Omega)$, where the $\tau_i$ form a basis for $H_{k+1}(\Omega,{\partial}\Omega)$. 3. The cup product algebras of $\Omega$, ${\bar\Omega}$ and ${\partial}\Omega$ obey $$s_i^* \cup \tau_j^* = \delta_{ij} [\Omega]^*, \quad t_i^* \cup \sigma_j^* = (-1)^{k+1} \delta_{ij}[{\bar\Omega}]^*$$ and $$s_i^* \cup s_j^* = 0, \quad t_i^* \cup s_j^* = \delta_{ij} [{\partial}\Omega]^*, \quad t_i^* \cup t_j^* = 0.$$ 4. The linking number ${\operatorname{Lk}}(s_i,t_j) = \delta_{ij}$. (Thus ${\operatorname{Lk}}(t_j,s_i) = (-1)^{(k+1)^2} \delta_{ij}$.) The Alexander duality isomorphism from $H_k(\Omega)$ to $H_k({\bar\Omega})$ maps $s_i$ to $t_i$. We will then study the effect of a homeomorphism on the Alexander basis, proving Suppose that $\Omega$ and $\Omega'$ are compact domains with smooth boundary in ${\mathbb{R}}^{2k+1}$ or $S^{2k+1}$ and that $f\co \Omega \rightarrow \Omega'$ is an orientation-preserving homeomorphism. Then if $\langle s_1, \dots, s_n \rangle$ is a basis for $H_k(\Omega)$ and $\langle s_1', \dots, s_n' \rangle$ is a corresponding basis for $H_k(\Omega')$ so that $f_*(s_i) = s_i'$, then we may build Alexander bases $\langle s_1,\dots,s_n,t_1,\dots,t_n \rangle$ for $H_k({\partial}\Omega)$ and $\langle s_1',\dots,s_n',t_1',\dots,t_n' \rangle$ for $H_k({\partial}\Omega')$. For these bases, we have $f_*(\tau_i) = \tau_i'$ and ${\partial}f_*(t_i) = t_i'$ so that the map ${\partial}f_*\co H_k({\partial}\Omega) \rightarrow H_k({\partial}\Omega')$ can be written as the $2n \times 2n$ matrix $${\partial}f_* = \left[ \begin{array}{c|c} I & 0 \\ \hline (c_{ij}) & I \end{array} \right], \tag{\ref{mmatrix}}$$ where each block represents an $(n \times n)$ matrix. If $k$ is odd, the block matrix $\left(c_{ij}\right)$ is symmetric, while if $k$ is even, the block matrix $\left(c_{ij}\right)$ is skew-symmetric. Since these theorems are somewhat complicated, we give an example in Figure \[fig:alexanderbasis\]. [alexanderbasis]{} (22,21.5)[$\sigma_1$]{} (37.5,21.5)[$\tau_1$]{} (28.5,12)[$s_1$]{} (38,15.5)[$t_1$]{} (22,69)[$\sigma_2$]{} (28.5,60)[$s_2$]{} (37,69.7)[$\tau_2$]{} (37,64)[$t_2$]{} (1,1)[$\Omega$]{} (55,1)[$\Omega'$]{} (49,50)[$f$]{} (14,46)[$\alpha$]{} (27.5,86)[$\beta$]{} (77,21.5)[$\sigma_1'$]{} (90.7,21.5)[$\tau_1'$]{} (84,12)[$s_1'$]{} (89,15.5)[$t_1'$]{} (77,69)[$\sigma_2'$]{} (90.7,70)[$\tau_2'$]{} (83,59)[$s_2'$]{} (89,64)[$t_2'$]{} (72.5,88.5)[$f(s_2)$]{} (73,6)[$f(s_1)$]{} Fluxless case. {#sec:fluxless} -------------- A closed Dirichlet $(k+1)$-form $\alpha$ on a domain $\Omega$ in ${\mathbb{R}}^{2k+1}$ is called *fluxless* if the integral $\int_S \alpha = 0$ over every oriented $(k+1)$-cycle $S \subset \Omega$ with ${\partial}S \subset {\partial}\Omega$. We note that since the $(k+1)$-form $\alpha$ represents a de Rham cohomology class in the relative $k$-homology of $\Omega$, the integral $\int_S \alpha$ depends only on the homology class represented by $S$ in $H_{k+1}(\Omega,{\partial}\Omega)$. Since $H_{k+1}(\Omega,{\partial}\Omega) = H_{k}(\Omega)$ by Poincaré duality, if $\Omega$ has no $k$-homology then every $(k+1)$-form $\alpha$ is fluxless. Let $\alpha$ be fluxless. We will utilize some facts about the cohomology and homology of the boundary of a domain in ${\mathbb{R}}^{2k+1}$. See Appendix \[hom\] for details. First, $$\label{pdecomp} H^k({\partial}\Omega) = H^k(\Omega) \oplus H^k({\mathbb{R}}^{2k+1} - \Omega).$$ We claim that $\gamma$ and ${\tilde{\gamma}}$ represent classes entirely in $H^k(\Omega)$. Suppose we have a $k$-cycle $c$ in ${\partial}\Omega$ which represents a class in $H_k({\mathbb{R}}^{2k+1} - \Omega)$. Such a cycle bounds in $\Omega$. Since $\gamma$ and $\beta$ differ by $d\phi$, they have the same integral over $c$. Further, by Stokes’ Theorem, the integral of $\beta$ over $c$ is equal to the integral of $\alpha$ on the $(k+1)$-cycle bounded by $c$ in $\Omega$. Since $\alpha$ is fluxless, this integral is zero. Thus $\int_c \gamma = 0$ for every $k$-cycle in ${\partial}\Omega$ which represents in ${\mathbb{R}}^{2k+1}-\Omega$, and (in terms of ), $\gamma \in H^k(\Omega)$. The same argument shows that ${\tilde{\gamma}}\in H^k(\Omega)$. However, in the cup product algebra of $H^k({\partial}\Omega)$, the only pairs of $k$-forms with nontrivial cup products have one member in $H^k(\Omega)$ and one in $H^k({\mathbb{R}}^{2k+1}-\Omega)$. Hence, $\int_{\partial\Omega} \gamma \wedge {\tilde{\gamma}}= 0$; likewise, $\int_{{\partial}\Omega} \gamma \wedge \gamma = \int_{{\partial}\Omega} {\tilde{\gamma}}\wedge {\tilde{\gamma}}= 0$. Thus, both cases of are zero, so we have proven \[fluxless-invariance\] If $f \co \Omega \rightarrow \Omega'$ is a diffeomorphism between compact domains in ${\mathbb{R}}^{2k+1}$ with smooth boundary, then for any fluxless $(k+1)$-form $\alpha$ on $\Omega$, its helicity is invariant under $f$, i.e., $${\operatorname{H}}(\alpha) = {\operatorname{H}}\left( {\left( f^{-1} \right)^*\alpha}\right) .$$ We note that for fluxless forms, it is not necessary to use the Biot-Savart operator in order to define helicity; replacing it with any primitive of $\alpha$ will produce an integral equivalent to helicity (see Definition \[arnoldhelicity\]). But what about closed Dirichlet $(k+1)$-forms $\alpha$ which are not fluxless? To understand the effect of a diffeomorphism on their helicity, we will have to compute the right hand side of  directly. We do so first for a solid torus before proceeding in general. Solid torus example. -------------------- We start with $\Omega$, a solid torus in ${\mathbb{R}}^3$. Let $f\co\Omega \rightarrow \Omega'$ be a diffeomorphism, homotopic to $j$ Dehn twists on a spanning disk of $\Omega$. Then $f$ induces isomorphisms of $H_*(\Omega)$ and $H_*({\partial}\Omega)$. By , the boundary homology decomposes as $H_1({\partial}\Omega) = H_1(\Omega) \oplus H_1({\mathbb{R}}^{3} - \Omega)$. We choose an *Alexander basis* (defined in Theorem \[alexander basis existence\]) $\langle s, t \rangle$: $t$ is a meridian on ${\partial}\Omega$, i.e., $t$ generates $H_1({\mathbb{R}}^3 - \Omega)$; $s$ is a longitude on ${\partial}\Omega$, i.e., $s$ generates $H_1(\Omega)$. Choose $s', t'$ similarly on ${\partial}\Omega'$. Let $\sigma$ be a surface in ${\mathbb{R}}^3-\Omega$ bounded by $s$; let $\tau$ be a surface in $\Omega$ bounded by $t$; similarly define $\sigma'$ and $\tau'$. Since $f$ applies $j$ Dehn twists to the solid torus $\Omega$, we have $f_*(s) = s' + jt'$ and $f_*(t) =t'$. We consider a closed Dirichlet 2-form $\alpha$ on $\Omega$. Following the argument above, we utilize the 1-forms $\beta = {\operatorname{BS}}(\alpha)$ and ${\tilde{\beta}}= f^*\left({\operatorname{BS}}(\alpha')\right)$, both primitives for $\alpha$. Choose suitable 1-forms as above, $\gamma$ and ${\tilde{\gamma}}$, which represent in terms of 1-cohomology classes. From (\[helicity-change-gamma\]), the change in helicity under $f$ is $$\begin{aligned} {\operatorname{H}}(\alpha') - {\operatorname{H}}(\alpha) & = & \int_{{\partial}\Omega} \gamma \wedge {\tilde{\gamma}}.\end{aligned}$$ We recognize this integral as a cup product pairing in $H^1({\partial}\Omega)$, since both forms in the integrand can be viewed as 1-cohomology classes. The cup product pairing is straightforward on the torus. If we write the cohomology classes dual to $s$ and $t$ as $s^*$ and $t^*$, then by Theorem \[alexander basis existence\] $$s^* \cup s^* = 0, \qquad t^* \cup t^* = 0, \qquad t^* \cup s^* = [{\partial}\Omega]^*,$$ where $[{\partial}\Omega]$ is the top class of the boundary in $H_2({\partial}\Omega)$ and $[{\partial}\Omega]^*$ its dual in $H^2({\partial}\Omega)$. Now we write $\gamma$ and ${\tilde{\gamma}}$ in terms of the cohomology classes they represent, $$[\gamma] = a s^* + b t^* \qquad [ {\tilde{\gamma}}] = \tilde{a} s^* + \tilde{b} t^*,$$ and find the coefficients by integrating. For example, $$\begin{aligned} \tilde{a} &= \int_{s} {\tilde{\gamma}}= \int_{s} f^*{\operatorname{BS}}\left({\left( f^{-1} \right)^*\alpha}\right) - \int_{s} d{\tilde{\phi}}= \int_{f(s)} {\operatorname{BS}}\left({\left( f^{-1} \right)^*\alpha}\right) - 0 \\ &= \int_{s'} {\operatorname{BS}}(\alpha') + j \int_{t'} {\operatorname{BS}}(\alpha') = \int_{\sigma'} \alpha' + j \int_{\tau'} \alpha'\\ &= \int_{\sigma'} \alpha' + j {\operatorname{Flux}}(\alpha',\tau'). \end{aligned}$$ Since $\alpha'$ is identically zero on ${\mathbb{R}}^3-\Omega$, the first term $\int_{\sigma'} \alpha' = 0$. We also note that ${\operatorname{Flux}}(\alpha,\tau) = {\operatorname{Flux}}(\alpha',\tau')$. Thus, $\tilde{a} = j {\operatorname{Flux}}(\alpha)$. By similar computations, we obtain $$[ \gamma ] = {\operatorname{Flux}}(\alpha) t^*, \qquad [ {\tilde{\gamma}}] = {\operatorname{Flux}}(\alpha) t^* + j {\operatorname{Flux}}(\alpha) s^*.$$ Thus, we can view $\int_{{\partial}\Omega'} \gamma \wedge {\tilde{\gamma}}$ as a cup product evaluated by integration on the top class of ${\partial}\Omega$: $${\operatorname{Flux}}(\alpha) \, t^* \cup \left( {\operatorname{Flux}}(\alpha) \, t^* + j {\operatorname{Flux}}(\alpha) \, s^* \right) = j {\operatorname{Flux}}(\alpha)^2 \, [{\partial}\Omega]^*.$$ In summary, we have proven the following theorem. \[torus-h\] Let $\Omega$ be a solid torus in ${\mathbb{R}}^3$. Let $f\co\Omega \rightarrow \Omega'$ be an orientation-preserving map which takes $\Omega$ diffeomorphically to a subset of ${\mathbb{R}}^3$ and is homotopic to applying $j$ Dehn twists to $\Omega$. Given a closed Dirichlet 2-form $\alpha$, the change in the helicity of $\alpha$ under $f$ is $${\operatorname{H}}({\left( f^{-1} \right)^*\alpha}) - {\operatorname{H}}(\alpha) = j \cdot {\operatorname{Flux}}(\alpha)^2,$$ where the flux is measured over a spanning surface in $\Omega$ which generates $H_2(\Omega,{\partial}\Omega)$. This theorem lets us classify the helicity-preserving diffeomorphisms on the solid torus. We know from Proposition \[homotopic-invariance\] that a map from the solid torus to itself preserves helicity for all $2$-forms if it is homotopic to the identity through diffeomorphisms. This theorem lets us prove an (almost) converse result: \[torus-g\] If $f\co\Omega \rightarrow \Omega$ is a diffeomorphism of the solid torus to itself, then $f$ preserves helicity for all closed Dirichlet $2$-forms $\alpha$ if and only if $f$ is homotopic to the identity through homeomorphisms. Wainryb [@wain Theorem 14] showed that the mapping class group of a solid torus is isomorphic to ${\mathbb{Z}}\oplus {\mathbb{Z}}_2$, where the ${\mathbb{Z}}$ counts Dehn twists and the ${\mathbb{Z}}_2$ detects change of orientation. Thus a map is homotopic to the identity through homeomorphisms if and only if it preserves orientation and has no Dehn twists. By Theorem \[torus-h\], such a map preserves helicity for any form $\alpha$. On the other hand, a map which reverses orientation reverses the sign of helicity (for any form $\alpha$ with nonzero helicity) and by Theorem \[torus-h\] a map which is homotopic to a nonzero number of Dehn twists changes the helicity of any form $\alpha$ with nonzero flux. We now check this theorem with an explicit example. Suppose that $\Omega$ is the solid torus of revolution in ${\mathbb{R}}^3$ whose core circle has radius 1 and whose tube has radius $R$. We set up (standard) toroidal coordinates $(r, \theta, \phi)$ on the torus where $\theta$ parametrizes the core circle and $(r,\phi)$ are polar coordinates on the cross-sections of the tube. Consider the diffeomorphism $f(r,\theta,\phi) = (r,\theta,\theta+\phi)$ on $\Omega$. This is a volume-preserving diffeomorphism which applies one Dehn twist to $\Omega$. We will compute the helicity of the 2-form $\alpha = *d\theta$ ($*$ is the Hodge star with respect to the standard form ${\operatorname{dvol}}_\Omega$) dual to the vector field ${\frac{\partial}{\partial{\theta}}}$ on $\Omega$ before and after the diffeomorphism $f$, as shown in Figure \[fig:straight and twisted\]. [straight\_field.eps]{} [twisted\_field.eps]{} It is easy to see that $f_* {\frac{\partial}{\partial{\theta}}} = {\frac{\partial}{\partial{\theta}}} + {\frac{\partial}{\partial{\phi}}}$. So we must compute the helicity of these two fields. It is convenient to do this via the ergodic definition of helicity given by Arnol’d [@MR1612569 p.146]: The *asymptotic linking number* of the pair of trajectories $g^t x_1$ and $g^t x_2$ ($x_1, x_2 \in \Omega)$ of a field $V$ is defined to be the limit $$\lambda_V(x_1,x_2) := \lim_{t_1, t_2 \rightarrow \infty} \frac{ {\textrm{lk}}_V(x_1,x_2;t_1,t_2) }{t_1 t_2}$$ where ${\textrm{lk}}_V(x_1,x_2;t_1,t_2)$ is the linking number of the closures of the trajectories extending from $x_1$ and $x_2$ for times $t_1$ and $t_2$. The definition of the asymptotic linking number requires that the trajectories be closed by a “system of short paths” joining any given pair of points on $\Omega$ and obeying certain mild technical hypotheses. Luckily, in the cases of interest to us, all of the orbits of our fields are closed with period $2\pi$, so we can ignore these details and let $$\lambda_V(x_1,x_2) = \lim_{p, q \in \mathbb{N} \rightarrow \infty} \frac{ {\textrm{lk}}_V(x_1,x_2;2\pi p, 2\pi q) }{4 \pi^2 pq}. \label{eq:asymp}$$ Now Arnol’d’s ergodic definition of helicity proves that The average asymptotic linking number of a divergence-free field tangent to the boundary of a closed domain in ${\mathbb{R}}^3$ is equal to the helicity of the field. That is, $${\operatorname{H}}(V) = \iint_{\Omega {\times}\Omega} \lambda_V(x_1,x_2) \, {\operatorname{dvol}}_{x_1} {\operatorname{dvol}}_{x_2}.$$ We can now compute the helicity of our fields. For the field ${\frac{\partial}{\partial{\theta}}}$, the orbits are all circles parallel to the $xy$ plane. These never link, so the helicity of this field is zero. For the field $V = {\frac{\partial}{\partial{\theta}}} + {\frac{\partial}{\partial{\phi}}}$, the orbits are all $(1,1)$ curves on a family of nested tori foliating the solid torus $\Omega$. Any pair of such curves has linking number $1$. Now the trajectories for times $2\pi p$ and $2 \pi q$ cover one of these curves $p$ times and the other $q$ times, so the linking number of the trajectories is $pq$. Taking the limit in , we get $\lambda_V(x_1,x_2) = 1/4 \pi^2$ for all $x_1$, $x_2$ in $\Omega$. We now compute the helicity of the field $${\operatorname{H}}(V) = \iint_{\Omega {\times}\Omega} \lambda_V(x_1,x_2) {\operatorname{dvol}}_{x_1} {\operatorname{dvol}}_{x_2} = \frac{{\operatorname{vol}}(\Omega)^2}{4\pi^2} = \frac{((2\pi)(\pi R^2))^2}{4\pi^2} = \pi^2 R^4. \label{eq:direct}$$ Here the volume of the tube is the product of the length of the core curve and the cross-sectional area by the Tube Formula. Now we compare the prediction of Theorem \[torus-h\] that ${\operatorname{H}}(V) = {\operatorname{Flux}}(V)^2$. We must compute the flux of $V$ across a cross-sectional disk of $\Omega$. Since ${\frac{\partial}{\partial{\phi}}}$ is tangent to such a disk, the flux is the same as the flux of ${\frac{\partial}{\partial{\theta}}}$. A computation shows that this flux is $\pi R^2$, but this is not hard to see: the flux is the rate at which the disk sweeps out volume when rotated around the axis. Since this rate is constant and the disk sweeps out the entire volume $2\pi^2 R^2$ of the tube after rotation through $2\pi$, the rate must be $\pi R^2$, as claimed. We conclude that ${\operatorname{H}}(V) = (\pi R^2)^2$, which agrees with our computation in . We note that a similar example would be easy to work out for a different number of Dehn twists, as a pair of closed orbits of the field after $j$ Dehn twists would have linking number $j$. General formula for change of helicity -------------------------------------- With Theorem \[torus-h\] in hand, we now show that a strikingly similar formula holds for general domains $\Omega^{2k+1}$. We begin with the same setup and compute the change of helicity via (\[helicity-change-gamma\]). Throughout this section, we will make use of the Einstein summation convention. Again, we choose an *Alexander basis* (Theorem \[alexander basis existence\]) $\langle s_1, \dots, s_n, t_1, \dots, t_n \rangle$ for $H_k({\partial}\Omega)$. With respect to this basis, we recall that Theorem \[alexander basis transformations\] tells us that there is a corresponding Alexander basis $\langle s_1',\dots,s_n', t_1', \dots, t_n' \rangle$ for $H_k({\partial}\Omega')$ so that the map ${\partial}f_*\co H_k({\partial}\Omega) \rightarrow H_k({\partial}\Omega)$ looks like a $2n \times 2n$ block matrix $${\partial}f_* = \left[ \begin{array}{c|c} I & 0 \\ \hline (c_{ij}) & I \end{array} \right].$$ We now write the classes $[\gamma]$ and $[{\tilde{\gamma}}]$ in terms of this basis. In terms of the Alexander basis, the cohomology classes represented by the forms $\gamma$ and ${\tilde{\gamma}}$ are \[coeffs\] $$[\gamma ] = {\operatorname{Flux}}(\alpha,\tau_i) t_i^*, \quad [{\tilde{\gamma}}] = {\operatorname{Flux}}(\alpha,\tau_i) t_i^* + c_{ij} {\operatorname{Flux}}(\alpha,\tau_i) s_j^*$$ where the $c_{ij}$ come from the expression of ${\partial}f_*$ as a matrix above. We obtain the general change in helicity formula. \[general-change\] Let $\Omega^{2k+1}$ be a subdomain of ${\mathbb{R}}^{2k+1}$, and let $f\co\Omega \rightarrow \Omega'$ be an orientation-preserving diffeomorphism. Consider a closed Dirichlet $(k+1)$-form $\alpha$ on $\Omega$. The change in the helicity of $\alpha$ under $f$ is $$\label{eq:general-change} {\operatorname{H}}\left(\alpha' \right) - {\operatorname{H}}(\alpha) = \sum_{i,j} c_{ij} \cdot {\operatorname{Flux}}(\alpha, \tau_i) {\operatorname{Flux}}(\alpha, \tau_j)$$ where the constants $c_{ij}$ arise from the homology isomorphism induced by $f$ on $H_k({\partial}\Omega)$ as above. The $(2m+2)$-form $\alpha'$ is the ‘push-forward’ of $\alpha$ under $f$; more precisely, $\alpha'={\left( f^{-1} \right)^*\alpha}$ is the pullback of $\alpha$ under the inverse diffeomorphism. \[vf-change\] Let $\Omega^3$ be a subdomain of ${\mathbb{R}}^3$, $f\co\Omega \rightarrow \Omega'$ be a volume-preserving diffeomorphism, and $V$ be a smooth vector field on $\Omega$. The change of helicity of $V$ under $f$ is calculated as above, $$\label{eq:vf-change} {\operatorname{H}}\left(f_* V \right) - {\operatorname{H}}(V) = \sum_{i,j} c_{ij} \cdot {\operatorname{Flux}}(V, \tau_i) {\operatorname{Flux}}(V, \tau_j).$$ This corollary is also true in general when considering the wedge-product of $k$-vectors, dual to a $(k+1)$-form, on $\Omega^{2k+1}\subset {\mathbb{R}}^{2k+1}$ under volume-preserving diffeomorphisms. The coefficients for $[\gamma]$ and $[{\tilde{\gamma}}]$ can be directly calculated. Write $[\gamma ] = a_i s_i + b_i t_i$ and $[{\tilde{\gamma}}] = \tilde{a}_i s_i + \tilde{b}_i t_i$. Then, $$a_i = \int_{s_i} \gamma = \int_{s_i = {\partial}\sigma_i} {\operatorname{BS}}(\alpha ) - \int_{s_i} d\phi \\ = \int_{\sigma_i} d{\operatorname{BS}}(\alpha) - 0 = \int_{\sigma_i} \alpha$$ by Stokes’ Theorem. But $\alpha \equiv 0$ outside of $\Omega$, where $\sigma_i$ is located. Thus $a_i = 0$. We can similarly calculate $b_i$ $$b_i = \int_{\tau_i} d{\operatorname{BS}}(\alpha) - 0 = \int_{\tau_i} \alpha = {\operatorname{Flux}}(\alpha,\tau_i).$$ Calculating $\tilde{a_i}$ and $\tilde{b_i}$ is more involved. $$\tilde{a_j} = \int_{s_j} {\tilde{\gamma}}= \int_{s_j} f^*({\operatorname{BS}}({\left( f^{-1} \right)^*\alpha}) - \int_{s_j} d{\tilde{\phi}}= \int_{f(s_j)} {\operatorname{BS}}(\alpha' ) - 0.$$ By (\[mmatrix\]), the image $f(s_j)$ is homologous to $s_j' + c_{ij} t_i'$. Since ${\operatorname{BS}}({\left( f^{-1} \right)^*\alpha})$ is a closed form on ${\partial}\Omega$, this integral is equal to $$\begin{aligned} \tilde{a_j} &= \int_{s_j'} {\operatorname{BS}}(\alpha') + c_{ij} \int_{t_i'} {\operatorname{BS}}(\alpha' ) \\ &= \int_{\sigma_j'} d{\operatorname{BS}}(\alpha' ) + c_{ij} \int_{\tau_i'} d{\operatorname{BS}}(\alpha' ) \\ &= \int_{\sigma_j'} \alpha' + c_{ij} \int_{\tau_i'} \alpha'. \end{aligned}$$ Now $\alpha'$ vanishes outside of $\Omega'$, which is where $\sigma_j'$ is located, so the first term $\int_{\sigma_j'} \alpha'$ must be zero. We compute the second integral on the original $\Omega$, $$\int_{\tau_i'} \alpha' = \int_{f^{-1}\left(\tau_i' \right)} \alpha$$ We know from Theorem \[alexander basis transformations\] that $f^{-1}(\tau_i')$ is homologous to $\tau_i$. So we conclude $$\tilde{a_j} = c_{ij} \int_{f^{-1}\left( \tau_i' \right)} \alpha = c_{ij} \int_{\tau_i} \alpha = c_{ij} {\operatorname{Flux}}(\alpha,\tau_i).$$ We compute $\tilde{b_i}$ similarly: $$\tilde{b_i} = \int_{t_i} {\tilde{\gamma}}= \int_{t_i} f^*({\operatorname{BS}}({\left( f^{-1} \right)^*\alpha}) - \int_{t_j} d{\tilde{\phi}}= \int_{f(t_i)} {\operatorname{BS}}(\alpha' ).$$ Since $f(t_i)$ is homologous to $t_i'$ and ${\operatorname{BS}}(\alpha')$ is closed on ${\partial}\Omega'$, $$\tilde{b_i} = \int_{t_i'} {\operatorname{BS}}(\alpha' ) = \int_{\tau_i'} d{\operatorname{BS}}(\alpha' ) = \int_{\tau_i'} \alpha' = \int_{\tau_i} \alpha = {\operatorname{Flux}}(\alpha,\tau_i),$$ using our previous observation that $f^{-1}(\tau_i')$ is homologous to $\tau_i$. This completes the proof. We can now derive the change in helicity formula using these coefficients. This will depend on the parity of $k$, so we start with the case where $k$ is odd. The change in helicity formula (\[helicity-change-gamma\]) is $$([\gamma] \cup [{\tilde{\gamma}}])([{\partial}\Omega]) = b_j \tilde{a_j} + a_j \tilde{b_j} = c_{ij} {\operatorname{Flux}}(\alpha,\tau_i){\operatorname{Flux}}(\alpha,\tau_j) + 0.$$ For $k$ even, the change in helicity formula (\[helicity-change-gamma\]) is $\nicefrac{1}{2} \left( [{\tilde{\gamma}}] \cup [{\tilde{\gamma}}] - [\gamma] \cup [\gamma] \right) ([{\partial}\Omega])$. $$\begin{aligned} ([{\tilde{\gamma}}] \cup [{\tilde{\gamma}}])([{\partial}\Omega]) &= 2 \tilde{a_j} \tilde{b_j} = 2 c_{ij} {\operatorname{Flux}}(\alpha,\tau_i) {\operatorname{Flux}}(\alpha,\tau_j) \\ (\left[\gamma \right] \cup [\gamma])([{\partial}\Omega]) &= a_i b_i = 0.\end{aligned}$$ We have calculated the change of helicity to be $$\label{boxformula} \boxed{ {\operatorname{H}}\left(\alpha' \right) - {\operatorname{H}}(\alpha) = \sum_{i,j} c_{ij} {\operatorname{Flux}}(\alpha,\tau_i){\operatorname{Flux}}(\alpha,\tau_j). }$$ which proves Theorem \[general-change\]. By Theorem \[alexander basis transformations\], the matrix $(c_{ij})$ is skew-symmetric for $k$ even, so the double sum on the right-hand-side of (\[boxformula\]) vanishes in this case. Of course this is what we expect, since we know that in this case ${\operatorname{H}}(\alpha) = {\operatorname{H}}(\alpha') = 0$ by Proposition \[even-k\]. Classifying the helicity-preserving diffeomorphisms --------------------------------------------------- We can now classify completely the helicity-preserving maps $f\co\Omega \rightarrow \Omega$. For even $k$, $\Omega$ is $(4m+1)$-dimensional, helicity is the trivial invariant, and all maps are helicity-preserving. In the case where $k$ is odd, $\Omega$ is $(4m+3)$-dimensional, and the helicity-preserving maps are the orientation-preserving maps with $c_{ij} {\operatorname{Flux}}(\alpha,\tau_i){\operatorname{Flux}}(\alpha,\tau_j) = 0$ for all $\alpha$. These maps form a subgroup ${\operatorname{HP}}$ of the diffeomorphism group of $\Omega$. In Theorem \[alexander basis transformations\] the $c_{ij}$ were determined by the homotopy type of $f$. Since they surely vanish for the identity map, ${\operatorname{HP}}$ also forms a subgroup of the smooth mapping class group of $\Omega$. In this language, we can give a more standard description of the group ${\operatorname{HP}}$. For a surface, the Torelli subgroup of the mapping class group is the group of homeomorphisms which act trivially on homology [@MR718141]. Analogously, for any manifold $M$ we will call the group of diffeomorphisms which act trivially on homology the Torelli subgroup ${\operatorname{Torelli}}(M)$ of the smooth mapping class group of $M$. Note that maps in ${\operatorname{Torelli}}(M)$ are orientation-preserving. It is easy to see \[classification\] Since a homeomorphism from $\Omega$ to itself naturally maps ${\partial}\Omega$ to itself, there is a natural inclusion ${\operatorname{Torelli}}({\partial}\Omega) \subset {\operatorname{Torelli}}(\Omega)$. With respect to this inclusion, if $\Omega$ is a domain in ${\mathbb{R}}^{2k+1}$ with $k$ odd, $${\operatorname{HP}}(\Omega) \cap {\operatorname{Torelli}}(\Omega) = {\operatorname{Torelli}}({\partial}\Omega) . $$ If $f \in {\operatorname{Torelli}}(\Omega)$, then $f$ is orientation-preserving and Theorem \[general-change\] applies. The right-hand side of  is the action of the quadratic form defined by the matrix $(c_{ij})$ on the vector ${\operatorname{Flux}}(\alpha,\tau_i)$. Since any vector of fluxes can be obtained by choosing an appropriate $\alpha$, this vanishes for all $\alpha$ if and only if the matrix $(c_{ij})$ is skew-symmetric. But since $k$ is odd, the matrix $(c_{ij})$ is symmetric by Theorem \[alexander basis transformations\], so $f$ is helicity-preserving if and only if all the $c_{ij}$ are zero. Looking at our construction, we see that if $f \in {\operatorname{Torelli}}(\Omega)$, then $s_i' = s_i$. In fact, this means $t_i' = t_i$ as well. Thus $f$ acts trivially on $H_*({\partial}\Omega)$ if and only if all the $c_{ij}$ vanish. This completes the proof. We note that ${\operatorname{HP}}(\Omega)$ is generally somewhat larger than ${\operatorname{Torelli}}({\partial}\Omega)$: if $\Omega$ is a handlebody with $n$ handles any (orientation-preserving) permutation of the handles will surely preserve helicity, but not act trivially on $H_*(\Omega)$ or $H_*({\partial}\Omega)$. In our construction above, the matrix $M$ will still be the identity matrix due to our careful choice of basis. We have now given a full account of the interplay between the map $f$ and the form $\alpha$ in determining the effect of a mapping on the helicity of a form. Our previous theorems are now revealed as easy corollaries of Theorem \[general-change\]: Proposition \[fluxless-invariance\] states that if $\alpha$ is fluxless, then the helicity of $\alpha$ is preserved by any diffeomorphism $f$. Indeed, the ${\operatorname{Flux}}(\alpha,\tau_i)$ vanish for all $i$, which means that the right hand side of  vanishes and helicity is invariant. On the other hand, Proposition \[homotopic-invariance\] states that if the map $f$ is homotopic to the identity map, then helicity is invariant under $f$ for any form. In our new language, this is the trivial statement that the identity element in the mapping class group of $\Omega$ is in the subgroup ${\operatorname{HP}}$. We have seen above in Corollary \[torus-g\] that ${\operatorname{HP}}= \{e\}$ for the solid torus. But in general ${\operatorname{HP}}$ is much larger. Figure \[fig:doubletorus\] shows an explicit example of a helicity-preserving diffeomorphism of a domain in ${\mathbb{R}}^3$ which is not homotopic to the identity. Let $\Omega$ be the solid 2-holed torus, and let $\alpha$ be the curve around the “waist” of the torus. Our map $f$ will be a Dehn twist around $\alpha$ extended to the interior of $\Omega$ along the spanning disk for $\alpha$ shown in the picture. [doubletorus.eps]{} (22,43)[$\alpha$]{} The map $f$ of Figure \[fig:doubletorus\] is helicity-preserving but not homotopic to the identity map. To see the first part, by the corollary above we must show that the map $({\partial}f)_*\co H_1({\partial}\Omega) \rightarrow H_1({\partial}\Omega)$ is the identity. But we can take a set of generators for $H_1({\partial}\Omega)$ which are fixed by ${\partial}f$ as long as we stay away from $\alpha$. To show the second, observe that if $f$ was homotopic to the identity, then ${\partial}f$ would be as well. But ${\partial}f$ is a Dehn twist around an essential curve in ${\partial}\Omega$, so ${\partial}f$ is nontrivial in the mapping class group of ${\partial}\Omega$ [@fmprimer Proposition 2.1]. Future Directions ================= Our perspective on helicity has allowed us to observe three new kinds of invariance for ${\operatorname{H}}(\alpha)$: invariance under change of volume form on $S^{2k}$, invariance in the cohomology class $[\alpha_x \wedge \alpha_y]$ in $H^{2k+2}(C_2[\Omega],{\partial}C_2[\Omega])$, and invariance under diffeomorphisms of $\Omega$ which preserve the homology of ${\partial}\Omega$. We have made stronger the analogy between helicity for forms and finite-type invariants for knots and links. And we have explained the effect of any diffeomorphism of $\Omega$ on the helicity of a form $\alpha$. We devote the rest of the paper to observing some immediate consequences of our point of view, and to suggesting some future directions for further study. Submanifold helicities ---------------------- So far we have only considered the case where $\Omega$ is a top-dimensional subdomain of ${\mathbb{R}}^{2k+1}$. We can define an analogous helicity just as easily for closed Dirichlet $(k+1)$-forms on an $n$-dimensional submanifold $\Omega$ of ${\mathbb{R}}^{m}$ by $${\operatorname{H}}(\alpha) = \int_{C_2[\Omega]} \alpha_x \wedge \alpha_y \wedge g^* {\operatorname{dvol}}_{S^{m-1}}$$ as long as the integrand $\Phi_m=\alpha_x \wedge \alpha_y \wedge g^* {\operatorname{dvol}}_{S^{m-1}}$ is a $2n$-form. This requires that $2k + 2 + m-1 = 2n$, i.e., that $m = 2n - 2k - 1$. We refer to such an integral as a *$(k,n,m)$-helicity* and note that the helicity from Definition \[fthelicity\] is the $(k,2k+1,2k+1)$-helicity. \[questions\] Two questions arise immediately: 1. \[qinvt\] For which values of $k$, $n$ and $m$ is $(k,n,m)$-helicity an invariant? 2. \[qtrivial\] When is the invariant nontrivial? As before, we know that $(k,n,m)$-helicity will be an invariant if the closed form $\alpha_x \wedge \alpha_y \wedge g^* {\operatorname{dvol}}_{S^{m-1}}$ is Dirichlet, i.e., if it vanishes on the boundary of $C_2[\Omega]$. Following the proof of Lemma \[alpha\], we only have to worry about the face $(12)$ of this boundary, which is diffeomorphic to $\Omega {\times}S^{m-1}$. On this face, $\alpha_x \wedge \alpha_y$ pulls back to $\alpha \wedge \alpha$. Our previous argument depended on the observation that this was a $(2k+2)$-form $\alpha \wedge \alpha$ on the $n=(2k+1)$-manifold $\Omega$. In general, $2k + 2$ may not be greater than $n$, so we cannot depend on this argument. However, we note that if $k+1$ is odd, then $\alpha \wedge \alpha$ vanishes by antisymmetry, providing a partial answer to the first question above. A standard example here is $(0,2,3)$-helicity, which should measure the linking of a 1-form on a surface in ${\mathbb{R}}^3$. We do not know whether the $(1,5,7)$-helicity measuring the linking of a $2$-form on a 5-dimensional surface in ${\mathbb{R}}^7$ is an invariant. The $(-1,1,3)$-helicity of $0$-forms on a curve in ${\mathbb{R}}^3$ turns out to be precisely the writhing number of the curve, so we know that this helicity is not an invariant. What about the second question? For a contractible domain $\Omega = D^{n}$, the configuration space $C_2[\Omega]$ has the topology of $D^n {\times}D^n - \{ \text{pt} \} = D^{n+1} {\times}S^{n-1}$ as we saw above. In this case, only the cohomology groups $H^*(C_2[\Omega],{\partial}C_2[\Omega])$ where $* = 0, n+1, 2n$ are nontrivial. So for a $(k,n,m)$-helicity to be nontrivial in this case, we must have $2k+2 = n+1$, which only occurs for $(k,2k+1,2k+1)$-helicity. But if $\Omega$ has nontrivial homology, then $C_2[\Omega]$ has more homology and $(k,n,m)$-helicity might be nontrivial. For example, we conjecture that if $\Omega$ has $1$-dimensional homology, then the invariant $(0,2,3)$-helicity is nontrivial on $\Omega$. When $k$ was even, we showed in Proposition \[even-k\] that helicity could only extend to a function that was identically zero. There is a corresponding result for $(k,n,m)$-helicities: If $k+n$ is even, then the $(k,n,m)$-helicity is identically zero for any $(k+1)$-form $\alpha$. The argument is similar to that which proves Proposition \[even-k\]. We consider the automorphism $a$ of $C_2(\Omega)$ that interchanges $x$ and $y$; it extends naturally to $C_2[\Omega]$. It changes the orientation of $C_2[\Omega]$ by a factor of $(-1)^{n^2}$. Next, we take the pullback $a^*\Phi_m = \alpha_y \wedge \alpha_x \wedge a^*g^*{\operatorname{dvol}}_{S^{m-1}}$. The map $a$ induces an antipodal map on $S^{m-1}$; since $m$ is odd, such a map has degree $-1$. Hence, $a^*g^*{\operatorname{dvol}}_{S^{m-1}}= -g^*{\operatorname{dvol}}_{S^{m-1}}$. Also, $\alpha_y \wedge \alpha_x = (-1)^{(k+1)^2} \alpha_x \wedge \alpha_y$. Combining these results, $a^*\Phi = (-1)^k \Phi$. We then compute $$\begin{aligned} \int_{C_2[\Omega]} a^*\Phi & = \int_{a\left(C_2[\Omega] \right)} \Phi \\ \int_{C_2[\Omega]} (-1)^k \Phi & = \int_{(-1)^n C_2[\Omega]} \Phi \\ (-1)^k {\operatorname{H}}(\alpha) & = (-1)^n{\operatorname{H}}(\alpha)\end{aligned}$$ If $k$ and $n$ have the opposite parity, this implies that ${\operatorname{H}}(\alpha) = -{\operatorname{H}}(\alpha)$, i.e., that helicity is zero, and proves our proposition. If $k$ and $n$ have the same parity, the conclusion is a tautology: ${\operatorname{H}}(\alpha) = {\operatorname{H}}(\alpha)$. Helicity on 3-manifolds with boundary ------------------------------------- Arnol’d and Khesin give a definition for helicity for $2$-forms on a simply-connected $3$-manifold $M$ without boundary in [@MR1612569]. If the manifold is not simply connected, their method works for “fluxless” $2$-forms which do not represent nontrivial classes in $H^2(M)$, but fails for forms which do represent in $H^2(M)$. Our work so far allows us to define and understand helicity for all forms on $2k+1$-manifolds $M^{2k+1}$ with boundary with an embedding into ${\mathbb{R}}^{2k+1}$. As we have shown, the helicity of a form on such a domain depends on the embedding of the domain into ${\mathbb{R}}^{2k+1}$. However, it is easy to remove this dependence, allowing us to define a kind of helicity on some 3-manifolds with boundary which is independent of their embedding into Euclidean space. Let $M^{2k+1}$ be a compact, oriented manifold with smooth boundary which admits an orientation-preserving diffeomorphic embedding $f$ into ${\mathbb{R}}^{2k+1}$. Let $\alpha$ be a Dirichlet $(k+1)$-form on $M^{2k+1}$. Let $\tau_1, \dots, \tau_n$ be a basis for $H^{k+1}(M,{\partial}M)$, and let $F(\alpha)$ be the largest number so that ${\operatorname{Flux}}(\alpha,\tau_i) = k_i F(\alpha)$ for some $k_i \in {\mathbb{Z}}$ or $0$ if no such $F(\alpha)$ exists. s We define the *residual helicity* of $\alpha$ to be $$\operatorname{ResHel}(\alpha) = {\operatorname{H}}(f_*(\alpha)) \mod F(\alpha).$$ if $F(\alpha) > 0$ and $\operatorname{ResHel}(\alpha) = 0$ otherwise. We have immediately from Theorem \[general-change\] that The residual helicity $\operatorname{ResHel}(\alpha)$ does not depend on $f$ and is a diffeomorphism invariant of $\alpha$. On most domains, this helicity is fairly weak, since it can vanish if the fluxes of the form $\alpha$ are irrational multiples of one another. So this definition is really most useful when $H^{k+1}(M) = {\mathbb{R}}$, as in the solid torus. But it may point the way towards defining a more powerful version of helicity on an arbitrary $2k+1$-manifold with boundary. Cross-helicities ---------------- So far we have only considered configuration spaces of two points in a single domain in ${\mathbb{R}}^{2k+1}$. But we could also construct similar configuration spaces where the points are restricted to lie in different domains. For instance, consider the configuration space $X {\times}Y$ where $X$ and $Y$ are disjoint linked solid tori of the form $S^{k} {\times}D^{k+1}$ in ${\mathbb{R}}^{2k+1}$. This configuration space simply restricts one point to lie in each torus. As before, there is a Gauss map $g\co X {\times}Y \rightarrow S^{2k}$ and we can define the cross-helicity of a pair of closed Dirichlet $(k+1)$-forms $\alpha_x$ and $\alpha_y$ defined on $X$ and $Y$ by $${\operatorname{H}}(\alpha_x,\alpha_y) = \int_{X {\times}Y} \alpha_x \wedge \alpha_y \wedge g^* {\operatorname{dvol}}_{S^{2k}}.$$ As before, we observe that $\alpha_x \wedge \alpha_y$ is a closed Dirichlet $(2k+2)$-form on $X {\times}Y$. But this means that $\alpha_x \wedge \alpha_y$ represents a cohomology class in $H^{2k+2}(X {\times}Y)$. Now $\alpha_x$ and $\alpha_y$ represent classes in $H^{k+1}(X,{\partial}X) \simeq {\mathbb{R}}$ and $H^{k+1}(Y,{\partial}Y) \simeq {\mathbb{R}}$. Since $X \simeq Y \simeq S^{k} {\times}D^{k+1}$, $H_{k+1}(X,{\partial}X)$ and $H_{k+1}(Y,{\partial}Y)$ are generated by cycles spanning the $D^{k+1}$ and the classes represented by $\alpha_x$ and $\alpha_y$ are determined by their flux across these spanning cycles. Let us call the cohomology duals to the spanning cycles $[g_x]$ and $[g_y]$ so that $$[\alpha_x] = {\operatorname{Flux}}(\alpha_x) [g_x] \in H^{k+1}(X,{\partial}X), \qquad [\alpha_y] = {\operatorname{Flux}}(\alpha_y) [g_y] \in H^{k+1}(Y,{\partial}Y).$$ We note that $g_x$ and $g_y$ are the Poincaré duals of the generators $s_x$ and $s_y$ for the homology of the $S^k$ in $X$ and $Y$ with respect to the top classes in $H^{2k+1}(X)$ and $H^{2k+1}(Y)$ which integrate to $1$ on their respective domains. In $X {\times}Y$, the cohomology class represented by $\alpha_x \wedge \alpha_y$ is simply ${\operatorname{Flux}}(\alpha_x) {\operatorname{Flux}}(\alpha_y) [g_x] \wedge [g_y]$, which is the Poincaré dual in $X {\times}Y$ of $[s_x] \wedge [s_y]$. Now if we restrict the Gauss map to the core $S^k {\times}S^k$ of $X {\times}Y$, we see that the pullback of the volume form on $S^{2k}$ represents the cohomology class ${\operatorname{Lk}}(X,Y) [s_x] \wedge [s_y]$. (Here ${\operatorname{Lk}}$ is the linking number of $X$ and $Y$ in ${\mathbb{R}}^{2k+1}$.) This reproves the standard result that $${\operatorname{H}}(\alpha_x,\alpha_y) = {\operatorname{Flux}}(\alpha_x) {\operatorname{Flux}}(\alpha_y) {\operatorname{Lk}}(X,Y)$$ using our language for forms in linked tubes. As in the standard helicity integral, the pullback of the area form on $S^{2k}$ to our configuration space is a multiple of the Poincaré dual of $\alpha_x \wedge \alpha_y$. In the original helicity integral, where $\alpha_x$ and $\alpha_y$ were pulled back from the same $\alpha$ this multiple measured a new topological property of the form $\alpha$. We could see that we were measuring new information in this case because the homology of $C_2[\Omega]$ had a new class in $H^{2k}(C_2[\Omega])$ which was not generated by the topology of the domain $\Omega$. On the other hand, when we calculate the cross-helicity of linked tubes, all of the homology classes involved are generated by the topology of the original domains $X$ and $Y$. This means that the cross-helicity is really an invariant of the core spheres of $X$ and $Y$– the forms $\alpha_x$ and $\alpha_y$ are multiples of the Poincaré duals to the generators of these spheres, and contribute no interesting information other than their fluxes. Similarly, several authors have defined “triple-helicity” integrals for the case of three divergence-free vector fields defined on three solid tori $X$, $Y$, and $Z$ in space. The resulting vector field (or form) invariants turn out to be equal to $${\operatorname{Flux}}(\alpha_x) {\operatorname{Flux}}(\alpha_y) {\operatorname{Flux}}(\alpha_z) I(X,Y,Z)$$ where $I(X,Y,Z)$ is a topological invariant of the three tubes. For example, a theorem of this kind appears in Proposition 5.5 of the recent preprint of Komendarczyk [@komendarczyk-2008]. We can now see that while such theorems are appealing, none of these integrals is likely to easily generalize to a meaningful invariant of 2-forms on a contractible domain in ${\mathbb{R}}^3$ defined by integration over $C_3[D^3]$. We could repeat the procedure above and generate a closed Dirichlet $6$-form $\alpha_x \wedge \alpha_y \wedge \alpha_z$ on $C_3[D^3]$. Unfortunately, in the $9$-dimensional space $C_3[D^3]$, the cohomology group $H^{6}(C_3[D^3],{\partial}C_3[D^3]) \simeq H_3(C_3[D^3]) \simeq 0$, since the (absolute) cohomology of $C_3[D^3]$ is known to be generated by 2-forms coming from the three Gauss maps $g_{xy}(x,y,z) = \nicefrac{x-y}{{\left| x-y \right|}}$, $g_{yz}(x,y,z) = \nicefrac{y-z}{{\left| y-z \right|}}$ and $g_{zx}(x,y,z) = \nicefrac{z-x}{{\left| z-x \right|}}$. Thus any such triple-helicity integral must be zero. It remains an important open problem to construct a nontrivial triple-helicity integral for forms on a contractible domain. Helicity and finite-type invariants ----------------------------------- On a last and somewhat speculative note, we wonder whether the finite-type invariants (expressed as integrals over certain configuration spaces of points on a knot) could be used to obtain integral invariants for divergence-free fields tangent to the boundary of a single knotted flux tube $\Omega$. The values of invariants would not be interesting– we expect each to have the value ${\operatorname{Flux}}(\alpha) I(\Omega)$ where $I(\Omega)$ is the corresponding finite-type invariant of the tube $\Omega$– but the integrals could in principle be used to obtain sharper energy bounds for such vector fields than the classical results of Freedman and He [@fh1]. The major obstacle here seems to be that the construction of the finite-type invariants as integrals depends on the fact that the configuration spaces of circles are disconnected (the order of points on the circle cannot change in a connected component), allowing different components to be attached to one another to form more complicated spaces. We do not yet understand the analogous constructions for configuration spaces of points in solid tori. Acknowledgements ================ The authors are grateful for many colleagues and friends with whom we have had fascinating and productive conversations. We are particularly indebted to Fred Cohen, Elizabeth Denne, Dennis DeTurck, Herman Gluck, Martha Holmes, Jamie Jorgensen, Will Kazez, Tom Kephart, Rafal Komendarczyk, Robert Kotiuga, Clint McCrory, Paul Melvin, Clay Shonkwiler, Jim Stasheff, and Shea Vela-Vick. Stokes’ Theorem for Product Manifolds {#gv} ===================================== Let $X^n$ and $Y^m$ be manifolds with boundary, where $n$ is finite but $m$ may be infinite. Consider the product manifold $M = X \times Y$. Let $\alpha$ be a smooth $(n+k)$-form on $M$, for $k \geq 0$. By integrating $\alpha$ over $X$, we can construct a map $$\begin{array}{lccl} \pi_* \co & \Lambda^{n+k}(X \times Y) & \rightarrow & \Lambda^{k}(Y), \\ \pi_* \co & \alpha & \mapsto & \int_X{ \alpha }. \end{array}$$ Here we are following Volic’s notation [@MR2300426] of $\pi_*$, even though this map is not a push-forward of forms; rather we map merely via integration. [**Stokes’ Theorem.**]{} Via this setup, the differential of the $k$-form $\pi_* \alpha$ on $Y$ is $$\begin{array}{lcccc} d\pi_* \alpha & = & \pi_* d\alpha & - & (\partial \pi)_* \alpha \\ d \int_X { \alpha } & = & \int_X { d\alpha } & - & \int_{\partial X} {\alpha} \end{array}$$ [*Rationale.*]{} Express $\alpha$ as the sum of three smooth forms: $\alpha = \alpha_n + \alpha_{n-1} + \beta$, where $\alpha_n = {\operatorname{dvol}}_x \wedge \cdots$ includes $n$ elementary $dx_i$ forms, $\alpha_{n-1}$ includes $n-1$ elementary $dx_i$ forms, and $\beta$ has less than $n-1$ elementary $dx_i$ forms. We consider these three forms in Stokes’ Theorem above. Sample terms include: $$\begin{array}{|c|lccc|} \hline \mathrm{Form} && \mathrm{Sample\ Term} && \\ \hline \alpha_n & f(x,y) \; & {\operatorname{dvol}}_x & \; \wedge \; & dy_{i_1} \wedge \cdots \wedge dy_{i_k} \\ \alpha_{n-1} & f(x,y) \; & dx_1 \wedge \cdots \wedge \widehat{dx_j} \wedge dx_n & \; \wedge \; & dy_{i_1} \wedge \cdots \wedge dy_{i_{k+1}} \\ \beta & f(x,y) \; & dx_1 \wedge \cdots \wedge \widehat{dx_j} \cdots \widehat{dx_k} \wedge dx_n & \; \wedge \; & dy_{i_1} \wedge \cdots \wedge dy_{i_{k+2}} \\ \hline \end{array}$$ Here the hat on $\widehat{dx_j}$ reports that term does not appear in the wedge product. Consider the form $\alpha_n$ first. Note that $\int_{\partial X} {\alpha_n} = 0$ since $\alpha_n$ has greater dimension in $x$ variables than the dimension of $\partial X$. Since $\alpha_n$ already contains the volume form on $X$, its differential $d\alpha_n$ will only introduce $y$ terms, so we may differentiate under the integral sign: $$d \int_X { \alpha_n } = \int_X { d\alpha_n } \qquad.$$ The form $\alpha_{n-1}$ does not contain the entire volume form on $X$, so $ \int_X { \alpha_{n-1} } = 0$. When computing $\int_{\partial X} {\alpha_{n-1}(x,y)}$, we may hold $y$ fixed and apply Stokes’ Theorem on $X$ to each elementary form of $(n-1)$ $dx$ terms; thus $\int_{\partial X} {\alpha_{n-1}} = \int_X { d\alpha_{n-1} }$. Finally, the third term $\beta$ is smooth and does not yield a top-dimensional form (in terms of $x$ variables) in any of the three integrals, so they all vanish: $$d \int_X { \beta } = \int_X { d\beta } = \int_{\partial X} {\beta} = 0 \qquad.$$ For our purposes usually $k=0$, and it is difficult to calculate $d\int_X \alpha$ directly. Thus, we shall invoke Stokes’ Theorem. Usually, either $\alpha$ is closed or the integral of $d\alpha$ is straightforward, so our efforts concentrate upon computing $\int_{\partial X} {\alpha}$. Its important terms have $(n-1)$ $dx$ terms and include a $dy$ term. When $Y$ is the space of embeddings or knots, we may view this term as dual to a variational vector field. The cohomology of domains in ${\mathbb{R}}^{2k+1}$ (or $S^{2k+1}$) {#hom} ================================================================== The purpose of this section is to give a self-contained exposition of some basic facts about the homology of domains in ${\mathbb{R}}^{2k+1}$ (or $S^{2k+1}$). In dimension 3, we gave an exposition of much of this material (without detailed proofs) in [@MR1901496]. But in this dimension, this material is certainly not original. For instance, between [@MR2067778 Chapter 3] and [@hiptmair-2008], almost all of the three-dimensional version of Theorem \[alexander basis existence\] has been published before. We do not yet know a reference for Theorem \[alexander basis transformations\]. In what follows, we will take coefficients for homology to lie in ${\mathbb{R}}$. In this case[^3], the Universal Coefficient Theorem yields a natural duality isomorphism which pairs a homology class $x$ with the dual cohomology class $x^*$. We will let $[\Omega] \in H_{2k+1}(\Omega,{\partial}\Omega)$ denote the top class of this orientable manifold and $[\Omega]^*$ denote the dual class in $H^{2k+1}(\Omega,{\partial}\Omega)$. Similarly, $[{\partial}\Omega] \in H_{2k}({\partial}\Omega)$ will be the top class of ${\partial}\Omega$ and $[{\partial}\Omega]^*$ its dual. We will let $\tilde{H}$ denote reduced homology. We will take the linking number of two $k$-cycles in ${\mathbb{R}}^{2k+1}$ to be given by ${\operatorname{Lk}}(a,b) = {\operatorname{Int}}(a,B)$ where $b = {\partial}B$ and ${\operatorname{Int}}$ is the intersection number. We recall that for $k$-cycles, ${\operatorname{Lk}}(b,a) = (-1)^{(k+1)^2} {\operatorname{Lk}}(a,b)$ [@MR1454127 Proposition 11.13]. We will also need a lemma. \[whocares\] For $n < 2k$, if we think of ${\mathbb{R}}^{2k+1}$ as $S^{2k+1} - \{x\}$ with $x \in {\operatorname{Int}}{\bar\Omega}$ then for a compact domain with boundary $\Omega \subset {\mathbb{R}}^{2k+1}$, $$H_n({\mathbb{R}}^{2k+1} - \Omega) = H_n(S^{2k+1} - \Omega).$$ Taking an open ball $D^{2k+1}$ around the point $x$, we have written $S^{2k+1}$ as a union of open sets. Then the (reduced) Mayer-Vietoris sequence yields an exact sequence $$\rightarrow H_n(S^{2k}) \rightarrow H_n(D^{2k+1}) \oplus H_n({\mathbb{R}}^{2k+1} - \Omega) \rightarrow H_n(S^{2k+1} - \Omega) \rightarrow H_{n-1}(S^{2k}) \rightarrow$$ Since $D^{2k+1}$ is contractible (and we are in reduced homology), this provides the desired isomorphism immediately since the first and last homology groups in the sequence vanish. We start with an existence theorem for a special basis for the $k$-th homology of ${\partial}\Omega$: \[alexander basis existence\] Let $\Omega$ be a compact domain with smooth boundary in ${\mathbb{R}}^{2k+1}$ or $S^{2k+1}$ (with $k > 0$) and ${\bar\Omega}$ be the complementary domain ${\mathbb{R}}^{2k+1} - \Omega$ or $S^{2k+1} - \Omega$. Then if we take coefficients in ${\mathbb{R}}$, $H_k({\partial}\Omega) = H_k(\Omega) \oplus H_k({\bar\Omega})$. Further, given any basis $\langle s_1, \dots, s_n \rangle$ for $H_k(\Omega)$ there is a corresponding basis $\langle s_1, \dots, s_n, t_1, \dots, t_n \rangle$ for $H_k({\partial}\Omega)$ which we call the *Alexander basis* corresponding to $\langle s_1, \dots, s_n \rangle$ so that: 1. The inclusion ${\partial}\Omega \hookrightarrow \Omega$ maps $\langle s_1, \dots, s_n \rangle \in H_k({\partial}\Omega)$ to the original basis $\langle s_1, \dots, s_n \rangle$ for $H_k(\Omega)$ and the inclusion ${\partial}\Omega \hookrightarrow {\bar\Omega}$ maps $\langle t_1, \dots, t_n \rangle$ to a basis for $H_k({\bar\Omega})$. 2. \[defs\] $s_i = {\partial}\sigma_i$ for $\sigma_i \in H_{k+1}({\bar\Omega},{\partial}{\bar\Omega})$, where the $\sigma_i$ form a basis for $H_{k+1}({\bar\Omega},{\partial}{\bar\Omega})$. Similarly, $t_i = {\partial}\tau_i$ for $\tau_i \in H_{k+1}(\Omega,{\partial}\Omega)$, where the $\tau_i$ form a basis for $H_{k+1}(\Omega,{\partial}\Omega)$. 3. \[cup structure\] The cup product algebras of $\Omega$, ${\bar\Omega}$ and ${\partial}\Omega$ obey $$s_i^* \cup \tau_j^* = \delta_{ij} [\Omega]^*, \quad t_i^* \cup \sigma_j^* = (-1)^{k+1} \delta_{ij}[{\bar\Omega}]^*$$ and $$s_i^* \cup s_j^* = 0, \quad t_i^* \cup s_j^* = \delta_{ij} [{\partial}\Omega]^*, \quad t_i^* \cup t_j^* = 0.$$ 4. \[linking\] The linking number ${\operatorname{Lk}}(s_i,t_j) = \delta_{ij}$. (Thus ${\operatorname{Lk}}(t_j,s_i) = (-1)^{(k+1)^2} \delta_{ij}$.) The Alexander duality isomorphism from $H_k(\Omega)$ to $H_k({\bar\Omega})$ maps $s_i$ to $t_i$. We will then study the effect of a homeomorphism on the Alexander basis, proving \[alexander basis transformations\] Suppose that $\Omega$ and $\Omega'$ are compact domains with smooth boundary in ${\mathbb{R}}^{2k+1}$ or $S^{2k+1}$ and that $f\co \Omega \rightarrow \Omega'$ is an orientation-preserving homeomorphism. Then if $\langle s_1, \dots, s_n \rangle$ is a basis for $H_k(\Omega)$ and $\langle s_1', \dots, s_n' \rangle$ is a corresponding basis for $H_k(\Omega')$ so that $f_*(s_i) = s_i'$, then we may build Alexander bases $\langle s_1,\dots,s_n,t_1,\dots,t_n \rangle$ for $H_k({\partial}\Omega)$ and $\langle s_1',\dots,s_n',t_1',\dots,t_n' \rangle$ for $H_k({\partial}\Omega')$. For these bases, we have $f_*(\tau_i) = \tau_i'$ and ${\partial}f_*(t_i) = t_i'$ so that the map ${\partial}f_*\co H_k({\partial}\Omega) \rightarrow H_k({\partial}\Omega')$ can be written as the $2n \times 2n$ matrix $$\label{mmatrix} M = \left[ \begin{array}{c|c} I & 0 \\ \hline (c_{ij}) & I \end{array} \right],$$ where each block represents an $(n \times n)$ matrix. If $k$ is odd, the block matrix $c_{ij}$ is symmetric, while if $k$ is even, the block matrix $c_{ij}$ is skew-symmetric. An example of these theorems was shown in Figure \[fig:alexanderbasis\]. We now restate some of our main tools for this theorem. The first is a form of Poincaré duality [@hatcher Theorem 3.4.3, p.254]: \[lefdual\] Suppose $M$ is a compact orientable $n$-manifold with boundary ${\partial}M$. Then cap product with a top class $[M] \in H^n(M, {\partial}M)$ gives isomorphisms $D_M \co H^i(M) \rightarrow H_{n-i}(M, {\partial}M)$ for all $k$. We now begin the proof. We are essentially reproving the Alexander duality theorem while recording additional information along the way. Let us restrict our attention to the case where $\Omega \subset S^{2k+1}$ for convenience. (Lemma \[whocares\] shows that the same proof works in both cases.) We observe that since ${\partial}\Omega$ is compact and smooth, it has a open tubular neighborhood which deformation retracts to ${\partial}\Omega$. Using this, the reduced Mayer-Vietoris sequence [@hatcher p.150] yields a long exact sequence including $$\label{mvseq} H_{k+1}(S^{2k+1}) = 0 \rightarrow H_k({\partial}\Omega) \xrightarrow{\Phi} H_k(\Omega) \oplus H_k({\bar\Omega}) \xrightarrow{\Psi} H_{k}(S^{2k+1}) = 0.$$ This proves that the map $\Phi$ given by the inclusions of ${\partial}\Omega$ into $\Omega$ and ${\bar\Omega}$ is an isomorphism between $H_k({\partial}\Omega)$ and $H_k(\Omega) \oplus H_k({\bar\Omega})$, proving the first statement in our theorem[^4] Using this isomorphism, we see that our original basis $s_1, \dots, s_n \in H_k(\Omega)$ is the $\Phi$-image of a linearly independent set of classes $s_1, \dots, s_n \in H_k({\partial}\Omega)$ which vanish when included in $H_k({\bar\Omega})$. By Lefschetz Duality (\[lefdual\]), $$[\Omega] \cap \co H^k(\Omega) \rightarrow H_{k+1}(\Omega,{\partial}\Omega),$$ is an isomorphism. Let $\tau_1, \dots, \tau_n$ denote the images of the cohomology duals of $s_1, \dots, s_n$ under this isomorphism. By construction, $[\Omega] \cap s_i^* = \tau_i$, or $\tau_j^* ( [\Omega] \cap s_i^*) = \delta_{ij}$. Now for $\alpha \in H_{k+l}(X,{\partial}X)$, $\phi \in H^k(X)$ and $\psi \in H^{l}(X,{\partial}X)$, the cup and cap product are related by the formula [@hatcher p.249], $$\label{capandcup} \psi( \alpha \cap \phi ) = (\phi \cup \psi) (\alpha).$$ Taking $\psi = \tau_j^*$, $\alpha = [\Omega]$, and $\phi = s_i^*$, we get $$1 = \tau_j^* ( [\Omega] \cap s_i^*) = (s_i^* \cup \tau_j^*) ([\Omega]).$$ This proves our statement about the cup structure of $\Omega$. We now map the $\tau_i$ to classes in $H_k({\bar\Omega})$. If we take a small open neighborhood ${\partial}\Omega_o$ of ${\partial}\Omega$ which deformation retracts to ${\partial}\Omega$, we can construct $\Omega_o = \Omega \cup {\partial}\Omega_o$ so that the closure of ${\bar\Omega}_o = S^{2k+1} - \Omega_o$ is contained in the interior of ${\bar\Omega}$. Clearly $\Omega_o = S^{2k+1} - {\bar\Omega}_o$ and $${\bar\Omega}- {\bar\Omega}_o = (S^{2k+1} - \Omega) - (S^{2k+1} - (\Omega \cup {\partial}\Omega_o)) = {\partial}\Omega_o.$$ This means that the pair $(\Omega_o,{\partial}\Omega_o) = (S^{2k+1} - {\bar\Omega}_o,{\bar\Omega}- {\bar\Omega}_o)$. Further, since the closure of ${\bar\Omega}_o$ is contained in the interior of ${\bar\Omega}$, the inclusion of $(S^{2k+1} - {\bar\Omega}_o,{\bar\Omega}- {\bar\Omega}_o) \hookrightarrow (S^{2k+1},{\bar\Omega})$ induces the homology isomorphism $$H_{k+1}(\Omega,{\partial}\Omega) = H_{k+1}(\Omega_o,{\partial}\Omega_o) = H_{k+1}(S^{2k+1},{\bar\Omega}).$$ by deformation retraction of $(\Omega_o,{\partial}\Omega_o)$ onto $(\Omega,{\partial}\Omega)$ and excision [@hatcher Theorem 2.20, p.119]. But the exact sequence of the pair $(S^{2k+1},{\bar\Omega})$ contains $$H_{k+1}(S^{2k+1}) = 0 \rightarrow H_{k+1}(S^{2k+1},{\bar\Omega}) \xrightarrow{{\partial}} H_k({\bar\Omega}) \rightarrow H_k(S^{2k+1}) = 0$$ So the boundary map carries the $\tau_i$ to a set of generators $t_i$ for $H_k({\bar\Omega})$. The entire map we have built from $s_i$ (as a basis for $H_k(\Omega)$) to $t_i$ (as a basis for $H_k({\bar\Omega})$) is the Alexander duality isomorphism. Since we can pull these $t_i$ back to $H_k({\partial}\Omega)$ under the isomorphism $\Phi\co H_k({\partial}\Omega) \rightarrow H_k(\Omega) \oplus H_k({\bar\Omega})$, we can regard the $t_i$ as a linearly independent set of elements in $H_k({\partial}\Omega)$ which complete the Alexander basis $\langle s_1, \dots, s_n, t_1, \dots, t_n \rangle$ for $H_k({\partial}\Omega)$. In fact, we can choose representatives for the $t_i$ so that $t_i = {\partial}\tau_i$. We now observe that ${\operatorname{Lk}}(s_i,t_j) = {\operatorname{Int}}(s_i,\tau_j) = (s_i^* \cup \tau_j^*)[\Omega] = \delta_{ij}$. We now work out the cup product of $s_i^*$ and $t_j^*$. The relation between the cap product and the boundary operator for an $i$-chain $\alpha$ in $C_i(X,A)$ and a cochain $\beta \in C^l(X)$ is given by $$\label{capandbdy} {\partial}(\alpha \cap \beta) = (-1)^l ({\partial}\alpha \cap \beta - \alpha \cap \delta \beta)$$ where $\delta$ is the coboundary operator [@hatcher p.240]. We now compute $$\label{ocapsi} t_i = {\partial}\tau_i = {\partial}( [\Omega] \cap s_i^* ) = (-1)^k ({\partial}[\Omega] \cap s_i^* - [\Omega] \cap \delta s_i^*) = (-1)^k ([{\partial}\Omega] \cap s_i^*),$$ where $\delta s_i^* = 0$ because $s_i^*$ is a cocycle. We can compute $$\label{sicupti} \delta_{ij} = t_j^*(t_i) = (-1)^k t_j^* ([{\partial}\Omega] \cap s_i^*) = (-1)^k (s_i^* \cup t_j^*) [{\partial}\Omega].$$ Now we recall that the cup product of $\alpha \in H^i(X)$ and $\beta \in H^j(X)$ obeys the (anti)commutativity relation [@hatcher Theorem 3.14, p.215]: $$\label{cupcommutes} \alpha \cup \beta = (-1)^{ij} \beta \cup \alpha.$$ Using this equation, we see immediately that  yields $(t_j^* \cup s_i^*)[{\partial}\Omega] = \delta_{ij}$ whether $k$ is even or odd. Thus we have $$\delta_{ij} = (t_j^* \cup s_i^*)[{\partial}\Omega] = s_i^*([{\partial}\Omega] \cap t_j^*),$$ and $[{\partial}\Omega] \cap t_i^* = s_i + d_l t_l$. (We will shortly argue that the $d_l$ are all zero.) As above, we know $[{\bar\Omega}] \cap \co H^k({\bar\Omega}) \rightarrow H_{k+1}({\bar\Omega},{\partial}{\bar\Omega})$ is an isomorphism. We let $\sigma_i = (-1)^{k+1} [{\bar\Omega}] \cap t_i^*$. We now show ${\partial}\sigma_i \in H_k({\partial}\Omega) = s_i$. We notice first that ${\partial}\sigma_i$ is a linear combination of $s_j$ since it is certainly the case that ${\partial}\sigma_i$ bounds in ${\bar\Omega}$, so ${\partial}\sigma_i$ must be contained in the $H_k(\Omega)$ summand of $H_k({\partial}\Omega) = H_k(\Omega) \oplus H_k({\bar\Omega})$. Now we compute $$\label{ocapti} \begin{split} {\partial}\sigma_i &= {\partial}((-1)^{k+1} [{\bar\Omega}] \cap t_i^*) = (-1)^{2k+1} ({\partial}[{\bar\Omega}] \cap t_i^* - [{\bar\Omega}] \cap \delta t_i^*) \\ &= (-1)^{2k+1}(-[{\partial}\Omega] \cap t_i^*) = [{\partial}\Omega] \cap t_i^*. \end{split}$$ We already know that $[{\partial}\Omega] \cap t_i^* = s_i + d_l t_l$. Since ${\partial}\sigma_i$ is a linear combination of $s_j$, we have shown that ${\partial}\sigma_i = s_i$, as desired. Notice that we needed the $(-1)^{k+1}$ in our definition of $\sigma_i = (-1)^{k+1}[{\bar\Omega}] \cap t_i^*$ to make the signs come out correctly in this last sequence of arguments. To verify this sign, observe that we can directly compute $$\begin{aligned} {\operatorname{Lk}}(t_j,s_i) &= {\operatorname{Int}}(t_j,\sigma_i) = (t_j^* \cup \sigma_i^*)([{\bar\Omega}]) \\ &= \sigma_i^* ([{\bar\Omega}] \cap t_j^*) = (-1)^{k+1} \sigma_i^*(\sigma_j) = (-1)^{k+1} \delta_{ij},\end{aligned}$$ which agrees with our previous computation $${\operatorname{Lk}}(t_j,s_i) = (-1)^{(k+1)^2} {\operatorname{Lk}}(s_i,t_j) = (-1)^{(k+1)^2} \delta_{ij}.$$ Now consider the cup product $s_i^* \cup s_j^*$. Using and , we have $$(s_i^* \cup s_j^*)[{\partial}\Omega] = s_j^* ([{\partial}\Omega] \cap s_i^*) = (-1)^k s_j^* (t_i) = 0.$$ Similarly, using  and , we have $$(t_i^* \cup t_j^*)[{\partial}\Omega] = t_j^* ([{\partial}\Omega] \cap t_i^*) = t_j^* (s_i) = 0. \qedhere$$ We now prove our second theorem about the Alexander basis. Since $f$ is an orientation-preserving homeomorphism, we have $f_*[{\partial}\Omega] = [{\partial}\Omega']$. We also know that in $H_k(\Omega')$ and $H_k(\Omega)$, $f^*({s_i'^*}) = s_i^*$ since we have $f_*(s_i) = {s_i'}$. This means that in $H_k({\partial}\Omega)$, ${\partial}f_*(s_i) = {s_i'}+ c_{ji} t_j'$ for some coefficients $c_{ji}$, since the $t_j'$ span the kernel of the inclusion homomorphism from $H_k({\partial}\Omega')$ to $H_k(\Omega')$. We now compute $f^*(\tau_j'^*) \in H^{k+1}(\Omega,{\partial}\Omega)$. We know $$f^*({s_i'^*}\cup {(\tau_j')^*}) = f^*({s_i'^*}) \cup f^*({(\tau_j')^*}) = s_i^* \cup f^*({(\tau_j')^*}).$$ On the other hand by conclusion (\[cup structure\]) of Theorem \[alexander basis existence\], we know $$f^*({s_i'^*}\cup {(\tau_j')^*}) = f^*(\delta_{ij} [\Omega']^*) = \delta_{ij} [\Omega]^*.$$ Thus, using our construction of $\tau_i = [\Omega] \cap s_i$, $$\delta_{ij} = (s_i^* \cup f^*({(\tau_j')^*}))[\Omega] = f^*({(\tau_j')^*}) ([\Omega] \cap s_i^*) = f^*({(\tau_j')^*})(\tau_i).$$ and we may conclude that $f^*({(\tau_j')^*}) = \tau_j^*$. Hence $f_*(\tau_j) = \tau_j'$, and $f_*(t_i) = t_i'$, since $t_i = {\partial}\tau_i$. This proves that the map ${\partial}f_*\co H_k({\partial}\Omega) \rightarrow H_k({\partial}\Omega')$ can be written in the matrix form above. Since ${\partial}f_*(s_i) = s_i' + c_{ji} t_j'$, it follows that $s_i^* = {\partial}f^*(s_i'^* + c_{ji} t_j'^*)$. Of course, using conclusion (\[cup structure\]) of Theorem \[alexander basis existence\] again, we know $$0 = s_i^* \cup s_j^* = {\partial}f^*(s_i'^* + c_{ki} t_k'^*) \cup {\partial}f^*(s_j'^* + c_{jl} t_l'^*).$$ In addition, we know ${\partial}f^*$ is an isomorphism and $t_i^* \cup s_j^* = \delta_{ij} [{\partial}\Omega']^*$ so $$\begin{aligned} 0 &= ({s_i'^*}+ c_{ik} t_k'^*) \cup (s_j'^* + c_{jl} t_l'^*) = c_{jl} s_i'^* \cup t_l'^* + c_{ik} t_k'^* \cup s_j'^* \\ &= (c_{jl} (-1)^{k^2} \delta_{il} + c_{ik} \delta_{kj}) [{\partial}\Omega']^* = ((-1)^{k^2} c_{ji} + c_{ij}) [{\partial}\Omega']^*.\end{aligned}$$ If $k$ is even, this equation becomes $c_{ji} + c_{ij} = 0$ and the matrix is skew-symmetric, while if $k$ is odd, this equation becomes $-c_{ji} + c_{ij} = 0$ and the matrix is symmetric, as claimed. It is tempting to wonder what happens if one computes the cup products $s_i^* \cup t_j^* = {\partial}f^*(s_i'^* + c_{ki} t_k'^*) \cup {\partial}f^*(t_j')$ or $t_i^* \cup t_j^* = {\partial}f^*(t_i'^*) \cup {\partial}f^*(t_j'^*)$. It is certainly possible to do so, yielding expansions similar to those above, but it turns out to be the case that this procedure yields no additional information about the $c_{ij}$. Thus we believe that Theorems \[alexander basis existence\] and \[alexander basis transformations\] summarize all of the cohomological information available for an arbitrary compact domain with boundary in ${\mathbb{R}}^{2k+1}$. The Hodge-Morrey-Friedrichs Decomposition for Manifolds with Boundary {#hodge} ===================================================================== The purpose of this section is to explain why every closed Dirichlet form on a domain $\Omega$ in ${\mathbb{R}}^{2k+1}$ is exact using a decomposition of the differential forms on $\Omega$, which for now we view as a smooth, compact, Riemannian $n$-manifold with non-empty boundary. As with the previous appendices, this section is expository; see Schwarz [@MR1367287] for full details. Non-experts may prefer the well-written treatment of this subject in Chapter 2 of Clayton Shonkwiler’s thesis [@shonk]. We let ${\Lambda^{p}(\Omega)}$ denote the vector space of smooth $p$-forms on $\Omega$. The exterior derivative $d$ and the codifferential $\delta$ map to ${\Lambda^{p+1}(\Omega)}$ and ${\Lambda^{p-1}(\Omega)}$, respectively. The Laplacian on $p$-forms is defined as $\Delta = d\delta + \delta d$. We can define natural subspaces of ${\Lambda^{p}(\Omega)}$ by looking at the kernel and image of $d$ and $\delta$: The kernel of $d$ is the space of *closed* forms. The kernel of $\delta$ is the space of *co-closed* forms. The intersection of these spaces is the space of *harmonic $p$-fields* ${\mathcal{H}^{p}(\Omega)}$. These form a subset of the *harmonic $p$-forms* ${\widehat{\mathcal{H}}^{p}(\Omega)}$, defined as the kernel of $\Delta$. We use the following notation for the image of $d$ and $\delta$: - *exact* $p$-forms: ${\mathcal{E}^{p}(\Omega)} \subset {\Lambda^{p}(\Omega)}$ is the image of $d \co {\Lambda^{p-1}(\Omega)} \rightarrow {\Lambda^{p}(\Omega)}$. - *co-exact* $p$-forms: ${c\mathcal{E}^{p}(\Omega)} \subset {\Lambda^{p}(\Omega)}$ is the image of $\delta \co {\Lambda^{p+1}(\Omega)} \rightarrow {\Lambda^{p}(\Omega)}$. We can take the intersections of appropriate subspaces to yield the exact harmonic $p$-fields ${\mathcal{EH}^{p}(\Omega)}$ and the co-exact harmonic $p$-fields ${c\mathcal{EH}^{p}(\Omega)}$. Since the boundary of $\Omega$ is nonempty, we can also classify forms into subspaces by their behavior on the boundary. A $p$-form $\omega \in {\Lambda^{p}(\Omega)}$ obeys *Dirichlet boundary conditions* if $\omega$ vanishes when restricted to ${\partial}\Omega$, i.e., if $V_1, \ldots, V_p$ all lie in $T_x{{\partial}\Omega}$, then $\omega(V_1, \ldots, V_p)=0$. We say $\omega$ obeys *Neumann boundary conditions* when the Hodge dual $\star \omega$ of $\omega$ obeys Dirichlet boundary conditions. We apply boundary conditions to the exact and co-exact forms and harmonic fields by attaching a subscript $D$ or $N$ to the notation above. For the harmonic fields, ${\mathcal{H}^{p}_D(\Omega)}$ and ${\mathcal{H}^{p}_N(\Omega)}$ are just the intersections of harmonic fields with Dirichlet and Neumann forms. For the exact and co-exact forms, the boundary condition applies also to the *primitives* of the forms in ${\mathcal{E}^{p}_N(\Omega)}$, ${c\mathcal{E}^{p}_N(\Omega)}$, ${\mathcal{E}^{p}_D(\Omega)}$, and ${c\mathcal{E}^{p}_D(\Omega)}$. We can now give \[hmf\] Let $\Omega$ be a compact, oriented, smooth Riemannian manifold with non-empty boundary ${\partial}\Omega$. Then ${\Lambda^{p}(\Omega)}$ admits the $L^2$ orthogonal decompositions $$\begin{aligned} {\Lambda^{p}(\Omega)} &= {c\mathcal{E}^{p}_N(\Omega)} \oplus \; {\mathcal{H}^{p}_N(\Omega)} \; \oplus {\mathcal{EH}^{p}(\Omega)} \oplus {\mathcal{E}^{p}_D(\Omega)} \label{hmf1}\\ {\Lambda^{p}(\Omega)} &= {c\mathcal{E}^{p}_N(\Omega)} \oplus {c\mathcal{EH}^{p}(\Omega)} \oplus {\mathcal{H}^{p}_D(\Omega)} \, \oplus {\mathcal{E}^{p}_D(\Omega)}. \label{hmf2}\end{aligned}$$ Further, ${\mathcal{H}^{p}_N(\Omega)} \simeq H^p(\Omega;{\mathbb{R}})$ while ${\mathcal{H}^{p}_D(\Omega)} \simeq H^p(\Omega,{\partial}\Omega; {\mathbb{R}})$. Using the first decomposition, we now make an observation about closed Dirichlet forms on $\Omega$. \[alphaisexact\] Let $0<p < 2k+1$. The space of closed Dirichlet $p$-forms is ${\mathcal{H}^{p}_D(\Omega)} \oplus {\mathcal{E}^{p}_D(\Omega)}$. If $\Omega$ is a compact subdomain of ${\mathbb{R}}^{2k+1}$ with smooth boundary, then every closed Dirichlet form $\alpha \in {\Lambda^{p}(\Omega)}$ is exact. Our notation has the unfortunate consequence of making it easy to misread this proposition. For Euclidean subdomains, the proposition does imply that every form in ${\mathcal{H}^{p}_D(\Omega)}$ must be exact. It is tempting to conclude that this means that ${\mathcal{H}^{p}_D(\Omega)} \subset {\mathcal{E}^{p}_D(\Omega)}$, contradicting the orthogonality of the Hodge-Morrey-Friedrichs decomposition. This is not the case. For a form $\alpha$ to be in ${\mathcal{E}^{p}_D(\Omega)}$ it is necessary but *not* sufficient that $\alpha$ be exact and Dirichlet: $\alpha$ must also have a Dirichlet primitive. The exact forms that are Dirichlet but fail to have a Dirichlet primitive all lie in ${\mathcal{H}^{p}_D(\Omega)}$. We begin by showing that the subspace ${c\mathcal{E}^{p}_N(\Omega)}$ is the orthogonal complement of the closed forms on $\Omega$. Any form $\alpha \in {c\mathcal{E}^{p}_N(\Omega)}$ is co-exact, hence co-closed; if $\alpha$ were also closed, it would be a harmonic field lying in ${\mathcal{H}^{p}_N(\Omega)}$ which has trivial intersection with ${c\mathcal{E}^{p}_N(\Omega)}$ by . Since harmonic fields and exact forms are both closed, all of the other three summands in are closed, which shows that ${\Lambda^{p}(\Omega)} = {c\mathcal{E}^{p}_N(\Omega)} \oplus \{\text{closed\ } p\text{-forms}\}$. We now can apply the second decomposition to say that the closed Dirichlet forms must lie inside ${c\mathcal{EH}^{p}(\Omega)} \oplus {\mathcal{H}^{p}_D(\Omega)} \oplus {\mathcal{E}^{p}_D(\Omega)}$. No $p$-form in ${c\mathcal{EH}^{p}(\Omega)}$ can be Dirichlet, since it is also a harmonic field and the harmonic Dirichlet fields comprise ${\mathcal{H}^{p}_D(\Omega)}$, which lies orthogonal to ${c\mathcal{EH}^{p}(\Omega)}$. Thus we conclude that the subspace of closed Dirichlet $p$-forms is precisely ${\mathcal{H}^{p}_D(\Omega)} \oplus {\mathcal{E}^{p}_D(\Omega)}$; all of these forms are both closed and Dirichlet. Now we turn toward the second statement of the proposition. Let $\Omega$ now be a compact subdomain of ${\mathbb{R}}^{2k+1}$ with smooth boundary. Using , we can decompose any closed Dirichlet form $\alpha \in {\Lambda^{p}(\Omega)}$ as $$\alpha = \alpha_{\mathcal{H}_N} + \alpha_{\mathcal{EH}} + \alpha_{\mathcal{E}_D},$$ since it has no component in ${c\mathcal{E}^{p}_N(\Omega)}$. In other words, $\alpha$ is the sum of a form in ${\mathcal{H}^{p}_N(\Omega)}$ and an exact form. Further, since ${\mathcal{H}^{p}_N(\Omega)} \simeq H^{p}(\Omega;{\mathbb{R}})$, we can determine $\alpha_{\mathcal{H}_N}$ by finding the cohomology class represented by $\alpha$ in $H^{p}(\Omega)$. We claim that this cohomology class is zero since $\alpha$ is Dirichlet. To see this, recall that if ${\bar\Omega}$ is the complement of $\Omega$ in ${\mathbb{R}}^{2k+1}$, the Mayer-Vietoris sequence for ${\mathbb{R}}^{2k+1} = \Omega \cup {\bar\Omega}$ includes $$0 = H_{p-1}({\mathbb{R}}^{2k+1}) \rightarrow H_{p}({\partial}\Omega) \xrightarrow{i_* \oplus i'_*} H_{p}(\Omega) \oplus H_{p}({\bar\Omega}) \rightarrow H_{p}({\mathbb{R}}^{2k+1}) = 0$$ where $i$ is the inclusion ${\partial}\Omega \hookrightarrow \Omega$. Thus in particular $i_* \co H_{p}({\partial}\Omega) \rightarrow H_{p}(\Omega)$ is onto. Now this means that for any cycle $[X] \in H_{p}(\Omega)$, we can find $Y \in {\partial}\Omega$ so that $i_*([Y]) = [X]$. Then, since a Dirichlet form vanishes on the boundary, $$\int_X \alpha = \int_Y i^*(\alpha) = \int_Y 0 = 0.$$ This means that $\alpha_{\mathcal{H}_N} = 0$, and so $\alpha$ is exact. One more subspace appears in our next decomposition. Define ${\mathcal{E}c\mathcal{E}^{p}(\Omega)}$ to be all $p$-forms that are both exact and co-exact; all such forms are clearly harmonic $p$-fields. For $\Omega$ is a compact subdomain of ${\mathbb{R}}^{2k+1}$ with smooth boundary, the following decomposition holds $$\label{hmfdg} {\Lambda^{p}(\Omega)} = {c\mathcal{E}^{p}_N(\Omega)} \oplus {\mathcal{E}c\mathcal{E}^{p}(\Omega)} \oplus {\mathcal{H}^{p}_N(\Omega)} \oplus {\mathcal{H}^{p}_D(\Omega)} \oplus {\mathcal{E}^{p}_D(\Omega)}.$$ The proposition above shows all forms in ${\mathcal{H}^{p}_D(\Omega)}$ are exact, hence ${\mathcal{H}^{p}_D(\Omega)} \subset {\mathcal{EH}^{p}(\Omega)}$, which lies orthogonal to ${\mathcal{H}^{p}_N(\Omega)}$. The orthogonal complement of ${\mathcal{H}^{p}_D(\Omega)}$ within ${\mathcal{EH}^{p}(\Omega)}$ are those harmonic fields that are both exact and co-exact. Hence, the harmonic $p$-fields decompose as $${\mathcal{H}^{p}(\Omega)} = {\mathcal{E}c\mathcal{E}^{p}(\Omega)} \oplus {\mathcal{H}^{p}_N(\Omega)} \oplus {\mathcal{H}^{p}_D(\Omega)},$$ which along with the Hodge-Morrey-Friedrichs decomposition proves this corollary. We conclude with one more decomposition, for vector fields on a domain in ${\mathbb{R}}^3$; see [@MR1901496] for full details. \[thm:hodgevf\] Let $\Omega$ be a compact subdomain of ${\mathbb{R}}^3$ with smooth $\partial\Omega$. Then, the space of smooth vector fields $VF(\Omega)$ decomposes into five mutually orthogonal subspaces, $$\label{hodgevf} VF(\Omega)=FK \oplus HK \oplus CG \oplus HG \oplus GG,$$ where, $$\begin{array}{cclcl} FK & = & \mathrm{fluxless\ knots} & = & \left\{{\nabla\cdot {V}}=0, \; V \cdot n =0, \; \mathrm{all\ interior\ fluxes} = 0\right\}\\ HK & = & \mathrm{harmonic\ knots} & = & \{{\nabla\cdot {V}}=0, \; V \cdot n =0, \; {\nabla\times {V}}=0\} \\ CG & = & \mathrm{curly\ gradients} & = & \{V=\nabla\phi, \; {\nabla\cdot {V}}=0, \; \mathrm{all\ boundary\ fluxes} = 0\} \\ HG & = & \mathrm{harmonic\ gradients} & = & \{V=\nabla\phi, \; {\nabla\cdot {V}}=0, \; \phi \mathrm{\ locally\ const.\ on\ } \partial\Omega\} \\ GG & = & \mathrm{grounded\ gradients} & = & \{V=\nabla\phi, \; \phi|_{\partial\Omega}=0\} \end{array}$$ The Hodge Decomposition Theorem for vector fields precisely corresponds to the five-term decomposition for 2-forms on $\Omega$. The five summands in each decomposition are isomorphic as vector spaces, and pair as follows: $$\label{five-terms} c\mathcal{E}_N \cong GG, \quad \mathcal{E}c\mathcal{E} \cong CG, \quad \mathcal{H}_N \cong HG, \quad \mathcal{H}_D \cong HK, \quad \mathcal{E}_D \cong FK .$$ To prove this corollary, we consider a 2-form $\alpha$ on a subdomain $\Omega \subset {\mathbb{R}}^3$ and its dual vector field $V$. We translate our definitions regarding forms into the statements about vector fields. - $\alpha$ is [*closed*]{}, i.e., $d\alpha = 0$ $\Longleftrightarrow$ $V$ is divergence-free. - $\alpha$ is [*exact*]{}, i.e., $\alpha = d\beta$ for a 1-form $\beta$ $\Longleftrightarrow$ $V$ lies in the image of curl. - $\alpha$ is [*co-closed*]{}, i.e., $\delta\alpha = \star d \star \alpha = 0$. Here $d \star \alpha$ amounts to taking the curl of $V$, so co-closed corresponds to $V$ lying in the kernel of curl. - $\alpha$ is [*co-exact*]{}, i.e., $\alpha = \delta\gamma = \star d \star \gamma $, for some 3-form $\gamma$. Any 3-form can be written as $\gamma = f {\operatorname{dvol}}$, so $\star \gamma = f$. Thus, co-exact corresponds to $V$ equaling a gradient $\nabla f$. - $\alpha$ is [*Dirichlet*]{} $\Longleftrightarrow$ $V$ is tangent to the boundary. - $\alpha$ is [*Neumann*]{}, meaning $\star \alpha$ is Dirichlet $\Longleftrightarrow$ $V$ is normal to the boundary. - $\alpha \in {c\mathcal{E}^{2}_N(\Omega)} \Longleftrightarrow V = \nabla f$ and $f=0$ on the boundary Now, let us examine each piece of . The first piece, ${c\mathcal{E}^{2}_N(\Omega)}$, corresponds to gradients that vanish on ${\partial}\Omega$; this defines the subspace $GG$. The exact and co-exact forms correspond to gradients that lie in the image of curl, namely $CG$. The third piece, ${\mathcal{H}^{2}_N(\Omega)}$ are closed and co-closed Neumann forms. (Combining and , we see that forms in ${\mathcal{H}^{p}_N(\Omega)}$ actually co-exact.) The forms in ${\mathcal{H}^{2}_N(\Omega)}$ are the only closed forms that are not exact. These correspond to vector fields that are divergence-free but not in the image of curl comprise $HG$. The fourth piece, ${\mathcal{H}^{2}_D(\Omega)}$ are exact, co-closed Dirichlet forms. These correspond to divergence-free fields which are tangent to the boundary and lie in the kernel of curl, which characterizes $HK$. As an additional check, ${\mathcal{H}^{2}_N(\Omega)} \simeq H^2(\Omega; {\mathbb{R}}) \simeq HG$ and ${\mathcal{H}^{2}_D(\Omega)} \simeq H^1(\Omega; {\mathbb{R}}) \simeq HK$. The final piece ${\mathcal{E}^{2}_D(\Omega)}$ consists of exact forms whose primitives obey a Dirichlet condition. The corresponding vector field then must be divergence-free and tangent to the boundary; furthermore, the Dirichlet condition on the primitive implies that all fluxes of the vector field over surfaces $\Sigma \subset \Omega$ with ${\partial}\Sigma \subset {\partial}\Omega$ must vanish. This precisely characterizes $FK$. [^1]: A corresponding theorem for the Godbillon-Vey invariant appears in a paper of G. Raby from 1988 [@MR949006]. [^2]: For domains in ${\mathbb{R}}^3$ with boundary, it is the fluxless condition, and not the null-homologous condition, which guarantees that helicity remains unchanged under all volume-preserving diffeomorphisms. We will give a homological interpretation of this fact in section \[sec:invariance\]. [^3]: In dimension 3, [@MR2067778 p.119] shows that since there is no torsion in any of the homology or cohomology groups, we could take coefficients in $\mathbb{Z}$ and get the same conclusions. [^4]: In fact, a version of this observation actually predates homology theory itself! The statement that (essentially) the dimension of $H_1({\partial}\Omega)$ was equal to the sum of the dimensions of $H_1(\Omega)$ and $H_1({\bar\Omega})$ was published by James Clerk Maxwell in 1891 in his *Treatise on Electricity and Magnetism* [@maxwell]. Steenrod gives the result in a form essentially the same as ours as a theorem of Hopf [@MR0145525 Theorem 2.1], while Kauffman proves that for any three manifold with boundary, half of the homology of the boundary is in the kernel of the inclusion of the boundary into the interior [@MR907872 Lemma 8.1].

--- abstract: | The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of *canonicity* — that every closed term computes to a canonical form. A computation becomes ‘stuck’ when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the computations are expressed using a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are *terms* which inhabit *types*, which are the simple types over $\Omega$. There are *proofs* which inhabit *propositions*, which are the terms of type $\Omega$. The canonical propositions are those constructed from $\bot$ by implication $\supset$. Thirdly, there are *paths* which inhabit *equations* $M =_A N$, where $M$ and $N$ are terms of type $A$. There are two ways to prove an equality: reflexivity, and *propositional extensionality* — logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda. author: - Robin Adams - Marc Bezem - Thierry Coquand bibliography: - '../../../../type.bib' title: 'A Normalizing Computation Rule for Propositional Extensionality in Higher-Order Minimal Logic ' --- Introduction ============ The *univalence axiom* of Homotopy Type theory (HoTT) [@hottbook] postulates a constant $${\ensuremath{\mathsf{isotoid}}}: A \simeq B \rightarrow A = B$$ that is an inverse to the obvious function $A = B \rightarrow A \simeq B$. However, if we simply add this constant to Martin-Löf type theory, then we lose the important property of *canonicity* — that every closed term of type $A$ computes to a unique canonical object of type $A$. When a computation reaches a point where we eliminate a path (proof of equality) formed by ${\ensuremath{\mathsf{isotoid}}}$, it gets ’stuck’. As possible solutions to this problem, we may try to do with a weaker property than canonicity, such as *propositional canonicity*: that every closed term of type $\mathbb{N}$ is *propositionally* equal to a numeral, as conjectured by Voevodsky. Or we may attempt to change the definition of equality to make ${\ensuremath{\mathsf{isotoid}}}$ definable [@Polonsky14a], or add a nominal extension to the syntax of the type theory (e.g. Cubical Type Theory [@cchm:cubical]). We could also try a more conservative approach, and simply attempt to find a reduction relation for a type theory involving ${\ensuremath{\mathsf{isotoid}}}$ that satisfies all three of the properties above. There seems to be no reason *a priori* to believe this is not possible, but it is difficult to do because the full Homotopy Type Theory is a complex and interdependent system. We can tackle the problem by adding univalence to a much simpler system, finding a well-behaved reduction relation, then doing the same for more and more complex systems, gradually approaching the full strength of HoTT. In this paper, we present a system we call PHOML, or predicative higher-order minimal logic. It is a type theory with three kinds of typing judgement. There are *terms* which inhabit *types*, which are the simple types over $\Omega$. There are *proofs* which inhabit *propositions*, which are the terms of type $\Omega$. The canonical propositions are those constructed from $\bot$ by implication $\supset$. Thirdly, there are *paths* which inhabit *equations* $M =_A N$, where $M$ and $N$ are terms of type $A$. There are two canonical forms for proofs of $M =_\Omega N$. For any term $\varphi : \Omega$, we have ${\ensuremath{\mathrm{ref} \left( {\varphi} \right)}} : \varphi =_\Omega \varphi$. We also add univalence for this system, in this form: if $\delta : \varphi \supset \psi$ and $\epsilon : \psi \supset\varphi$, then ${\ensuremath{\mathrm{univ}_{{\varphi}, {\psi}} \left({\delta} , {\epsilon} \right)}} : \varphi =_\Omega \psi$. This entails that in PHOML, two propositions that are logically equivalent are equal. Every function of type $\Omega \rightarrow \Omega$ that can be constructed in PHOML must therefore respect logical equivalence. That is, for any $F$ and logically equivalent $x$, $y$ we must have that $Fx$ and $Fy$ are logically equivalent. Moreover, if for $x:\Omega$ we have that $Fx$ is logically equivalent to $Gx$, then $F =_{\Omega\to\Omega} G$. Every function of type $(\Omega \rightarrow \Omega) \rightarrow \Omega$ must respect this equality; and so on. This is the manifestation in PHOML of the principle that only homotopy invariant constructions can be performed in homotopy type theory. (See Section \[section:exampletwo\].) We present a call-by-name reduction relation for this system, and prove that every typable term reduces to a canonical form. From this, it follows that the system is consistent. For the future, we wish to include the equations in $\Omega$, allowing for propositions such as $M =_A N \supset N =_A M$. We wish to expand the system with universal quantification, and expand it to a 2-dimensional system (with equations between proofs). We then wish to add more inductive types and more dimensions, getting ever closer to full homotopy type theory. Another system with many of the same aims is cubical type theory [@cchm:cubical]. The system PHOML is almost a subsystem of cubical type theory. We can attempt to embed PHOML into cubical type theory, mapping $\Omega$ to the universe $U$, and an equation $M =_A N$ to either the type ${\ensuremath{\mathsf{Path} \, {A} \, {M} \, {N}}}$ or to $\mathrm{Id}\ A\ M\ N$. However, PHOML has more definitional equalities than the relevant fragment of cubical type theory; that is, there are definitionally equal terms in PHOML that are mapped to terms that are not definitionally equal in cubical type theory. In particular, ${\ensuremath{\mathrm{ref} \left( {x} \right)}}^+ p$ and $p$ are definitionally equal, whereas the terms $\mathrm{comp}^i x [] p$ and $p$ are not definitionally equal in cubical type theory (but they are propositionally equal). See Section \[section:cubical\] for more information. The proofs in this paper have been formalized in Agda. The formalization is available at `https://github.com/radams78/TYPES2016`. Predicative Higher-Order Minimal Logic with Extensional Equality ================================================================ We call the following type theory PHOML, or *predicative higher-order minimal logic with extensional equality*. Syntax ------ Fix three disjoint, infinite sets of variables, which we shall call *term variables*, *proof variables* and *path variables*. We shall use $x$ and $y$ as term variables, $p$ and $q$ as proof variables, $e$ as a path variable, and $z$ for a variable that may come from any of these three sets. The syntax of PHOML is given by the grammar: $$\begin{array}{lrcl} \text{Type} & A,B,C & ::= & \Omega \mid A \rightarrow B \\ \text{Term} & L,M,N, \varphi,\psi,\chi & ::= & x \mid \bot \mid \varphi \supset \psi \mid \lambda x:A.M \mid MN \\ \text{Proof} & \delta, \epsilon & ::= & p \mid \lambda p:\varphi.\delta \mid \delta \epsilon \mid P^+ \mid P^- \\ \text{Path} & P, Q & ::= & e \mid {\ensuremath{\mathrm{ref} \left( {M} \right)}} \mid P \supset^* Q \mid {\ensuremath{\mathrm{univ}_{{\varphi}, {\psi}} \left({P} , {Q} \right)}} \mid \\ & & & {\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_A y. P \mid P_{MN} Q \\ \text{Context} & \Gamma, \Delta, \Theta & ::= & \langle \rangle \mid \Gamma, x : A \mid \Gamma, p : \varphi \mid \Gamma, e : M =_A N \\ \text{Judgement} & \mathbf{J} & ::= & \Gamma {\ensuremath{\vdash \mathrm{valid}}}\mid \Gamma \vdash M : A \mid \Gamma \vdash \delta : \varphi \mid \\ & & & \Gamma \vdash P : M =_A N \end{array}$$ In the path ${\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_A y . P$, the term variables $x$ and $y$ must be distinct. (We also have $x \not\equiv e \not\equiv y$, thanks to our stipulation that term variables and path variables are disjoint.) The term variable $x$ is bound within $M$ in the term $\lambda x:A.M$, and the proof variable $p$ is bound within $\delta$ in $\lambda p:\varphi.\delta$. The three variables $e$, $x$ and $y$ are bound within $P$ in the path ${\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e:x =_A y.P$. We identify terms, proofs and paths up to $\alpha$-conversion. We write $E[z:=F]$ for the result of substituting $F$ for $z$ within $E$, using $\alpha$-conversion to avoid variable capture. We shall use the word ’expression’ to mean either a type, term, proof, path, or equation (an equation having the form $M =_A N$). We shall use $E$, $F$, $S$ and $T$ as metavariables that range over expressions. Note that we use both Roman letters $M$, $N$ and Greek letters $\varphi$, $\psi$, $\chi$ to range over terms. Intuitively, a term is understood as either a proposition or a function, and we shall use Greek letters for terms that are intended to be propositions. Formally, there is no significance to which letter we choose. Note also that the types of PHOML are just the simple types over $\Omega$; therefore, no variable can occur in a type. The intuition behind the new expressions is as follows (see also the rules of deduction in Figure \[fig:lambdaoe\]). For any object $M : A$, there is the trivial path ${\ensuremath{\mathrm{ref} \left( {M} \right)}} : M =_A M$. The constructor $\supset^*$ ensures congruence for $\supset$ — if $P : \varphi =_\Omega \varphi'$ and $Q : \psi =_\Omega \psi'$ then $P \supset^* Q : \varphi \supset \psi =_\Omega \varphi' \supset \psi'$. The constructor $\mathsf{univ}$ gives ’univalence’ (propositional extensionality) for our propositions: if $\delta : \varphi \supset \psi$ and $\epsilon : \psi \supset \varphi$, then ${\ensuremath{\mathrm{univ}_{{\varphi}, {\psi}} \left({\delta} , {\epsilon} \right)}}$ is a path $\varphi =_\Omega \psi$. The constructors $^+$ and $^-$ are the converses, which denote the action of transport along a path: if $P$ is a path of type $\varphi =_\Omega \psi$, then $P^+$ is a proof of $\varphi \supset \psi$, and $P^-$ is a proof of $\psi \supset \varphi$. The constructor ${\ensuremath{\lambda \!\! \lambda \!\! \lambda}}$ gives functional extensionality. Let $F$ and $G$ be functions of type $A \rightarrow B$. If $F x =_B G y$ whenever $x =_A y$, then $F =_{A \rightarrow B} G$. More formally, if $P$ is a path of type $Fx =_B Gy$ that depends on $x : A$, $y : A$ and $e : x =_A y$, then ${\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_A y . P$ is a path of type $F =_{A \rightarrow B} G$. The proofs $P^+$ and $P^-$ represent transport along the path $P$. Finally, if $P$ is a path $M =_{A \rightarrow B} M'$, and $Q$ is a path $N =_A N'$, then $P_{MN} Q$ is a path $MN =_B M'N'$. ### Substitution and Path Substitution Intuitively, if $N$ and $N'$ are equal then $M[x:=N]$ and $M[x:=N']$ should be equal. To handle this syntactically, we introduce a notion of *path substitution*. If $N$, $M$ and $M'$ are terms, $x$ a term variable, and $P$ a path, then we shall define a path $N \{ x := P : M = M' \}$. The intention is that, if $\Gamma \vdash P : M =_A M'$ and $\Gamma, x : A \vdash N : B$ then $\Gamma \vdash N \{ x := P : M = M' \} : N [ x:= M ] =_B N [ x := M' ]$ (see Lemma \[lm:pathsub\]). Given terms $M_1$, …, $M_n$ and $N_1$, …, $N_n$; paths $P_1$, …, $P_n$; term variables $x_1$, …, $x_n$; and a term $L$, define the path $$L \{ x_1 := P_1 : M_1 = N_1 , \ldots, x_n := P_n : M_n = N_n \}$$ as follows. $$\begin{aligned} x_i \{ \vec{x} := \vec{P} : \vec{M} = \vec{N} \} & {\mathrel{\smash{\stackrel{\text{def}}{=}}}}P_i \\ y \{ \vec{x} := \vec{P} : \vec{M} = \vec{N} \} & {\mathrel{\smash{\stackrel{\text{def}}{=}}}}{\ensuremath{\mathrm{ref} \left( {y} \right)}} \qquad (y \not\equiv x_1, \ldots, x_n) \\ \bot \{ \vec{x} := \vec{P} : \vec{M} = \vec{N} \} & {\mathrel{\smash{\stackrel{\text{def}}{=}}}}{\ensuremath{\mathrm{ref} \left( {\bot} \right)}} \\ (LL') \{ \vec{x} := \vec{P} : \vec{M} = \vec{N} \} \\ \omit\rlap{\qquad \qquad ${\mathrel{\smash{\stackrel{\text{def}}{=}}}}L \{ \vec{x} := \vec{P} : \vec{M} = \vec{N} \}_{L' [\vec{x} := \vec{M}] L' [\vec{x} := \vec{N}]} L' \{ \vec{x} := \vec{P} : \vec{M} = \vec{N} \}$} \\ (\lambda y:A.L) \{ \vec{x} := \vec{P} : \vec{M} = \vec{N} \} & \\ \omit\rlap{\qquad\qquad ${\mathrel{\smash{\stackrel{\text{def}}{=}}}}{\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : a =_A a' . L \{ \vec{x} := \vec{P} : \vec{M} = \vec{N} , y := e : a = a' \}$} \\ (\varphi \supset \psi) \{ \vec{x} := \vec{P} : \vec{M} = \vec{N} \} & {\mathrel{\smash{\stackrel{\text{def}}{=}}}}\varphi \{ \vec{x} := \vec{P} : \vec{M} = \vec{N} \} \supset^* \psi \{ \vec{x} := \vec{P} : \vec{M} = \vec{N} \}\end{aligned}$$ We shall often omit the endpoints $\vec{M}$ and $\vec{N}$. The case $n = 0$ is permitted, and we shall have that, if $\Gamma \vdash M : A$ then $\Gamma \vdash M \{\} : M =_A M$. There are thus two paths from a term $M$ to itself: ${\ensuremath{\mathrm{ref} \left( {M} \right)}}$ and $M \{\}$. There are not always equal; for example, $(\lambda x:A.x) \{\} \equiv {\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_A y. e$, which (after we define the reduction relation) will not be convertible with ${\ensuremath{\mathrm{ref} \left( {\lambda x:A.x} \right)}}$. The following lemma shows how substitution and path substitution interact. \[lm:subpathsub\] Let $\vec{y}$ be a sequences of variables and $x$ a distinct variable. Then 1. \[lm:subpathsubi\] $ \begin{aligned}[t] & M [ x:= N ] \{ \vec{y} := \vec{P} : \vec{L} = \vec{L'} \} \\ & \equiv M \{ x := N \{ \vec{y} := \vec{P} : \vec{L} = \vec{L'} \} : N [ \vec{y}:= \vec{L} ] = N [ \vec{y} := \vec{L'} ], \vec{y} := \vec{P} : \vec{L} = \vec{L'} \} \end{aligned} $ 2. \[lm:subpathsubii\] $ \begin{aligned}[t] & M \{ \vec{y} := \vec{P} : \vec{L} = \vec{L'} \} [ x := N ] \\ & \equiv M \{ \vec{y} := \vec{P} [x := N] : \vec{L} [x := N] = \vec{L'} [x := N], x := {\ensuremath{\mathrm{ref} \left( {N} \right)}} : N = N \} \end{aligned} $ An easy induction on $M$ in all cases. The familiar substitution lemma also holds as usual: $t [\vec{z_1} := \vec{s_1}] [\vec{z_2} := \vec{s_2}] \equiv t [\vec{z_1} := \vec{s_1}[\vec{z_2} := \vec{s_2}], \vec{z_2} := \vec{s_2}]$. We cannot form a lemma about the fourth case, simplifying $M \{ \vec{x} := \vec{P} \} \{ \vec{y} := \vec{Q} \}$, because $M \{ \vec{x} := \vec{P} \}$ is a path, and path substitution can only be applied to a term. We introduce a notation for simultaneous substitution and path substitution of several variables: A *substitution* is a function that maps term variables to terms, proof variables to proofs, and path variables to paths. We write $E[\sigma]$ for the result of substituting the expression $\sigma(z)$ for $z$ in $E$, for each variable $z$ in the domain of $\sigma$. A *path substitution* $\tau$ is a function whose domain is a finite set of term variables, and which maps each term variable to a path. Given a path substitution $\tau$ and substitutions $\rho$, $\sigma$ with the same domain $\{ x_1, \ldots, x_n \}$, we write $$M \{ \tau : \rho = \sigma \} \text{ for } M \{ x_1 := \tau(x_1) : \rho(x_1) = \sigma(x_1), \ldots, \tau(x_n) : \rho(x_n) = \sigma(x_n) \} \enspace .$$ ### Call-By-Name Reduction Define the relation of *call-by-name reduction* $\rightarrow$ on the expressions. The inductive definition is given by the rules in Figure \[fig:reduction\] #### Reduction on Terms {#reduction-on-terms .unnumbered} $$\infer{(\lambda x:A.M)N \rightarrow M[x:=N]}{} \quad \infer{MN \rightarrow M'N}{M \rightarrow M'}$$ $$\infer{\varphi \supset \psi \rightarrow \varphi' \supset \psi}{\varphi \rightarrow \varphi'} \quad \infer{\varphi \supset \psi \rightarrow \varphi \supset \psi'}{\psi \rightarrow \psi'}$$ #### Reduction on Proofs {#reduction-on-proofs .unnumbered} $$\infer{(\lambda p : \varphi . \delta)\epsilon \rightarrow \delta [ p := \epsilon ]}{} \quad \infer{{\ensuremath{\mathrm{ref} \left( {\varphi} \right)}}^+ \rightarrow \lambda p : \varphi . p}{} \quad \infer{{\ensuremath{\mathrm{ref} \left( {\varphi} \right)}}^- \rightarrow \lambda p : \varphi . p}{}$$ $$\infer{\delta \epsilon \rightarrow \delta' \epsilon}{\delta \rightarrow \delta'} \quad \infer{{\ensuremath{\mathrm{univ}_{{\varphi}, {\psi}} \left({\delta} , {\epsilon} \right)}}^+ \rightarrow \delta}{} \quad \infer{{\ensuremath{\mathrm{univ}_{{\varphi}, {\psi}} \left({\delta} , {\epsilon} \right)}}^- \rightarrow \epsilon}{}$$ $$\infer{P^+ \rightarrow Q^+}{P \rightarrow Q} \quad \infer{P^- \rightarrow Q^-}{P \rightarrow Q}$$ #### Reduction on Paths {#reduction-on-paths .unnumbered} $$\infer{({\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e:x =_A y.P)_{MN} Q \rightarrow P [x := M, y := N, e := Q]}{}$$ $$\infer{{\ensuremath{\mathrm{ref} \left( {\lambda x:A.M} \right)}}_{N N'} P \rightarrow M \{ x:=P : N = N' \}}{}$$ $$\infer{{\ensuremath{\mathrm{ref} \left( {\varphi} \right)}} \supset^* {\ensuremath{\mathrm{ref} \left( {\psi} \right)}} \rightarrow {\ensuremath{\mathrm{ref} \left( {\varphi \supset \psi} \right)}}}{}$$ $$\infer{{\ensuremath{\mathrm{ref} \left( {\varphi} \right)}} \supset^* {\ensuremath{\mathrm{univ}_{{\psi}, {\chi}} \left({\delta} , {\epsilon} \right)}} \rightarrow {\ensuremath{\mathrm{univ}_{{\varphi \supset \psi}, {\varphi \supset \chi}} \left({\lambda p:\varphi \supset \psi. \lambda q : \varphi. \delta (pq)} , {\lambda p : \varphi \supset \chi. \lambda q : \varphi. \epsilon (pq)} \right)}}}{}$$ $$\infer{{\ensuremath{\mathrm{univ}_{{\varphi}, {\psi}} \left({\delta} , {\epsilon} \right)}} \supset^* {\ensuremath{\mathrm{ref} \left( {\chi} \right)}} \rightarrow {\ensuremath{\mathrm{univ}_{{\varphi \supset \chi}, {\psi \supset \chi}} \left({\lambda p : \varphi \supset \chi. \lambda q : \psi. p (\epsilon q)} , {\lambda p:\psi \supset \chi. \lambda q : \varphi. p(\delta q)} \right)}}}{}$$ $$\infer{\begin{array}{l} {\ensuremath{\mathrm{univ}_{{\varphi}, {\psi}} \left({\delta} , {\epsilon} \right)}} \supset^* {\ensuremath{\mathrm{univ}_{{\varphi'}, {\psi'}} \left({\delta'} , {\epsilon'} \right)}} \\ \rightarrow {\ensuremath{\mathrm{univ}_{{\varphi \supset \varphi'}, {\psi \supset \psi'}} \left({\lambda p : \varphi \supset \varphi'. \lambda q : \psi. \delta' (p(\epsilon q))} , {\lambda p : \psi \supset \psi'. \lambda q : \varphi. \epsilon' (p (\delta q))} \right)}} \end{array}}{}$$ $$\infer{P_{MN} Q \rightarrow P'_{MN} Q}{P \rightarrow P'} \quad \infer{{\ensuremath{\mathrm{ref} \left( {M} \right)}}_{NN'}P \rightarrow {\ensuremath{\mathrm{ref} \left( {M'} \right)}}_{NN'}P}{M \rightarrow M'}$$ $$\infer{P \supset^* Q \rightarrow P' \supset^* Q}{P \rightarrow P'} \quad \infer{P \supset^* Q \rightarrow P \supset^* Q'}{Q \rightarrow Q'}$$ \[lm:diamond\] If $E \twoheadrightarrow F$ and $E \twoheadrightarrow G$, then there exists $H$ such that $F \twoheadrightarrow H$ and $G \twoheadrightarrow H$. The proof is given in Appendix \[section:confluence\]. \[lm:resp-sub\] If $M \rightarrow N$ then $M \{ \tau : \rho = \sigma \} \rightarrow N \{ \tau : \rho = \sigma \}$. Induction on $M \rightarrow N$. The only difficult case is $\beta$-contraction. We have $$\begin{aligned} & ((\lambda x:A.M)N)\{ \tau : \rho = \sigma \} \\ \equiv & ({\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_A x' . M \{ \tau : \rho = \sigma , x := e : x = x' \})_{N [ \rho ] N [ \sigma ]} N \{ \tau : \rho = \sigma \} \\ \rightarrow & M \{ \tau : \rho = \sigma, x := N \{ \tau \} : N [ \rho ] = N [ \sigma ] \} \\ \equiv & M [ x := N ] \{ \tau : \rho = \sigma \} & (\text{Lemma \ref{lm:pathsubsub}})\end{aligned}$$ We write $\rightarrow^?$ for the reflexive closure of $\rightarrow$, we write $\twoheadrightarrow$ for the reflexive transitive closure of $\rightarrow$, and we write $\simeq$ for the reflexive symmetric transitive closure of $\rightarrow$. We say an expression $E$ is in *normal form* iff there is no expression $F$ such that $E \rightarrow F$. $ $ 1. Reduction on proofs and paths does *not* respect substitution. For example, let $M \equiv \lambda x:\Omega.x$. Then we have $$\begin{aligned} {\ensuremath{\mathrm{ref} \left( {\lambda y : \Omega. y'} \right)}}_{\bot \bot} {\ensuremath{\mathrm{ref} \left( {\bot} \right)}} & \rightarrow y' \{ y:= {\ensuremath{\mathrm{ref} \left( {\bot} \right)}} : \bot = \bot \} \equiv {\ensuremath{\mathrm{ref} \left( {y'} \right)}} \nonumber \\ ({\ensuremath{\mathrm{ref} \left( {\lambda y : \Omega.y'} \right)}}_{\bot \bot} {\ensuremath{\mathrm{ref} \left( {\bot} \right)}}) [y' := M] & \equiv {\ensuremath{\mathrm{ref} \left( {\lambda x : \Omega.M} \right)}}_{\bot \bot} {\ensuremath{\mathrm{ref} \left( {\bot} \right)}} \label{eq:exp1} \\ \label{eq:exp2} {\ensuremath{\mathrm{ref} \left( {y'} \right)}}[y':=M] & \equiv {\ensuremath{\mathrm{ref} \left( {M} \right)}} \equiv {\ensuremath{\mathrm{ref} \left( {\lambda x : \Omega.x} \right)}}\end{aligned}$$ Expression (\[eq:exp1\]) does not reduce to (\[eq:exp2\]). Instead, (\[eq:exp1\]) reduces to $$M \{ y := {\ensuremath{\mathrm{ref} \left( {\bot} \right)}} : \bot = \bot \} \equiv {\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_\Omega x'. e \enspace .$$ 2. Reduction on terms does respect substitution: if $M \rightarrow N$ then $M[x:=P] \rightarrow N[x:=P]$, as is easily shown by induction on $M \rightarrow N$. Rules of Deduction ------------------ The rules of deduction of PHOML are given in Figure \[fig:lambdaoe\]. #### Contexts {#contexts .unnumbered} $$(\langle \rangle) \quad \vcenter{\infer{\langle \rangle {\ensuremath{\vdash \mathrm{valid}}}}{}} \qquad (\mathrm{ctx}_T) \quad \vcenter{\infer{\Gamma, x : A {\ensuremath{\vdash \mathrm{valid}}}}{\Gamma {\ensuremath{\vdash \mathrm{valid}}}}} \qquad (\mathrm{ctx}_P) \quad \vcenter{\infer{\Gamma, p : \varphi {\ensuremath{\vdash \mathrm{valid}}}}{\Gamma \vdash \varphi : \Omega}}$$ $$(\mathrm{ctx}_E) \quad \vcenter{\infer{\Gamma, e : M =_A N {\ensuremath{\vdash \mathrm{valid}}}}{\Gamma \vdash M : A \quad \Gamma \vdash N : A}}$$ $$(\mathrm{var}_T) \quad \vcenter{\infer[(x : A \in \Gamma)]{\Gamma \vdash x : A}{\Gamma {\ensuremath{\vdash \mathrm{valid}}}}} \qquad (\mathrm{var}_P) \quad \vcenter{\infer[(p : \varphi \in \Gamma)]{\Gamma \vdash p : \varphi}{\Gamma {\ensuremath{\vdash \mathrm{valid}}}}}$$ $$(\mathrm{var}_E) \quad \vcenter{\infer[(e : M =_A N \in \Gamma)]{\Gamma \vdash e : M =_A N}{\Gamma {\ensuremath{\vdash \mathrm{valid}}}}}$$ #### Terms {#terms .unnumbered} $$(\bot) \quad \vcenter{\infer{\Gamma \vdash \bot : \Omega}{\Gamma {\ensuremath{\vdash \mathrm{valid}}}}} \qquad (\supset) \quad \vcenter{\infer{\Gamma \vdash \varphi \supset \psi : \Omega}{\Gamma \vdash \varphi : \Omega \quad \Gamma \vdash \psi : \Omega}}$$ $$(\mathrm{app}_T) \quad \vcenter{\infer{\Gamma \vdash M N : B} {\Gamma \vdash M : A \rightarrow B \quad \Gamma \vdash N : A}} \qquad (\lambda_T) \quad \vcenter{\infer{\Gamma \vdash \lambda x:A.M : A \rightarrow B}{\Gamma, x : A \vdash M : B}}$$ #### Proofs {#proofs .unnumbered} $$(\mathrm{app}_P) \quad \vcenter{\infer{\Gamma \vdash \delta \epsilon : \psi} {\Gamma \vdash \delta : \varphi \supset \psi \quad \Gamma \vdash \epsilon : \varphi}} \qquad (\lambda_P) \quad \vcenter{\infer{\Gamma \vdash \lambda p : \varphi . \delta : \varphi \supset \psi}{\Gamma, p : \varphi \vdash \delta : \psi}}$$ $$(\mathrm{conv}_P) \quad \vcenter{\infer[(\varphi \simeq \psi)]{\Gamma \vdash \delta : \psi}{\Gamma \vdash \delta : \varphi \quad \Gamma \vdash \psi : \Omega}}$$ #### Paths {#paths .unnumbered} $$(\mathrm{ref}) \quad \vcenter{\infer{\Gamma \vdash {\ensuremath{\mathrm{ref} \left( {M} \right)}} : M =_A M}{\Gamma \vdash M : A}} \qquad (\supset^*) \quad \vcenter{\infer{\Gamma \vdash P \supset^* Q : \varphi \supset \psi =_\Omega \varphi' \supset \psi'}{\Gamma \vdash P : \varphi =_\Omega \varphi' \quad \Gamma \vdash Q : \psi =_\Omega \psi'}}$$ $$(\mathrm{univ}) \quad \vcenter{\infer{\Gamma \vdash {\ensuremath{\mathrm{univ}_{{\varphi}, {\psi}} \left({\delta} , {\epsilon} \right)}} : \varphi =_\Omega \psi}{\Gamma \vdash \delta : \varphi \supset \psi \quad \Gamma \vdash \epsilon : \psi \supset \varphi}}$$ $$(\mathrm{plus}) \quad \vcenter{\infer{\Gamma \vdash P^+ : \varphi \supset \psi}{\Gamma \vdash P : \varphi =_\Omega \psi}} \qquad (\mathrm{minus}) \quad \vcenter{\infer{\Gamma \vdash P^- : \psi \supset \varphi}{\Gamma \vdash P : \psi =_\Omega \psi}}$$ $$({\ensuremath{\lambda \!\! \lambda \!\! \lambda}}) \quad \vcenter{\infer{\Gamma \vdash {\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_A y . P : M =_{A \rightarrow B} N} {\begin{array}{c} \Gamma, x : A, y : A, e : x =_A y \vdash P : M x =_B N y \\ \Gamma \vdash M : A \rightarrow B \quad \Gamma \vdash N : A \rightarrow B \end{array}}}$$ $$(\mathrm{app}_E) \quad \vcenter{\infer{\Gamma \vdash P_{NN'}Q : MN =_B M' N'}{\Gamma \vdash P : M =_{A \rightarrow B} M' \quad \Gamma \vdash Q : N =_A N' \quad \Gamma \vdash N : A \quad \Gamma \vdash N' : A}}$$ $$(\mathrm{conv}_E) \quad \vcenter{\infer[(M \simeq M', N \simeq N')]{\Gamma \vdash P : M' =_A N'}{\Gamma \vdash P : M =_A N \quad \Gamma \vdash M' : A \quad \Gamma \vdash N' : A}}$$ ### Metatheorems {#section:meta} In the lemmas that follow, the letter $\mathcal{J}$ stands for any of the expressions that may occur to the right of the turnstile in a judgement, i.e. $\mathrm{valid}$, $M : A$, $\delta : \varphi$, or $P : M =_A N$. Every derivation of $\Gamma, \Delta \vdash \mathcal{J}$ has a subderivation of $\Gamma {\ensuremath{\vdash \mathrm{valid}}}$. Induction on derivations. If $\Gamma \vdash \mathcal{J}$, $\Gamma \subseteq \Delta$ and $\Delta {\ensuremath{\vdash \mathrm{valid}}}$ then $\Delta \vdash \mathcal{J}$. Induction on derivations. $ $ 1. If $\Gamma \vdash \delta : \varphi$ then $\Gamma \vdash \varphi : \Omega$. 2. If $\Gamma \vdash P : M =_A N$ then $\Gamma \vdash M : A$ and $\Gamma \vdash N : A$. Induction on derivations. The cases where $\delta$ or $P$ is a variable use Context Validity. $ $ 1. If $\Gamma \vdash x : A$ then $x : A \in \Gamma$. 2. If $\Gamma \vdash \bot : A$ then $A \equiv \Omega$. 3. If $\Gamma \vdash \varphi \supset \psi : A$ then $\Gamma \vdash \varphi : \Omega$, $\Gamma \vdash \psi : \Omega$ and $A \equiv \Omega$. 4. If $\Gamma \vdash \lambda x:A.M : B$ then there exists $C$ such that $\Gamma, x : A \vdash M : C$ and $B \equiv A \rightarrow C$. 5. If $\Gamma \vdash MN : A$ then there exists $B$ such that $\Gamma \vdash M : B \rightarrow A$ and $\Gamma \vdash N : B$. 6. If $\Gamma \vdash p : \varphi$, then there exists $\psi$ such that $p : \psi \in \Gamma$ and $\varphi \simeq \psi$. 7. If $\Gamma \vdash \lambda p:\varphi.\delta : \psi$, then there exists $\chi$ such that $\Gamma, p : \varphi \vdash \delta : \chi$ and $\psi \simeq (\varphi \supset \chi)$. 8. If $\Gamma \vdash \delta \epsilon : \varphi$ then there exists $\psi$ such that $\Gamma \vdash \delta : \psi \supset \varphi$ and $\Gamma \vdash \epsilon : \psi$. 9. If $\Gamma \vdash e : M =_A N$, then there exist $M'$, $N'$ such that $e : M' =_A N' \in \Gamma$ and $M \simeq M'$, $N \simeq N'$. 10. If $\Gamma \vdash {\ensuremath{\mathrm{ref} \left( {M} \right)}} : N =_A P$, then we have $\Gamma \vdash M : A$ and $M \simeq N \simeq P$. 11. If $\Gamma \vdash P \supset^* Q : \varphi =_A \psi$, then there exist $\varphi_1$, $\varphi_2$, $\psi_1$, $\psi_2$ such that $\Gamma \vdash P : \varphi_1 =_\Omega \psi_1$, $\Gamma \vdash Q : \varphi_2 =_\Omega \psi_2$, $\varphi \simeq (\varphi_1 \supset \psi_1)$, $\psi \simeq (\varphi_2 \supset \psi_2)$, and $A \equiv \Omega$. 12. If $\Gamma \vdash {\ensuremath{\mathrm{univ}_{{\varphi}, {\psi}} \left({\delta} , {\epsilon} \right)}} : \chi =_A \theta$, then we have $\Gamma \vdash \delta : \varphi \supset \psi$, $\Gamma \vdash \epsilon : \psi \supset \varphi$, $\chi \simeq \varphi$, $\theta \simeq \psi$ and $A \equiv \Omega$. 13. If $\Gamma \vdash {\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_A y. P : M =_B N$ then there exists $C$ such that $\Gamma, x : A, y : A, e : x =_A y \vdash P : M x =_C N y$ and $B \equiv A \rightarrow C$. 14. If $\Gamma \vdash P_{M M'} Q : N =_A N'$, then there exist $B$, $F$ and $G$ such that $\Gamma \vdash P : F =_{B \rightarrow A} G$, $\Gamma \vdash Q : M =_B M'$, $N \simeq F M$ and $N' \simeq G M'$. 15. If $\Gamma \vdash P^+ : \varphi$, then there exist $\psi$, $\chi$ such that $\Gamma \vdash P : \psi =_\Omega \chi$ and $\varphi \simeq (\psi \supset \chi)$. 16. If $\Gamma \vdash P^- : \varphi$, there exist $\psi$, $\chi$ such that $\Gamma \vdash P : \psi =_\Omega \chi$ and $\varphi \simeq (\chi \supset \psi)$. Induction on derivations. ### Substitutions Let $\Gamma$ and $\Delta$ be contexts. A *substitution from $\Delta$ to $\Gamma$*[^1], $\sigma : \Delta \Rightarrow \Gamma$, is a substitution whose domain is ${\ensuremath{\operatorname{dom}}}\Gamma$ such that: - for every term variable $x : A \in \Gamma$, we have $\Delta \vdash \sigma(x) : A$; - for every proof variable $p : \varphi \in \Gamma$, we have $\Delta \vdash \sigma(p) : \varphi [ \sigma ]$; - for every path variable $e : M =_A N \in \Gamma$, we have $\Delta \vdash \sigma(e) : M [ \sigma ] =_A N [ \sigma ]$. If $\Gamma \vdash \mathcal{J}$, $\sigma : \Delta \Rightarrow \Gamma$ and $\Delta {\ensuremath{\vdash \mathrm{valid}}}$, then $\Delta \vdash \mathcal{J} [\sigma]$. Induction on derivations. If $\rho, \sigma : \Delta \Rightarrow \Gamma$ and $\tau$ is a path substitution whose domain is the term variables in ${\ensuremath{\operatorname{dom}}}\Gamma$, then we write $\tau : \sigma = \rho : \Delta \Rightarrow \Gamma$ iff, for each variable $x : A \in \Gamma$, we have $\Delta \vdash \tau(x) : \sigma(x) =_A \rho(x)$. \[lm:pathsub\] If $\tau : \sigma = \rho : \Delta \Rightarrow \Gamma$ and $\Gamma \vdash M : A$ and $\Delta {\ensuremath{\vdash \mathrm{valid}}}$, then $\Delta \vdash M \{ \tau : \sigma = \rho \} : M [ \sigma ] =_A M [ \rho ]$. Induction on derivations. If $\Gamma \vdash s : T$ and $s \twoheadrightarrow t$ then $\Gamma \vdash t : T$. It is sufficient to prove the case $s \rightarrow t$. The proof is by a case analysis on $s \rightarrow t$, using the Generation, Well-Typed Substitution and Path Substitution Lemmas. ### Canonicity $ $ - The *canonical propositions*, are given by the grammar $$\theta ::= \bot \mid \theta \supset \theta$$ - A *canonical proof* is one of the form $\lambda p : \varphi . \delta$. - A *canonical path* is one of the form ${\ensuremath{\mathrm{ref} \left( {M} \right)}}$, ${\ensuremath{\mathrm{univ}_{{\phi}, {\psi}} \left({\delta} , {\epsilon} \right)}}$ or ${\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_A y.P$. \[lm:compat-beta\] Suppose $\varphi$ reduces to a canonical proposition $\theta$, and $\varphi \simeq \psi$. Then $\psi$ reduces to $\theta$. This follows from the fact that $\rightarrow$ satisfies the diamond property, and every canonical proposition $\theta$ is a normal form. ### Neutral Expressions $ $ The *neutral* terms, paths and proofs are given by the grammar $$\begin{array}{lrcl} \text{Neutral term} & M_n & ::= & x \mid M_n N \\ \text{Neutral proof} & \delta_n & ::= & p \mid P_n^+ \mid P_n^- \mid \delta_n \epsilon \\ \text{Neutral path} & P_n & ::= & e \mid P_n \supset^* Q \mid Q \supset^* P_n \mid (P_n)_{MN} Q \end{array}$$ Examples ======== We present two examples illustrating the way that proofs and paths behave in PHOML. In each case, we compare the example with the same construction performed in cubical type theory. Functions Respect Logical Equivalence {#section:exampletwo} ------------------------------------- As discussed in the introduction, every function of type $\Omega \rightarrow \Omega$ that can be constructed in PHOML must respect logical equivalence. This fact can actually be proved in PHOML, in the following sense: there exists a proof $\delta$ of $$f : \Omega \rightarrow \Omega, x : \Omega, y : \Omega, p : x \supset y, q : y \supset x \vdash \delta : f x \supset f y$$ and a proof of $f y \supset f x$ in the same context. Together, these can be read as a proof of ’if $f : \Omega \rightarrow \Omega$ and $x$ and $y$ are logically equivalent, then $fx$ and $fy$ are logically equivalent’. Specifically, take $$\delta {\mathrel{\smash{\stackrel{\text{def}}{=}}}}({\ensuremath{\mathrm{ref} \left( {f} \right)}}_{xy} {\ensuremath{\mathrm{univ}_{{x}, {y}} \left({p} , {q} \right)}})^+ \enspace .$$ Note that this is not possible in Martin-Löf Type Theory. In cubical type theory, we can construct a term $\delta$ such that $$f : {\mathsf{Prop}}\rightarrow {\mathsf{Prop}}, x : {\mathsf{Prop}}, y : {\mathsf{Prop}}, p : x.1 \rightarrow y.1, q : y.1 \rightarrow x.1 \vdash \delta : (f x).1 \rightarrow (f y).1$$ In fact, we can go further and prove that equality of propositions is equal to logical equivalence. That is, we can prove $${\ensuremath{\mathsf{Path} \, {U} \, {({\ensuremath{\mathsf{Path} \, {{\mathsf{Prop}}} \, {x} \, {y}}})} \, {((x.1 \rightarrow y.1) \times (y.1 \rightarrow x.1))}}} \enspace .$$ Computation with Paths ---------------------- Let $\top {\mathrel{\smash{\stackrel{\text{def}}{=}}}}\bot \supset \bot$. Using propositional extensionality, we can construct a path of type $\top = \top \supset \top$, and hence a proof of $\top \supset (\top \supset \top)$. But which of the canonical proofs of $\top \supset (\top \supset \top)$ have we constructed? We define $$\top := \bot \supset \bot, \quad \iota := \lambda p:\bot.p, \quad I := \lambda x:\Omega.x, \quad F := \lambda x:\Omega.\top \supset x, \quad H := \lambda h.h \top \enspace .$$ Let $\Gamma$ be the context $$\Gamma {\mathrel{\smash{\stackrel{\text{def}}{=}}}}x : \Omega, y : \Omega, e : x =_\Omega y \enspace .$$ Then we have $$\begin{aligned} \Gamma & \vdash \lambda p:\top \supset x. e^+ (p \iota) & : & (\top \supset x) \supset y \\ \Gamma & \vdash \lambda m : y. \lambda n : \top. e^- m & : & y \supset (\top \supset x) \\ \Gamma & \vdash {\ensuremath{\mathrm{univ}_{{*}, {\lambda p:\top \supset x. e^- m}} \left({\lambda m:y. \lambda n:\top. e^- m} , {&} \right)}} : & (\top \supset x) =_\Omega y\end{aligned}$$ Let $P \equiv \univ*{\lambda p:\top \supset x. e^+ (p \iota)}{\lambda m:y. \lambda n:\top. e^- m}$. Then $$\begin{aligned} \therefore & \vdash {\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e:x =_\Omega y. P & : & F =_{\Omega \rightarrow \Omega} I \label{eq:llleP} \\ \therefore & \vdash ({\ensuremath{\mathrm{ref} \left( {H} \right)}})_{FI}({\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e:x =_\Omega y. P) & : & (\top \supset \top) =_\Omega \top \label{eq:llleP2} \\ \therefore & \vdash (({\ensuremath{\mathrm{ref} \left( {H} \right)}})_{FI}({\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e:x =_\Omega y.P))^- & : & \top \supset (\top \supset \top) \label{eq:llleP3}\end{aligned}$$ And now we compute: $$\begin{aligned} & (({\ensuremath{\mathrm{ref} \left( {H} \right)}})_{FI}({\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e:x =_\Omega y.P))^- \\ \twoheadrightarrow & ((h \top) \{ h := {\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_\Omega y.P : F = I \})^- \\ \equiv & (({\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_\Omega y.P)_{\top \top} ({\ensuremath{\mathrm{ref} \left( {\top} \right)}}))^- \\ \rightarrow & (P [ x := \top, y := \top, e := {\ensuremath{\mathrm{ref} \left( {\top} \right)}} ])^- \\ \equiv & {\ensuremath{\mathrm{univ}_{{*}, {\lambda p : \top \supset \top. {\ensuremath{\mathrm{ref} \left( {\top} \right)}}^+ (p \iota)}} \left({\lambda m:\top. \lambda n:\top. {\ensuremath{\mathrm{ref} \left( {\top} \right)}}^- m} , {^} \right)}}- \\ \rightarrow & \lambda m : \top. \lambda n : \top. {\ensuremath{\mathrm{ref} \left( {\top} \right)}}^- m\end{aligned}$$ Therefore, given proofs $\delta, \epsilon : \top$, we have $$(({\ensuremath{\mathrm{ref} \left( {H} \right)}})_{FI}({\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e:x =_\Omega y.P))^- \delta \epsilon \twoheadrightarrow \delta \enspace .$$ Thus, the construction gives a proof of $\top \supset (\top \supset \top)$ which, given two proofs of $\top$, selects the first. We could have anticipated this: consider the context $\Delta {\mathrel{\smash{\stackrel{\text{def}}{=}}}}X : \Omega, Y : \Omega, p : X$. By replacing in our example some occurrences of $\top$ with $X$ and others with $Y$, and replacing $\iota$ with $p$, we can obtain a path $$Y =_\Omega X \supset Y$$ and hence a proof of $Y \supset (X \supset Y)$. By parametricity, any proof that we can construct in the context $\Delta$ of this proposition must return the left input. ### Comparison with Cubical Type Theory {#section:cubical} In cubical type theory, we say that a type $A$ is a *proposition* iff any two terms of type $A$ are propositionally equal; that is, there exists a path between any two terms of type $A$. Let $${\mathsf{isProp} \left( {A} \right)} {\mathrel{\smash{\stackrel{\text{def}}{=}}}}\Pi x,y:A. {\ensuremath{\mathsf{Path} \, {A} \, {x} \, {y}}}$$ and let ${\mathsf{Prop}}$ be the type of all types in $U$ that are propositions: $${\mathsf{Prop}}{\mathrel{\smash{\stackrel{\text{def}}{=}}}}\Sigma X : U. {\mathsf{isProp} \left( {X} \right)} \enspace .$$ Let $\bot$ be any type in the universe $U$ that is a proposition; that is, there exists a term of type ${\mathsf{isProp} \left( {\bot} \right)}$. ($\bot$ may be the empty type, but we do not require this in what follows.) Define $$\top := \bot \rightarrow \bot$$ Then there exists a term $\top_{\mathsf{Prop}}$ of type ${\mathsf{isProp} \left( {\top} \right)}$ (we omit the details). Define $$I := \lambda X:{\mathsf{Prop}}.X.1, \quad F := \lambda X : {\mathsf{Prop}}.\top \rightarrow X.1, \quad H := \lambda h.h (\top , \top_{\mathsf{Prop}})$$ Then we have $$\vdash \top : U \quad \vdash I : {\mathsf{Prop}}\rightarrow U \quad \vdash F : {\mathsf{Prop}}\rightarrow U \quad \vdash H : ({\mathsf{Prop}}\rightarrow U) \rightarrow U$$ From the fact that univalence is provable in cubical type theory [@cchm:cubical], we can construct a term $Q$ such that $$\vdash Q : {\ensuremath{\mathsf{Path} \, {({\mathsf{Prop}}\rightarrow U)} \, {I} \, {F}}} \enspace .$$ Hence we have $$\vdash \langle i \rangle H (Q i) : {\ensuremath{\mathsf{Path} \, {U} \, {HI} \, {HF}}}$$ which is definitionally equal to $$\vdash \langle i \rangle H (Q i) : {\ensuremath{\mathsf{Path} \, {U} \, {\top \rightarrow \top} \, {\top}}}$$ From this, we can apply transport to create a term $Q' : \top \rightarrow \top \rightarrow \top$. Applying this to any terms $\delta, \epsilon : \top$ gives a term that is definitionally equal to $$Q' \delta \epsilon = {\mathsf{mapid}_{\top} \, {{\mathsf{mapid}_{\top} \, {\delta}}}}$$ where ${\mathsf{mapid}_{} \, {}}$ represents transport across the trivial path: $${\mathsf{mapid}_{A} \, {t}} {\mathrel{\smash{\stackrel{\text{def}}{=}}}}{\mathsf{comp}}^i \, A \, [] \, t \qquad (i \text{ does not occur in } A) \enspace .$$ (For the details of the calculation, see Appendix \[appendix:cubical\].) In the version of cubical type theory given in [@bch:cubical], we have ${\mathsf{mapid}_{X} \, {x}}$ is definitionally equal to $x$, and therefore $Q' \delta \epsilon = \delta$, just as in PHOML. This is no longer true in the version of cubical type theory given in [@cchm:cubical]. Computable Expressions {#section:computable} ====================== We now proceed with the proof of canonicity for PHOML. Our proof follows the lines of the Girard-Tait reducibility method [@tait1967]: we define what it means to be a *computable* term (proof, path) of a given type (proposition, equation), and prove: (1) every typable expression is computable (2) every computable expression reduces to either a neutral or a canonical expression. In particular, every closed computable expression reduces to a canonical expression. In this section, we use $E$, $F$, $S$ and $T$ as metavariables that range over expressions. In each case, either $E$ and $F$ are terms and $S$ and $T$ are types; or $E$ and $F$ are proofs and $S$ and $T$ are propositions; or $E$ and $F$ are paths and $S$ and $T$ are equations. We define the relation $\models E : T$, read ‘$E$ is a computable expression of type $T$’, as follows. - $\models \delta : \bot$ iff $\delta$ reduces to a neutral proof. - For $\theta$ and $\theta'$ canonical propositions, $\models \delta : \theta \supset \theta'$ iff, for all $\epsilon$ such that $\models \epsilon : \theta$, we have $\models \delta \epsilon : \theta'$. - If $\varphi$ reduces to the canonical proposition $\theta$, then $\models \delta : \varphi$ iff $\models \delta : \theta$. - $\models P : \varphi =_\Omega \psi$ iff $\models P^+ : \varphi \supset \psi$ and $\models P^- : \psi \supset \varphi$. - $\models P : M =_{A \rightarrow B} M'$ iff, for all $Q$, $N$, $N'$ such that $\models N : A$ and $\models N' : A$ and $\models Q : N =_A N'$, then we have $\models P_{NN'}Q : MN =_B M'N'$. - $\models M : A$ iff $\models M \{\} : M =_A M$. Note that the last three clauses define $\models M : A$ and $\models P : M =_A N$ simultaneously by recursion on $A$. Let $\sigma$ be a substitution with domain ${\ensuremath{\operatorname{dom}}}\Gamma$. We write $\models \sigma : \Gamma$ and say that $\sigma$ is a *computable* substitution on $\Gamma$ iff, for every entry $z : T$ in $\Gamma$, we have $\models \sigma(z) : T [ \sigma ]$. We write $\models \tau : \rho = \sigma : \Gamma $, and say $\tau$ is a *computable* path substitution between $\rho$ and $\sigma$, iff, for every term variable entry $x : A$ in $\Gamma$, we have $\models \tau(x) : \rho(x) =_A \sigma(x)$. \[lm:conv-compute\] If $\models E : S$ and $S \simeq T$ then $\models E : T$. This follows easily from the definition and Lemma \[lm:compat-beta\]. \[lm:expansion\] If $\models F : T$ and $E \rightarrow F$ then $\models E : T$. An easy induction, using the fact that call-by-name reduction respects path substitution (Lemma \[lm:resp-sub\]). \[lm:reduction\] If $\models E : T$ and $E \rightarrow F$ then $\models F : T$. An easy induction, using the fact that call-by-name reduction is confluent (Lemma \[lm:diamond\]). We introduce a closed term $c_A$ for every type $A$ such that $\models c_A : A$. $$\begin{aligned} c_\Omega & {\mathrel{\smash{\stackrel{\text{def}}{=}}}}\bot \\ c_{A \rightarrow B} & {\mathrel{\smash{\stackrel{\text{def}}{=}}}}\lambda x:A.c_B\end{aligned}$$ $\models c_A : A$ An easy induction on $A$. \[lm:neutral-canon\] $ $ 1. If $\models \delta : \phi$ then $\delta$ reduces to either a neutral proof or canonical proof. 2. If $\models P : M =_A N$ then $P$ reduces either to a neutral path or canonical path. 3. If $\models M : A$ then $M$ reduces either to a canonical proposition or a $\lambda$-term. We prove by induction on the canonical proposition $\theta$ that, if $\models \delta : \theta$, then $\delta$ reduces to a neutral proof or a canonical proof of $\theta$. If $\models \delta : \bot$ then $\delta$ reduces to a neutral proof. Now, suppose $\models \delta : \theta \supset \theta'$. Then $\models \delta p : \theta'$, so $\delta p$ reduces to either a neutral proof or canonical proof by the induction hypothesis. This reduction must proceed either by reducing $\delta$ to a neutral proof, or reducing $\delta$ to a $\lambda$-proof then $\beta$-reducing. We then prove by induction on the type $A$ that, if $\models P : M =_A N$, then $P$ reduces to a neutral path or a canonical path. The two cases are straightforward. Now, suppose $\models M : A$, i.e. $\models M \{\} : M =_A M$. Let $A \equiv A_1 \rightarrow \cdots \rightarrow A_n \rightarrow \Omega$. Then $$\models M \{\}_{c_{A_1} c_{A_1}} c_{A_1} \{\}_{c_{A_2} c_{A_2}} \cdots c_{A_n} \{\} : M c_{A_1} \cdots c_{A_n} =_\Omega M c_{A_1} \cdots c_{A_n} \enspace .$$ Therefore, $M c_{A_1} \cdots c_{A_n}$ reduces to a canonical proposition. The reduction must consist either in reducing $M$ to a canonical proposition (if $n = 0$), or reducing $M$ to a $\lambda$-expression then performing a $\beta$-reduction. \[lm:pre-ref-compute\] If $\models M : A \rightarrow B$ then $M$ reduces to a $\lambda$-expression. Similar to the last paragraph of the previous proof. \[lm:ref-compute-Omega\] For any term $\varphi$ that reduces to a canonical proposition, we have $\models {\ensuremath{\mathrm{ref} \left( {\varphi} \right)}} : \varphi =_\Omega \varphi$. In fact we prove that, for any terms $M$ and $\varphi$ such that $\varphi$ reduces to a canonical proposition, we have $\models {\ensuremath{\mathrm{ref} \left( {M} \right)}} : \varphi =_\Omega \varphi$. It is sufficient to prove the case where $\varphi$ is a canonical proposition. We must show that $\models {\ensuremath{\mathrm{ref} \left( {M} \right)}}^+ : \varphi \supset \varphi$ and $\models {\ensuremath{\mathrm{ref} \left( {M} \right)}}^- : \varphi \supset \varphi$. So let $\models \delta : \varphi$. Then $\models {\ensuremath{\mathrm{ref} \left( {M} \right)}}^+ \delta : \varphi$ and $\models {\ensuremath{\mathrm{ref} \left( {M} \right)}}^- \delta : \varphi$ by Expansion (Lemma \[lm:expansion\]), as required. \[lm:models-canon\] $\models \varphi : \Omega$ if and only if $\varphi$ reduces to a canonical proposition. If $\models \varphi : \Omega$ then $\models \varphi \{\}^+ : \varphi \supset \varphi$. Therefore $\varphi \supset \varphi$ reduces to a canonical proposition, and so $\varphi$ must reduce to a canonical proposition. Conversely, suppose $\varphi$ reduces to a canonical proposition $\theta$. We have $\varphi \{\} \twoheadrightarrow \theta \{\}$, and $\theta \{\} \twoheadrightarrow {\ensuremath{\mathrm{ref} \left( {\theta} \right)}}$ for every canonical proposition $\theta$. Therefore, $\models \varphi \{\} : \varphi =_\Omega \varphi$ by Expansion (Lemma \[lm:expansion\]). Hence $\models \varphi : \Omega$. \[lm:neutral-proof\] If $\delta$ is a neutral proof and $\varphi$ reduces to a canonical proposition, then $\models \delta : \varphi$. It is sufficient to prove the case where $\varphi$ is a canonical proposition. The proof is by induction on $\varphi$. If $\varphi \equiv \bot$, then $\models \delta : \bot$ immediately from the definition. If $\varphi \equiv \psi \supset \chi$, then let $\models \epsilon : \psi$. We have that $\delta \epsilon$ is neutral, hence $\models \delta \epsilon : \chi$ by the induction hypothesis. \[lm:neutral-path\] Let $\models M : A$ and $\models N : A$. If $P$ is a neutral path, then $\models P : M =_A N$. The proof is by induction on $A$. For $A \equiv \Omega$: we have that $P^+$ and $P^-$ are neutral proofs, and $M$ and $N$ reduce to canonical propositions (by Lemma \[lm:models-canon\]), so $\models P^+ : M \supset N$ and $\models P^- : N \supset M$ by Lemma \[lm:neutral-proof\], as required. For $A \equiv B \rightarrow C$: let $\models L : B$, $\models L' : B$ and $\models Q : L =_B L'$. Then we have $\models ML : C$, $\models NL' : C$ and $P_{LL'} Q$ is a neutral path, hence $\models P_{L L'} Q : ML =_C NL'$ by the induction hypothesis, as required. \[lm:ref-compute\] If $\models M : A$ then $\models {\ensuremath{\mathrm{ref} \left( {M} \right)}} : M =_A M$. If $A \equiv \Omega$, this is just Lemma \[lm:ref-compute-Omega\]. So suppose $A \equiv B \rightarrow C$. Using Lemma \[lm:pre-ref-compute\], Reduction (Lemma \[lm:reduction\]) and Expansion (Lemma \[lm:expansion\]), we may assume that $M$ is a $\lambda$-term. Let $M \equiv \lambda y:D.N$. Let $\models L : B$ and $\models L' : B$ and $\models P : L =_B L'$. We must show that $$\models {\ensuremath{\mathrm{ref} \left( {\lambda y:D.N} \right)}}_{L L'} P : (\lambda y:D.N)L =_C (\lambda y:D.N)L' \enspace .$$ By Expansion and Conversion, it is sufficient to prove $$\models N \{ y := P : L = L' \} : N [ y:= L ] =_C N [y := L'] \enspace .$$ We have that $\models (\lambda y:D.N)\{\} : \lambda y:D.N =_{B \rightarrow C} \lambda y:D.N$, and so $$\models ({\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : y =_D y' . N \{ y := e : y = y' \})_{L L'} P : (\lambda y:D.N)L =_C (\lambda y:D.N)L' \enspace ,$$ and the result follows by Reduction and Conversion. \[lm:compute-supset\*\] If $\models P : \varphi =_\Omega \varphi'$ and $\models Q : \psi =_\Omega \psi'$ then $\models P \supset^* Q : \varphi \supset \psi =_\Omega \varphi' \supset \psi'$. By Reduction (Lemma \[lm:reduction\]) and Expansion (Lemma \[lm:expansion\]), we may assume that $P$ and $Q$ are either neutral, or have the form ${\ensuremath{\mathrm{ref} \left( {-} \right)}}$ or ${\ensuremath{\mathrm{univ}_{{-}, {-}} \left({-} , {-} \right)}}$ or ${\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_A y.-$. We cannot have that $P$ reduces to a ${\ensuremath{\lambda \!\! \lambda \!\! \lambda}}$-path; for let $\varphi'$ reduce to the canonical proposition $\theta_1 \supset \cdots \supset \theta_n \supset \bot$. Then we have $$\models P^+ p q_1 \cdots q_n : \bot$$ and so $P^+ p q_1 \cdots q_n$ must reduce to a neutral path. Similarly, $Q$ cannot reduce to a ${\ensuremath{\lambda \!\! \lambda \!\! \lambda}}$-path. If either $P$ or $Q$ is neutral then $P \supset^* Q$ is neutral, and the result follows from Lemma \[lm:neutral-path\]. Otherwise, let $\models \delta : \varphi \supset \psi$ and $\epsilon \models \varphi'$. We must show that $\models (P \supset^* Q)^+ \delta \epsilon : \psi'$. If $P \equiv {\ensuremath{\mathrm{ref} \left( {M} \right)}}$ and $Q \equiv {\ensuremath{\mathrm{ref} \left( {N} \right)}}$, then we have $$(P \supset^* Q)^+ \delta \epsilon \rightarrow {\ensuremath{\mathrm{ref} \left( {M \supset N} \right)}}^+ \delta \epsilon \rightarrow \delta \epsilon \enspace .$$ Now, $\models P^- \epsilon : \varphi$, hence $\models \epsilon : \varphi$ by Reduction, and so $\models \delta \epsilon : \psi$. Therefore, $\models Q^+ (\delta \epsilon) : \psi'$, and hence by Reduction $\models \delta \epsilon : \psi'$ as required. If $P \equiv {\ensuremath{\mathrm{ref} \left( {M} \right)}}$ and $Q \equiv {\ensuremath{\mathrm{univ}_{{N}, {N'}} \left({\chi} , {\chi'} \right)}}$, then we have $$\begin{aligned} (P \supset^* Q)^+ \delta \epsilon & \rightarrow {\ensuremath{\mathrm{univ}_{{M \supset N}, {M \supset N'}} \left({\lambda pq.\chi(pq)} , {\lambda pq.\chi'(pq)} \right)}}^+ \delta \epsilon \\ & \rightarrow (\lambda pq.\chi(pq)) \delta \epsilon \\ & \twoheadrightarrow \chi (\delta \epsilon)\end{aligned}$$ We have $\models P^- \epsilon : \varphi$, hence $\models \epsilon : \varphi$ by Reduction, and so $\models \delta \epsilon : \psi$. Therefore, $\models Q^+ (\delta \epsilon) : \psi'$, and hence by Reduction $\models \chi (\delta \epsilon) : \psi'$ as required. The other two cases are similar. \[lm:univ-compute\] If $\models \delta : \phi \supset \psi$ and $\models \epsilon : \psi \supset \phi$ then $\models {\ensuremath{\mathrm{univ}_{{\phi}, {\psi}} \left({\delta} , {\epsilon} \right)}} : \phi =_\Omega \psi$. We must show that $\models {\ensuremath{\mathrm{univ}_{{\phi}, {\psi}} \left({\delta} , {\epsilon} \right)}}^+ : \phi \supset \psi$ and $\models {\ensuremath{\mathrm{univ}_{{\phi}, {\psi}} \left({\delta} , {\epsilon} \right)}}^- : \psi \supset \phi$. These follow from the hypotheses, using Expansion (Lemma \[lm:expansion\]). Proof of Canonicity =================== $ $ 1. If $\Gamma \vdash \mathcal{J}$ and $\models \sigma : \Gamma$, then $\models \mathcal{J} [ \sigma ]$. 2. If $\Gamma \vdash M : A$ and $\models \tau : \rho = \sigma : \Gamma$, then $\models M \{ \tau : \rho = \sigma \} : M [ \rho ] =_A M [ \sigma ]$. The proof is by induction on derivations. Most cases are straightforward, using the lemmas from Section \[section:computable\]. We deal with one case here, the rule ($\lambda_T$). $$\infer{\Gamma \vdash \lambda x:A.M : A \rightarrow B}{\Gamma, x : A \vdash M : B}$$ 1. We must show that $$\models \lambda x:A.M[\sigma] : A \rightarrow B \enspace .$$ So let $\models Q : N =_A N'$. Define the path substitution $\tau$ by $$\tau(x) \equiv Q, \qquad \tau(y) \equiv {\ensuremath{\mathrm{ref} \left( {\sigma(y)} \right)}} \ (y \in {\ensuremath{\operatorname{dom}}}\Gamma)$$ Then we have $\models \tau : (\sigma, x:=N) = (\sigma, x:=N') : \Gamma , x : A$, and so the induction hypothesis gives $$\models M \{ \tau \} : M[\sigma, x:=N] =_B M [ \sigma, x:= N' ]$$ We observe that $M \{ \tau \} \equiv M [ \sigma ] \{ x:=Q:N=N' \}$ (Lemma \[lm:pathsubsub\]), and so by Expansion (Lemma \[lm:expansion\]) and Conversion (Lemma \[lm:conv-compute\]) we have $$\models (\lambda x:A.M[\sigma])\{\}_{N N'} Q : (\lambda x:A.M[\sigma])N =_B (\lambda x:A.M[\sigma])N'$$ as required. 2. We must show that $$\models {\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_A y. M \{ \tau : \rho = \sigma, x := e : x = y \} : \lambda x:A.M [ \rho ] =_{A \rightarrow B} \lambda x:A.M [ \sigma ] \enspace .$$ So let $ \models P : N =_A N'$. The induction hypothesis gives $$\models M \{ \tau : \rho = \sigma, x := P : N = N' \} : M [\rho, x := N] =_B M [\sigma, x := N'] \enspace ,$$ and so we have $$\begin{aligned} \models & ({\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_A y. M \{ \tau : \rho = \sigma, x := e : x = y \})_{N N'} P \\ : & (\lambda x:A.M [ \rho ])N =_B (\lambda x:A.M [ \sigma ])N'\end{aligned}$$ by Expansion and Conversion, as required. Let $\Gamma$ be a context in which no term variables occur. 1. If $\Gamma \vdash \delta : \phi$ then $\delta$ reduces to a neutral proof or canonical proof. 2. If $\Gamma \vdash P : M =_A N$ then $P$ reduces to a neutral path or canonical path. Let ${\mathsf{id}}$ be the substitution $\Gamma \Rightarrow \Gamma$ such that ${\mathsf{id}}(x) {\mathrel{\smash{\stackrel{\text{def}}{=}}}}x$. If $\Gamma {\ensuremath{\vdash \mathrm{valid}}}$ then $\models {\mathsf{id}}: \Gamma$ using Lemmas \[lm:neutral-proof\] and \[lm:neutral-path\]. Therefore, if $\Gamma \vdash E : T$ then $\models E [ {\mathsf{id}}] : T [ {\mathsf{id}}]$, that is, $\models E : T$. Hence $E$ reduces to a neutral expression or canonical expression. Let $\Gamma$ be a context with no term variables. 1. If $\Gamma \vdash \delta : \bot$ then $\delta$ reduces to a neutral proof. 2. If $\Gamma \vdash \delta : \phi \supset \psi$ then $\delta$ reduces either to a neutral proof, or a proof $\lambda p : \phi' . \epsilon$ where $\phi \simeq \phi'$ and $\Gamma, p : \phi \vdash \epsilon : \psi$. 3. If $\Gamma \vdash P : \phi =_\Omega \psi$ then $P$ reduces either to a neutral path; or to ${\ensuremath{\mathrm{ref} \left( {\chi} \right)}}$ where $\phi \simeq \psi \simeq \chi$; or to ${\ensuremath{\mathrm{univ}_{{\phi'}, {\psi'}} \left({\delta} , {\epsilon} \right)}}$ where $\phi \simeq \phi'$, $\psi \simeq \psi'$, $\Gamma \vdash \delta : \phi \supset \psi$ and $\Gamma \vdash \epsilon : \psi \supset \phi$. 4. If $\Gamma \vdash P : M =_{A \rightarrow B} M'$ then $P$ reduces either to a neutral path; or to ${\ensuremath{\mathrm{ref} \left( {N} \right)}}$ where $M \simeq M' \simeq N$; or to ${\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_A y. Q$ where $\Gamma, x : A, y : A, e : x =_A y \vdash Q : Mx =_B M'y$. A closed expression cannot be neutral, so from the previous corollary every typed closed expression must reduce to a canonical expression. We now apply case analysis to the possible forms of canonical expression, and use the Generation Lemma. There is no $\delta$ such that $\vdash \delta : \bot$. We have not proved canonicity for terms. However, we can observe that PHOML restricted to terms and types is just the simply-typed lambda calculus with one atomic type $\Omega$ and two constants $\bot$ and $\supset$; and our reduction relation restricted to this fragment is head reduction. Canonicity for this system is already a well-known result (see e.g. [@Girard1989 Ch. 4]). Conclusion and Future Work ========================== We have presented a system with propositional extensionality, and shown that it satisfies the property of canonicity. This gives hope that it will be possible to find a computation rule for homotopy type theory that satisfies canonicity, and that does not involve extending the type theory, either with a nominal extension of the syntax as in cubical type theory or otherwise. We now intend to do the same for stronger and stronger systems, getting ever closer to full homotopy type theory. The next steps will be: - a system where the equations $M =_A N$ are objects of $\Omega$, allowing us to form propositions such as $M =_A N \supset N =_A M$. - a system with universal quantification over the types $A$, allowing us to form propositions such as $\forall x:A. x =_A x$ and $\forall x,y : A. x =_A y \supset y =_A x$ Ultimately, we hope to approach full homotopy type theory. The study of how the reduction relation and its properties change as we move up and down this hierarchy of systems should reveal facts about computing with univalence that might be lost when working in a more complex system such as homotopy type theory or cubical type theory. Calculation in Cubical Type Theory {#appendix:cubical} ================================== We can prove that, if $X$ is a proposition, then the type $\Sigma f:\top \rightarrow X. Path \, X \, x \, (f I)$ is contractible (we omit the details). Let $e[X, x, p]$ be the term such that $$\begin{aligned} & X : {\mathsf{Prop}}, x : X.1, p : \Sigma f:\top \rightarrow X.1. {\ensuremath{\mathsf{Path} \, {X.1} \, {x} \, {(fI)}}} \\ \vdash & e[X, x, p] : {\ensuremath{\mathsf{Path} \, {(\Sigma f:\top \rightarrow X.1. {\ensuremath{\mathsf{Path} \, {X.1} \, {x} \, {(f I)}}})} \, {\langle \lambda t : \top . x, 1_{X.1} \rangle} \, {p}}}\end{aligned}$$ Let ${\mathsf{step}_2}[X,x] {\mathrel{\smash{\stackrel{\text{def}}{=}}}}\langle \langle \lambda t : T.x, 1_{X.1} \rangle, \lambda p : \Sigma f : T \rightarrow X.1. {\ensuremath{\mathsf{Path} \, {X.1} \, {x} \, {(fI)}}}. e[X, x, p] \rangle$. Then $$X : {\mathsf{Prop}}, x : X.1 \vdash {\mathsf{step}_2}[X, x] : \mathsf{isContr}(\Sigma f:T \rightarrow X.1. {\ensuremath{\mathsf{Path} \, {X.1} \, {x} \, {(fI)}}}) \enspace .$$ Let ${\mathsf{step}_3}[X] \equiv \lambda x : X.1. {\mathsf{step}_2}[X, x]$. Then $$X : {\mathsf{Prop}}\vdash {\mathsf{step}_3}[X] : \mathsf{isEquiv} \, (T \rightarrow X.1) \, X.1 \, (\lambda f : T \rightarrow X.1. f I) \enspace .$$ Let $E[X] \equiv \langle \lambda f : \top \rightarrow X.1. f I, step3[X] \rangle$. Then $$X : {\mathsf{Prop}}\vdash E[X] : \mathsf{Equiv} \, (\top \rightarrow X.1) \, X.1$$ From this equivalence, we want to get a path from $\top \rightarrow X.1$ to $X.1$ in $U$. We apply the proof of univalence in [@cchm:cubical] Let $P[X] \equiv \langle i \rangle \mathsf{Glue} [(i = 0) \mapsto (\top \rightarrow X.1, E[X]), (i = 1) \mapsto (X.1, equiv^k X.1)] X.1$. Then $$X : {\mathsf{Prop}}\vdash P[X] : {\ensuremath{\mathsf{Path} \, {U} \, {(\top \rightarrow X.1)} \, {X.1}}}$$ Let $Q \equiv \langle i \rangle \lambda x : {\mathsf{Prop}}. P[X] i$. Then $$\vdash Q : {\ensuremath{\mathsf{Path} \, {({\mathsf{Prop}}\rightarrow U)} \, {F} \, {I}}}$$ This is the term in cubical type theory that corresponds to ${\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e : x =_\Omega y.P$ in PHOML (formula \[eq:llleP\]). We now construct terms corresponding to formulas (\[eq:llleP2\]) and (\[eq:llleP3\]): $$\vdash \langle i \rangle H (Q i) : {\ensuremath{\mathsf{Path} \, {U} \, {(\top \rightarrow \top)} \, {\top}}}$$ $$\vdash \lambda x : \top. {\mathsf{comp}}^i (H (Q (1 - i))) [] x : \top \rightarrow \top \rightarrow \top$$ Let us write ${\mathsf{output}}$ for this term: $${\mathsf{output}}{\mathrel{\smash{\stackrel{\text{def}}{=}}}}\lambda x : \top. {\mathsf{comp}}^i (H (Q (1 - i))) [] x \enspace .$$ And we calculate (using the notation from [@cchm:cubical] section 6.2): $$\begin{aligned} & \quad {\mathsf{output}}\\ & = \lambda x : \top. {\mathsf{comp}}^i (Q (1 - i) \top) [] x \\ & = \lambda x : \top. {\mathsf{comp}}^i (P[\top] (1 - i)) [] x \\ & = \lambda x : \top. {\mathsf{comp}}^i (\mathsf{Glue}[(i = 1) \mapsto (\top \rightarrow \top, E[\top]), (i = 0) \mapsto (\top, \mathsf{equiv}^k \top)] \top) [] x \\ & = \lambda x : \top. \mathsf{glue} [ 1_\mathbb{F} \mapsto t_1 ] a_1 \\ & = \lambda x : \top. t_1 \\ & = \lambda x : \top. (\mathsf{equiv} \, E[\top] \, [] \, {\mathsf{mapid}_{\top} \, {x}}).1 \\ & = \lambda x : \top. (\mathsf{contr} (step2[\top, {\mathsf{mapid}_{\top} \, {x}}]) []).1 \\ & = \lambda x : \top. ({\mathsf{comp}}^i \\ & \qquad (\Sigma f : \top \rightarrow \top. {\ensuremath{\mathsf{Path} \, {\top} \, {({\mathsf{mapid}_{\top} \, {x}})} \, {(fI)}}}) \\ & \qquad [] \\ & \qquad \langle \lambda t : \top. {\mathsf{mapid}_{\top} \, {x}}, 1_{mapid_\top(x)} \rangle).1 \\ & = \lambda x : \top. {\mathsf{mapid}_{\top \rightarrow \top} \, {(\lambda y : \top. {\mathsf{mapid}_{\top} \, {x}})}}\end{aligned}$$ Therefore, $$\begin{aligned} & {\mathsf{output}}\, m \, n \\ & = {\mathsf{mapid}_{\top \rightarrow \top} \, {(\lambda y : \top. {\mathsf{mapid}_{\top} \, {m}})}} n \\ & \equiv ({\mathsf{comp}}^i (\top \rightarrow \top) [] (\lambda _ : \top. {\mathsf{mapid}_{\top} \, {m}})) n \\ & = {\mathsf{mapid}_{\top} \, {{\mathsf{mapid}_{\top} \, {m}}}}\end{aligned}$$ Proof of Confluence {#section:confluence} =================== The proof follows the same lines as the proof given in [@luo:car]. #### Reflexivity {#reflexivity .unnumbered} $$\infer{E \rhd E}{}$$ #### Reduction on Terms {#reduction-on-terms-1 .unnumbered} $$\infer{(\lambda x:A.M)N \rhd M[x:=N]}{} \quad \infer{MN \rhd M'N}{M \rhd M'} \quad \infer{\varphi \supset \psi \rhd \varphi' \supset \psi'}{\varphi \rhd \varphi' \quad \psi \rhd \psi'}$$ #### Reduction on Proofs {#reduction-on-proofs-1 .unnumbered} $$\infer{(\lambda p : \varphi . \delta)\epsilon \rhd \delta [ p := \epsilon ]}{} \quad \infer{{\ensuremath{\mathrm{ref} \left( {\varphi} \right)}}^+ \rhd \lambda p : \varphi . p}{} \quad \infer{{\ensuremath{\mathrm{ref} \left( {\varphi} \right)}}^- \rhd \lambda p : \varphi . p}{}$$ $$\infer{{\ensuremath{\mathrm{univ}_{{\varphi}, {\psi}} \left({\delta} , {\epsilon} \right)}}^+ \rhd \delta}{} \quad \infer{{\ensuremath{\mathrm{univ}_{{\varphi}, {\psi}} \left({\delta} , {\epsilon} \right)}}^- \rhd \epsilon}{}$$ $$\infer{\delta \epsilon \rhd \delta' \epsilon}{\delta \rhd \delta'} \quad \infer{P^+ \rhd Q^+}{P \rhd Q} \quad \infer{P^- \rhd Q^-}{P \rhd Q}$$ #### Reduction on Paths {#reduction-on-paths-1 .unnumbered} $$\infer{({\ensuremath{\lambda \!\! \lambda \!\! \lambda}}e:x =_A y.P)_{MN} Q \rhd P [x := M, y := N, e := Q]}{}$$ $$\infer{{\ensuremath{\mathrm{ref} \left( {\lambda x:A.M} \right)}}_{N N'} P \rhd M \{ x:=P : N = N' \}}{}$$ $$\infer{{\ensuremath{\mathrm{ref} \left( {\varphi} \right)}} \supset^* {\ensuremath{\mathrm{ref} \left( {\psi} \right)}} \rhd {\ensuremath{\mathrm{ref} \left( {\varphi \supset \psi} \right)}}}{}$$ $$\infer{{\ensuremath{\mathrm{ref} \left( {\varphi} \right)}} \supset^* {\ensuremath{\mathrm{univ}_{{\psi}, {\chi}} \left({\delta} , {\epsilon} \right)}} \rhd {\ensuremath{\mathrm{univ}_{{\varphi \supset \psi}, {\varphi \supset \chi}} \left({\lambda p:\varphi \supset \psi. \lambda q : \varphi. \delta (pq)} , {\lambda p : \varphi \supset \chi. \lambda q : \varphi. \epsilon (pq)} \right)}}}{}$$ $$\infer{{\ensuremath{\mathrm{univ}_{{\varphi}, {\psi}} \left({\delta} , {\epsilon} \right)}} \supset^* {\ensuremath{\mathrm{ref} \left( {\chi} \right)}} \rhd {\ensuremath{\mathrm{univ}_{{\varphi \supset \chi}, {\psi \supset \chi}} \left({\lambda p : \varphi \supset \chi. \lambda q : \psi. p (\epsilon q)} , {\lambda p:\psi \supset \chi. \lambda q : \varphi. p(\delta q)} \right)}}}{}$$ $$\infer{\begin{array}{l} {\ensuremath{\mathrm{univ}_{{\varphi}, {\psi}} \left({\delta} , {\epsilon} \right)}} \supset^* {\ensuremath{\mathrm{univ}_{{\varphi'}, {\psi'}} \left({\delta'} , {\epsilon'} \right)}} \\ \rhd {\ensuremath{\mathrm{univ}_{{\varphi \supset \varphi'}, {\psi \supset \psi'}} \left({\lambda p : \varphi \supset \varphi'. \lambda q : \psi. \delta' (p(\epsilon q))} , {\lambda p : \psi \supset \psi'. \lambda q : \varphi. \epsilon' (p (\delta q))} \right)}} \end{array}}{}$$ $$\infer{P_{MN} Q \rhd P'_{MN} Q}{P \rhd P'} \quad \infer{{\ensuremath{\mathrm{ref} \left( {M} \right)}}_{NN'}P \rhd {\ensuremath{\mathrm{ref} \left( {M'} \right)}}_{NN'}P}{M \rhd N} \quad \infer{P \supset^* Q \rhd P' \supset^* Q}{P \rhd P' \quad Q \rhd Q'}$$ Define the notion of *parallel one-step reduction* $\rhd$ by the rules given in Figure \[fig:POSR\]. Let $\rhd^*$ be the transitive closure of $\rhd$. \[lm:rhdiff\] 1. If $E \rightarrow F$ then $E \rhd F$. 2. If $E \twoheadrightarrow F$ then $E \rhd^* F$. 3. If $E \rhd^* F$ then $E \twoheadrightarrow F$. These are easily proved by induction. Our reason for defining $\rhd$ is that it satisfies the diamond property: If $E \rhd F$ and $E \rhd G$ then there exists an expression $H$ such that $F \rhd H$ and $G \rhd H$. The proof is by case analysis on $E \rhd F$ and $E \rhd G$. We give the details for one case here: $${\ensuremath{\mathrm{ref} \left( {\phi} \right)}} \supset^* {\ensuremath{\mathrm{ref} \left( {\psi} \right)}} \rhd {\ensuremath{\mathrm{ref} \left( {\phi \supset \psi} \right)}} \mbox{ and } {\ensuremath{\mathrm{ref} \left( {\phi} \right)}} \supset^* {\ensuremath{\mathrm{ref} \left( {\psi} \right)}} \rhd {\ensuremath{\mathrm{ref} \left( {\phi'} \right)}} \supset^* {\ensuremath{\mathrm{ref} \left( {\psi'} \right)}}$$ where $\phi \rhd \phi'$ and $\psi \rhd \psi'$. In this case, we have ${\ensuremath{\mathrm{ref} \left( {\phi \supset \psi} \right)}} \rhd {\ensuremath{\mathrm{ref} \left( {\phi' \supset \psi'} \right)}}$ and ${\ensuremath{\mathrm{ref} \left( {\phi'} \right)}} \supset^* {\ensuremath{\mathrm{ref} \left( {\psi'} \right)}} \rhd {\ensuremath{\mathrm{ref} \left( {\phi' \supset \psi'} \right)}}$. If $E \rhd^* F$ and $E \rhd^* G$ then there exists $H$ such that $F \rhd^* H$ and $G \rhd^* H$. If $E \twoheadrightarrow F$ and $E \twoheadrightarrow G$ then $F \twoheadrightarrow H$ and $G \twoheadrightarrow H$. Immediate from the previous corollary and Lemma \[lm:rhdiff\]. [^1]: These have also been called *context morphisms*, for example in Hoffman [@Hofmann97syntaxand].

--- author: - 'A. Rothkegel' - 'A. Rothkegel' - 'K. Lehnertz' title: 'Synchronization in populations of sparsely connected pulse-coupled oscillators' --- The collective dynamics of interacting oscillatory systems has been studied in many different contexts in the natural and life sciences [@Winfree1967; @Kuramoto1984; @Pikovsky_Book2001; @Arenas2008]. In the thermodynamic limit, evolution equations for the population density proved to be a useful description [@Desai1978; @Omurtag2000; @Acebron2005], in particular to characterize the stability of synchronous and asynchronous states (see, e.g., [@Mirollo1990; @Strogatz1991; @Treves1993; @Abbott1993; @Strogatz2000; @Vreeswijk2000; @Gerstner2000; @Ly2010; @Newhall2010; @Louca2013]). Usually, dense or all-to-all-coupled networks are considered for these descriptions. Motivated by natural systems in which constituents interact with few others only, investigations of complex networks have revealed a large influence of the degree and sparseness of connectivity on network dynamics [@Hopfield1995; @Golomb2000; @Boergers2003; @Zillmer2006; @Zillmer2009; @Rothkegel2011; @Luccioli2012; @Tessone2012]. Especially when the knowledge about the connection structure is limited, it suggests itself to assume random connections (as in [Erdős-Rényi ]{}networks) or random interactions (where excitations are assigned randomly to target oscillators [@Omurtag2000; @Sirovich2006; @Dumont2013; @Nicola2013]). Both approaches often yield comparable dynamics (e.g. [@Ferreira2012; @Tattini2012]) whereas random interactions represents a substantial simplification from a mathematical point of view, allowing one to describe the networks in terms of evolution equations for the phase density. These equations are usually posed as starting point for the commonly applied mean- or the fluctuation-driven limits. However, rarely are they studied in full although it can be expected that sparseness largely influences the collective dynamics as has been discussed for excitable systems [@Sirovich2006]. In this Letter, we propose a population model of $\delta$-pulse coupled oscillators with sparse connectivity, derive the governing equations from a general definition of the density flux, and characterize existence and uniqueness of stationary solutions. For integrate-and-fire-like oscillators, the latter may either disappear with diverging firing rate or lose stability at a supercritical Andronov-Hopf bifurcation (AHB). This is in contrast to the global convergence to complete synchrony for all-to-all coupling that has been shown for finite [@Mirollo1990] and for infinite [@Mauroy2013] number of oscillators. Consider a population of oscillators $n \in N$ with cyclic phases $\phi_n(t) \in [ 0,1 )$ and intrinsic dynamics $\dot{\phi}_n(t) = 1$. If for some $t_f$ and some oscillator $n$ the phase reaches 1, the oscillator fires and we introduce a phase jump in all oscillators $n'$ with probability $p= m/N$ [@DeVille2008; @Olmi2010]. Here, $m$ is the number of recurrent connections per oscillator. The height of the phase jump is defined by the phase response curve $\Delta(\phi)$ (PRC) (or equivalently by the phase transition curve $R(\phi)$): $$\label{eq:interactionSingleOsci} \phi_{n'}(t_f^+) = \phi_{n'}(t_f) + \Delta\left(\phi_{n'}(t_f)\right) = R( \phi_{n'} ( t_f)).$$ The model can be interpreted as an all-to-all coupled network in which connections are not reliable and mediate interactions between oscillators only with a small probability ($p$). It can also be interpreted as an approximation to an [Erdős-Rényi ]{}network in which the quenched disorder, imposed by its construction, is replaced by a dynamic coupling structure which takes the form of an ongoing random influence. For the limit of large sparse networks ($N \rightarrow \infty, m = \mbox{const.}$), we represent the network dynamics by a continuity equation for the phase density $\rho(\phi,t)$ $$\label{eq:continuity} \partial_t \rho(\phi,t) + \partial_\phi J(\phi,t) = 0$$ with $\rho (\phi,t) \geq 0$ and $\int_0^1 \rho(\phi,t) d\phi = 1$. We assume the probability flux $J(\phi,t)$ to be continuous and define both $\rho$ and $J$ at phases $\phi \in [0,1)$. Evaluations at $\phi = 1$ are meant as left-sided limits towards $\phi = 1$. $J(1,t)$ is the firing rate. Every oscillator is subject to Poisson excitations $\eta_{\lambda(t)}$ with inhomogeneous rate $\lambda(t) = m J(1,t)$ and we can describe its phase variable by the stochastic differential equation $\partial_t \phi(t) = 1 + \eta_{\lambda(t)}$. To shorten our notation, we will omit in the following the time $t$ as argument of $\rho$, $\lambda$, and $J$. As we expect $R(\phi)$ to be non-invertible and to map intervals to a single phase, we have to take care in which way $\rho$ and $J$ are interpreted at these phases. Given some distribution of oscillators phases, we consider $\rho(\phi,t) d\phi$ as the fraction of oscillators which are contained in a small interval whose left boundary is fixed to $\phi$. With this definition, $\rho(\phi,t)$ is continuous for right-sided limits and the corresponding $J(\phi,t)$ is defined by the oscillators which pass an imaginary boundary which is infinitely close to $\phi$ and right to $\phi$. The flux can be formalized in the following way: $$\label{eq:fluxGeneralForm} J(\phi) = \rho(\phi) + \lambda \left(\int \limits_{I_>(\phi)} \rho(\tilde{\phi}) d\tilde{\phi} - \int \limits_{I_\leq(\phi)} \rho(\tilde{\phi}) d\tilde{\phi} \right),$$ where $I_>(\phi) := \{ \tilde{\phi} < \phi | R(\tilde{\phi}) > \phi\}$ is the set of phases smaller than $\phi$ which is mapped by $R(\phi)$ to a phase larger than $\phi$, and $I_\leq(\phi) := \{ \tilde {\phi} > \phi | R(\tilde{\phi}) \leq\phi\} $ is defined analogously (the order relations in in these formulas are interpreted for unwrapped phases). The first term of the r.h.s. of represents convection due to the intrinsic dynamics of oscillators. The integrals represent the fractions of oscillators which are moved across phase $\phi$ by an excitation, either to smaller or larger values (cf. ). PRCs which are derived from limit cycle oscillators by phase reduction usually have invertible phase transition curves [@Brown2004a]. However, even holds if $R(\phi)$ is not invertible and has no or uncountably many inverse images. For phases $\phi$ at which $R(\phi)$ has at most countably many inverse images, we can represent the sets $I_>(\phi)$ and $I_\leq(\phi)$ by a product of two Heaviside functions and derive, differentiating the latter to $\delta$-functions, the following expression: $$\partial_\phi J(\phi) = \partial_\phi \rho(\phi) + \lambda \int_0^1 \rho(\tilde{\phi})\left( \delta ( \phi - \tilde{\phi} )- \delta (\phi - R(\tilde{\phi}) \right) d\tilde{\phi}.$$ Denoting with $(R_i^{-1}(\phi) | i \in I)$ an enumeration of the inverse images of $R(\phi)$ at phase $\phi$ for an appropriate index set $I$, the continuity equation reads: $$\label{eq:sparseLimit} \partial_t \rho(\phi) = - \partial_\phi \rho(\phi) - \lambda \rho(\phi) + \lambda \sum_{i \in I} \frac{\rho(R_i^{-1}(\phi)) }{R'(R^{-1}(\phi))}.$$ For uncountably many inverse images of some phase $\varphi$, they will be contained in $I_>(\varphi)$ or $I_\leq (\varphi)$ but not in $I_>(\varphi^-)$ and $I_\leq(\varphi^-)$. In this case, we obtain a discontinuity between $\rho(\varphi^-)$ and $\rho(\varphi)$ which can be expressed by requiring continuity of the flux for left-sided limits at $\phi = \varphi$ ($J(\varphi^-) = J(\varphi)$). Note that the definition in automatically ensures continuity for right-sided limits. Setting $\phi = 1$ in , we obtain the following relationship for the excitation rate $\lambda= m J(1)$ $$\label{eq:firingRate} \lambda = m \rho(1) /\left( 1 - m \int\limits_{I_>(1)} \rho(\tilde{\phi}) d\tilde{\phi} + m \int\limits_{I_\leq(1)} \rho(\tilde{\phi})d \tilde{\phi} \right).$$ Given a PRC and the number of recurrent connections $m$, the population model for oscillators with sparse connectivity is given by , , and by the requirement that $\rho(\phi)$ is normalized to allow for an interpretation as probability density function. The integral over $I_\leq ( 1)$ in corresponds to oscillators which pass the firing threshold in the *wrong* direction. Usually, it is not desirable that such oscillators decrease the firing rate, which can be prevented by requiring the PRC to be bounded by $-\phi$ from below. Note that the excitation rate as defined in may diverge or turn negative. In these cases every firing oscillator will make, on average, at least one other oscillator fire immediately and a macroscopic amount of oscillators fires in an instant. We will refer to this situation as an avalanche. Clearly, a numerical integration via some finite difference scheme will break down at this point [@Kovacic2009; @Dumont2013a]. Nevertheless, Monte-Carlo simulations may still be meaningful. Let us briefly consider the mean-driven limit, i.e., a sequence of PRCs indexed by $i$ and parameters $m_i$ such that $\Delta_i(\phi)$ vanishes as $i \rightarrow \infty$ and the product $\Delta_i(\phi) m_i$ converges point-wise to some function $Z(\phi)$. The flux $J_i(\phi)$ is then straightforwardly approximated by $$\label{eq:denseLimitFlux} J_i(\phi) \rightarrow \rho(\phi) \left( 1 + J(1) Z(\phi) \right)$$ as $i \rightarrow \infty$. Setting $\phi = 1$ in gives the expression for the firing rate $\nu = \rho(1) / \left(1 - Z(1) \rho(1)\right)$ of the non-linear evolution equation $$\label{eq:denseLimit} \partial_t \rho(\phi) = - \partial_\phi \left[ \left(1 + \frac {\rho(1)} {1 - Z(1)\rho(1)} Z(\phi)\right) \rho(\phi) \right ],$$ for which the continuity of the flux in leads to the following non-linear boundary condition $$\label{eq:denseLimitBoundary} \rho(0) ( 1 + \lambda Z(0)) = \rho(1) (1 + \lambda Z(1)).$$ The dynamics of the system defined by and is easily describable for monotonous PRCs [@Mauroy2013]. For increasing $Z(\phi)$ the probability density concentrates to a single phase in finite time for arbitrary initial distributions. For decreasing $Z(\phi)$ convergence to the stationary solution $\rho_0(\phi):= c/ ( 1 + c Z(\phi))$ can be observed. The question of synchronization in the population model with sparse connectivity can be addressed by investigating the existence and the stability of normalized stationary solutions of and . Stationary solutions of Eq.  (with $\partial_t \rho(\phi,t) = 0$) can be obtained by segmenting $[0,1]$ into intervals in which oscillators receive either phase advances or retardations. For each of these intervals, a solution can then be obtained with some solver for delay differential equations with state dependent delays, stepping towards either larger or smaller phases. Phases $\varphi$ at which $R(\phi)$ crosses the identity from below serve as suitable starting points for such a stepping approach because the sets $I_>(\varphi)$ and $I_\leq(\varphi)$ are empty at such points and we have $\rho(\varphi) = J(\varphi) = J(1) = \lambda /m$. Using this initial value, solutions fulfill . In this way we can obtain stationary solutions $\rho(\phi;\lambda)$ of and in sole dependence on $\lambda$. Let us denote $I(\lambda) := \int_0^1 \rho(\phi;\lambda) d \phi$. In order to allow for a stochastic interpretation of $\rho(\phi;\lambda)$, $\lambda$ must then be chosen with shooting in such a way that $I(\lambda) = 1$. However, depending on the PRC such a choice may not be possible. We can characterize the condition under which solutions exist by assuming that $R(\phi)$ is non-decreasing. Note that this assumption is valid for commonly considered PRCs including those of integrate-and-fire oscillators [@Brown2004a]. Under this assumption $I(\lambda)$ is strictly increasing in $\lambda$, which we will show at the end of this letter. Stationary solutions are thus unique, and to decide on their existence, it is thus sufficient to investigate the solutions $\rho(\phi;\lambda)$ for large $\lambda$. Near $\varphi$, the non-local term in vanishes, and $\rho(\phi)$ decays as $e^{ - \lambda \phi} (\lambda / m)$ which has an integral independent on $\lambda$ and thus concentrates to $\delta(\phi - \varphi) / m$ for large $\lambda$. Analogously, $\delta$-peaks are generated at phases $R(\varphi), R^2(\varphi), ...$, at which oscillators arrive after having received a certain amount of excitations. Integrating over such a sequence of $\delta$-peaks, we can express $I(\infty)$ and thus characterize the existence of asynchronous solutions by the following inequality: $$\label{eq:existenceSolutions} I(\infty) = \max_i \{ i | R^{i} (\varphi) \leq 1 + \varphi \} / m > 1.$$ We now report on findings of a dynamical analysis for the case of integrate-and-fire oscillators [@Peskin1975]. They are a popular model in many scientific fields ranging from physics and biology to the neurosciences [@Mirollo1990]. Our approach allows us to treat the non-invertible phase transition curves of excitatory and inhibitory oscillators and to study both dynamical regimes from a unified point of view. The PRC reads $$\label{eq:PRCmirollo} \Delta(\phi) = \max \{ \min \{ a\phi + b, 1 - \phi \}, -\phi \}.$$ The maximum and minimum bound $\Delta(\phi)$ by $-\phi$ from below and by $1 - \phi$ from above. The bound from above ensures that an excitation of oscillators cannot push them past the firing threshold ($I_>(0) = \emptyset)$ and leads to uncountably many inverse images $R(\phi)$ for $\phi = 1$. This assumption strongly favours synchronization; two oscillators adapt their phases completely after a suprathreshold excitation from one to the other. The bound form below prevents oscillators with small phases which receive an inhibitory excitation to attain a negative phase or a phase just below the firing threshold ($I_\leq(1) = \emptyset$). We obtain the following boundary condition from : $$\label{eq:boundaryCondition} \underbrace{\rho(0) - \lambda \int\limits_{I_\leq(0)} \rho(\tilde{\phi}) d\tilde{\phi}}_{J(0)} = \underbrace{\rho(1) + \lambda \int\limits_{I_>(1)} \rho(\tilde{\phi}) d\tilde{\phi}}_{J(1)}.$$ The parameters $a$ and $b$ control both leakiness and coupling strength. For $a, b > 0$, $\Delta(\phi)$ represents excitatory integrate-and-fire oscillators with concave-down charging function. For $a,b <0$ , $\Delta(\phi)$ represents inhibitory oscillators with concave-down charging function. For other parameter combinations, $\Delta(\phi)$ represents excitatory and inhibitory oscillators with concave-up charging function and dynamical systems with both positive and negative phase responses. In Fig. \[fig:convergence\] we show stationary solutions for the PRC given in . Given the PRC, phases in the interval $[0,b]$ are not reachable by excitations. The phase density $\rho(\phi)$ thus decays exponentially in this interval according to . Phases $\phi > b$ are reachable, and $\rho(\phi)$ exhibits excursions of decreasing amplitudes, in which smoothed versions of the initial exponential segment in $[0,b]$ are repeated (cf. [@Sirovich2006]). Analogously, inhibitory oscillators with phases near $\phi=1$ do not receive excitations, which leads to a sharp decrease of $\rho(\phi)$ close to the firing threshold (not shown). For the mean-driven limit, stationary solutions show oscillations near $\phi = 0$ with frequencies that diverge with increasing $m$ (cf. Fig. \[fig:convergence\] top). For large coupling strengths or mean degrees, stationary solutions converge to a series of $\delta$-peaks (cf. Fig \[fig:convergence\] bottom) and eventually disappear. For small $m$ and depending on oscillator parameters $a$ and $b$, we can distinguish different dynamics (cf. Fig. \[fig:parameterScan\]): asynchronous states with oscillator phases distributed according to the stationary solution, and (partially) synchronous states with oscillatory evolutions of the excitation rate $\lambda$. As for the mean-driven limit, large positive coupling strengths (above the black line in Fig. \[fig:parameterScan\]), do not allow for normalized stationary solutions with positive values which can be interpreted as probability density. In this regime, oscillators synchronize completely within a few collective oscillations. Near this boundary the excitation rate diverges, and we observe no partially synchronous states. For smaller coupling strengths, stationary solutions do exist and we now discuss their stability. We consider a small, localized perturbation of the stationary solution, which travels periodically around the phase circle. An oscillator represented by this perturbation is shifted towards larger phase values due to its intrinsic dynamics and due to excitations. Both contributions are reflected by the corresponding terms in . The uncertainty of the oscillator’s phase after some time leads to a broadening of the perturbation which increases with the strength of excitations and thus with $|b|$ (and to some degree with $a$). When the perturbation crosses the firing threshold, positive values of $a$ lead to a larger excitation of oscillators near the perturbation which leads to a sharpening. Consequently, the perturbation vanishes for large $|b|$ and small $a$ and increases otherwise. The boundary between both behaviors is characterized by a locus of Andronov-Hopf bifurcation (AHB) points. The AHB gives rise to oscillatory states with partial synchrony, in which a small perturbation of the stationary solution travels periodically around the phase circle. The amplitude of these oscillations increases with the distance to the AHB. For negative $b < a$, oscillators have negative phase responses near $\phi = 1$ and both integrals in the denominator in (\[eq:firingRate\]) vanish. Consequently, we observe no avalanches and no phase concentrations in $\rho(\phi)$ up to some value of $b$ for which complete synchrony is reached. For positive $a$ and $b$, the first integral in does not vanish. When the amplitude of the oscillations grows so large that the denominator in vanishes, an avalanche emerges. For larger values of $a$ , subsequent avalanches increase in size leading to complete synchrony after a few oscillations. For smaller values of $a$, these avalanches may, for finite networks, lead to complicated partially synchronous states with recurring avalanches, which, however, lie outside what can be described with the evolution equation. We will report on these states elsewhere (Rothkegel and Lehnertz, manuscript in preparation). Note that the aforementioned broadening is not present in the mean-driven limit, in which both intrinsic dynamics and excitations are represented by a single convection term. If we consider the PRC in with parameters $a = \alpha / m$ and $b = \beta / m$, we obtain $Z(\phi) = \alpha \phi + \beta$ in the mean-driven limit ($m_i \rightarrow \infty$). The phase density as determined by and thus converges to the stationary solution $\rho_0(\phi)$ for negative $a$ and concentrates for positive $a$ leading to complete synchrony of oscillators. In particular, the system does not allow for periodic solutions with partial synchrony of oscillators [@Mauroy2013]. In this case stable stationary solutions cannot be observed for $a > 0$. We have presented a population model of $\delta$-pulse-coupled oscillators with sparse connectivity. Interactions between oscillators are defined by a phase response curve (PRC). We have defined the model in such a way that allowed us to treat non-invertible PRCs which lead to discontinuous distributions of oscillator phases. We have demonstrated the uniqueness of asynchronous solutions and characterized their existence. Finally, we have shown—using integrate-and-fire-like oscillators—two different mechanism which may lead to loss of asynchronous states. Stationary solutions may lose stability, giving rise to oscillations and partially synchronous states, or they may disappear completely, leading to avalanche-like synchronization and a fast convergence to synchrony. We are confident that the model may further the understanding of the dynamics of sparsely coupled oscillatory networks. Systems that can be modelled as such appear ubiquitously in Nature. In this last part of the letter, we will show that $I(\lambda)$, the norm of stationary solutions of (5) and (6), is strictly increasing in $\lambda$, provided that the phase transition curve $R(\phi)$ is increasing in $\phi$ and crosses the identity at one or more points from below. For the sake of simplicity, we will shift the phases in such a way, that the crossing occurs at $\phi = 0$ such that we have $\rho(0) = J(0) = \lambda / m$. As first step, we relate $I(\lambda)$ defined for some PRC to an exit-time problem for the stochastic dynamics of oscillators $\partial_t \phi(t) = 1 + \eta_{\lambda(t)}$ which is determined by convection with velocity 1 and by Poissonian excitations $\eta_{\lambda(t)}$ with inhomogeneous rate $\lambda(t)$. To this end, we consider the interval $(0,1)$ to be empty at $t = 0$. If we now inject a constant flux $J(0)$, oscillators will pass the interval and exit at $\phi = 1$ after some variable time $t_\mathrm {E}$. We consider the distribution $P(t_\mathrm{E})$ of these exit times. The flux $J(1,t)$ will increase from $J(1,0) = 0$ and will eventually approach the injected amount $J(0)$, at which time the same amount of oscillators enter and exit the interval. If we inject $J(0) = \lambda /m$, then the number of oscillators which are in the interval at a large time $t$, is given by $I(\lambda)$ and can be expressed by integrating over the difference of incoming and outgoing fluxes: $$I(\lambda) = \lim_{\kappa \rightarrow \infty} \int_0^\kappa \frac {\lambda}{m} - J(1,t) dt.$$ Given the distribution of exit times $P(t_\mathrm{E})$, we can express the outgoing flux by integrating over the time $t_0$ at which oscillators are injected into the interval: $$I(\lambda) = \lim_{\kappa \rightarrow \infty} \left(\frac{\lambda}{m} \kappa - \int_0^\kappa dt \int_0^t dt_0 P(t - t_0) \frac{\lambda}{m} \right).$$ The domain of the integral is the area of the quadrant $(t_0, t) \in [0,\kappa] \times [0,\kappa]$ which lies above the diagonal. If we parametrise this domain by $t_\mathrm{E}:= t - t_0$ and $t_0$, we obtain, using substitution for multiple variables, $$I(\lambda) = \frac{\lambda}{m} \lim_{\kappa \rightarrow \infty} \left( \kappa - \int_0^\kappa dt_\mathrm{E} \int_0^{\kappa - t_\mathrm{E}} dt_0 P(t_\mathrm{E}) \right ).$$ As every oscillator eventually reaches $\phi = 1$, we have $\int_0^\infty P(t_\mathrm{E}) dt_\mathrm{E} = 1$, and we obtain a surprisingly simple relationship, which says that the norm of $I(\lambda)$ is given by the product of the injected flux and the mean exit time: $$\label{eq:increasingI} I(\lambda) = \frac{\lambda}{m} \int_0^\infty { t_\mathrm{E} P(t_\mathrm{E}) d t_\mathrm{E} }.$$ Let us represent Eq.  by an integral equation. We define $M(\varphi)$ as the mean time an oscillator with phase $1 - \varphi$ remains in the unit interval before it reaches $\phi = 1$. With this definition, the mean exit time for the entire interval is $M(1)$. Oscillators outside of the interval have a vanishing exit time: $M(\varphi) = 0$ for $\varphi < 0$. $M(\varphi)$ can now be expressed by an average over the time of the next excitation. Assuming an exponential distribution $\iota(t) := \lambda e^{-\lambda t}$ for the times between excitations, we can relate these times to probabilities. With probability $\bar{\iota}(\varphi) = 1 - \int_0^\varphi \iota(t) dt$, the oscillator will leave the interval without receiving another excitation. For the case that the oscillator receives an excitation at time $t$ after injection, it has a phase of $1-\varphi + t + \Delta(1 - \varphi + t)$ afterwards. The mean time the oscillator needs to pass the remaining phase distance can again be expressed by $M(\varphi)$ which results in the following integral equation for $M(\varphi)$: $$\label {eq:volterraM} M(\varphi) = \bar{\iota}(\varphi) \varphi + \int_0^\varphi \iota(t) \left[ t + M(\varphi - t - \Delta(1 - \varphi + t)) \right ] dt . $$ Inserting $\iota(t)$ and multiplying Eq.  by $\lambda / m$, we obtain a similar equation for $I(\lambda,\varphi) : = \int_0^\varphi \rho(\phi;\lambda/m,\lambda)$ which we define as generalization of $I(\lambda)$ with $I(\lambda,1) = I(\lambda)$: $$\label{eq:volterraMeanExitTimes2} I (\lambda, \varphi) = \frac{1 - e^{-\lambda \varphi}}{m} + \int_0^\varphi \lambda e^{-\lambda(\varphi -t)} I (\lambda, t- \Delta(1-t)) dt.$$ For convenience, we will use the abbreviations $G(\lambda,\varphi):= (1- e^{-\lambda \varphi}) / m$ and $z(\varphi, u) := \varphi + u - \Delta(1 - \varphi - u)$. $G(\lambda,\varphi)$ is strictly increasing in both arguments. Using our assumption about the PRC, we see that $z(\varphi,u)$ is increasing in both arguments but not necessarily strictly increasing. Eq.  takes the following form: $$\label{eq:volterraMeanExitTime3} I (\lambda, \varphi) = G(\lambda, \varphi) + \int_0^\infty \lambda e^{-\lambda u} I (\lambda, z ( \varphi, -u)) du.$$ We have extended the integral from $\varphi$ to $\infty$ using $I(\lambda,\varphi) = 0$ for $\varphi < 0$. The equation is a Volterra integral equation of the second kind. Note that $z(\varphi, -u) \leq \varphi$ such that the equation defines $I(\lambda, \varphi)$ in a hierarchical way. $I(\lambda, \varphi_0)$ is obtained by taking some value $G(\lambda, \varphi_0)$ and by adding an weighted average over previous values $I(\lambda, \varphi), \varphi < \varphi_0$. We can thus conclude that $I(\lambda,\varphi)$ is positive, if $G(\lambda,\varphi)$ is positive for all $\varphi$. We demonstrate that $I(\lambda)$ is strictly increasing in $\lambda$ for PRCs with $\partial_\phi R(\phi) \geq 0$. We first argue that $I(\lambda,\varphi)$ is strictly increasing in its second argument for every $\lambda > 0$. Differentiating by $\varphi$, we obtain an integral equation for $\partial_\varphi I(\lambda, \varphi)$ which is of the same kind as and has a non-negative kernel $\lambda e^{-\lambda u} \partial_\varphi z(\varphi, -u)$ and a positive function $\partial_\varphi G(\lambda, \varphi)$ outside of the integral. Analogously, we can thus conclude that $\partial_\varphi I(\lambda, \varphi)$ is positive. Finally, we argue that $I(\lambda, \varphi) $ is increasing in $\lambda$ for every $\varphi \leq 1$. Taking the derivative of with respect to $\lambda$, we obtain three terms according to the dependences on $\lambda$ of $G$, of the kernel, and of $I$. The first term $\partial_\lambda G(\lambda, \varphi)$ is positive as G is strictly increasing in its arguments. As second term, we obtain $$\int_0^\infty I(\lambda, z(\varphi, -u)) \partial_\lambda \lambda e^{-\lambda u} du.$$ Using $\partial_\lambda (\lambda e^{-\lambda u}) = \partial_u ( u e^{-\lambda u})$ and integrating by parts, we obtain also a positive contribution as $I$ and $z$ are increasing in their second arguments. The third term contains a weighted average via a positive kernel of $\partial_\lambda I(\lambda, \varphi)$. As before, we infer that $\partial_\lambda I(\lambda,\varphi)$ is strictly increasing which gives for $\varphi = 1$ the desired proposition. We are grateful to Stefano Cardanobile for fruitful discussions and Gerrit Ansmann for careful revision of an earlier version of the manuscript. This work was supported by the Deutsche Forschungsgemeinschaft (LE 660/4-2). [10]{} url\#1[`#1`]{} . (Springer Verlag, Berlin) 1984. (Cambridge University Press, Cambridge, UK) 2001. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (Courant Institute of Mathematical Sciences, New York) 1975. . .

--- abstract: 'We consider distributed and dynamic caching of coded content at small base stations (SBSs) in an area served by a macro base station (MBS). Specifically, content is encoded using a maximum distance separable code and cached according to a time-to-live (TTL) cache eviction policy, which allows coded packets to be removed from the caches at periodic times. Mobile users requesting a particular content download coded packets from SBSs within communication range. If additional packets are required to decode the file, these are downloaded from the MBS. We formulate an optimization problem that is efficiently solved numerically, providing TTL caching policies minimizing the overall network load. We demonstrate that distributed coded caching using TTL caching policies can offer significant reductions in terms of network load when request arrivals are bursty. We show how the distributed coded caching problem utilizing TTL caching policies can be analyzed as a specific single cache, convex optimization problem. Our problem encompasses static caching and the single cache as special cases. We prove that, interestingly, static caching is optimal under a Poisson request process, and that for a single cache the optimization problem has a surprisingly simple solution.' author: - | Jesper Pedersen, Alexandre Graell i Amat, , Jasper Goseling, ,\ Fredrik Brännström, , Iryna Andriyanova, , and Eirik Rosnes,  [^1] [^2] [^3] [^4] [^5] bibliography: - 'confs-jrnls.bib' - 'IEEEabrv.bib' - 'library.bib' title: Dynamic Coded Caching in Wireless Networks --- Caching, content delivery networks, erasure correcting codes, TTL. Introduction ============ Distributed wireless caching has attracted a significant amount of attention in the last few years as a promising technology to alleviate the load on backhaul links [@Boccardi2014]. Content may be cached in a distributed fashion across small base stations (SBSs) such that users can download requested content directly from them. For distributed caching, the use of erasure correcting codes has been shown to reduce the download delay as well as the network load [@Shanmugam2013; @Bioglio2015]. Content may also be cached directly in mobile devices such that users can download content from neighboring devices using device-to-device communication. Similar to the SBS caching case, the use of erasure correcting codes has been demonstrated to reduce the network load also for this scenario [@Pedersen2016; @Pedersen2019; @Wang2017]. Caching furthermore facilitates index-coded broadcasts to multiple users requesting different content, which has been shown to drastically reduce the amount of data that has to be transmitted over the SBS-to-device downlink [@Maddah-Ali2014]. All these works consider the cached content to be static for a period of time (e.g., a day) according to a given file popularity distribution. Dynamic cache eviction policies, e.g., first-in-first-out (FIFO), least-recently-used (LRU), least-frequently-used (LFU), and random (RND), may be beneficial to use when the file library or file popularity profile is dynamic, or when users request content according to a renewal process [@Gelenbe1973]. Due to the complexity in analyzing such policies, timer-based policies that are significantly more tractable have been suggested. One such policy is time-to-live (TTL) where a request for a particular piece of content triggers it to be cached and then evicted after the expiration of a timer. The TTL policy has been shown to yield similar performance to FIFO, LRU, LFU, and RND policies in [@Che2002; @Fricker2012; @Bianchi2013; @Dehghan2019]. Goseling and Simeone extended the TTL policy to cache fractions of files, referred to as fractional TTL (FTTL), and showed that this can improve performance under a renewal request process [@Goseling2019]. Decreasing the fraction of a file that is cached over time, termed soft TTL (STTL), can further improve the performance. Optimal STTL caching policies are obtained through a convex optimization problem [@Goseling2019]. All previous works on TTL policies assume either a single cache or a number of caches, e.g., structured into lines or hierarchies, where users access a single cache. For these scenarios, coded caching does not bring any benefits. However, if users can access several caches, the use of erasure correcting codes can be beneficial. Hence, merging distributed coded caching with the TTL schemes in [@Goseling2019], which have both independently been shown to bring performance improvements, is an intriguing prospect. In this paper, we generalize the TTL policies in [@Goseling2019] to a distributed coded caching scenario. Specifically, we consider the scenario where content is encoded using a maximum distance separable (MDS) code and cached in a distributed fashion across several SBSs. Coded content is evicted from the caches in accordance with the TTL policies in [@Goseling2019]. Users requesting a particular piece of content download coded packets from SBSs within communication range and, if necessary, download additional packets from a macro base station (MBS). We formulate a network load minimization problem, where the network load is defined as a sum of data rates over various network links, weighted by a cost representing, e.g., transmission delay or energy consumption of transmitting data over these links. We then rewrite the optimization problem as a mixed integer linear program (MILP) that is efficiently solved numerically. We furthermore prove that the distributed coded caching problem can equivalently be analyzed as a single cache problem with a specific decreasing and convex cost function. This is an important result because it shows that such a function, previously studied for the single cache case due to its analytical tractability [@Goseling2019], arises naturally in a distributed caching scenario. For SBSs deployed according to a Poisson point process [@Chiu2013 Ch. 2.3], we derive the cost function explicitly. We analyze two important special cases of the network load minimization problem. In particular, we show that our problem has the static coded caching problem where content is never updated (considered in, e.g., [@Shanmugam2013; @Bioglio2015]), as a special case. We furthermore prove that static coded caching is optimal under the assumption of a Poisson request process. Moreover, for the special case of users accessing a single cache, we prove that the STTL problem is a fractional knapsack problem with a greedy optimal solution. The performance of TTL, FTTL, and STTL, in terms of network load, is evaluated for a renewal process, specifically when the times between requests follow a Weibull distribution. We show that distributed coded caching using TTL caching policies can offer significant reductions in network load, especially for bursty renewal request processes. Distributed caching of coded content utilizing TTL cache eviction policies was also investigated in [@Chen2019]. Compared to the problem studied in this paper, the work in [@Chen2019] is significantly different in a number of ways. Specifically, we consider an STTL policy with optimized TTL timers under a renewal request process, which was not considered in [@Chen2019]. Furthermore, a dynamic library of files with location-dependent popularity is considered in [@Chen2019], which is typically considered to be more general than a static file library and is not in the scope of our work. However, it is reasonable to consider scenarios where the file library remains fixed for a considerable amount of time, e.g., a day, and focus on an area with homogeneous file popularity. System Model {#sec:model} ============ We consider an area served by an MBS that always has access to a file library of $N$ files, where file ${i}= 1, 2, \ldots, N$ has size $s_i$. Mobile users request files from the library according to independent renewal processes. Specifically, we denote the independent and identically distributed times between requests for file ${i}$ by $X_i$, the cumulative distribution function (CDF) of $X_i$ by $${F_{X_i}(t)} \triangleq \Pr(X_i \le t),$$ and the request rate of file ${i}$ by $$\omega_i \triangleq \operatorname*{\mathbb{E}}[X_i]^{-1}.$$ We let $p_i = \omega_i/\omega$, for some aggregate request rate in the area, $\omega = \sum_{i=1}^N \omega_i$. For a Poisson request process, i.e., exponentially distributed $X_i$, $p_i$ can be interpreted as the probability that file ${i}$ is requested. The request rates $\omega_i$ are assumed to be constant over a sufficiently long period of time, e.g., not changing during the course of one day. For such scenarios, file popularity predictions and content allocation optimization can be carried out during periods of low network traffic, e.g., during night time. ${B}$ SBSs are deployed in the area and each SBS has a cache with storage capacity $C$. We assume that a user can download content from an SBS if it is within a range ${r_\text{SBS}}$ and we denote by $\gamma_b$ the probability that a user is within range of $b$ SBSs at any given time. The model considered in this paper is illustrated in Fig. \[fig:model\]. Caching Policy {#sec:policy} -------------- Each file ${i}$ of size $s_i$ is partitioned into $k_i$ packets, each of size $s_i/k_i$. The packets are encoded into $n_i$ coded packets (also of size $s_i/k_i$) using an $(n_i, k_i)$ MDS code. For analytical tractability, we assume that all SBSs cache the same content at all times, i.e., the caches are *synchronized*. With slight abuse of notation, we let $m_i(t) \le k_i$ denote the number of coded packets of file ${i}$ cached at each SBS at time $t$, where $t$ is the time since the last request for file ${i}$. We will use this interpretation of $t$ throughout the paper. The amount of file ${i}$ cached by each SBS at time $t$ is hence $m_i(t) s_i/k_i$. We normalize by the file size $s_i$ and let $$\mu_i(t) = m_i(t)/k_i$$ denote the fraction of file ${i}$ cached at time $t$. Similar to [@Goseling2019], we refer to $\mu_i(t)$ as the *caching policy*. We adopt an STTL cache eviction policy, shown to increase the amount of content that can be downloaded from a single cache under a renewal request process in [@Goseling2019]. Hence, coded packets of file ${i}$ may be evicted from the caches at periodic times with period $T$ after the last request for file ${i}$. We allow $K$ potential updates within a total time equal to $KT$, which we refer to as the *update window* length. For $K=0$, the caches are never updated. This corresponds to *static caching*, which is the type of caching considered in a big part of the literature [@Shanmugam2013; @Bioglio2015; @Pedersen2016; @Pedersen2019; @Wang2017; @Maddah-Ali2014]. The caching policies $\mu_i(t)$ are decreasing functions of $t$ given by [@Goseling2019] $$\mu_i(t) = \begin{cases} \mu_{i,0}, & \text{if}~t < T,\\ \mu_{i,j}, & \text{if}~jT \le t < (j+1)T,~j = 1, 2, \ldots, K-1,\\ \mu_{i,K}, & \text{if}~t \ge KT, \end{cases} \label{eq:mu}$$ where $$1 \ge \mu_{i,0} \ge \mu_{i,1} \ge \cdots \ge \mu_{i,K} \ge 0.$$ In practice, each $\mu_{i,j}$ has to be quantized to correspond to valid code parameters $k_i$ and $n_i$. In this work, we will assume that $\mu_{i,j} \in {\mathbb{R}}$ for simplicity. We refer to $f = 1/T$ as the update frequency and remark that static caching ($K=0$) corresponds to $f=0$. Content Download {#sec:down} ---------------- For an MDS code, any $k_i$ coded packets of file ${i}$ suffice to decode the file. All files requested by users can be decoded by downloading packets available at SBSs within communication range and, if necessary, retrieving additional packets from the MBS. Specifically, a user requesting file ${i}$ at time $t$ downloads $m_i(t)$ packets from the $b$ SBSs within communication range. If $bm_i(t) \ge k_i$, the user can decode the file. If $b m_i(t) < k_i$, the additional $k_i-bm_i(t)$ coded packets required to decode the file are downloaded from the MBS. Consequently, the fraction of file ${i}$ downloaded from SBSs can be expressed as $$\min\{1, b\mu_i(t)\} \label{eq:sbsfrac}$$ and the fraction of file ${i}$ downloaded from the MBS as $$\max\{0, 1-b\mu_i(t)\}. \label{eq:mbsfrac}$$ We assume that downloading one bit of data from the MBS and the SBSs comes at a cost ${{\theta}_\text{MBS}}$ and ${{\theta}_\text{SBS}}$ per bit, respectively. The cost represents, e.g., the transmission delay or energy consumption of transmitting one bit. Finally, the cost to send data to the caches, referred to as the cache update cost, is denoted by ${{\theta}_\text{C}}$. Preliminaries {#sec:prel} ============= The caching policies in correspond to STTL [@Goseling2019]. FTTL policies are obtained as a special case of , where the same fraction $\nu_i$ of file ${i}$ is cached for a time $LT$, defined by an integer $0 \le L \le K$, i.e., $\mu_{i,0} = \mu_{i,1} = \ldots = \mu_{i,L} = \nu_i$ and $\mu_{i,L+1} = \mu_{i,L+2} = \ldots = \mu_{i,K} = 0$ [@Goseling2019]. Furthermore, letting $\nu_i = 1$ we obtain TTL caching policies. In [@Goseling2019], the caching problem is framed as a utility maximization problem where a strictly concave and increasing utility function $g_i(\mu)$ measures the utility resulting from caching a fraction $\mu$ when file ${i}$ is requested. The choice of letting $g_i(\mu)$ be a strictly concave and increasing function of $\mu$ is due to analytical tractability. In practice, a linear utility function is more reasonable [@Neglia2018]. For the case of a single cache, the sum utility maximization solved in [@Goseling2019] is $$\begin{aligned} \underset{\substack{\mu_{i,j}, \nu_i \in {\mathbb{R}}\\ \beta_{i,j} \in \{0,1\}}}{\text{maximize}}~ & \sum_{i=1}^N \omega_i \sum_{j=0}^K g_i(\mu_{i,j}) F_{i,j}, \label{obj:goseling}\\ \text{subject to}~ & \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \mu_{i,j} A_{i,j} \le C, \label{cnstr:cache}\\ & 1 \ge \mu_{i,0} \ge \mu_{i,1} \ge \cdots \ge \mu_{i,K} \ge 0, \label{cnstr:mu}\\ & 0 \le \nu_i \le 1, \label{cnstr:nu}\\ & -\beta_{i,j} \le \mu_{i,j}\le \beta_{i,j}, \label{cnstr:beta1}\\ & \beta_{i,j}-1 \le \mu_{i,j}-\nu_i \le 1-\beta_{i,j}, \label{cnstr:beta2}\end{aligned}$$ where is a long-term average cache capacity constraint [@Goseling2019 Lem. 1], $$\label{eq:F} F_{i,j} = \begin{cases} {F_{X_i}((j+1)T)}-{F_{X_i}(jT)}, & \text{if}~j = 0,\ldots, K-1,\\ 1-{F_{X_i}(KT)}, & \text{if}~j = K \end{cases}$$ is the probability that file $i$ is requested in time-slot $j$, and $$\label{eq:A} A_{i,j} = \begin{cases} \displaystyle{\int}_{jT}^{(j+1)T} 1-{F_{X_i}(t)} {~\mathrm{d}}t, & \text{if}~j = 0,\ldots, K-1,\\[1em] \displaystyle{\int}_{KT}^\infty 1-{F_{X_i}(t)} {~\mathrm{d}}t, & \text{if}~j = K. \end{cases}$$ The ratio $F_{i,j}/A_{i,j} \approx h_i(jT)$, where $h_i(\cdot)$ is the hazard function of the request process, represents the probability to observe a request given the time since the last request, and the approximation follows by considering the continuous limit $T \to 0$ [@Goseling2019]. For the remainder of this paper, $h_i(jT)$ is assumed to be decreasing in $j$. The solution to – provides optimal FTTL caching policies [@Goseling2019]. Optimal TTL caching policies are achieved by letting $\nu_i = 1$ and removing the constraint , while STTL policies are achieved by removing the constraints – [@Goseling2019]. Distributed Coded TTL Caching {#sec:analysis} ============================= In this section, we formulate the average rate at which data is sent through the network described in Section \[sec:model\] and an optimization problem to minimize the network load for coded TTL caching. In particular, we generalize the optimization problem – to a distributed coded caching scenario, utilizing TTL caching policies. We propose an equivalent, more tractable formulation of the optimization problem that is efficiently solved numerically. The average rate at which data is downloaded from the SBSs and the MBS is denoted by ${R_\text{SBS}}$ and ${R_\text{MBS}}$, respectively. We choose the utility function $g_i(\mu_i(t)) = s_i \min\{1, b \mu_i(t)\}$, representing the amount of data that a user requesting file $i$ at time $t$ can download from the $b$ SBSs within communication range (see ). Using and also averaging over the number of SBSs within range of a user requesting a particular content in , we obtain $${R_\text{SBS}}= \sum_{b=0}^{{B}} \gamma_b \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \min\{1, b\mu_{i,j}\} F_{i,j}. \label{eq:ratesbs}$$ Similarly, substituting in , the MBS download rate is $$\label{eq:ratembsbeta} {R_\text{MBS}}= \sum_{b=0}^{{B}} \gamma_b \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \max\{0, 1-b\mu_{i,j}\} F_{i,j}.$$ Note that $\max\{0, 1-b\mu_{i,j}\} = 1-\min\{1, b\mu_{i,j}\}$ and that, using together with ${F_{X_i}(0)} = 0$, $$\label{eq:sumF} \sum_{j=0}^K F_{i,j} = {F_{X_i}(KT)}-{F_{X_i}(0)}+1-{F_{X_i}(KT)} = 1.$$ Hence, we may rewrite as $$\begin{aligned} {R_\text{MBS}}& = \sum_{b=0}^{{B}} \gamma_b \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K (1-\min\{1, b\mu_{i,j}\}) F_{i,j}\nonumber\\ & = \sum_{i=1}^N \omega_i s_i - {R_\text{SBS}}, \label{eq:ratembs}\end{aligned}$$ i.e., all requested content not downloaded from SBSs is downloaded from the MBS. The average data rate at which the SBSs caches are updated, denoted by ${R_\text{C}}$, is $${R_\text{C}}= {B}\sum_{i=1}^N \omega_i s_i \sum_{j=0}^K (\mu_{i,0}-\mu_{i,j}) F_{i,j}.$$ The above expression assumes that all caches are updated at each request in the area, which we refer to as *synchronous* updates. This simplification is due to analytical tractability. Obtaining optimal caching policies under *asynchronous* updates appears to be a formidable task. In Section \[sec:results\], we nonetheless simulate caching policies that are optimal under synchronous updates for an asynchronous cache updating scenario. We define the *network load* as $$W = {{\theta}_\text{MBS}}{R_\text{MBS}}+{{\theta}_\text{SBS}}{R_\text{SBS}}+{{\theta}_\text{C}}{R_\text{C}}, \label{eq:W}$$ where $${{\theta}_\text{MBS}}{R_\text{MBS}}+{{\theta}_\text{SBS}}{R_\text{SBS}}= {{\theta}_\text{MBS}}\sum_{i=1}^N \omega_i s_i - ({{\theta}_\text{MBS}}-{{\theta}_\text{SBS}}) {R_\text{SBS}}\label{eq:sumrate}$$ using . We want to minimize the network load over the caching policies $\mu_i(t)$ under the constraints –. The first term in is independent of $\mu_i(t)$. Hence, minimizing is equivalent to minimizing $${{\theta}_\text{C}}{R_\text{C}}- ({{\theta}_\text{MBS}}-{{\theta}_\text{SBS}}) {R_\text{SBS}}.$$ Thus, minimizing the network load corresponds to the optimization problem $$\begin{aligned} \underset{\substack{\mu_{i,j}, \nu_i \in {\mathbb{R}}\\ \beta_{i,j} \in \{0,1\}}}{\text{minimize}}~ & {{\theta}_\text{C}}{R_\text{C}}- ({{\theta}_\text{MBS}}-{{\theta}_\text{SBS}}) {R_\text{SBS}}, \label{obj:min}\\ \text{subject to}~ & \text{\eqref{cnstr:cache}--\eqref{cnstr:beta2}}. \nonumber\end{aligned}$$ Consider briefly the case of zero cache update cost, i.e., ${{\theta}_\text{C}}= 0$. For ${{\theta}_\text{MBS}}> {{\theta}_\text{SBS}}$, we see that represents a maximization of the SBS download rate ${R_\text{SBS}}$ and for ${{\theta}_\text{MBS}}\le {{\theta}_\text{SBS}}$, has a trivial solution $\mu_{i,j} = 0$, i.e., caching at the SBSs is turned off (${R_\text{SBS}}= 0$) and all data is fetched from the MBS, for which $$W = {{\theta}_\text{MBS}}\sum_{i=1}^N \omega_i s_i$$ using and . Next, we reformulate the optimization problem in a way that is more tractable. Using the epigraph formulation [@Boyd2009 Ch. 3.1.7], we introduce the auxiliary optimization variables $\xi_{b,i,j} \in {\mathbb{R}}$ and the constraints $$\begin{aligned} \xi_{b,i,j} & \le 1, \label{cnstr:xi1}\\ \xi_{b,i,j} & \le b \mu_{i,j}. \label{cnstr:xi2}\end{aligned}$$ Expressing the SBS download rate in as $$\label{eq:ratesbstilde} {\tilde{R}_\text{SBS}}= \sum_{b=0}^B \gamma_b \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \xi_{b,i,j} F_{i,j},$$ with the notation ${\tilde{R}_\text{SBS}}$ to emphasize that it corresponds to the download rate of the epigraph formulation, we formulate the MILP $$\begin{aligned} \underset{\substack{\mu_{i,j}, \nu_i, \xi_{b,i,j} \in {\mathbb{R}}\\ \beta_{i,j} \in \{0,1\}}}{\text{minimize}}~ & {{\theta}_\text{C}}{R_\text{C}}- ({{\theta}_\text{MBS}}-{{\theta}_\text{SBS}}) {\tilde{R}_\text{SBS}}, \label{obj:epi}\\ \text{subject to}~ & \text{\eqref{cnstr:cache}--\eqref{cnstr:beta2}, \eqref{cnstr:xi1}, \eqref{cnstr:xi2}},\nonumber\end{aligned}$$ which is equivalent to and efficiently solved using, e.g., Gurobi [@Gurobi2018]. The MILP provides optimal FTTL caching policies for the distributed coded caching scenario. Optimal coded TTL policies are achieved by letting $\nu_i = 1$ and removing the constraint , while coded STTL policies are attained by removing the constraints –. Note that the STTL optimization problem is a linear program. Analysis as Single Cache TTL {#sec:transform} ---------------------------- In this subsection, we will show that the distributed coded caching problem using TTL caching policies in can equivalently be analyzed as a single cache TTL problem using a particular decreasing and convex cost function. We also show how our distributed caching problem maps to the sum utility maximization for the single cache case. Let the random variable $Y$ denote the number of SBSs within range of a user, with $\Pr(Y = b) = \gamma_b$, $b = 0, 1, \ldots, {B}$. Changing the order of summation in yields $$\begin{aligned} {R_\text{SBS}}& = \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K F_{i,j} \operatorname*{\mathbb{E}}[\min\{1, \mu_{i,j} Y\}].\label{eq:sumb}\end{aligned}$$ Regarding the expectation in , we will need the following lemma in subsequent theorems. \[lem:exp\] For a nonnegative random variable $Y$ and $\mu \ge 0$, $$\operatorname*{\mathbb{E}}[\min\{1, \mu Y\}] = \int_0^1 1 - F_Y(z/\mu) {~\mathrm{d}}z.$$ See Appendix \[prf:lemexp\]. The following theorem gives some important properties of the expectation in , as a function of the caching policy $\mu_{i,j}$. \[th:exp\] For a nonnegative random variable $Y$, the expectation $\operatorname*{\mathbb{E}}[\min\{1, \mu Y\}]$ is an increasing and concave function of $\mu \ge 0$. See Appendix \[prf:exp\]. The result of Theorem \[th:exp\] is interesting because it proves that is convex. Furthermore, it shows how the cost minimization with link costs ${{\theta}_\text{MBS}}= 1$, ${{\theta}_\text{SBS}}= 0$, and no cache update cost (${{\theta}_\text{C}}= 0$), which corresponds to a distributed caching scenario, maps to the utility maximization , which assumes a single cache. The following theorem considers the important special case of SBSs distributed in an area according to a Poisson point process, in which case $Y$ corresponds to a Poisson random variable [@Chiu2013 Ch. 2.3]. \[th:poissexp\] For $Y\sim\text{Poisson}(\lambda)$, $$\label{eq:poissexp} \operatorname*{\mathbb{E}}[\min\{1, \mu Y\}] = 1+(\lambda\mu-1) Q({\lceil1/\mu\rceil}, \lambda)-\frac{{\mathrm{e}}^{-\lambda} \lambda^{{\lceil1/\mu\rceil}} \mu}{\Gamma({\lceil1/\mu\rceil})},$$ where $Q(\cdot, \cdot)$ is the regularized Gamma function and $\Gamma(\cdot)$ is the Gamma function. See Appendix \[prf:poissexp\]. The expression is an increasing and concave function of $\mu$, according to Theorem \[th:exp\]. Due to the ceiling function ${\lceil1/\mu\rceil}$ in , we see that we should set $1/\mu \in {\mathbb{N}}$ in order to minimize while not wasting cache capacity resources (see ). Special Cases ============= The distributed coded caching problem utilizing TTL caching policies has two interesting problems as special cases; static caching ($K = 0$), studied in [@Shanmugam2013; @Bioglio2015; @Pedersen2019; @Wang2017] for MDS codes, and single cache ($\gamma_1 = 1$), investigated in [@Goseling2019]. In this section, we show the connection between our problem and the special case of static caching, which we prove is optimal under a Poisson request process, and the special case of a single cache, which we prove has a particularly simple optimal solution. Static Coded Caching -------------------- Before showing that includes static caching as a special case, we have the following theorem. \[th:static\] For a Poisson request process, static caching minimizes . See Appendix \[prf:static\]. Under static caching, FTTL and STTL are identical as only the updates distinguish the two caching policies. In the following, we assume that all files are of equal size, i.e., $s_i = s$, and study the nontrivial case ${{\theta}_\text{MBS}}> {{\theta}_\text{SBS}}$. For static caching ($K=0$), reduces to $$F_{i,j} = F_{i,0} = 1 - {F_{X_i}(0)} = 1$$ and the objective function becomes $$\begin{aligned} -{R_\text{SBS}}& = -\sum_{b=0}^{{B}} \gamma_b \sum_{i=1}^N \omega_i s_i \min\{1, b\mu_{i,0}\} F_{i,0}\nonumber\\ & = -\omega s \sum_{b=0}^{{B}} \gamma_b \sum_{i=1}^N p_i \min\{1, b\mu_{i,0}\}.\label{obj:bioglio}\end{aligned}$$ Also, is $$A_{i,j} = A_{i,0} = \int_0^\infty 1-{F_{X_i}(t)} {~\mathrm{d}}t = \operatorname*{\mathbb{E}}[X_i] = \omega_i^{-1}.$$ Hence, the constraint simplifies to $$\sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \mu_{i,j} A_{i,j} = s \sum_{i=1}^N \mu_{i,0} \le C. \label{cnstr:bioglio}$$ Using in , under constraints and , the optimization problem is precisely the static caching problem considered in [@Bioglio2015], apart from additive and multiplicative constants. Hence, the static caching problem explored in [@Bioglio2015] is a special case of . Single Cache TTL ---------------- We proceed with the other interesting special case, i.e., the single cache problem. Letting $\gamma_1 = 1$ in , i.e., users access a single cache with probability 1, we see that the average rate at which data is downloaded from the SBSs is $$\begin{aligned} {R_\text{SBS}}& = \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \min\{1, \mu_{i,j}\} F_{i,j} \nonumber\\ & = \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \mu_{i,j} F_{i,j}, \label{obj:minsingle}\end{aligned}$$ since $\mu_{i,j} \le 1$ using . For the nontrivial case of ${{\theta}_\text{MBS}}> {{\theta}_\text{SBS}}$, and update cost ${{\theta}_\text{C}}= 0$, the minimization problem is equivalent to a maximization problem of the objective function . In particular, the STTL problem has a surprisingly simple solution given by the following theorem. \[th:fracknap\] For users accessing a single cache, link costs ${{\theta}_\text{MBS}}> {{\theta}_\text{SBS}}$ and ${{\theta}_\text{C}}= 0$, the STTL optimization problem, i.e., maximizing under constraints and , is a fractional knapsack problem with a greedy optimal solution equivalent to FTTL and TTL. See Appendix \[prf:fracknap\]. A similar result was proved in [@Goseling2019] for a single file, i.e., $N=1$. Numerical Results {#sec:results} ================= In the following, we will assume that the times between requests are distributed according to the Weibull distribution, which has been shown to accurately estimate inter-request times [@Costa2004], i.e., $X_i \sim \text{Weibull}(a, b_i)$, where $a$, $0 < a \le 1$, and $b_i$ are the shape and scale parameters of the distribution, respectively. Its CDF is $${F_{X_i}(t)} = 1 - \exp\left[-\left(\frac{t}{b_i}\right)^{a}\right].$$ Also, $$\omega_i^{-1} = \operatorname*{\mathbb{E}}[X_i] = b_i \Gamma(1+a^{-1}),$$ which implies $$b_i = \frac{1}{\omega_i \Gamma(1+a^{-1})}.$$ We assume the aggregate request rate per hour $\omega = 100$ and $$p_i = \frac{1/i^\alpha}{\sum_{\ell=1}^N 1/\ell^\alpha},$$ which is the Zipf probability mass function with parameter $\alpha \ge 0$. We remind that $p_i$ has the interpretation of file popularity under a Poisson request process. The area is defined by the communication range of the MBS, which is denoted by ${r_\text{MBS}}$ and assumed to be ${r_\text{MBS}}= 800$ meters (m), i.e., the considered area is $\pi {r_\text{MBS}}^2 \approx 2$ square kilometers. The SBSs are deployed in the area according to a Poisson point process. Let $\rho$ be the density of SBSs per square kilometer (km$^{-2}$), i.e., $\rho = {B}/(\pi {r_\text{MBS}}^2)$. The probability that a user is within range of $b$ SBSs is [@Chiu2013 Ch. 2.3] $$\gamma_b = {\mathrm{e}}^{-\lambda} \frac{\lambda^b}{b!},$$ where $\lambda = \rho \pi {r_\text{SBS}}^2 = {B}({r_\text{SBS}}/{r_\text{MBS}})^2$. Unless stated otherwise, we will assume the following setup for the remainder of this section. The library holds $N = 100$ files, each of normalized size $s_i = 1$. We set the Weibull shape $a = 0.6$, which is within the range specified in [@Costa2004]. Also, we set $\alpha = 0.7$, which has been shown to accurately capture the popularity of Youtube videos [@Cheng2008]. We assume that there are $B = 100$ SBSs in the area, corresponding to a density $\rho \approx 50$ km$^{-2}$ and that users can download content from SBSs within a range of ${r_\text{SBS}}= 100$ m. Each SBS has the capacity to cache $C = 10$ files or $10\%$ of the file library. We assume the link costs ${{\theta}_\text{SBS}}= 0$ and ${{\theta}_\text{MBS}}= 1$. Furthermore, we assume that ${{\theta}_\text{C}}\ll {{\theta}_\text{MBS}}$, which is a reasonable assumption since data can be transmitted to caches over high capacity fiber-optical or highly directional wireless backhaul links, while the MBS serves a large number of users over potentially large distances. Finally, we consider an update window length of $K/f = 1$ hour and update frequencies $f = 6$ per hour. ![The fraction of data downloaded from the MBS as a function of the SBS density $\rho$.[]{data-label="fig:density"}](density.pdf){width="\columnwidth"} We obtain optimal TTL, FTTL, and STTL caching policies by solving and plot the network load normalized by the aggregate request rate $\omega$. For ${{\theta}_\text{C}}= 0$, the network load is interpreted as the fraction of data downloaded from the MBS. Fig. \[fig:density\] shows this fraction as a function of the SBS density $\rho$ for no cache updates, i.e., $f = 0$ implying $\mu_{i,j} = \mu_{i,0}$, and cache update frequency per hour $f = 6$. The network load using FTTL and STTL overlap for $f = 0$, which is expected since only the cache updates distinguish the two policies. We also see that there is a reduction in network load when choosing the FTTL or STTL caching policies over the TTL policy and that the network load decreases with increasing SBS density. The reason for the performance loss when using the TTL caching policy is that users within range of $b>1$ SBSs will download superfluous data, which correspond to a wasteful use of cache memory resources. The gain for the static caching scenario ($f = 0$) was observed already in [@Bioglio2015]. Finally, we observe that, for $f=6$, there is only a small reduction in network load for STTL as compared to FTTL, but the load reduction is increasing for increasing $\rho$. ![The fraction of data downloaded from the MBS as a function of the Weibull shape $a$ for an update frequency $f=6$.[]{data-label="fig:shape"}](shape.pdf){width="\columnwidth"} Fig. \[fig:shape\] shows the fraction of data downloaded from the MBS versus the Weibull shape $a$, for update frequency per hour $f=6$, and no cache update cost (${{\theta}_\text{C}}=0$). We also include curves for static caching ($f=0$), which do not depend on $a$ as is shown in and , for comparison. For bursty request arrivals, i.e., small values of $a$, we see that the use of TTL caching policies reduces the fraction of data downloaded from the MBS significantly with respect to static caching. Furthermore, we observe that, for very small values of $a$, all TTL policies have similar performance. This is because, with high probability, the times between requests are less than the period $T$, i.e., $F_{i,0} \approx 1$ for all $i$, and TTL is an optimal caching policy. For $a=1$, corresponding to a Poisson request process, FTTL and STTL yield the same network load as proved in Theorem \[th:static\], which is, however, lower than the network load using TTL. A similar effect was shown in [@Goseling2019] for the single cache case. Fig. \[fig:updatefreq\] shows the normalized minimum network load as a function of the update frequency $f$ for the case of no cache update cost (${{\theta}_\text{C}}= 0$) and ${{\theta}_\text{C}}= 10^{-3}$. For both cases, updating content on the caches is seen to be beneficial for all TTL policies. For example, using STTL and assuming ${{\theta}_\text{C}}= 0$, the reduction is roughly 10% as compared to static caching. We observe that the decrease in network load for an increase in update frequency saturates for moderately large $f$. Hence, cache updates need not be very frequent to reap the benefits of the TTL, FTTL, and STTL caching policies. The sufficient update frequency of course depends on several parameter values, in particular, the Weibull shape $a$ is a key parameter when deciding update frequencies. Finally, in Fig. \[fig:updatecost\], the normalized minimum network load is plotted versus the cache update cost ${{\theta}_\text{C}}$ for an update frequency per hour $f=6$. The load for static caching ($f=0$) is also shown in the figure. As previously described, the network load when using FTTL and STTL is the same for static caching. It is interesting to note that all TTL policies revert to static caching for sufficiently large values of ${{\theta}_\text{C}}$. Also included in Fig. \[fig:updatecost\] is a simulation of the optimal (under synchronous cache updates) STTL and TTL caching policies for asynchronous cache updates, i.e., only the SBSs within range of a user placing a request update cached content. The considered caching policies do not exhibit a better performance under asynchronous updates, which is to be expected for two reasons. Firstly, since the file request process is homogenous over the considered area, the spatial average cached content is important and the same average cached content can be achieved by both synchronous and asynchronous cache updates. Secondly, the request rate within the communication range of an SBS is smaller than the request rate in the entire area, implying less content to be cached over time using asynchronous updates, i.e., the caches are underutilized. ![Normalized network load as a function of the cache update frequency $f$.[]{data-label="fig:updatefreq"}](updatefreq.pdf){width="\columnwidth"} Conclusion ========== We optimized time-to-live (TTL) caching policies with periodic eviction of coded content to minimize the overall network load for a scenario where content is encoded using a maximum distance separable code and cached in a distributed fashion across small base stations. The proposed optimization problem is efficiently solved numerically. Interestingly, we show that the problem can equivalently be analyzed as a single cache optimization problem under a specific decreasing and convex cost function. For small base stations deployed according to a Poisson point process, we provide the cost function explicitly. The analyzed scenario encompasses static caching and single caching as important special cases. We proved that, interestingly, static caching is optimal under a Poisson request process. We also proved that the single cache problem has a simple greedy solution. We showed that TTL caching policies can offer substantial reductions in network load compared with static caching under a request renewal process, in particular when the request process is bursty. Conversely, for sufficiently large cache update cost, dynamic caching is futile, i.e., static caching is optimal. Finally, although we consider a wireless network scenario, the results are general in the sense that they can be applied to any distributed caching scenario. ![Normalized network load versus the update cost ${{\theta}_\text{C}}$ when using the various caching policies under synchronous and asynchronous cache updates.[]{data-label="fig:updatecost"}](updatecost.pdf){width="\columnwidth"} Proof of Lemma \[lem:exp\] {#prf:lemexp} ========================== We represent $1$ as a random variable with degenerate distribution $\delta(z-1)$, where $\delta(\cdot)$ is the Dirac delta function, and let $Z = \min\{1, \mu Y\}$, for which the CDF of $Z$ is $$\begin{aligned} F_Z(z) & \triangleq \Pr(Z \le z)\\ & = \Pr(\min\{1, \mu Y\} \le z)\\ & = 1-\Pr(1>z, \mu Y > z)\\ & = 1-H(1-z) (1-F_Y(z/\mu)),\end{aligned}$$ where $H(\cdot)$ is the heavyside function. The expected value of $Z$ is $$\begin{aligned} \operatorname*{\mathbb{E}}[Z] & = \int_0^\infty 1-F_Z(z) {~\mathrm{d}}z\\ & = \int_0^\infty H(1-z) (1-F_Y(z/\mu)) {~\mathrm{d}}z\\ & = \int_0^1 1- F_Y(z/\mu) {~\mathrm{d}}z.\end{aligned}$$ Proof of Theorem \[th:exp\] {#prf:exp} =========================== Since $F_Y(y)$ is an increasing function of $y$, $1-F_Y(z/\mu)$ is an increasing function of $\mu$, and $$\int_0^1 1-F_Y(z/\mu) {~\mathrm{d}}z$$ is an increasing function of $\mu$. Using Lemma \[lem:exp\], the expectation $\operatorname*{\mathbb{E}}[\min\{1, \mu Y\}]$ is an increasing function of $\mu$. For $\mu_1 \ge 0$, $\mu_2 \ge 0$, $Y \ge 0$, and $0 \le \alpha \le 1$, the following inequalities hold, $$\begin{aligned} (1-\alpha) \mu_1 Y & \ge (1-\alpha) \min\{1, \mu_1 Y\},\\ \alpha \mu_2 Y & \ge \alpha \min\{1, \mu_2 Y\}.\end{aligned}$$ Hence, $$((1-\alpha) \mu_1 + \alpha \mu_2) Y \ge (1-\alpha) \min\{1, \mu_1 Y\} + \alpha \min\{1, \mu_2 Y\}. \label{eq:conv1}$$ Similarly, using $$\begin{aligned} (1-\alpha) & \ge (1-\alpha) \min\{1, \mu_1 Y\},\\ \alpha & \ge \alpha \min\{1, \mu_2 Y\},\end{aligned}$$ we have that $$1 = 1-\alpha+\alpha \ge (1-\alpha) \min\{1, \mu_1 Y\} + \alpha \min\{1, \mu_2 Y\}. \label{eq:conv2}$$ Using and , we get $$\begin{aligned} & \min\{1, ((1-\alpha) \mu_1 + \alpha \mu_2) Y\}\\ & \hspace{4em} \ge (1-\alpha) \min\{1, \mu_1 Y\} + \alpha \min\{1, \mu_2 Y\}.\end{aligned}$$ Taking the expectation of both sides yields $$\begin{aligned} & \operatorname*{\mathbb{E}}[\min\{1, ((1-\alpha) \mu_1 + \alpha \mu_2) Y\}]\\ & \hspace{4em} \ge \operatorname*{\mathbb{E}}[(1-\alpha) \min\{1, \mu_1 Y\} + \alpha \min\{1, \mu_2 Y\}]\\ & \hspace{4em} = (1-\alpha) \operatorname*{\mathbb{E}}[\min\{1, \mu_1 Y\}] + \alpha \operatorname*{\mathbb{E}}[\min\{1, \mu_2 Y\}],\end{aligned}$$ which concludes the proof. Proof of Theorem \[th:poissexp\] {#prf:poissexp} ================================ For a Poisson random variable $Y$ with rate $\lambda$, $$\label{eq:ycdf} F_Y(y) = \sum_{i=0}^{{\lfloory\rfloor}} \frac{\lambda^i}{i!} {\mathrm{e}}^{-\lambda}.$$ For a positive integer $x$, let $\Gamma(x)$ and $Q(x, \lambda)$ denote the Gamma function and the regularized Gamma function, i.e., $\Gamma(x) = (x-1)!$ and $$\label{eq:qrec} Q(x+1, \lambda) = \int_\lambda^\infty \frac{t^x {\mathrm{e}}^{-t}}{\Gamma(x+1)} {~\mathrm{d}}t \overset{(a)}{=} \frac{\lambda^x {\mathrm{e}}^{-\lambda}}{\Gamma(x+1)} + Q(x, \lambda),$$ respectively, where $(a)$ is obtained after integration by parts. Unfolding the recursion in , using $Q(1, \lambda) = {\mathrm{e}}^{-\lambda}$ yields $$\label{eq:qsum} Q(x+1, \lambda) = \sum_{i=0}^x \frac{\lambda^i}{\Gamma(i+1)} {\mathrm{e}}^{-\lambda} = \sum_{i=0}^x \frac{\lambda^i}{i!} {\mathrm{e}}^{-\lambda}$$ and $$\label{eq:qcdf} Q({\lfloory\rfloor}+1, \lambda) = F_Y(y),$$ using . Using , $$\begin{aligned} & \int_0^1 F_Y(y/\mu) {~\mathrm{d}}y = \int_0^1 Q({\lfloory/\mu\rfloor}+1, \lambda) {~\mathrm{d}}y\nonumber\\ & \hspace{2em} = \mu \sum_{i=1}^{{\lfloor1/\mu\rfloor}} Q(i, \lambda) + (1-{\lfloor1/\mu\rfloor}\mu) Q({\lfloor1/\mu\rfloor}+1, \lambda),\label{eq:cdfint}\end{aligned}$$ where the integral is a summation due to the floor function in the argument of $Q(\cdot, \cdot)$. Applying the recursion repeatedly yields $$\begin{aligned} \sum_{i=1}^{{\lfloor1/\mu\rfloor}} Q(i, \lambda) & = {\lfloor1/\mu\rfloor} Q({\lfloor1/\mu\rfloor}+1, \lambda) - \sum_{i=1}^{{\lfloor1/\mu\rfloor}} i \frac{\lambda^i {\mathrm{e}}^{-\lambda}}{\Gamma(i+1)}\\ & = {\lfloor1/\mu\rfloor} Q({\lfloor1/\mu\rfloor}+1, \lambda) - \lambda \sum_{j=0}^{{\lfloor1/\mu\rfloor}-1} \frac{\lambda^{j} {\mathrm{e}}^{-\lambda}}{\Gamma(j+1)}\\ & \overset{(b)}{=} {\lfloor1/\mu\rfloor} Q({\lfloor1/\mu\rfloor}+1, \lambda) - \lambda Q({\lfloor1/\mu\rfloor}, \lambda),\end{aligned}$$ where we have used in $(b)$. Inserting this expression in , one obtains $$\begin{aligned} \int_0^1 F_Y(y/\mu) {~\mathrm{d}}y & = Q({\lfloor1/\mu\rfloor}+1, \lambda) - \mu \lambda Q({\lfloor1/\mu\rfloor}, \lambda)\nonumber\\ & \overset{(c)}{=} (1-\lambda\mu) Q({\lceil1/\mu\rceil}, \lambda) + \frac{\lambda^{{\lceil1/\mu\rceil}} \mu {\mathrm{e}}^{-\lambda}}{\Gamma({\lceil1/\mu\rceil})},\label{eq:cdfint2}\end{aligned}$$ where we have used and $${\lceil1/\mu\rceil}-{\lfloor1/\mu\rfloor} = \begin{cases} 0, & \text{if}~1/\mu \in \mathbb{Z}\\ 1, & \text{if}~1/\mu \notin \mathbb{Z} \end{cases}$$ in $(c)$. Combining with the result of Lemma \[lem:exp\] yields the desired result. Proof of Theorem \[th:static\] {#prf:static} ============================== For proving that static caching minimizes (and hence ) it is sufficient to show that it maximizes ${\tilde{R}_\text{SBS}}$ (see ), as for static caching $R_C = 0$. Maximizing under constraints , , , and , is equivalent to $$\begin{aligned} \underset{\mu_{i,j}, \xi_{b,i,j}, C_i \in {\mathbb{R}}}{\text{maximize}}~ & \sum_{b=0}^B \gamma_b \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \xi_{b,i,j} F_{i,j},\label{obj:sbs}\\ \text{subject to}~ & s_i \sum_{j=0}^K \mu_{i,j} F_{i,j} \le C_i,\label{cnstr:cache1}\\ & \sum_{i=1}^N C_i = C,\label{cnstr:cache2}\\ & \mu_{i,j} - \mu_{i,j-1} \le 0,~\mu_{i,-1} = 1,\\ & -\mu_{i,j} \le 0,\\ & \xi_{b,i,j} \le 1,\\ & \xi_{b,i,j} \le b \mu_{i,j},\label{cnstr:xi2sbs}\end{aligned}$$ where $C_i$ may be regarded as the size of the cache partition reserved for file $i$, and we used the fact that for exponentially distributed inter-request times $F_{i,j}/A_{i,j} = \omega_i$ (from and ), i.e., the hazard function is constant for a Poisson request process. For fixed $C_i$’s, the maximization problem is separable in $i$. Thus, we can consider the following optimization problem $$\begin{aligned} \underset{\mu_{i,j}, \xi_{b,i,j} \in {\mathbb{R}}}{\text{maximize}}~ & \sum_{b=0}^B \gamma_b \sum_{j=0}^K \xi_{b,i,j} F_{j}, \label{obj:single}\\ \text{subject to}~ & s_i \sum_{j=0}^K \mu_{i,j} F_{i,j} \le C_i, \label{cnstr:cachesingle}\\ & \mu_{i,j} - \mu_{i,j-1} \le 0,~\mu_{i,-1} = 1, \label{cnstr:mu1single}\\ & -\mu_{i,j} \le 0, \label{cnstr:mu2single}\\ & \xi_{b,i,j} \le 1, \label{cnstr:xi1single}\\ & \xi_{b,i,j} \le b \mu_{i,j}, \label{cnstr:xi2single}\end{aligned}$$ for each file $i=1, \ldots, N$ separately. We can now prove the following lemma. \[lem:kkt\] Static caching is an optimal solution to –. For ease of exposition, we drop the subindex $i$ in the proof. Introducing the dual variables $\lambda$, $\phi_j$, $\psi_j$, $\delta_{b,j}$, and $\epsilon_{b,j}$, the Karush-Kuhn-Tucker (KKT) conditions of – are $$\begin{aligned} -\gamma_b F_j + \delta_{b,j} + \epsilon_{b,j} & = 0,\\ \lambda s F_j + \phi_j - \phi_{j+1} - \psi_j - \sum_{b=0}^B \epsilon_{b,j} b & = 0,~\phi_{K+1} = 0,\\ \lambda \left( - C + s \sum_{j=0}^K \mu_j F_j \right) & = 0,\\ \phi_j (\mu_j-\mu_{j-1}) & = 0,~\mu_{-1} = 1,\\ \psi_j (-\mu_j) & = 0,\\ \delta_{b,j} (\xi_{b,j}-1) & = 0,\\ \epsilon_{b,j} (\xi_{b,j}-b\mu_j) & = 0,\\ \lambda & \ge 0,\\ \phi_j & \ge 0,\\ \psi_j & \ge 0,\\ \delta_{b,j} & \ge 0,\\ \epsilon_{b,j} & \ge 0,\end{aligned}$$ and –. Let $$\begin{aligned} \mu_j & = C/s,\\ \xi_{b,j} & = \min\{1, b C/s\},\end{aligned}$$ which corresponds to static caching utilizing completely the given cache partition, and largest possible values of the variables $\xi_{b,j}$. Furthermore, let $$\begin{aligned} \phi_j & = 0,\\ \psi_j & = 0,\\ \delta_{b,j} & = \begin{cases} 0, & \text{if}~b \le s/C\\ \gamma_b F_j, & \text{if}~b > s/C \end{cases},\\ \epsilon_{b,j} & = \begin{cases} \gamma_b F_j, & \text{if}~b \le s/C\\ 0, & \text{if}~b > s/C \end{cases},\label{eq:epsilon}\\ \lambda & = \frac{1}{s} \sum_{j=0}^K \sum_{b=0}^B \epsilon_{b,j} b \overset{(a)}{=} \frac{1}{s} \sum_{b=0}^{{\lfloors/C\rfloor}} \gamma_b b,\end{aligned}$$ where we have used and in $(a)$. It is readily verified that the choice of optimization and dual variables satisfy the KKT conditions and are hence optimal since the problem is convex [@Boyd2009 Ch. 5.5.3]. Therefore, static caching maximizes . It remains to optimize over the $C_i$, but since, by Lemma \[lem:kkt\], static caching is optimal for any assignment of $C_i$’s, it is optimal for –. Proof of Theorem \[th:fracknap\] {#prf:fracknap} ================================ Letting $\gamma_1 = 1$, ${{\theta}_\text{C}}= 0$, and ${{\theta}_\text{MBS}}> {{\theta}_\text{SBS}}$, the STTL problem, i.e., maximizing under constraints and , is equivalent to $$\begin{aligned} \underset{\mu_{i,j} \in {\mathbb{R}}}{\text{maximize}}~ & \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \mu_{i,j} F_{i,j},\\ \text{subject to}~ & \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \mu_{i,j} A_{i,j} \le C,\\ & 1 \ge \mu_{i,0} \ge \mu_{i,1} \ge \cdots \ge \mu_{i,K} \ge 0.\label{cnstr:mufracknap}\end{aligned}$$ Relaxing the constraint , replacing it with $0 \le \mu_{i,j} \le 1$, and letting $x_{i,j} = \omega_i s_i \mu_{i,j} A_{i,j}$, we obtain $$\begin{aligned} \underset{x_{i,j} \in {\mathbb{R}}}{\text{maximize}}~ & \sum_{i=1}^N \sum_{j=0}^K x_{i,j} \frac{F_{i,j}}{A_{i,j}},\\ \text{subject to}~ & \sum_{i=1}^N \sum_{j=0}^K x_{i,j} \le C,\\ & 0 \le x_{i,j} \le \omega_i s_i A_{i,j},\end{aligned}$$ which is recognized as the fractional knapsack problem [@Danzig1957]. The optimal solution to this problem is obtained by setting $x_{i,j} = \omega_i s_i A_{i,j}$, i.e., $\mu_{i,j} = 1$, greedily with respect to the fractions $F_{i,j}/A_{i,j}$ [@Danzig1957]. We observe that, since $F_{i,j}/A_{i,j}$ is decreasing in $j$ as explained in Sec. \[sec:prel\], the constraint is met and we have a valid STTL caching policy. Apart from the possibility that one $\mu_{i,j}<1$ depending on the value of $C$, the constraints – are also satisfied by this policy and, hence, the optimal STTL caching policy is equivalent to the optimal FTTL and TTL caching policies. [^1]: This work was funded by the Swedish Research Council under grant 2016-04253 and by the National Center for Scientific Research in France under grant CNRS-PICS-2016-DISCO. [^2]: J. Pedersen, A. Graell i Amat, and F. Brännström are with the Department of Electrical Engineering, Chalmers University of Technology, SE-41296 Gothenburg, Sweden (e-mail: {jesper.pedersen, alexandre.graell, fredrik.brannstrom}@chalmers.se). [^3]: J. Goseling is with the Department of Applied Mathematics, University of Twente, 7522 Enschede, The Netherlands (e-mail: j.goseling@utwente.nl). [^4]: I. Andriyanova is with the ETIS-UMR8051 group, ENSEA/University of Cergy-Pontoise/CNRS, 95015 Cergy, France (e-mail: iryna.andriyanova@ensea.fr). [^5]: E. Rosnes is with Simula UiB, N-5020 Bergen, Norway (e-mail: eirikrosnes@simula.no).

--- abstract: 'Blebs are cell protrusions generated by local membrane–cortex detachments followed by expansion of the plasma membrane. Blebs are formed by some migrating cells, for example primordial germ cells of the zebrafish. While blebs occur randomly at each part of the membrane in unpolarized cells, a polarization process guarantees the occurrence of blebs at a preferential site and thereby facilitates migration towards a specified direction. Little is known about the factors involved in development and maintenance of a polarized state, yet recent studies revealed the influence of an intracellular flow and the stabilizing role of the membrane-cortex linker molecule Ezrin. Based on this information, we develop and analyse a coupled bulk-surface model describing a potential cellular mechanism by which a bleb could be induced at a controlled site. The model rests upon intracellular Darcy flow and a diffusion-advection-reaction system, describing the temporal evolution from an unpolarized to a stable polarized Ezrin distribution. We prove the well-posedness of the mathematical model and show that simulations qualitatively correspond to experimental observations, suggesting that indeed the interaction of an intracellular flow with membrane proteins can be the cause of the cell polarization.' author: - Carolin Dirks - Paul Striewski - Benedikt Wirth - Anne Aalto - 'Adan Olguin-Olguin' - Erez Raz bibliography: - 'mybibliography.bib' title: A mathematical model for cell polarization in zebrafish primordial germ cells --- Introduction ============ Several recent studies investigated the directional cell migration process via local membrane protrusions, so-called blebs. While the mechanisms of the actual bleb formation are quite well understood, the process of cell polarization leading to a stable *directional* blebbing remains still unexplained. In some recent works (such as [@Ref:PaluchRaz2013 Paluch, Raz, 2013], [@Ref:FritzscheEtAl2014 Fritzsche, Thorogate et al., 2014]), researchers suggested the role of the membrane-cortex linker Ezrin in inhibiting the probability for bleb formation in regions with a high Ezrin concentration. In addition, a directed intracellular flow has been observed during cell polarization that seems to be related to the occurrence of so-called actin brushes, filamentous actin structures forming at the front side of the cell [@Ref:KardashEtAl2010 Kardash, Reichmann-Fried, 2010]. In this article, we take up these observations and hypothesize that shear stresses induced by the intracellular flow may lead to a local destabilization of the Ezrin linkages between membrane and cortex, resulting in a redistribution of membranous Ezrin and bleb formation. This hypothesis is tested using a mathematical model for the time interval between actin brush formation and the onset of blebbing. The model incorporates an intracellular flow driven by actin brushes and a description of the flow-controlled membranous Ezrin concentration including turnover rates from active (membrane-bound) and inactive (cytosolic) Ezrin. The experimentally observed Ezrin depletion in the front and accumulation in the back of the cell can be reproduced by the model. Thereby our model positively answers the question whether there could be a mechanical basis for Ezrin polarization, in our case an actin-induced flow. This work is organized as follows. We start with providing a brief overview of the biological context and the related work, and we introduce the notation used throughout this article. In \[sec:Model\], we describe our model used to simulate the temporal behaviour of the active Ezrin. The corresponding model analysis is presented in \[sec:Analysis\], where we prove well-posedness of the surface equation. Finally, we describe the numerical treatment of the coupled bulk-surface equation system and compare simulation results to experiments in \[sec:Experiments\]. Biological setting ------------------ The process of directional cell migration is an important and extensively studied mechanism in early embryonic development. A widely used model for in vivo studies are primordial germ cells (PGCs). These cells are specified within the embryo and have to travel a certain distance to reach their destination, namely the site where the gonad develops [@Ref:DoitsidouEtAl2002 Doitsidou, Reichmann-Fried et al., 2002]. This migration process is performed via blebs, local detachments of the cell membrane from the cortex which move the cell to a certain direction specified by a chemical gradient. Little is known about the signaling process within the cell in the time interval between the arrival of the chemical signal and the actual directed movement, in which the cell changes from an unpolarized to a polarized state. However, several factors have been shown to play a role in the polarization process [@Ref:PaluchRaz2013 Paluch, Raz, 2013]. Blebbing is produced by an increase in the intracellular pressure coupled to detachment of the cell membrane from the cell cortex. While migrating, PGCs go through two different phases, named “run” and “tumble”. During the “tumble” state, PGCs are apolar and blebs are formed at random sites around the cell perimeter. When the PGCs are in the “run” state the cells are polarized such that blebs form predominantly in one direction which is defined as the leading edge [@Ref:KardashEtAl2010 Kardash, Reichmann-Fried, 2010], [@Ref:PaksaRaz2015 Paksa, Raz, 2015]. Although the entire process of PGC polarization has not yet been fully understood, some factors have been identified to wield a strong influence. In the polarized state, a preferential polymerization of filamentous actin structures, so-called actin brushes, at the front edge of cell was reported, whereas such structures are absent in unpolarized cells [@Ref:KardashEtAl2010 Kardash, Reichmann-Fried, 2010]. The actin brushes are considered to be responsible for a recruitment of myosin, that leads to an increase of the contractility, favouring the corresponding side of the cell as the leading edge [@Ref:PaluchRaz2013 Paluch, Raz, 2013]. The accumulation of actin brushes is furthermore assumed to be correlated with a flow of cytoplasm towards the expanding bleb on the one hand and a strong retrograde flow of cortical actin on the other [@Ref:KardashEtAl2010 Kardash, Reichmann-Fried, 2010], [@Ref:ReigEtAl2014 Reig, Pulgar et al., 2014]. Moreover, a frequently reported feature of polarized blebbing cells is a local decrease of the membrane-cortex attachment at the front edge in combination with an increase at the back. Hence, a negative correlation between the propensity for blebbing and the stability of membrane-cortex attachment is assumed. A presumable candidate for regulating the membrane-cortex attachment is the linker molecule Erzin [@Ref:PaluchRaz2013 Paluch, Raz, 2013]. Experiments have shown that in polarized cells, Ezrin accumulates at the back [@Lorentzen1256 Lorentzen, Bamber et al.,2011]. Besides, the linker molecule is able to switch between an active and an inactive state. During its active form, it links the cell cortex to the membrane via two binding terminals (the membrane-binding N-terminal and the actin-binding C-terminal), whereas in its inactive form, these terminals interact with each other causing the molecule to diffuse within the cytoplasm. Ezrin constantly keeps turning from one state to the other, resulting in a frequent change between binding to and detaching from the membrane [@Ref:FritzscheEtAl2014 Fritzsche, Thorogate et al., 2014], [@Ref:BruecknerEtAl2015 Brückner, Pietuch et al., 2015]. To get a better understanding of the intracellular events involved in the polarization processes, we develop a mathematical model expressing the interaction of different factors which are known to play a role in the emergence and maintenance of a polarized state. The model focuses on the role and regulation of the active and inactive Ezrin concentration, including the influence of the cytoplasmic flow driven by localised actin-myosin contraction. In particular, we present a potential model for the binding and unbinding dynamics along the cell membrane by incorporating the reported information together with reviewing some additional hypotheses. Related work ------------ A variety of models for cell polarization have been proposed, many of them based on reaction-diffusion equations, suggesting that diffusive instabilities are involved in the process of cell polarization [@Ref:Levine Levine et al. 2006], [@Ref:OnsumRao Onsum, Rao, 2007], [@Ref:RaetzRoeger2012 Rätz, Röger, 2012], [@Ref:RaetzRoeger2014 Rätz, Röger, 2014]. Cell polarization induced by active transport of polarization markers was for example studied by [@Ref:Hawkins Hawkins, Bénichou et al., 2009], [@Ref:Calvez Calvez, Hwakins et al., 2012]. In either article, the presented models account for active transport of polarization markers along the cytoskeleton. [@hausberg2018well Hausberg, Röger, 2018] described the activity of GTPases by a system of three coupled bulk-surface advection-reaction-diffusion equations. The system models the interconversion of active and inactive GTPase, lateral drift and diffusion of molecules along the membrane and also the diffusion of inactive molecules into the cytosol. In contrast to our approach, Hausberg and Röger suggest flow-independent reaction terms and assume the geometry of the cell to be more regular than we do. [@Ref:GarckeKampmann2015 Garcke, Kampmann et al., 2015] proposed a model for lipid raft formation in cellular membranes and their interaction with intracellular cholesterol. Although not directly linked to cell polarization, their model comprises phase separation and interaction energies, which are similar to those presented in this article. In their work, a diffusion equation is to account for the intracellular diffusion of cholesterol, whereas a Cahn–Hilliard equation coupled with a reaction-diffusion-type equation models the formation of rafts and cholesterol binding and unbinding dynamics on the membrane. In a recent work, [@Ref:BurgerPietschmann2019 Burger, Pietschmann et al., 2019] presented a model for bleb formation based on a combination of intracellular flow and interactions of linker molecules. Their model includes a variable domain, where a bleb is initiated by a certain threshold distance between membrane and cortex. The flow inside and outside of the cell domain is modelled via Stokes equations, interactions of active and inactive linker molecules are described by a reaction-diffusion system. Although the main model ingredients are similar, the model details (such as the type of flow or our resulting phase field equation) as well as the mathematical analysis differ, where in both cases we aim to impose a minimum amount of extra assumptions. Additionally, the numerical simulations in [@Ref:BurgerPietschmann2019 Burger, Pietschmann et al., 2019] target the membrane-cortex disruption, while the aim of our approach is to investigate the effect of different model parameters on the global distribution of the linker molecules and to compare the results directly to biological experiments. Outside of the scope of this work are models that describe the process of the actual bleb formation, as presented for example in [@Ref:YoungMitran Young, Mitran, 2010] or [@Ref:StrychalskiGuy Strychalsky, Guy, 2012]. Notation -------- Throughout the article, we assume that $\Omega$ is an open and convex bounded domain in ${\mathbf{R}}^n$ with boundary $\Gamma = \partial \Omega$, modelling a cell and its membrane. For a function $g$ defined on $(0,T)\times\Omega$, we denote the temporal derivative and the gradient with respect to the spatial variable by $\partial_t g$ and $\nabla g$ respectively. The symbols $\nabla_{\Gamma}$, $\Delta_{\Gamma}$ and $\mathrm{div}_{\Gamma}$ are used to denote the surface gradient, Laplace-Beltrami operator and surface divergence on $\Gamma$. Additionally, we make use of the following conventions (the units refer to a three-dimensional cell): ------------------------------------------------- ------------------------------------------------------ $w:(0,T)\times\Omega\rightarrow {\mathbf{R}}^n$ intracellular flow \[$\mu$m/s\] $p:(0,T)\times\Omega\rightarrow {\mathbf{R}}$ intracellular pressure \[Pa\] $f:(0,T)\times\Omega\rightarrow {\mathbf{R}}^n$ flow-inducing force per unit volume \[N/m$^3$\] $u:(0,T)\times\Gamma\rightarrow [0,1]$ density of active Ezrin \[mol/$\mu$m$^2$\] $v:(0,T)\times\Omega\rightarrow [0,1]$ density of inactive Ezrin \[mol/$\mu$m$^3$\] $\kappa:\Omega\rightarrow{\mathbf{R}}$ local permeability of cytoplasmatic matrix \[m$^2$\] $\rho$ density of cytoplasm \[g/cm$^3$\] $\lambda$ kinematic viscosity of cytoplasm \[cm$^2$/s\] $\nu$ diffusion coefficient in cytoplasm \[$\mu$m$^2$/s\] ------------------------------------------------- ------------------------------------------------------ Model derivation {#sec:Model} ================ In this chapter, we present and describe a coupled bulk-surface model for the contribution of the linker-molecule Ezrin to the polarized state of a blebbing cell. The model covers the time interval in between the formation of the actin brushes, creating an intracellular flow, and the point where the cell arrives in a stable polarized state. The key points of the model can be summarized as follows: We assume a circular intracellular flow driven by actin-myosin motors, which creates a shear stress along the cell membrane. This stress destabilizes membrane-bound Ezrin, which as a consequence undergoes a retrograde flow to the back side of the cell. In addition, a nonlinear self-enhancing effect is modelled by a stronger affinity of inactive Ezrin to bind in regions with a high active Ezrin concentration and vice versa. The system variables of our model are ----- ------------------------------------------ $w$ intracellular flow $p$ intracellular pressure $u$ density of active (membrane-bound) Ezrin $v$ density of inactive (unbound) Ezrin. ----- ------------------------------------------ Model description ----------------- The model presented in this section aims at incorporating all biological information assumed to play a role in bleb formation, while trying to be as simple as possible on the other side. In detail it is based on the following observations: 1. Very recent experiments [@Ref:OlguinEtAl2019 Olguin-Olguin, Aalto et al., 2019] suggest that the membrane-cortex linker Ezrin becomes localised to the cell back where it functions in inhibiting bleb formation. 2. Experiments observed high cytosolic diffusion rates of inactive Ezrin in the range of 30$\mu$m$^2$/s [@Ref:Coscoy Coscoy et al.]. We show (see \[sec:Nondimensionalisation\]) that as a consequence, on the time scales of interest cytosolic concentrations of inactive Ezrin are therefore likely to be spatially constant throughout the cell. On the other hand, [@Ref:FritzscheEtAl2014 Fritzsche et al.] reported a slow diffusion of membranous Ezrin of around 0.003$\mu$m$^2$/s, caused by differences between the binding characteristics of its N- and C-terminal. 3. Experiments suggest the existence of an intracellular flow [@goudarzi2017bleb Goudarzi, Tarbashevich et al., 2017], presumably mediated by myosin motors in the actin brushes, resulting in a fountain-like motion pattern of cytoplasmic particles (also modelled in [@Ref:StrychalskiGuy Strychalsky, Guy, 2012], for instance). We hypothesize that beyond mere passive transport of Ezrin molecules, hydrodynamic shear stresses might lead to alteration of the binding and unbinding dynamics of membrane-cortex linkers, especially to an increased dissociation in regions of high stress. 4. As different studies revealed [@Ref:Berryman Berryman et al], [@Ref:Gautreau Gautreau et al.], inactive Ezrin can form oligomers in the cytoplasm and at the plasma membrane, and the article [@Ref:Herrig Herrig et al.] suggests that Ezrin binds cooperatively to membrane regions with a high concentration of certain types of phospholipids (PIP(2)). Thus, the binding of Ezrin between cortex and membrane might be reinforced by higher active Ezrin concentrations. On the other hand, a lack of membrane-cytoskeleton linkers could result in delamination of the membrane from the cortex, thereby preventing the binding of Ezrin to the cortex-membrane complex so that low active Ezrin concentrations may have an inhibitory effect. Finally, a certain maximum density of linker molecules between cortex and membrane cannot be exceeded. Altogether this suggests a nonlinear influence of the active Ezrin concentration on the Ezrin binding dynamics. A generic primordial germ cell (PGC) is expressed by a time-independent domain $\Omega \subset {\mathbf{R}}^n$ with a smooth boundary $\Gamma = \partial \Omega$ (which is temporally static throughout our work since we are only interested in the cell behaviour *before* bleb formation). To simulate a realistic environment, one would choose $n=3$, while for the sake of simplicity, we set $n=2$ in our simulations, since no qualitative difference is expected in the behaviour between a two- or a three-dimensional cell. The density of active Ezrin (in mol/$\mu$m$^2$) is then described by a function $u:(0,T)\times\Gamma \to {\mathbf{R}}$, the density of inactive Ezrin (in mol/$\mu$m$^3$) as $v:(0,T)\times\Omega\rightarrow{\mathbf{R}}$. As the cytoplasm can be described as a poroelastic material (see e.g. [@Ref:Moeendarbary Moeendarbary et al.]), cytoskeleton and organelles behaving like elastic solids and the cytosol like a fluid, the cytoplasmic flow $w$ is described by the incompressible Brinkman–Navier–Stokes equation $$\begin{aligned} \rho\partial_t w + \rho(w\cdot\nabla)w + \frac{\rho\lambda}{\kappa}w &= -\nabla p + \rho\lambda\Delta w + f &&\text{ in } \Omega, \label{Eq:BNSEquation} \\ \mathrm{div}\, w &= 0\, ,&&\text{ in } \Omega\label{Eq:incompressibility}\end{aligned}$$ with no-outflow boundary conditions. Here, $\kappa$ denotes the (spatially varying) permeability, $p$ denotes the intracellular pressure and $f$ a body force induced by actin-myosin contraction in the actin brushes. The parameter $\lambda$ represents the kinematic viscosity and $\rho$ the density of the cytoplasm. The density $u$ of active Ezrin on $\Gamma$ is obtained as a solution of the reaction-advection-diffusion equation $$\begin{aligned} \partial_t u &= -\underbrace{\mathrm{div}_{\Gamma}(u\, w)}_{\mathrm{advection}} + \underbrace{\vphantom{()}\nu\, \Delta_{\Gamma} u}_{\mathrm{diffusion}} - \underbrace{D(w,u,v)}_{\mathrm{desorption}} + \underbrace{A(w,u,v)}_{\mathrm{adsorption}} & \text{in }\ \Gamma\, . \label{Eq:SurfaceEquation}\end{aligned}$$ The two functions $A,D:{\mathbf{R}}^n\times{\mathbf{R}}\times{\mathbf{R}}\to {\mathbf{R}}^{\geq 0}$ describe the adsorption and desorption kinetics of membranous Ezrin, which have been studied for instance in [@Ref:Bosk Bosk et al.], but whose exact form is not known. In order to derive reasonable rate descriptions, we model the turnover between the two concentrations via classical reaction kinetics. To this end, denote by $u_{\max}$ the theoretical concentration of Ezrin binding sites on the cell membrane-cortex complex and by $b=u_{\max}-u$ the concentration of free sites. Due to potential local membrane-cortex detachments, it might happen that some binding sites are actually not available for inactive Ezrin since the distance between membrane and cortex is too large. Hence, we denote by $\tilde{b}\leq b$ the binding site concentration where new Ezrin is actually allowed to bind. Since a locally higher active Ezrin concentration is associated with a smaller distance between membrane and cortex, there is a (at least to first order) linear relation between $b$ and $\tilde{b}$ via $$\tilde{b} = (\tfrac{u}{u_{\max}})^\alpha b$$ for some exponent $\alpha>0$, which for lack of experimental data we will choose as $\alpha=1$ (as in the case of membrane-binding experiments for the small Rho GTPase Cdc42 in [@Ref:GoryachevPokhilko2008 Goryachev, Pokhilko 2008]; the choice does not affect the qualitative model behaviour). The binding rates follow the classical reaction kinetics inactive Ezrin $v$ + available binding-sites $\tilde{b}$ active Ezrin $u$, where for simplicity we assume a constant adsorption rate $k_1$ and a flow-dependent desorption rate $k_2(w)$. By the law of mass action this implies $$\frac{du}{dt} = k_1 v \tilde{b} - k_2(w)u = k_1(\tfrac{u}{u_{\max}})^\alpha bv - k_2(w)u = k_1(\tfrac{u}{u_{\max}})^\alpha(u_{\max}-u)v - k_2(w)u\,,$$ where in the last step we used $b = u_{\max}-u$. Thus, with $\gamma=k_1/u_{\max}^\alpha$ we set $$A(w,u,v) = a(u,v) = \begin{cases} \gamma u^\alpha(u_{\max}-u)v & \text{ if } u\leq u_{\max}, \\ 0 & \text{ otherwise.} \end{cases}$$ The desorption rate $k_2(w)$ on the other hand is taken to depend affinely (which can be thought of as the first order expansion) on the tangential cytoplasm flow velocity along the membrane, modelling a shear-stress induced Ezrin destabilization. For Ezrin concentrations above the saturation concentration $u_{\max}$, which could in principle occur within the membrane due to passive transport by the flow, we assume a strong nonlinear desorption rate increase with exponent $\zeta>1$ (the exact form is not expected to have any qualitative effect). In summary, we set $$\begin{aligned} D(w,u,v) = d(w,u) = (\beta_1|w|+\beta_2)\begin{cases} u & \text{ if } u\leq u_{\max}, \\ \frac{u^\zeta+(\zeta-1)u_{\max}^\zeta}{\zeta u_{\max}^{\zeta-1}}& \text{otherwise,} \end{cases}\end{aligned}$$ where the parameter $\beta_1> 0$ controls the influence of hydrodynamical contributions in Ezrin dissociation. Thus, $D$ increases with a stronger flow as well as with an increasing concentration of active Ezrin. The cytosol concentration $v:\Omega\to{\mathbf{R}}$ of inactive Ezrin is governed by an advection-diffusion equation, $$\label{Eq:pdeV} \partial_t v = -\mathrm{div}(vw) + \mu\Delta v \qquad \text{ in } \Omega,$$ where Ezrin production or decay is negligible on the time-scale considered. The boundary conditions are dictated by the turnover rates between active and inactive Ezrin, $$\label{Eq:BCV} \mu\nabla v\cdot n = D(w,u,v)-A(w,u,v)= d(w,u) - a(u,v) \qquad \text{ on } \Gamma.$$ Summarizing, the full system of equations for the variables $w,p,u,v$ with their initial and boundary conditions reads $$\begin{aligned} \rho\partial_t w&= - \rho(w\cdot\nabla)w - \frac{\rho\lambda}{\kappa}w -\nabla p + \rho\lambda\Delta w + f &&\text{ on } (0,T)\times\Omega, \\ 0&=\mathrm{div}\, w &&\text{ on } (0,T)\times\Omega, \\ \partial_t u &= -\mathrm{div}_{\Gamma}(u\, w) + \nu\, \Delta_{\Gamma} u - d(w,u) + a(u,v) &&\text{ on } (0,T)\times\Gamma, \\ \partial_t v &= -\mathrm{div}(vw) + \mu\Delta v &&\text{ on } (0,T)\times\Omega, \\ w(0,\cdot) &= w_0 &&\text{ on } \Omega, \\ u(0,\cdot) &= u_0 &&\text{ on } \Gamma, \\ v(0,\cdot) &= v_0 &&\text{ on } \Omega, \\ w\cdot n &= 0 &&\text{ on } (0,T)\times\Gamma, \\ \mu\nabla v\cdot n &= d(w,u) - a(u,v) &&\text{ on } (0,T)\times\Gamma.\end{aligned}$$ Note that the (third) equation for $u$ can be seen as an Allen–Cahn equation with nonlinearity $W_{w,v}'(u)=d(w,u)-a(u,v)$, or equivalently with potential $$W_{w,v}(u) = \begin{cases} \frac{\beta_1|w|+\beta_2}{2}(u^2-u_{\max}^2) -\gamma v\left(\frac{u_{\max}u^{\alpha+1}}{\alpha+1}-\frac{u^{\alpha+2}}{\alpha+2}-\frac{u_{\max}^{\alpha+2}}{(\alpha+1)(\alpha+2)}\right) & \text{ if } u\leq u_{\max},\\ \left(\beta_1|w|+\beta_2\right)\left(\frac{u^{\zeta+1}}{\zeta(\zeta+1)u_{\max}^{\zeta-1}} + \frac{(\zeta-1)u_{\max}}{\zeta}u-\frac{\zeta u_{\max}^2}{\zeta+1}\right) & \text{ otherwise,} \end{cases}$$ and additional advection term. Illustrative sketches of the form and behaviour of $W_{w,v}$ are provided in \[fig:doubleWell1,fig:doubleWell2\]. Allen–Cahn type equations have seen frequent use in the description of phase separations in binary mixtures, see e.g. [@Ref:Cahn Cahn and Novick-Cohen], and we expect the above model to also exhibit a phase separation into a phase rich in active Ezrin in the rear and a phase poor ofactive Ezrin in the front. ![Sketches of the adsorption and desorption rate $a$ and $d$ (left), the corresponding turnover rate $W'$ and the potential $W$ as a function of membranous Ezrin concentration $u$ for different intensities of the velocity $w$ and $\alpha=1$. The potentials have the form of single wells.[]{data-label="fig:doubleWell1"}](DoubleWellPlot1.pdf){width="\linewidth"} ![Sketches of the adsorption and desorption rate $a$ and $d$ (left), the corresponding turnover rate $W'$ and the potential $W$ as a function of membranous Ezrin concentration $u$ for different intensities of the velocity $w$ and $\alpha>1$. The potentials have the form of double wells.[]{data-label="fig:doubleWell2"}](DoubleWellPlot2.pdf){width="\linewidth"} Nondimensionalisation and model reduction {#sec:Nondimensionalisation} ----------------------------------------- To reduce the number of parameters and to identify the predominant model mechanisms we transform the model equations into a dimensionless form using the following parameter values. --------------------------------------------- ------------------------------------ ------------------------ ------------------------------------------------------- $L$ cell diameter 10-20$\mu$m [@Ref:BraatElAl1999 Braat et al. 1999] $L_N$ diameter of nucleus 6-10$\mu$m [@Ref:BraatElAl1999 Braat et al. 1999] $T$ duration of polarization process 150s own experimental data $\rho$ density of cytoplasm 1.03-1.1g/cm$^3$ [@Ref:BarsantiGualtieri2005 Barsanti, Gualtieri 2005] $\rho\lambda$ dynamic viscosity of cytoplasm 10$^{-2}$-10$^{-1}$Pas [@Ref:MogilnerManhart2018 Mogilner, Manhart 2018] $c_w$ typical intracellular velocity 0.1$\mu$m/s [@Ref:MogilnerManhart2018 Mogilner, Manhart 2018] $\tilde{\kappa}=\frac{\kappa}{\rho\lambda}$ hydraulic permeability 0.1$\mu$m$^4$/(pNs) [@Ref:MogilnerManhart2018 Mogilner, Manhart 2018] $\nu$ diffusion rate of active Ezrin 0.003$\mu$m$^2$/s [@Ref:FritzscheEtAl2014 Fritzsche et al. 2014] $\mu$ diffusion rate of inactive Ezrin 30$\mu$m$^2$/s [@Ref:Coscoy Coscoy et al. 2002] $u_{\max}$ saturation density of active Ezrin unknown $c_v$ typical density of inactive Ezrin unknown --------------------------------------------- ------------------------------------ ------------------------ ------------------------------------------------------- Subsequently we will indicate dimensionless variables by a hat. Following the standard procedure, we choose $$x = L\hat{x}, \ w = c_w\hat{w}, \ t = \frac{L}{c_w}\hat{t}, \ p = \frac{c_w\rho\lambda L}{\kappa}\hat{p}, \ f = \frac{\rho\lambda c_w}{\kappa}\hat{f}$$ in the incompressible Brinkman–Navier–Stokes equations - and arrive at $$\begin{aligned} \frac{\kappa}{L^2}{{\mathrm{Re}}}\Big( \partial_{\hat{t}}\hat{w} + (\hat{w}\cdot\nabla)\hat{w} \Big) + \hat{w} &= -\nabla\hat{p} + \frac\kappa{L^2} \Delta\hat{w} + \hat f,\\ \operatorname{div}\hat w&=0.\end{aligned}$$ Using using the above-listed parameter values, the Reynolds number is ${{\mathrm{Re}}}=\frac{c_wL}{\lambda}\sim1\cdot10^{-8}$-$2\cdot10^{-7}$ and $\kappa/L^2\sim0.0025$-$0.1$. Thus, keeping only the terms with nonnegligible coefficients, the equations of fluid motion simplify to the Darcy flow equations $$\begin{aligned} \hat{w} &= -\nabla\hat{p} + \hat{f}, \\ \operatorname{div}\, \hat{w} &= 0.\end{aligned}$$ Note that the observed blebbing time scale $T$ roughly coincides with the time that the observed flow needs to traverse the length of the cell so that the dimensionless final time scales like $\hat T=\frac{Tc_w}L\sim$0.75-1.5. Further note that even though the bulk force $f$ generated in the actin brushes is unknown, an upper bound can be obtained by multiplying reasonable myosin concentrations (e.g. $\sim0.2\,\mu$mol/l in plant endoplasm [@Ref:Yamamoto2006]) with the force generated per myosin head ($\sim40$pN, see e.g. [@Ref:Lohner2018]), resulting in values of $f\sim5\cdot10^9$ or $\hat f\sim5\cdot10^5$-$5\cdot10^7$ if all myosin molecules were simultaneously active. Even if only a fraction of the myosin motors is active at any time, the generated force will thus still be able to drive an intracellular flow. Choosing the same temporal and spatial scales and additionally $$u = u_{\max}\hat{u}, \ v = c_v\hat{v},$$ the equation for the active Ezrin concentration turns into $$\partial_{\hat{t}}\hat{u} = -\operatorname{div}_\Gamma(\hat{u}\hat{w}) + \varepsilon\Delta_\Gamma\hat{u} - \frac{L}{c_wu_{\max}}d(c_w\hat{w},u_{\max}\hat{u}) + \frac{L}{c_wu_{\max}}a(u_{\max}\hat{u},c_v\hat{v})$$ with $\varepsilon=\frac{\nu}{Lc_w}\sim$0.0015-0.003. Setting $$\begin{aligned} \hat{d}(\hat{w},\hat{u}) &= (C_1|\hat w|+C_2)\cdot\begin{cases} \hat{u} &\text{ if } \hat{u}\leq 1, \\ \frac{\hat u^\zeta+\zeta-1}{\zeta} & \text{ otherwise,} \end{cases} \label{eqn:desorption}\\ \hat{a}(\hat{u},\hat{v}) &= \begin{cases} C_3\hat{u}^\alpha(1-\hat{u})\hat{v} & \text{ if } \hat{u}\leq 1, \\ 0 & \text{ otherwise,} \end{cases}\label{eqn:adsorption}\end{aligned}$$ for the dimensionless parameters $$C_1 = L\beta_1, \ C_2 = \frac{L}{c_w}\beta_2, \ C_3 = \frac{Lu_{\max}^\alpha c_v}{c_w}\gamma,$$ the equation finally reduces to $$\begin{aligned} \partial_{\hat{t}}\hat{u} &= -\operatorname{div}_\Gamma(\hat{u}\hat{w}) + \varepsilon\Delta_\Gamma\hat{u} - \hat{d}(\hat{w},\hat{u}) + \hat{a}(\hat{u},\hat{v}).\end{aligned}$$ We will keep the $\varepsilon$-weighted diffusion term as a regularization; from the theory of Allen–Cahn type equations it is known that it governs the width of the diffusive interface between phases of different Ezrin concentration. Due to a lack of experimental information about typical concentrations $u_{\max}$ and $c_v$ of active and inactive Ezrin as well as the exact form of the adsorption and desorption rate, the dimensionless model equations still involve some unknown parameters. We will present some possible choices and discuss the influence of different parameters within \[sec:ExperimentalResults\]. To reduce the equations for the inactive Ezrin concentration we will make the assumption that $$c_vL\gg u_{\max} \qquad\text{for the typical inactive Ezrin concentration } c_v=\frac1{|\Omega|}\int_\Omega v_0\,{{\mathrm d}}x,$$ where $|\Omega|$ denotes the Lebesgue measure of $\Omega$. Unfortunately, in the literature we were not able to find any measurements of Ezrin concentrations in cells in order to support this assumption. Nevertheless we believe the assumption to be reasonable and to represent the typical situation for most membrane-active proteins in a cell, since assuming $c_vL\sim u_{\max}$ would mean that the typical number of protein molecules in the cytosol equals the typical number in the membrane, meaning a (highly unlikely) perfect recruitment to the membrane. We now introduce the spatial concentration average and residual $$\bar{v} = \frac{1}{|\Omega|}\int_\Omega v \, {{\mathrm d}}x, \quad r = v-\bar{v}.$$ Using -, the functions $\bar v$ and $r$ are governed by the differential equations $$\begin{aligned} \frac{{{\mathrm d}}\bar{v}}{{{\mathrm d}}t} &= \frac{1}{|\Omega|} \int_\Omega \mu\Delta v- \operatorname{div}(vw) \, {{\mathrm d}}x = \frac{1}{|\Omega|} \int_\Gamma D-A \, d\mathcal{H}^2, &&\text{on }(0,T)\times\Omega,\\ \partial_t r &= \mu\Delta r - \operatorname{div}(rw) - \frac{1}{|\Omega|}\int_\Gamma D-A \, d\mathcal{H}^2 &&\text{on }(0,T)\times\Omega, \\ \mu\nabla r\cdot n &= D-A &&\text{on }(0,T)\times\Gamma, \\ r(0,\cdot) &= r_0 &&\text{on }\Omega, \\ \bar v(0)&=\bar v_0,\\ \int_\Omega r \, {{\mathrm d}}x &= 0\end{aligned}$$ for $\bar v_0=\int_\Omega v_0\,{{\mathrm d}}x/|\Omega|$ and $r_0=v_0-\bar v_0$, where we used integration by parts and $w\cdot n=0$ on $\Gamma$, and where ${\mathcal{H}}^2$ denotes the surface (two-dimensional Hausdorff) measure. Introducing $$r = c_v\hat{r},\ \bar v=c_v\hat v$$ as well as the Péclet number ${{\mathrm{Pe}}}=\frac{c_wL}{\mu}\sim$0.03-0.07$\ll1$, the differential equation for $\hat r$ becomes $$\begin{aligned} {{\mathrm{Pe}}}\,\partial_{\hat{t}}\hat{r} &= \Delta\hat{r} - {{\mathrm{Pe}}}\,\operatorname{div}(\hat{w}\hat{r}) - \frac{1}{|\hat{\Omega}|}\frac{L^2}{\mu}\frac{u_{\max}}{Lc_v}\int_{\hat{\Gamma}} \frac{D-A}{u_{\max}} \, d\mathcal{H}^2 &&\text{on }(0,T)\times\Omega, \\ \nabla\hat r\cdot n &= \frac{L^2}{\mu}\frac{u_{\max}}{Lc_v}\frac{D-A}{u_{\max}} &&\text{on }(0,T)\times\Gamma.\end{aligned}$$ Our biological experiments suggest that it takes roughly around 10s until a substantial Ezrin concentration establishes on an originally Ezrin-depleted membrane so that the Ezrin turnover rate can be estimated to satisfy $$\frac{|D-A|}{u_{\max}}\leq\frac1{10\,\text s}\,.$$ Together with $c_vL\gg u_{\max}$ we thus obtain $\frac{L^2}{\mu}\frac{u_{\max}}{Lc_v}\frac{|D-A|}{u_{\max}}\sim\frac{u_{\max}}{Lc_v}\ll1$ so that, by neglecting terms with small coefficients, the equations for $\hat r$ turns into $$\Delta\hat r = 0 \text{ on }(0,T)\times\Omega, \ \nabla\hat r\cdot n =0 \text{ on }(0,T)\times\Gamma, \ \int_\Omega\hat r \, {{\mathrm d}}x = 0,$$ and thus $\hat r=0$. Therefore, we may assume that $v$ remains constant in space, $v=\bar{v}$. Now it is obvious that the total Ezrin molecule number within the cell stays constant over time. Indeed, we have $$\frac{{\mathrm d}}{{{\mathrm d}}t}\left(|\Omega|\bar v+\int_\Gamma u\,{{\mathrm d}}{\mathcal{H}}^2\right)=\int_\Gamma D-A\,{{\mathrm d}}{\mathcal{H}}^2+\int_\Gamma\partial_tu\,{{\mathrm d}}{\mathcal{H}}^2=\int_\Gamma\nu\Delta_\Gamma u-\operatorname{div}_\Gamma(uw)\,{{\mathrm d}}{\mathcal{H}}^2=0$$ using integration by parts. Furthermore, if $u_0$ is nonnegative, $u$ will be so for all times so that $|\Omega|\bar v\leq\int_\Omega v_0\,{{\mathrm d}}x+\int_\Gamma u_0\,{{\mathrm d}}{\mathcal{H}}^2=|\Omega|c_v+\int_\Gamma u_0\,{{\mathrm d}}{\mathcal{H}}^2$. Likewise, $|\Omega|\bar v\geq|\Omega|c_v-\max_{t\in(0,T)}\int_\Gamma u\,{{\mathrm d}}{\mathcal{H}}^2$, where we can estimate $\int_\Gamma u\,{{\mathrm d}}{\mathcal{H}}^2\lesssim{\mathcal{H}}^2(\Gamma)u_{\max}$ due to the strong desorption of Ezrin for $u>u_{\max}$. In summary, $$\left|\frac{\bar v}{c_v}-1\right|=\frac1{|\Omega|c_v}||\Omega|\bar v-|\Omega|c_v|\lesssim\frac{{\mathcal{H}}^2(\Gamma)u_{\max}}{|\Omega|c_v}\sim\frac{u_{\max}}{c_vL}\ll1$$ so that we may well approximate $$\bar v=c_v\,.$$ In summary, the reduced dimensionless system of equations read $$\begin{aligned} \hat w &= -\nabla\hat p +\hat f&&\text{ on } (0,\hat T)\times\hat\Omega,\label{eqn:Darcy1}\\ \operatorname{div}\hat w &=0&&\text{ on } (0,\hat T)\times\hat\Omega,\label{eqn:Darcy2}\\ \hat w\cdot n &= 0 &&\text{ on } (0,\hat T)\times\hat\Gamma,\label{eqn:DarcyBC}\\ \partial_{t}\hat u &= -\operatorname{div}_{\hat\Gamma}(\hat u\hat w) + \varepsilon\Delta_{\hat\Gamma}\hat u - \hat{d}(\hat w,\hat u) + \hat{a}(\hat u,1)&&\text{ on } (0,\hat T)\times\hat\Gamma,\label{eqn:Ezrin}\\ \hat u(0,\cdot) &=\hat u_0 &&\text{ on } \hat\Gamma,\label{eqn:EzrinBC}\end{aligned}$$ in which all quantities can be expected to be roughly of size 1 and in which $\hat d$ and $\hat a$ are defined by -. Existence and regularity of the solution {#sec:Analysis} ======================================== For ease of notation, from now on we remove the hat on the dimensionless variables. The reduced model consists of two differential equation systems, one for the velocity field $w$, which is independent of $u$ and thus can be solved separately, and one for the membranous Ezrin concentration $u$, into which the velocity $w$ enters via advection and as a parameter in the desorption term. Both sets of equations are standard, and in this we briefly summarize the well-posedness of the model. For a bounded, relatively open domain $A$ we will denote by $L^q(A)$ and $W^{m,q}(A)$ the real Lebesgue and Sobolev spaces on $A$ with exponent $q$ and (potentially fractional or negative) differentiability order $m$; in case of $X$-valued function spaces for some vector space $X$ we indicate the image space via $L^q(A;X)$ and $W^{m,q}(A;X)$, respectively (the same notation is used for the following function spaces). The corresponding Hilbert spaces are called $H^m(A)=W^{m,2}(A)$. The space of $m$ times continuously differentiable functions on $\overline A$ is denoted $C^m(\overline A)$ and of the corresponding Hölder-differentiable functions with exponent $\alpha$ by $C^{m,\alpha}(\overline A)$. Finally, the space of Radon measures on $\overline A$ (the dual space to $C^0(\overline A)$) is denoted ${{\mathcal{M}}}(\overline A)$. We will either use $A=\Omega$ or $A=\Gamma$. In the latter case, recall that the function spaces can be defined on geodesically complete compact manifolds via charts, that is, we consider a finite atlas $(U_i, x_i)_{1\leq i \leq N}$ of $\Gamma$ and a smooth partition $\psi_i:U_i\to[0,1]$, $i=1,\ldots,N$, of unity on $\Gamma$ so that the function space $X(\Gamma)$ is defined as the set of all functions $g:\Gamma\to{\mathbf{R}}$ such that $(\psi_ig)\circ x_i^{-1}\in X(x_i(U_i))$, and the corresponding norm is defined as $$\|g\|_{X(\Gamma)}=\sum_{i=1}^N \|(\psi_i v)\circ x^{-1}_i\|_{X(x_i(U_i))}\,.$$ As long as the charts are bi-Lipschitz (which is the case for Lipschitz $\Gamma$), all spaces $C^{0,\alpha}(\Gamma)$ and $W^{m,p}(\Gamma)$ with $m\leq1$ are well-defined by this procedure, that is, the norm depends on the chosen atlas, but all atlases lead to equivalent norms. The definition of more regular function spaces on $\Gamma$ then requires correspondingly smoother $\Gamma$. Influence of the bulk force on Darcy flow ----------------------------------------- For the well-posedness of the differential equation for membranous Ezrin we will have to bound the influence of the advection term. For this we will require a sufficiently smooth flow $w$ tangentially to the membrane. Using standard results for elliptic differential equations, this could be achieved by requiring sufficiently high regularity of the bulk force $f$ and the domain boundary $\Gamma$. However, recalling that the bulk force is produced by the actin brushes in the cell interior, we believe it more natural to obtain the necessary flow regularity from the positive distance of $f$ to the cell boundary. As for the regularity of $\Gamma$, we only assume that the cell $\Omega$ is convex, which before blebbing is certainly a reasonable assumption. The convexity entails that $\Gamma$ is Lipschitz, the minimum requirement to make sense of (weak) gradients of functions on $\Gamma$ and thus of the differential equation for Ezrin. For simplicity, we will sometimes consider the second-order form of Darcy’s equations, $$\begin{aligned} \Delta p&=\operatorname{div}f &&\text{in }\Omega,\label{eqn:DarcyP}\\ \nabla p\cdot n&=f\cdot n &&\text{on }\Gamma,\label{eqn:DarcyPBC}\end{aligned}$$ which is equivalent to - by taking the divergence in . Let $f=D\delta_z$ for $D\in{\mathbf{R}}^n$ and $\delta_z$ the Dirac distribution in $z\in{\mathbf{R}}^n$, $n>1$. Then is solved in the distributional sense by $p=D\cdot \Psi(\cdot-z)$ with $$\Psi(x)=\frac1{\omega_n}\frac{x}{|x|^n}\,,$$ where $\omega_n$ denotes the surface area of the $n$-dimensional unit ball. This can be checked by straightforward calculation or by noticing that $\Psi$ is nothing else than the gradient of the funcamental solution to Poisson’s equation, however, it is also well-known in electro-encephalography research, where $D\cdot\Psi$ is the electric field induced by a voltage dipole as they occur in the human brain [@Ref:Sarvas1987 (12)]. \[thm:existenceDarcy\] Let $\Omega\subset{\mathbf{R}}^n$ be convex, let $f\in{{\mathcal{M}}}(\overline\Omega;{\mathbf{R}}^n)$ be compactly supported in $\Omega$, $n>1$. Then there exists a (unique up to a constant) distributional solution $p$ to -, which satisfies $$p(x)=\tilde p(x)+\int_\Omega\Psi(x-z)\cdot{{\mathrm d}}f(z)$$ for some $\tilde p\in C^{0,1}(\overline\Omega)$, and for any $s>n$ there exists a constant $C(s,n,\Omega)$ with $$\|\tilde p\|_{C^{0,1}(\overline\Omega)} \leq C(s,n,\Omega){{\mathrm{dist}}}({{\mathrm{spt}}}f,\Gamma)^{-s}\|f\|_{{{\mathcal{M}}}(\overline\Omega)}\,.$$ As a consequence, the unique distributional solution $(w,p)$ to - satisfies $$\|w\|_{L^\infty(\Gamma)}\leq C(s,n,\Omega){{\mathrm{dist}}}({{\mathrm{spt}}}f,\Gamma)^{-s}\|f\|_{{{\mathcal{M}}}(\overline\Omega)}\,.$$ Abbreviate $R={{\mathrm{dist}}}({{\mathrm{spt}}}f,\Gamma)$, let $\chi:{\mathbf{R}}^n\to[0,1]$ be a smooth radially symmetric function with $\chi(x)=1$ for $|x|<1/2$ and $\chi(x)=0$ for $|x|>1$, and set $\chi_R(x)=\chi(x/R)$ as well as $$P(x)=\int_\Omega[(1-\chi_R)\Psi](x-z)\cdot{{\mathrm d}}f(z)\,.$$ As a convolution of a measure with an infinitely smooth function, $P$ is smooth. Furthermore, let $\hat p\in W^{1,2}(\Omega)$ denote the unique weak solution with zero mean to $$\Delta\hat p=\Delta P\quad\text{in }\Omega,\qquad \nabla\hat p\cdot n=0\quad\text{on }\Gamma.$$ Then $$p(x)=\hat p(x)+\int_\Omega[\chi_R\Psi](x-z)\cdot{{\mathrm d}}f(z)$$ solves (uniquely up to a constant) -. Now by the Lipschitz regularity result [@Ref:Mazya2009 §2] for the Neumann problem on convex domains we have $$\|\hat p\|_{C^{0,1}(\overline\Omega)}\leq\tilde C\left\|\Delta P\right\|_{L^q(\Omega)}$$ for any $q>n$, where the constant $\tilde C>0$ depends on $\Omega$, $n$, and $q$. Now by Young’s convolution inequality we can estimate $$\begin{aligned} \left\|\Delta P\right\|_{L^q(\Omega)} &=\left[\int_\Omega\left|\int_\Omega\Delta\left[(1-\chi_R)\Psi\right](x-z)\cdot{{\mathrm d}}f(z)\right|^q{{\mathrm d}}x\right]^{\frac1q}\\ &\leq\left[\int_\Omega\left(\int_\Omega\left|\Delta\left[(1-\chi_R)\Psi\right](x-z)\right|{{\mathrm d}}\frac{|f|}{\|f\|_{{{\mathcal{M}}}(\overline\Omega)}}(z)\right)^q{{\mathrm d}}x\right]^{\frac1q}\|f\|_{{{\mathcal{M}}}(\overline\Omega)}\\ &\leq\left[\int_\Omega\int_\Omega\left|\Delta\left[(1-\chi_R)\Psi\right](x-z)\right|^q{{\mathrm d}}\frac{|f|}{\|f\|_{{{\mathcal{M}}}(\overline\Omega)}}(z)\,{{\mathrm d}}x\right]^{\frac1q}\|f\|_{{{\mathcal{M}}}(\overline\Omega)}\\ &\leq\left[\int_\Omega\int_\Omega\left|\Delta\left[(1-\chi_R)\Psi\right](x-z)\right|^q{{\mathrm d}}x\,{{\mathrm d}}\frac{|f|}{\|f\|_{{{\mathcal{M}}}(\overline\Omega)}}(z)\right]^{\frac1q}\|f\|_{{{\mathcal{M}}}(\overline\Omega)}\\ &=\|\Psi\Delta\chi_R+2\nabla\chi_R\cdot\nabla\Psi\|_{L^q({\mathbf{R}}^n)}\|f\|_{{{\mathcal{M}}}(\overline\Omega)}\\ &\leq\|\chi\|_{C^2({\mathbf{R}}^n)}\left(R^{-2}\|\Psi\|_{L^q({\mathbf{R}}^n\setminus B_R(0))}+R^{-1}\|\nabla\Psi\|_{L^q({\mathbf{R}}^n\setminus B_R(0))}\right)\|f\|_{{{\mathcal{M}}}(\overline\Omega)}\\ &\leq\hat CR^{-n-1+n/q}\|f\|_{{{\mathcal{M}}}(\overline\Omega)}\end{aligned}$$ for some constant $\hat C$ depending only on $\chi$, $\Omega$, $n$, and $q$. Summarizing, we have $$\|\hat p\|_{C^{0,1}(\overline\Omega)}\leq\tilde C\hat CR^{-n-1+n/q}\|f\|_{{{\mathcal{M}}}(\overline\Omega)}\,.$$ Now $\tilde p(x)=p(x)-\int_\Omega\Psi(x-z)\cdot{{\mathrm d}}f(z)=\hat p(x)-P(x)$ so that $$\|\tilde p\|_{C^{0,1}(\overline\Omega)} \leq\|\hat p\|_{C^{0,1}(\overline\Omega)}+\|P\|_{C^{0,1}(\overline\Omega)} \leq\tilde C\hat CR^{-n-1+n/q}\|f\|_{{{\mathcal{M}}}(\overline\Omega)}+\bar CR^{-n}\|f\|_{{{\mathcal{M}}}(\overline\Omega)}$$ for some $\bar C>0$ depending on $\chi$. Due to the arbitrariness of $q>n$ this proves the first claim. As for the estimate on $w$, we also have $$\|p\|_{W^{1,\infty}(\Gamma)}=\|p\|_{C^{0,1}(\Gamma)}=\|\hat p\|_{C^{0,1}(\Gamma)}\leq\|\hat p\|_{C^{0,1}(\Omega)}\leq CR^{-n-1+n/q}\|f\|_{{{\mathcal{M}}}(\overline\Omega)}\,,$$ from which the second claim follows by noting $\|w\|_{L^\infty(\Gamma)}=\|\nabla p\|_{L^\infty(\Gamma)}\leq\|p\|_{W^{1,\infty}(\Gamma)}$ (recall that $\nabla p$ is tangential to $\Gamma$). \[rem:DarcyRegularity\] Due to $w=-\nabla p+f$, the above result shows that $w$ is in $L^\infty(\Omega\setminus B_\delta({{\mathrm{spt}}}f))$ for $B_\delta$ the $\delta$-neighbourhood, and the boundary flux is its trace on $\Gamma$ (in the sense that it is the weak-$\ast$ limit of $w$ on surfaces $\Gamma_n\subset\Omega$ approaching $\Gamma$). By another regularity result for the Neumann problem on convex domains we additionally have $p\in W^{2,2}(\Omega\setminus B_\delta({{\mathrm{spt}}}f))$ so that for $n=3$ we have $w\in H^{1/2}(\Gamma)$ [@AdolfssonJerison]. If higher regularity of $\Gamma$ is assumed, then the boundary flux $w|_\Gamma$ may be even smoother. In our exposition, though, we will stick to minimal regularity requirements on $\Gamma$. On a ball one can explicitly compute a Green’s function and thus state an analytical solution, which can provide some intuition on the flow behaviour. \[thm:DarcyGreenBall\] Equations - for $f=D\delta_z$ on $\Omega=B_1(0)\subset{\mathbf{R}}^3$ the unit ball are solved by $p=D\cdot G_z$ for the Green’s function $$\begin{gathered} G_z(x) =\frac1{4\pi}\left[\left(\frac{x\cdot e-|z|}{|x-z|^3}-\frac{x\cdot e-\frac1{|z|}}{|z|^3|x-\zeta|^3}\right)e\right.\\ \left.+\frac1{|z||x-\zeta|}\left(1+\frac1{|z|^2|x-\zeta|^2}+\frac{|z||x-\zeta|}{|x-z|^3}+\frac{x\cdot e}{\frac1{|z|}-x\cdot e+|x-\zeta|}\right)\left(I-e\otimes e\right)x\right]\,,\end{gathered}$$ where $\zeta=z/|z|^2$ and $e=z/|z|$. Thus, the solution for general $f$ is given by $$p(x)=\int_\Omega G_z(x)\cdot{{\mathrm d}}f(z)\,.$$ One can check by explicit calculation that $G_z$ solves the desired system. Alternatively, one can exploit $G_z(x)=-\nabla_z\tilde G_z(x)$, where $\tilde G_z$ is the Green’s function for the Neumann problem, $\Delta\tilde G_z=\delta_z-\frac3{4\pi}$ in $\Omega$ with $\nabla\tilde G_z\cdot n=0$ on $\partial\Omega$, which is for instance stated in [@Ref:DiBenedetto2010 §7.1.2c] (note that the first term in that reference should be corrected to have a negative sign). provides an upper bound on the maximum boundary velocity, which diverges as the forces $f$ approach the boundary $\Gamma$. Of course, this does not automatically imply the blow-up of the boundary velocity as the forces get closer to $\Gamma$ (indeed, the boundary velocity will for instance be zero if the forces are arranged within a small ball such that the net force outside is zero). However, within a cell we do not expect force cancellation so that a rough intuition of the flow can be obtained by approximating the actin brushes with a point force $f$. Then, approximating the cell by a ball $\Omega=B_1(0)$ with the cell front being the northpole $e_3=(0\,0\,1)^T$ and the actin brush $f=e_3\delta_{ce_3}$ underneath for some $c\in(0,1)$, the analytical solution from \[thm:DarcyGreenBall\] simplifies to $$p(x) =\frac1{4\pi}\left(\frac{x_3-c}{|x-e_3c|^3}-\frac{x_3-1/c}{c^3|x-e_3/c|^3}\right)\,.$$ The resulting velocity $w$ at the boundary $\partial B_1(0)$ is given by $$w(x)=\frac3{4\pi}\frac{1-c^2}{|x-z|^5}(x_3x-e_3)\,,$$ pointing south along great circles. Its magnitude is given by $$|w(x)|=\frac3{4\pi}\frac{1-c^2}{|x-z|^5}\sqrt{1-x_3^2} =\frac3{4\pi}\frac{1-c^2}{\sqrt{(x_3-c)^2+1-x_3^2}^5}\sqrt{1-x_3^2}\,,$$ where $1-x_3^2$ is the squared distance of $x$ to the vertical axis. This is maximized at $x_3=(\sqrt{(c^2+1)^2+60c^2}-(c^2+1))/6c\sim1-(c-1)^2/8=1-{{\mathrm{dist}}}({{\mathrm{spt}}}f,\Gamma)^2/8$, for which $\sqrt{1-x_3^2}\sim{{\mathrm{dist}}}({{\mathrm{spt}}}f,\Gamma)/\sqrt8$. In other words, the point of maximal boundary velocity occurs roughly ${{\mathrm{dist}}}({{\mathrm{spt}}}f,\Gamma)/\sqrt8$ away from the cell front, and the velocity there indeed scales like ${{\mathrm{dist}}}({{\mathrm{spt}}}f,\Gamma)^{-3}$. Well-posedness of the Ezrin equation ------------------------------------ The well-posedness of - can be obtained via classical semigroup theory (we will later briefly discuss other approaches as well). To this end we abbreviate $$\begin{aligned} L&=\varepsilon\Delta_\Gamma \qquad\text{and}\\ F_w(u) &= -\mathrm{div}_{\Gamma}(uw) - d(w,u) + a(u,1)\end{aligned}$$ so that - turn into $$\begin{aligned} \partial_t u&=Lu+F_w(u)&&\text{on }(0,T)\times\Gamma\,,\label{eqn:reactionAdvectionDiffusion}\\ u(0,\cdot)&=u_0&&\text{on }\Gamma\,.\label{eqn:reactionAdvectionDiffusionBC}\end{aligned}$$ Furthermore, the bounded operator which maps an element $v_0\in L^q(\Gamma)$ to the unique solution $v$ of the Cauchy problem $$\partial_t v = Lv,\quad v(0,\cdot) = v_0,$$ will be denoted by $e^{tL}$, thus $v(t,\cdot)=e^{tL}v_0$. The short-term existence of a solution to - is based on the following Lipschitz property of $F_w$. \[thm:LipschitzProperty\] Let $\Gamma=\partial\Omega\subset{\mathbf{R}}^3$ be bounded and Lipschitz and $w\in L^\infty(\Gamma)$. $F_w$ is locally Lipschitz from $L^q(\Gamma)$ into $W^{-1,q}(\Gamma)$ for any $q\geq\max\{\zeta,2\zeta-2\}$, where $\zeta>1$ is the exponent in . By Hölder’s inequality we have $$\|uw\|_{L^q(\Gamma)}\leq\|u\|_{L^q(\Gamma)}\|w\|_{L^\infty(\Gamma)}$$ so that $\|\operatorname{div}(uw)\|_{W^{-1,q}(\Gamma)}\leq n\|w\|_{L^\infty(\Gamma)}\|u\|_{L^q(\Gamma)}$. Furthermore, letting $\ell$ denote the Lipschitz constant of $a(\cdot,1)$, we obtain $$\|a(u_1,1)-a(u_2,1)\|_{W^{-1,q}(\Gamma)} \leq\|a(u_1,1)-a(u_2,1)\|_{L^q(\Gamma)} \leq\ell\|u_1-u_2\|_{L^q(\Gamma)}\,.$$ Finally, abbreviating $r=\frac q\zeta\geq1$ we have by Hölder’s inequality $$\|d(w,u)\|_{L^r(\Gamma)} \leq\|C_1w+C_2\|_{L^\infty(\Gamma)}\|1+|u|^\zeta\|_{L^r(\Gamma)} \leq\|C_1w+C_2\|_{L^\infty(\Gamma)}\left(\|1\|_{L^r(\Gamma)}+\|u\|^\zeta_{L^q(\Gamma)}\right)\,,$$ and the directional derivative of $d(w,u)$ in direction $\phi\in L^q(\Gamma)$ satisfies $$\begin{aligned} \|\partial_ud(w,u)(\phi)\|_{L^r(\Gamma)} &=\left\|(C_1|w|+C_2)\max\{1,\mathrm{sign}(u)|u|^{\zeta-1}\}\phi\right\|_{L^r(\Gamma)}\\ &\leq\|C_1w+C_2\|_{L^\infty(\Gamma)}\|\phi\|_{L^q(\Gamma)}\|1+|u|^{\zeta-1}\|_{L^{\frac{q}{\zeta-1}}(\Gamma)}\\ &\leq\|C_1w+C_2\|_{L^\infty(\Gamma)}\|\phi\|_{L^q(\Gamma)}\left(\|1\|_{L^{\frac{q}{\zeta-1}}(\Gamma)}+\|u\|_{L^q(\Gamma)}^{\zeta-1}\right)\,.\end{aligned}$$ Consequently, $d(w,\cdot)$ lies in $L^r(\Gamma)$ and is locally Lipschitz (even Gâteaux differentiable) from $L^q(\Gamma)$ into $L^r(\Gamma)$. Due to the Sobolev embedding of $L^r(\Gamma)$ into $W^{-1,q}(\Gamma)$ we arrive at the desired result. \[Thm:ShortTermExistence\] Let $\Gamma$, $w$, and $q>2$ as in \[thm:LipschitzProperty\] and $u_0\in L^q(\Gamma)$. Then there exists $\hat t > 0$ depending on $\Gamma$ and $\|u_0\|_{L^q(\Gamma)}$ such that the initial value problem $$\label{Eq:ProblemAbstract} \partial_t u = Lu + F_w(u), \quad u(0,\cdot) = u_0$$ on $\Gamma$ has a unique weak solution $u\in C^0([0,\hat t]; L^q(\Gamma))\cap C^0((0,\hat t]; C^{0,\alpha}(\Gamma))$ for any $\alpha\in[0,1-2/q)$. Following the exposition in [@Ref:Taylor Taylor, Ch.15.1], the result is obtained by studying the integral equation $$\label{Eq:IntegralEquationSolution} \Psi(u(t,\cdot)) = u(t,\cdot) \quad\text{ with } \Psi(u(t,\cdot)) = e^{tL}u_0 + \int_0^t e^{(t-s)L}F_w(u(s,\cdot))\, {{\mathrm d}}s$$ and treating it as a fixed point equation. Analogously to the proof of [@Ref:Taylor Taylor, Ch.15.1, Prop.1.1], the existence of a fixed point can be ascertained via the contraction mapping principle, given that for appropriate Banach spaces $X$ and $Y$ $$\begin{aligned} e^{tL}&:X \to X &&\text{is a }C^0\text{-semigroup for }t \geq 0, \\ F_w&:X \to Y &&\text{is locally Lipschitz,} \\ e^{tL}&:Y \to X&&\text{with }\|e^{tL}\|_{\mathcal{L}(Y,X)} \leq Ct^{-\xi} \text{ for some } C>0,\,\xi <1\text{ and all } t \in (0, 1] .\hspace*{-3ex} \end{aligned}$$ Above, $\|\cdot\|_{\mathcal L(Y,X)}$ denotes the operator norm of a linear operator from $Y$ to $X$. For our choice $L = \varepsilon \Delta$, standard results for linear parabolic equations (see e.g. [@Ref:TaylorFirstVolume Ch.6.1]) guarantee that the semigroup generated by $L$ is well-defined and strongly continuous. The appropriate Banach spaces $X$ and $Y$ are determined by the properties of $F_w$ and the obtainable bounds on the operator norm of $e^{tL}$. For $X = W^{s,p}(\Gamma)$ and $Y=W^{r,q}(\Gamma)$ with $q \leq p$ and $r\leq s$, [@Ref:TaylorFirstVolume Ch.6.1] states $$\label{Eq:NormEstimateOperatorNorm} \|e^{tL}\|_{\mathcal{L}(Y,X)} \leq C t^{ -\frac{1}{2}(\frac{1}{q}-\frac{1}{p})-\frac{1}{2}(s-r)}\,.$$ In our case we pick $X = L^q(\Gamma)$, $Y = W^{-1,q}(\Gamma)$ so that $\xi = \frac12$ and the map $F_w:X\to Y$ is well-defined and locally Lipschitz continuous by \[thm:LipschitzProperty\]. For small enough $\hat t > 0$, which might still depend on the Lipschitz constant of $F_w$, $\xi$ and $\|u_0\|_{L^q(\Gamma)}$, we then obtain the existence and uniqueness of a solution $u \in C([0,\hat t]; X)$ to . Higher spatial regularity can be derived from the estimate $$\|u(t,\cdot)\|_{V} \leq \|e^{(t-t_0)L}\|_{\mathcal{L}(W,V)} \|u(t_0,\cdot)\|_{W} + \int_{t_0}^t \|e^{(t-s)L}\|_{\mathcal{L}(W,V)} \|F_w(u(s,\cdot))\|_{W}\, {{\mathrm d}}s$$ with appropriate Banach function spaces $V$ and $W$ on $\Gamma$ and any $t_0\in[0,\hat t]$. Choosing $W=Y=W^{-1,q}(\Gamma)$ and $V=W^{r,q}(\Gamma)$ with $r<1$ we obtain $\|e^{tL}\|_{\mathcal{L}(Y,X)} \leq Ct^{-\xi}$ with $\xi=\frac{1}{2}(r+1)<1$ so that $u\in C^0((0,\hat t];W^{r,q}(\Gamma))$. In particular, by the continuous embedding $W^{r,q}(\Gamma)\subset C^{0,r-2/q}(\Gamma)$ [@Ref:NezzaPalatucciValdinoci2012 Thm.8.2] we obtain $u\in C^0((0,\hat t];C^{0,r-2/q}(\Gamma))$ for any $r\in(\frac2q,1)$. For smoother $\Gamma$ (and thus also smoother $w$, see \[rem:DarcyRegularity\]), the spatial regularity of $u$ can be improved by consecutively using different pairs $V$, $W$ in the above argument. For instance, one could study the action of $F_w$ and $e^{tL}$ on the following sequence of spaces, $$\begin{gathered} L^q(\Gamma) \stackrel{F_w}{\longrightarrow} H^{-1}(\Gamma) \stackrel{ e^{tL} }{\longrightarrow} H^{\frac{1}{2}} (\Gamma) \stackrel{F_w}{\longrightarrow} H^{-\frac{1}{2}} (\Gamma) \stackrel{e^{tL}}{\longrightarrow} H^{1}(\Gamma) \ldots \\ \ldots \stackrel{ F_w }{\longrightarrow} L^2(\Gamma) \stackrel{e^{tL}}{\longrightarrow} H^{\frac{3}{2}}(\Gamma) \stackrel{F_w}{\longrightarrow} H^{\frac{1}{2}}(\Gamma) \stackrel{e^{tL}}{\longrightarrow} H^{2}(\Gamma)\,,\end{gathered}$$ which is for instance valid for twice differentiable $\Gamma$ and $w$ and would yield $u\in C^0((0,\hat t];H^2(\Gamma))$. \[thm:globalBound\] Let $\Gamma$, $w$, $q$, $\alpha$, $u_0$ and $\hat t$ as in \[Thm:ShortTermExistence\], assume $u_0>0$, and let $u$ be the weak solution to - on $[0,\hat t]\times\Gamma$. Then there exists some $M>0$ only depending on $\Gamma$, $w$, and $q$ with $u(t,\cdot)\geq0$ and $\|u(t,\cdot)\|_{L^q(\Gamma)}\leq\max\{\|u_0\|_{L^q(\Gamma)},M\}$ for all $t\in(0,\hat t)$. We would like to prove the above via a priori estimates obtained from testing the differential equation with different functions, however, for this we need $\partial_t u$ to be sufficiently regular. We achieve this by mollification in time. Let $G:{\mathbf{R}}\to[0,\infty)$ be a smooth mollifier with support on $[-1,1]$ and mass $\int_{\mathbf{R}}G(t)\,{{\mathrm d}}t=1$. For $\delta>0$ define $G_\delta(t)=G(t/\delta)/\delta$, and let $u_\delta(\cdot,x)=G_\delta\ast u(\cdot,x)$ for $x\in\Gamma$ be the mollification of the solution $u$ in time so that $u_\delta\in C^\infty((\delta,\hat t-\delta),C^{0,\alpha}(\Gamma))$. Note that $u_\delta$ satisfies $$\partial_tu_\delta=\operatorname{div}_\Gamma(\varepsilon\nabla_\Gamma u_\delta-u_\delta w)+G_\delta\ast(a(u,1)-d(w,u))\,.$$ Now assume that $u$ does not stay positive. Since $u$ is continuous, there is a time $$t_\pm=\inf\left\{t\in(0,\hat t)\,\middle|\,\min u(t,\cdot)<0\right\}$$ when $u$ changes sign for the first time. Take $\delta<\min\{t_\pm,\hat t-t_\pm\}/2$ small enough such that also $u_\delta$ changes sign at some time $t_\delta\in(2\delta,\hat t-2\delta)$. Then necessarily, $\int_\Gamma u_\delta(t,x)^{-2/\alpha}\,{{\mathrm d}}{\mathcal{H}}^2(x)\to\infty$ as $t\to t_\delta$, since this integral must be infinite for any nonnegative $C^{0,\alpha}(\Gamma)$ function taking the value $0$ at some $x\in\Gamma$. However, $$\begin{aligned} \frac\alpha2\frac{{\mathrm d}}{{{\mathrm d}}t}\int_\Gamma u_\delta^{-\frac2\alpha}\,{{\mathrm d}}{\mathcal{H}}^2 &=-\int_\Gamma u_\delta^{-\frac2\alpha-1}\partial_t u_\delta\,{{\mathrm d}}{\mathcal{H}}^2\\ &=-\int_\Gamma u_\delta^{-\frac2\alpha-1}\left(\operatorname{div}_\Gamma(\varepsilon\nabla_\Gamma u_\delta-u_\delta w)+G_\delta\ast(a(u,1)-d(w,u))\right)\,{{\mathrm d}}{\mathcal{H}}^2\\ &=\int_\Gamma\left(1+\frac2\alpha\right)u_\delta^{-\frac2\alpha-2}\nabla_\Gamma u_\delta\cdot\left(u_\delta w-\varepsilon\nabla_\Gamma u_\delta\right)+u_\delta^{-\frac2\alpha-1}G_\delta\ast(d(w,u)-a(u,1))\,{{\mathrm d}}{\mathcal{H}}^2\,.\end{aligned}$$ Now we can find some $C_\delta>0$ sufficiently large such that $d(w,u)-a(u,1)\leq C_\delta u$ for all times $t\in(\delta,\hat t-\delta)$, since $u$ is continuous on $[\delta,\hat t-\delta]$ and $w$ is bounded. Thus we can estimate $G_\delta\ast(d(w,u)-a(u,1))\leq C_\delta u_\delta$ on $(2\delta,t_\delta]$. Additionally using Young’s inequality we arrive at $$\begin{aligned} \frac\alpha2\frac{{\mathrm d}}{{{\mathrm d}}t}\int_\Gamma u_\delta^{-\frac2\alpha}\,{{\mathrm d}}{\mathcal{H}}^2 &\leq\int_\Gamma\left(1+\frac2\alpha\right)u_\delta^{-\frac2\alpha-2}\left(u_\delta w\nabla_\Gamma u_\delta-\varepsilon|\nabla_\Gamma u_\delta|^2\right)+C_\delta u_\delta^{-\frac2\alpha}\,{{\mathrm d}}{\mathcal{H}}^2\\ &\leq\int_\Gamma\left(1+\frac2\alpha\right)u_\delta^{-\frac2\alpha-2}\left(|u_\delta|^2\frac{\|w\|_{L^\infty(\Gamma)}^2}{4\varepsilon}+\varepsilon|\nabla_\Gamma u_\delta|^2-\varepsilon|\nabla_\Gamma u_\delta|^2\right)+C_\delta u_\delta^{-\frac2\alpha}\,{{\mathrm d}}{\mathcal{H}}^2\\ &=\left(C_\delta+\frac{\|w\|_{L^\infty(\Gamma)}^2}{4\varepsilon}\left(1+\frac2\alpha\right)\right)\int_\Gamma u_\delta^{-\frac2\alpha}\,{{\mathrm d}}{\mathcal{H}}^2\,.\end{aligned}$$ The above implies boundedness of $\int_\Omega u_\delta^{-2/\alpha}\,{{\mathrm d}}{\mathcal{H}}^2$ for all times $t\in[2\delta,t_\delta]$, thereby contradicting the blowup of the integral at $t_\delta$. Consequently, $u$ does not change sign in $(0,\hat t)$. The next a priori estimate tests equation with $u(t,\cdot)^{q-1}$, only again we mollify $u$ in time. In more detail, we have $$\begin{aligned} \frac1q\frac{{\mathrm d}}{{{\mathrm d}}t}\|u_\delta\|_{L^q(\Gamma)}^q &=\int_\Gamma u_\delta^{q-1}\,\partial_t u_\delta\,{{\mathrm d}}{\mathcal{H}}^2\\ &=\int_\Gamma u_\delta^{q-1}\left(\operatorname{div}_\Gamma(\varepsilon\Delta_\Gamma u_\delta-u_\delta w)+G_\delta\ast(a(u,1)-d(w,u))\right)\,{{\mathrm d}}{\mathcal{H}}^2\,.\end{aligned}$$ This time we estimate $G_\delta\ast(a(u,1)-d(w,u))\leq G_\delta\ast(C_3-C_2u^\zeta/\zeta)\leq C_3-C_2u_\delta^\zeta/\zeta$ by Jensen’s inequality. Again using Young’s inequality we arrive at $$\begin{aligned} \frac1q\frac{{\mathrm d}}{{{\mathrm d}}t}\|u_\delta\|_{L^q(\Gamma)}^q &\leq\int_\Gamma u_\delta^{q-1}\operatorname{div}_\Gamma(\varepsilon\Delta_\Gamma u_\delta-u_\delta w)+C_3u_\delta^{q-1}-\frac{C_2}\zeta u_\delta^{q-1+\zeta}\,{{\mathrm d}}{\mathcal{H}}^2\\ &\leq\int_\Gamma(q-1)u_\delta^{q-2}\left(|u_\delta|^2\frac{\|w\|_{L^\infty(\Gamma)}^2}{4\varepsilon}\right)+C_3u_\delta^{q-1}-\frac{C_2}\zeta u_\delta^{q-1+\zeta}\,{{\mathrm d}}{\mathcal{H}}^2\\ &=(q-1)\frac{\|w\|_{L^\infty(\Gamma)}^2}{4\varepsilon}\|u_\delta\|_{L^q(\Gamma)}^q+C_3\|u_\delta\|_{L^{q-1}(\Gamma)}^{q-1}-\frac{C_2}\zeta\|u_\delta\|_{L^{q-1+\zeta}(\Gamma)}^{q-1+\zeta}\\ &\leq c_1\|u_\delta\|_{L^q(\Gamma)}^q+c_2\|u_\delta\|_{L^q(\Gamma)}^{q-1}-c_3\|u_\delta\|_{L^q(\Gamma)}^{q+\zeta-1}\end{aligned}$$ for some positive constants $c_1,c_2,c_3$. The right-hand side is concave in $\|u_\delta\|_{L^q(\Gamma)}^q$ and changes sign from positive to negative at some $M^q>0$. Thus by linearizing the right-hand side in $\|u_\delta\|_{L^q(\Gamma)}^q$ about $M^q$ we can estimate $$\frac{{\mathrm d}}{{{\mathrm d}}t}\|u_\delta\|_{L^q(\Gamma)}^q \leq-c_4\left(\|u_\delta\|_{L^q(\Gamma)}^q-M^q\right)$$ for some $c_4>0$. Setting $e=\|u_\delta\|_{L^q(\Gamma)}^q-\max\{\|u_\delta(\delta,\cdot)\|_{L^q(\Gamma)}^q,M^q\}$, we thus arrive at $\frac{{\mathrm d}}{{{\mathrm d}}t}e\leq-c_4e$ with $e(\delta)\leq0$, which by Grönwall’s inequality implies $e(t)\leq0$ for all $t\in[\delta,\hat t-\delta]$. Letting now $\delta\to0$ we arrive at the desired estimate for $\|u\|_{L^q(\Gamma)}$. A direct consequence is the long-term existence of solutions to the equation. Let $\Gamma$, $w$, $q$, $\alpha$, and $u_0$ as in \[thm:globalBound\], then for any $T>0$ there exists a unique nonnegative weak solution $u\in C^0([0,T]; L^q(\Omega))\cap C^0((0,T]; C^{0,\alpha}(\Gamma))$ to - with $\|u(t,\cdot)\|_{L^q(\Gamma)}$ bounded uniformly in $t$. The short-term solution from \[Thm:ShortTermExistence\] exists up to some time $\hat t$ depending on $\|u_0\|_{L^q(\Gamma)}$. By \[thm:globalBound\] this solution is nonnegative and satisfies a uniform $L^q$-bound. Thus, by again appealing to \[Thm:ShortTermExistence\] for the solution of with initial condition $u(\hat t,\cdot)$ we can continue the solution to time $2\hat t$. Again, \[thm:globalBound\] yields the nonnegativity and the same $L^q$-bound. After a finite number of repetitions of this argument we have continued the solution up to time $T$. As already mentioned previously, our system - represents an advective Allen–Cahn type equation. Such a system has already been analysed in [@Ref:LiuBertozziKolokolnikov2012], however, for a simpler nonlinearity and under much higher regularity conditions. In particular, the velocity field $w$ of [@Ref:LiuBertozziKolokolnikov2012] is differentiable, and the domain is smooth enough (in their case it is an open set rather than an embedded lower-dimensional manifold) to allow classical solutions. In more detail, the authors use the Galerkin method to show existence of solutions to the advective Cahn–Hilliard and Allen–Cahn equation with an odd polynomial as nonlinearity. The odd polynomial allows a simpler a priori estimate than in our case where the nonlinearity behaves qualitatively different on the positive and the negative real line, which is why in our analysis we showed nonnegativity of the solution before an $L^q$ estimate. The authors also consider the nonlocal advective Allen–Cahn equation with mass conservation. Here they apply the same semigroup approach as we do, also putting the advective term into the nonlinearity. Again, the analysis is simplified by the higher regularity of domain, initial value, and velocity field. Global boundedness and nonnegativity of the solution then follow from a maximum principle exploiting the smoothness of the solution and the velocity field. (In fact, the authors use an Allen–Cahn well at $0$ and at $1$ and claim that the solution $u$ thus stays within $[0,1]$. However, this is only true for velocity fields with low enough compression. In general the argument can only provide a bound depending on the size of $\operatorname{div}w$.) In contrast, our argument makes use of as little regularity requirements as possible (note that with little modification it even works for $w\in L^r$ with $r<\infty$ large enough); in particular, function spaces with more than one weak derivative do not even make sense on our domain $\Gamma$. The authors of [@Ref:LiuBertozziKolokolnikov2012] also study (in one spatial dimension) how droplets (regions with high values of $u$) are affected by the advection term. In the smooth case they show that droplets cannot be broken up by an expansive flow as long as the expansion rate does not increase towards the droplet centre. However, for general flow fields droplets can indeed be broken into smaller ones as their numerical experiments show. In our setting we expect the flow field $w$ to be strongly expansive at the cell front and contractive in the back, thus assisting in the accumulation of Ezrin at the back of the cell. Phase separation ---------------- Without the transport term $\operatorname{div}_\Gamma(wu)$, is a classical Allen–Cahn type equation, that is, the $L^2$-gradient flow of the energy $$\begin{gathered} E[u]=\int_\Gamma\frac\varepsilon2|\nabla u|^2+W_{w,1}(u)\,{{\mathrm d}}x\\ \text{with } W_{w,1}(u)=\begin{cases} \tfrac{C_1|w|+C_2}2(u^2-1)-C_3\left(\tfrac{u^{\alpha+1}-1}{\alpha+1}-\tfrac{u^{\alpha+2}-1}{\alpha+2}\right)&\text{if }u\leq1,\\ \tfrac{C_1|w|+C_2}\zeta\left(\tfrac{u^{\zeta+1}-1}{\zeta+1}+(\zeta-1)(u-1)\right)&\text{else.} \end{cases}\end{gathered}$$ It is well-known that this gradient flow results in a separation of the domain into different phases (a phase rich in Ezrin and one without) which correspond to the stable steady states of the corresponding reaction equation $$\label{eqn:reaction} \partial_tu=-W_{w,1}'(u)=a(u,1)-d(w,u)=\begin{cases}C_3u^\alpha(1-u)-(C_1|w|+C_2)u&\text{if }u\leq1,\\-(C_1|w|+C_2)\frac{u^\zeta-1}\zeta+1&\text{else.}\end{cases}$$ In this paragraph we briefly describe the involved phases. In the case $\alpha=1$, a transcritical bifurcation happens at $|w|=\bar w=\frac{C_3-C_2}{C_1}$: - If $|w|\geq\bar w$, has the only nonnegative steady state $u=0$, which is stable. - If $|w|<\bar w$, has two nonnegative steady states, an instable one at $u=0$ and a stable one at $u=\frac{C_3-C_2-C_1|w|}{C_3}$. In the case $\alpha>1$, a saddle node bifurcation occurs at $|w|=\bar w=\frac{C_3}{C_1}\frac{(1-1/\alpha)^{\alpha-1}}\alpha-\frac{C_2}{C_1}$: - If $|w|\geq\bar w$, has the only nonnegative steady state $u=0$, which is stable. - If $|w|<\bar w$, has exactly two stable nonnegative steady states, one at $u=0$ and one in $[\frac{\alpha-1}\alpha,1-\frac{C_2}{C_3}]$. The case of $\alpha=1$ follows directly from noting that for $|w|\geq\bar w$ the flow field $-W_{w,1}'$ is monotonically decreasing and has a single zero in $0$, while for $|w|<\bar w$ the two zeros of $-W_{w,1}$ are given by $u=0$ (around which $-W_{w,1}'$ is increasing) and $u=\frac{C_3-C_2-C_1|w|}{C_3}$ (around which $-W_{w,1}'$ is decreasing). In the case $\alpha>1$ we have $-W_{w,1}'(0)=0$ and $-W_{w,1}''(0)<0$ so that $u=0$ is always a stable steady state. From the shape of $-W_{w,1}'$ (cf. \[fig:doubleWell2\]) it is then obvious that, depending on $|w|$, this is either the only zero of $W_{w,1}'$ or there are two more (thus the smaller one must be instable and the larger stable). That the larger one lies within $[\frac{\alpha-1}\alpha,1-\frac{C_2}{C_3}]$ can easily be shown by checking $-W_{w,1}'(1-\frac{C_2}{C_3})<0$ and $-W_{w,1}'(\frac{\alpha-1}\alpha)>0$ for $|w|<\bar w$. That $w=\bar w$ is critical can be seen from noticing that $-W_{\bar w,1}'$ has a maximum in $u=\bar u=1-\frac1\alpha$, at which $-W_{\bar w,1}'(\bar u)=0$. Thus, the expected final configuration of the evolving system is as follows: In vicinity of the actin brushes the flow $w$ is strong so that there the phase $u=0$ is the only stable state of the reaction part in . Thus, all throughout that region $u$ quickly decreases to zero, and the transport term $\operatorname{div}_\Gamma(wu)$ has little influence in that region due to $u\approx0$ so that after a while the equation indeed almost behaves like the gradient flow of $E$. Further away from the actin brushes, where the flow $w$ becomes sufficiently weak, the Ezrin-rich phase with $u>0$ suddenly becomes stable. Here we have to distinguish between the behaviour for $\alpha=1$ and $\alpha>1$. - For $\alpha=1$, the stable state of the Ezrin-reaction is directly coupled to the flow by $u=\max\{0,\frac{C_3-C_2-C_1|w|}{C_3}\}$. In particular, towards the back of the cell the flow will be so weak that $u$ will approximately take the value $u\approx1-\frac{C_2}{C_3}$. Again the transport term $\operatorname{div}_\Gamma(wu)$ becomes negligible in that region due to the smallness of $w$ and $\operatorname{div}_\Gamma w$ and the constancy of $u$ so that after a while the equation roughly behaves like a gradient flow of $E$. Now if the transition from high boundary velocity $|w|$ at the cell front to low velocity in the back is gradual, then also $u$ will change gradually along the cell membrane from almost $0$ to almost $1-\frac{C_2}{C_3}$. If on the other hand the flow velocity changes rather abruptly (for instance in the wake of the nucleus), then the transition width from Ezrin-low to Ezrin-rich phase is governed by the diffusion $\epsilon\Delta_\Gamma u$. In that case it is known since the work of Modica and Mortola [@Ref:ModicaMortola1977] that this width roughly behaves like $$\sqrt{\tfrac{\varepsilon}{W_{w,1}(u)-W_{w,1}(1-C_2/C_3)}}(1-\tfrac{C_2}{C_3})\sim\sqrt{\tfrac\varepsilon{C_3}}(1-\tfrac{C_2}{C_3})^{-3/2}$$ for $u$ between the two phases. - For $\alpha>1$ there will be no gradual change between the Ezrin-rich and Ezrin-low phase since there is a concentration gap of $\frac{\alpha-1}\alpha$ between both phases so that $u$ will more or less jump from $u\approx0$ to $u>\frac{\alpha-1}\alpha$. As in the case $\alpha=1$, this jump will be regularized by the diffusion and happen across a width of $\sqrt{\tfrac{\varepsilon}{W_{w,1}(u)-W_{w,1}(1-1/\alpha)}}(1-\tfrac1\alpha)$. Comparison and discussion of numerical and biological experiments {#sec:Experiments} ================================================================= Verification of the model is nontrivial since inside a germ cell the model parameters can only be influenced indirectly, cannot be properly quantified, and typically also govern other processes that can completely alter the cell behaviour. In this section we vary parameters in numerical simulations of the model and compare the results to biological experiments in which the parameters are subjected to qualitatively similar changes. For simplicity (and since a quantitative comparison to biological experiments is ruled out anyway) we discretize and simulate the equations in two rather than three spatial dimensions. Again we will drop the hats on all variables. Finite Element Discretization {#sec:Discretization} ----------------------------- The weak form of - (we will replace by the equivalent ) as well as consists in finding $p\in H^1(\Omega)$, $w\in L^2(\Omega;{\mathbf{R}}^2)$ and $u\in C([0,T];H^1(\Gamma))\cap C^1([0,T];H^{-1}(\Gamma))$ such that $$\begin{aligned} \int_\Omega \nabla p\cdot\nabla\psi \,{{\mathrm d}}x &= \int_\Omega f\cdot\nabla\psi \,{{\mathrm d}}x &&\forall\psi\in H^1(\Omega)\,,\\ \int_\Omega w\cdot q \,{{\mathrm d}}x &= \int_\Omega p\,\operatorname{div}q + f\cdot q \,{{\mathrm d}}x &&\forall q\in L^2(\Omega)\,,\\ \int_\Gamma \partial_tu \, \varphi \,{{\mathrm d}}x &= \int_\Gamma uw\cdot\nabla_\Gamma\varphi - \varepsilon\nabla_\Gamma u\cdot\nabla_\Gamma\varphi - d(w,u)\varphi + a(u,1)\varphi \,{{\mathrm d}}x &&\forall \varphi\in H^1(\Gamma),t\in[0,T]\,.\end{aligned}$$ Note that here we use the standard spaces for the Ezrin equation on $\Gamma$ rather than the minimum regularity spaces from the previous section, since in our numerical experiments the domains and thus also solutions will be smooth. We will use semi-implicit finite differences in time, using a step length $\Delta t = \frac{T}{M}$ for some $M\in{\mathbf{N}}$, and finite elements in space. In order to allow a potential generalization of the model from Darcy to viscous Darcy or Stokes–Brinkmann flow we use Taylor–Hood elements. Thus, given a triangulation $\mathcal T_h$ of a polygonal approximation $\Omega_h$ of our domain and a discretization $\mathcal S_h$ of $\Gamma_h$ into line segments we use the finite element spaces $$\begin{aligned} \mathbf W_h&=\left\{w\in C(\Omega_h;{\mathbf{R}}^2)\,\middle|\,w|_T\in\mathcal P_2\text{ for all }T\in\mathcal T_h\right\}\,,\\ \mathbf P_h&=\left\{p\in C(\Omega_h)\,\middle|\,p|_T\in\mathcal P_1\text{ for all }T\in\mathcal T_h\right\}\,,\\ \mathbf U_h&=\left\{w\in C(\Gamma_h)\,\middle|\,w|_T\in\mathcal P_2\text{ for all }S\in\mathcal S_h\right\}\,,\end{aligned}$$ where $\mathcal P_k$ is the space of polynomials of degree no larger than $k$. In summary, the discretized version of the equations is to find $p_h\in\mathbf P_h$, $w_h\in\mathbf W_h$ and $u_h^0,\ldots,u_h^M\in\mathbf U_h$ such that $$\begin{aligned} \int_{\Omega_h} \nabla p_h^{k+1}\cdot\nabla\psi_h\, {{\mathrm d}}x &= \int_{\Omega_h} f\cdot\nabla\psi_h \, {{\mathrm d}}x &&\forall\psi_h\in\mathbf P_h\,,\\ \int_{\Omega_h} w_h\cdot q_h \, {{\mathrm d}}x &= \int_{\Omega_h} p_h\operatorname{div}_\Gamma\, q_h + f\cdot q_h \, {{\mathrm d}}x &&\forall q_h\in\mathbf W_h\,,\\ \int_{\Gamma_h} \frac{u_h^{k+1}-u_h^k}{\Delta t}\varphi_h \, {{\mathrm d}}x &= \int_{\Gamma_h} u_h^{k+1}w_h\cdot\nabla_{\Gamma_h}\varphi_h - \varepsilon\nabla_{\Gamma_h} u_h^{k+1}\cdot\nabla_{\Gamma_h}\varphi_h\\ &\qquad- d(w_h,u_h^{k})\varphi_h + a(u_h^{k},1)\varphi_h \, {{\mathrm d}}x &&\forall\varphi_h\in\mathbf U_h,k=0,\ldots,M-1\,.\end{aligned}$$ In principle one might let $f$ also depend on time in which case $p_h$ and $w_h$ would become the corresponding time-dependent quasistationary solutions. Since a higher spatial resolution for $u$ was desired, we chose to work with a finer polygonal approximation $\Gamma_h$ of $\Gamma$ than $\partial\Omega_h$ and used nearest neighbour interpolation to approximate $w|_{\Gamma_h}$ by $w|_{\partial\Omega_h}$. Denote by $M,S,B$ the mass, stiffness and mixed mass-stiffness matrix and by $M^\omega$ the weighted mass matrix with weight function $\omega$. Using capital letters for the vectors of degrees of freedom in $p_h$, $w_h$, and $u_h^k$, the discrete system can be rewritten as $$\begin{aligned} SP_h &= B^TF\,, \\ MW_h &= -BP_h+MF\,, \\ \left(\frac{1}{\Delta t}M-A^{w_h}+\varepsilon S\right)U_h^{k+1} &= \frac{1}{\Delta t}MU_h^k - M^{d(w_h,u_h^k)}\mathbf{1} + M^{a(u_h^k,1)}\mathbf{1}\,,\end{aligned}$$ where $\mathbf 1$ denotes the vector representing the constant function $1$. Here, the matrix $A^\omega$ is defined as $A_{ij}^\omega = \int_{\Gamma_h} \varphi_j\omega\cdot\nabla_{\Gamma_h}\varphi_i\,{{\mathrm d}}x$ for the basis functions $\varphi_i$ of $\mathbf U_h$. For a fixed $\varepsilon$ and small enough time step $\Delta t$, the matrix $\left( \frac{1}{\Delta t}M-A+\varepsilon S \right)$ is positive definite so that the last equation can be solved. Likewise, $M$ is invertible and $S$ is invertible on the subspace $\{p\in\mathbf P_h\,|\,\int_{\Omega_h}p\,{{\mathrm d}}x=0\}$ so that the first two equations can be solved as well. Experimental study {#sec:ExperimentalResults} ------------------ We implemented the scheme from the previous section in MATLAB$^{\tiny{\textcircled{c}}}$ and performed simulations on an elliptical cell $\Omega$ with a circular impermeable nucleus $N\subset\Omega$. Our chosen default parameters are provided in \[tab:parameters\]. In dimensional variables these correspond to the values from \[sec:Nondimensionalisation\] as well as the following. --------------------------------- ------------------------------- width and length of the cell 12$\mu$m and 18$\mu$m diameter of the nucleus 6$\mu$m cytoplasmic Ezrin concentration $10^{-18}$mol/$\mu$m$^3$ membraneous Ezrin concentration $10^{-22}$mol/$\mu$m$^2$ $(\gamma,\beta_1,\beta_2)$ $(3\cdot 10^{38},3.3,0.0007)$ --------------------------------- ------------------------------- parameter value ------------------ ----------------------------------------------------------- $T$ 1 $\Omega$ $\{x\in{\mathbf{R}}^2\,|\,(x_1/1.2)^2+(x_2/0.8)^2\leq1\}$ nucleus $N$ $\{x\in{\mathbf{R}}^2\,|\,|x-(0.2 0)|<0.4\}$ $|f|$ $20$ $\varepsilon$ $0.002$ $(\alpha,\zeta)$ $(1,2)$ $(C_1,C_2,C_3)$ $(50,0.1,5)$ : Default parameter values for the numerical simulation of the cytoplasmic flow and Ezrin concentration in a primordial germ cell.[]{data-label="tab:parameters"} The force $f$ which represents the actin brushes is a smoothed point force, $f=\vec f G$ with a smooth nonnegative, compactly supported kernel $G$ of mass $1$, located close to the cell front and pointing forward. Its strength is chosen as $|\vec f|=20$ ($20$ pN/$\mu$m$^2$ in physical dimensions) so as to achieve an average cytoplasmic velocity corresponding to $0.1\,\mu$m/s, which is the value estimated from a few biological microscopy videos. The resulting simulated intracellular flow is shown in \[fig:IntracellularFlowSimulation1\]. at (0,0) [![Intracellular flow resulting from -. Left: Streamlines, colour-coded according to velocity (logarithmic scale; location and direction of the actin brush-induced forces are indicated by the black arrow). Middle: Intracellular pressure distribution. Right: Magnitude of boundary velocity (logarithmic scale). []{data-label="fig:IntracellularFlowSimulation1"}](FlowPlot.pdf "fig:"){width="100.00000%"}]{}; (-154,57) ; (-154,-64) ; (-33,57) ; (-33,-64) ; (91,57) ; (91,-64) ; As for the numerical simulation of -, we typically started from a random Ezrin distribution on $\Gamma$ at time $0$ (that is, $u(0,x)$ for each grid point $x$ was sampled according to the uniform distribution on $[0,1]$). A stationary state was then typically reached towards the end of the simulated time interval (see \[fig:timeEvolution\]). Note that even though exactly at the cell front there is no shear stress, the diffusion is strong enough to remove Ezrin there as well. at (0,0) [![Time evolution of the active Ezrin distribution during actin-brush-induced polarization. Towards the end of the simulation a stationary state is reached. Location and direction of the actin brush-induced forces are indicated in black. []{data-label="fig:timeEvolution"}](EzrinPlotDefault.pdf "fig:"){width="110.00000%"}]{}; (-179,-58) (-116,-58) (-47,-58) (20,-58) (93,-58) (-183,-68) (-120,-68) (-50,-68) (15,-68) (85,-68) In the remainder of this section we try to compare the model behaviour to the observed biological cell behaviour. To this end we varied those model parameters in our simulations which we expect to strongly determine the resulting Ezrin concentration and which could be influenced also experimentally. Subsequently we tried to reproduce these model changes in biological experiments so as to compare the biological result to our simulation. To prepare the experiments, the zebrafish embryos were injected at 1-cell stage into the yolk with 1nl of the sense mRNA and translation blocking morpholino antisense oligonucleotide against chemoattractant Cxcl12a (MOs; GeneTools). Messenger RNA was synthesized using the mMessageMachine kit (Ambion). To express proteins preferentially in germ cells the corresponding coding region was cloned upstream of the 3’UTR of nanos3 gene [@Ref:KoeprunnerEtAl2001 Köprunner, Thisse et al., 2001]. Embryos were incubated at 25 degrees prior to imaging. The representative biological images were acquired with the VisiView software using a widefield fluorescence microscope (Carl Zeiss Axio imager Z1 with a Retiga R6 camera (experiments 1-3)). Confocal microscopy imaging was done using a Carl Zeiss LSM 710 microscope and time lapse movies were taken using the Zen software (experiment 4). Photoactivation was performed using a Carl Zeiss LSM 710 confocal microscope in the bleaching menu of the Zen software. At the beginning, $5$ frames (every $7.75$ seconds) were imaged before the first round of photoactivation with a $456$nm laser ($120-240$ iterations, 100% laser power). Photoactivation was performed near the cell boundaries, opposing existing actin brushes in a circular region of interest (diameter of $5\mu$m) and repeated every $4$ frames. In the following we describe the single experiments; the corresponding numerical and biological results are shown in \[fig:Experiments\]. 1. *Changes in total Ezrin concentration.* The total Ezrin concentration enters in the adsorption strength $C_3$ at the membrane. Its increase will thus increase the amount of active Ezrin on the membrane. Biologically, an Ezrin overexpression (experiment 1(a)) was achieved by injection of Ypet.Ezrin.nos3’UTR RNA construct ($250$pg). A comparable decrease of the Ezrin concentration is more difficult to achieve, thus this was only performed numerically. , 1(a) shows the numerical result for an increased value of the default $C_3$ by factor $5$ alongside with a microscopy image of the biological experiment. While active Ezrin still accumulates in the back, its concentration is elevated on a much larger portion of the cell membrane than usual. , 1(b) shows the numerical result for a decreased value of $C_3$ by factor $1/10$, resulting in an almost vanishing Ezrin concentration on the membrane. 2. *Changes in actin activity.* Numerically, an increased (or decreased) actin activity can be achieved by an in- (or de-)creased force strength $|\vec f|$, resulting in a higher (or lower) intracellular velocity (in experiment 2(a) we use a $5$-fold increase, in experiment 2(b) a decrease by the factor $1/10$). Biologically, the cell contractility was increased (experiment 2(a)) by injection of CA-RhoA.nos3’UTR RNA construct ($15$pg) [@Ref:Paterson1990 Paterson, Self et al., 1990] and decreased (experiment 2(b)) by injecting DN-ROCK.nos3’UTR RNA construct ($150$pg) [@Ref:BlaserEtAl2006 Blaser, Reichmann-Fried et al., 2006]. Both numerical and experimental results are displayed in \[fig:Experiments\], 2(a)-(b). As the microscopy image in 2(a) shows, an increased contractility quickly leads to the formation of a huge bleb, consistent with the complete Ezrin depletion over a large part of the cell membrane observed in the simulation. In case of a decreased actin activity, both the experimental and numerical results in 2(b) show a high active Ezrin concentration over the major part of the membrane. 3. *Changes in cell shape and nucleus position.* The intracellular flow is strongly influenced by the cell shape as well as the position of the nucleus, which shields the back of the cell from the flow. We tested the effect of the cell shape on the equilibrium configuration of active Ezrin by decreasing the semi-minor and increasing the semi-major axis of the cell to $5/18$ and $9/5$, while keeping the cell volume constant. Additionally, we changed the position of the nucleus by shifting it closer to one side of the cell. In order to observe the same behaviour in the microscopy images, the zebrafish embryos were injected with a moderate amount of Ypet.Ezrin.nos3’UTR RNA construct ($80$pg). The cells were followed by time lapse imaging, and the position of the nucleus was examined. The results are displayed in \[fig:Experiments\], 3(a)-(c). While in the simulation results for the long and thin cell, the flow was partially blocked by the nucleus, the high density of active Ezrin in the microscopy image seems to be restricted to the back side of the cell. Reasons for the different behaviour could be the restriction to a two-dimensional region in the simulations or the different proportions of the nucleus and actual cell size. The simulation results for experiments 3(b) and 3(c) however fit the biological images. 4. *Counteracting flows.* Numerically it is straightforward to add a second layer of actin brushes that counteracts the flow of the first actin brushes. The result is a flow-induced removal of active Ezrin in the cell front as well as the back so that active Ezrin only remains at the cell sides (\[fig:Experiments\], 4). Biologically, the embryos were injected with Ypet.Ezrin.nos3’UTR ($80$pg) and photoactivatable Rac1 construct ($150$pg, [@Ref:WuEtAl2009 Wu, Frey et al., 2009]), which is a version of small GTPase Rac1 that becomes active when excited with the $456$nm light, promoting actin polymerization. Thereby, it is possible to generate actin brushes in a second location within a cell. In the mathematical simulations as well as in the corresponding microscopy images, active Ezrin accumulates at the cell sides, in case of the biological experiment it remains restricted to one side, presumably due to the slight asymmetry of the cell. ![All experiments described in section \[sec:ExperimentalResults\] next to each other. The top shows the numerical simulation result and the bottom a corresponding microscopy image.[]{data-label="fig:Experiments"}](ExperimentPlot2.pdf){width="110.00000%"} Overall we find good qualitative agreement between numerical and experimental results, supporting the hypothesis that Ezrin destabilization at the cell membrane by an actin brush-induced intracellular flow may be the mechanism finally leading to bleb formation. Acknowledgement {#acknowledgement .unnumbered} =============== The work was supported by the Alfried Krupp Prize for Young University Teachers awarded by the Alfried Krupp von Bohlen und Halbach-Stiftung, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Germany’s Excellence Strategy through the Clusters of Excellence “Cells-in-Motion” (EXC 1003, particularly project PP-2017-07) and “Mathematics Münster: Dynamics – Geometry – Structure” (EXC 2044) at the University of Münster as well as by the European Research Council (ERC, CellMig) and the German Research Foundation (DFG RA 863/11-1).

--- abstract: 'Let $\mathbb{N}$ be a set of the natural numbers. Symmetric inverse semigroup $R_\infty$ is the semigroup of all infinite 0-1 matrices $\left[ g_{ij}\right]_{i,j\in \mathbb{N}}$ with at most one 1 in each row and each column such that $g_{ii}=1$ on the complement of a finite set. The binary operation in $R_\infty$ is the ordinary matrix multiplication. It is clear that infinite symmetric group $\mathfrak{S}_\infty$ is a subgroup of $R_\infty$. The map $\star:\left[ g_{ij}\right]\mapsto\left[ g_{ji}\right]$ is an involution on $R_\infty$. We call a function $f$ on $R_\infty$ positive definite if for all $r_1, r_2, \ldots, r_n\in R_\infty$ the matrix $\left[ f\left( r_ir_j^\star\right)\right]$ is Hermitian and non-negatively definite. A function $f$ said to be indecomposable if the corresponding $\star$-representation $\pi_f$ is a factor-representation. A class of the $R_\infty$-central functions (characters) is defined by the condition $f(rs)=f(sr)$ for all $r,s\in R_\infty$. In this paper we classify all factor-representations of $R_\infty$ that correspond to the $R_\infty$-central positive definite functions.' author: - 'N.I. Nessonov' --- Introduction ============ Let $R_n$ be the set of all $n\times n$ matrices that contain at most one entry of one in each column and row and zeroes elsewhere. Under matrix multiplication, $R_n$ has the structure of a semigroup, a set with an associative binary operation and an identity element. The number of ${\rm rank}\,k$ matrices in $R_n$ is ${{\displaystyle{{{n}\choose{k}}}^2}}k!$ and hence $R_n$ has a total of $\sum\limits_{k=0}^n{{\displaystyle{{{n}\choose{k}}}^2}}k!$ elements. Note that the set of ${\rm rank}\,n$ matrices in the semigroup $R_n$ is isomorphic to $\mathfrak{S}_n$, the symmetric group on $n$ letters. The semigroup $R_\infty$ is the inductive limit of the chain $R_n$, $ n = 1, 2, . . .$ , with the natural embeddings: $ R_n\ni r=\left[ r_{ij} \right]\mapsto \hat{r}=\left[ \hat{r}_{ij} \right]\in R_{n+1}$, where $r_{ij}=\hat{r}_{ij}$ for all $i,j\leq n$ and $\hat{r}_{n+1\,n+1}=1$. Respectively, the group $\mathfrak{S}_\infty\subset R_\infty$ is the inductive limit of the chain $\mathfrak{S}_n$, $n=1,2,\ldots$. For convenience we will use the matrix representation of the elements of $R_\infty$. Namely, if $r=\left[ r_{ij} \right]\in R_\infty$ then the matrix $\left[ r_{ij} \right]$ contains at most one entry of one in each column and row and $r_{nn}=1$ for all sufficiently large $n$. Denote by $D_\infty\subset R_\infty$ the abelian subsemigroup of the diagonal matrices. For any subset $\mathbb{A}\subset\mathbb{N}$ denote by $\epsilon_\mathbb{A}$ the matrix $\left[ \epsilon_{ij} \right]\in D_\infty$ such that $\epsilon_{ii} =\left\{ \begin{array}{rl} 0, &\text{ if } i\in \mathbb{A}\\ 1, &\text{ if } i\notin \mathbb{A}. \end{array}\right.$ For example, $\epsilon_{\{2\}}= \left[\begin{matrix} 1&0&0&\cdots&\\ 0&0&0&\cdots\\ 0&0&1&\cdots\\ \cdots&\cdots&\cdots&\cdots \end{matrix}\right]$. The ordinary transposition of matrices define an involution on $R_\infty:\left[ r_{ij} \right]^{\star}=\left[ r_{ji} \right]$. Let $\mathcal{B}(\mathcal{H})$ be the algebra of all bounded operators in a Hilbert space $\mathcal{H}$. By a $\star$-representation of $R_\infty$ we mean a homomorphism $\pi$ of $R_\infty$ into the multiplicative semigroup of the algebra $\mathcal{B}(\mathcal{H})$ such that $\pi(r^*)=\left( \pi(r) \right)^*$, where $\left( \pi(r) \right)^*$ is the Hermitian adjoint of operator $\pi(r) $. It follows immediately that $\pi(s)$ is an unitary operator, when $s\in\mathfrak{S}_\infty$, and $\pi(d)$ is self-adjoint projection for $d\in D_\infty$. Recall the notion of the quasiequivalent representation. Let $\mathcal{N}_1$ and $\mathcal{N}_2$ be the $w^*$-algebras generated by the operators of the representations $\pi_1$ and $\pi_2$, respectively, of group or semigroup $G$. $\pi_1 $ and $\pi_2$ are quasiequivalent if there exists isomorphism $\theta:\mathcal{N}_1\to\mathcal{N}_2$ such that $\theta\left(\pi_1(g)\right)=\pi_2(g)$ for all $g\in G$. \[support\_of\_el\_semigroup\] Given an element $r=[r_{mn}]\in R_\infty$, let ${\rm supp}\,r$ be the complement of a set $\left\{n\in \mathbb{N}:r_{nn}=1\right\}$. By definition of $R_\infty$, ${\rm supp}\,r$ is a finite set. Let $c=\left( n_1\;n_2,\;\cdots\;n_k\right)$ be the cycle from $\mathfrak{S}_\infty$. If $\mathbb{A}\subseteq{\rm supp}\,c$, then $q=c\cdot \epsilon_{\mathbb{A}}$ we call as a [*quasicycle*]{}. Notice, that $\epsilon_{\{k\}}$ is a quasicycle. Two quasicycles $q_1$ and $q_2$ are called [*independent*]{}, if $({\rm supp}\,q_1)\cap({\rm supp}\,q_2)=\emptyset$. Each $r\in R_\infty$ can be decomposed in the product of the independent quasicycles\[quasicycle\]: $$\begin{aligned} \label{decomposition_into_product} r=q_1\cdot q_2\cdots q_k, \text{ where } {\rm supp}\, q_i\,\cap \,{\rm supp}\, q_j=\emptyset \text{ for all } i\neq j.\end{aligned}$$ In general, this decomposition is not unique. In this paper we study the $\star$-representations of the $R_\infty$. The main results are the construction of the list of ${\rm II}_1$-factor representations and the proof of its fullness. Finite semigroup $R_n$, its semigroup algebra and the corresponding representations theory was investigated by various authors in [@Munn_1; @Munn_2; @Solomon_1]. The irreducible representations of $R_n$ are indexed by the set of all Young diagrams with at most $n$ cells. An analog of the Specht modules for finite symmetric semigroup was built by C. Grood in [@Grood]. The main motivation of this paper is due to A. M. Vershik and P. P. Nikitin. Using the branching rule for the representations of the semigroups $R_n$, they found some class of the characters of $R_\infty$ [@VN]. Our elementary approach is based on the study of limiting operators, proposed by A. Okounkov [@Ok1; @Ok2]. The examples of $\star$-representations of $R_\infty$. ------------------------------------------------------ Given $r\in R_\infty$ define in the space $l^2(\mathbb{N})$ the map $$\begin{aligned} l^2(\mathbb{N})\ni(c_1,c_2,\ldots,c_n,\ldots)\stackrel{\mathfrak{N}(r)}{\mapsto} (c_1,c_2,\ldots,c_n,\ldots)r\in l^2(\mathbb{N}). \end{aligned}$$ It is easy to check that the next statement holds. The operators $\mathfrak{N}(r)$ generate the [*irreducible*]{} $\star$-representation of $R_\infty$. The next important representation is called the [*left regular representation* ]{} of $R_\infty$ [@APat]. The formula for the action of the corresponding operators in the space $l^2(R_\infty)$ is given by $$\begin{aligned} \label{Left_Reg_Repr} \mathfrak{L}^{reg}(r)\left(\sum\limits_{t\in R_\infty} c_t\cdot t\right)=\sum\limits_{t\in R_\infty:(tt^*)(r^*r)=tt^*} c_t\cdot r\cdot t,\;\;c_t\in\mathbb{C}.\end{aligned}$$ The operators of the [*right regular representation* ]{} of $R_\infty$ act by $$\begin{aligned} \label{Right_Reg_Repr} \mathfrak{R}^{reg}(r)\left(\sum\limits_{t\in R_\infty} c_t\cdot t\right)=\sum\limits_{t\in R_\infty:(t^*t)(rr^*)=t^*t} c_t\cdot t\cdot r,\;\;c_t\in\mathbb{C}.\end{aligned}$$ It is obvious that $\mathfrak{L}^{reg}$ and $\mathfrak{R}^{reg}$ are $\star$-representations of $R_\infty$. Denote by $l^2_k$ the subspace of $l^2(R_\infty)$ generated by the elements of the view $\sum\limits_{t\in R_\infty:{\rm rank}\,(I-tt^*)=k }c_t\cdot t$. By definition, the subspaces $l^2_k$ are pairwise orthogonal. It follows from (\[Left\_Reg\_Repr\]) and (\[Right\_Reg\_Repr\]) that $\mathfrak{L}^{reg}(r)l^2_k=l^2_k$ and $\mathfrak{R}^{reg}(r)l^2_k=l^2_k$ for all $r\in R_\infty$. Denote by $\mathfrak{L}^{reg}_k$ $\left(\mathfrak{R}^{reg}_k \right)$ the restriction of $\mathfrak{L}^{reg}$ $\left( \mathfrak{R}^{reg} \right)$ to $l^2_k$. Since $\mathfrak{R}^{reg}(r_1)\cdot \mathfrak{L}^{reg}(r_2)=\mathfrak{L}^{reg}(r_2)\cdot \mathfrak{R}^{reg}(r_1)$ for all $r_1,r_2\in R_\infty$, the operators $T_k^{(2)}(r_1,r_2)=\mathfrak{L}^{reg}_k(r_1)\cdot\mathfrak{R}^{reg}_k(r_2)$ $\left( r_1,r_2\in R_\infty \right)$ define $\star$-representation of the semigroup $R_\infty\times R_\infty$. The next properties hold: - [**a**]{}) the representation $T_k^{(2)}$ is irreducible for each $k=0,1,\ldots,m,\ldots$; - [**b**]{}) the representation $\mathfrak{L}^{reg}_0$ $\left( \mathfrak{R}^{reg}_0\right)$ is ${\rm II}_1$-factor-representation of $R_\infty$; in particular, $\mathfrak{L}^{reg}_0\left( \epsilon_{\{j\}} \right)=0$ $\left( \mathfrak{R}^{reg}_0\left( \epsilon_{\{j\}} \right)=0 \right)$; - [**c**]{}) for each $k\geq1$ the representation $\mathfrak{L}^{reg}_k$ $\left( \mathfrak{R}^{reg}_k\right)$ is ${\rm II}_\infty$-factor-representation of $R_\infty$; in particular, if $l>k$ then $\mathfrak{L}^{reg}_k\left( \epsilon_{\{j_1\}}\cdots\epsilon_{\{j_l\}} \right)=0$ $\left(\mathfrak{R}^{reg}_k\left( \epsilon_{\{j_1\}}\cdots\epsilon_{\{j_l\}} \right)=0\right)$; - [**d**]{}) for each $k\geq 0$ the restriction of $\mathfrak{L}^{reg}_k$ $\left( \mathfrak{R}^{reg}_k\right)$ to the subgroup $\mathfrak{S}_\infty\subset R_\infty$ is ${\rm II}_1$-factor-representation quasiequivalent to the regular one. We do not use this proposition below and leave its proof to the reader. The results ----------- Let $\pi$ be ${\rm II}_1$-factor $\star$-representation of $R_\infty$ in Hilbert space $\mathcal{H}$. Set $\left(\pi(R_\infty)\right)^\prime=\left\{A\in\mathcal{B}(\mathcal{H}): \pi(r)\cdot A=A\cdot\pi(r) \text{ for all } r\in R_\infty\right\}$, $\left(\pi(R_\infty)\right)^{\prime\prime}=\left(\left(\pi(R_\infty)\right)^\prime\right)^\prime$. Throughout this paper we will denote by $\left[v_1,v_2,\ldots, \right]$ the closure of the linear span of the vectors $v_1,v_2,\ldots\in \mathcal{H}$. Let ${\rm tr}$ be the unique faithful normal trace on the factor $\left(\pi(R_\infty)\right)^{\prime\prime}$. Replacing if needed $\pi$ by the quasi-equivalent representation, we will suppose below that there exists the unit vector $\xi\in \mathcal{H}$ such that $$\begin{aligned} \label{bicyclic_1} \left[\pi\left(R_\infty \right)\xi \right]=\left[ \left(\pi(R_\infty)\right)^\prime\xi\right]=\mathcal{H};\\ \label{vector_trace} {\rm tr}(A)=(A\xi,\xi)\text{ for all } A\in \left(\pi(R_\infty)\right)^{\prime\prime}.\end{aligned}$$ The function $f$ on $R_\infty$, defined by $f(r) =\left(\pi(r)\xi,\xi \right)$, satisfy the next conditions: - ([**1**]{}) central, that is, $f(rs)=f(sr)$ for all $s,r\in R_\infty$; - ([**2**]{}) positive definite, that is, for all $r_1,r_2, \ldots, r_n\in R_\infty$ the matrix $\left[ f\left( r_i^*r_j \right) \right]$ is Hermitian and non-negative definite; - ([**3**]{}) [*indecomposable*]{}, that is, it cannot be presented as a sum of two linearly independent functions, satisfying ([**1**]{}) and ([**2**]{}); - ([**4**]{}) normalized by $f(e)=1$, where $e$ is unit of $R_\infty$. The functions with such properties are called the [*finite characters*]{} of semigroup or group. \[Mult\_th\] Let $f$ be the [*indecomposable*]{} character on $R_\infty$, and let $r=q_1\cdot q_2\cdots q_k$ be its decomposition into the product of the independent quasicycles (see (\[decomposition\_into\_product\])). Then $f(r)=\prod\limits_{j=1}^k f\left(q_j\right)$. Since $({\rm supp}\,q_k)\cap \left(\bigcup\limits_{j=1}^{k-1}{\rm supp}\,q_j\right)=\emptyset$, there exist the sequence $\left\{s_n \right\}\subset \mathfrak{S}_\infty$ such that $$\begin{aligned} \label{supp_infinity} {\rm supp}\,s_nq_ks_n^{-1}\subset(n,\infty ) \text{ and } s_nl=l \text{ for all } l\in\bigcup\limits_{j=1}^{k-1}{\rm supp}\,q_j.\end{aligned}$$ Let us prove that a sequence $\left\{\pi\left( s_nq_ks_n^{-1}\right)\xi \right\}\subset\mathcal{H}$ converges in the weak topology to a vector $f(q_k)\xi$. Indeed, using Multiplicativity Theorem and (\[supp\_infinity\]), we have $\lim\limits_{n\to\infty}\left(\pi( s_nq_ks_n^{-1})\xi,\eta \right)=\left(\pi\left(q_k \right)\xi,\xi \right)\cdot\left(\xi,\eta \right)$ for any $\eta=\pi(r)\xi$, where $r\in R_\infty$. Now we conclude from (\[bicyclic\_1\]) that $\lim\limits_{n\to\infty}\left(\pi( s_nq_ks_n^{-1})\xi,\eta \right)=\left(\pi\left(q_k \right)\xi,\xi \right)\cdot\left(\xi,\eta \right)$ for all $\eta\in\mathcal{H}$. Again using (\[bicyclic\_1\]), we obtain that there exists $\lim\limits_{n\to\infty}\pi( s_nq_ks_n^{-1})=f(q_k)\cdot I_\mathcal{H}$ in the weak operator topology. Therefore, $f(r)= f(s_n rs_n^{-1})$ $ \stackrel{(\ref{supp_infinity})}{=} \lim\limits_{n\to\infty}f\left(s_n q_ks_n^{-1}\prod\limits_{j=1}^{k-1} q_j\right)$ $=f(q_k)\cdot f\left(\prod\limits_{j=1}^{k-1} q_j \right)$ Let us consider any cycle $c=\left(n_1\,n_2\,\ldots\,,n_k \right)$ of the length $k$. If $\mathbb{A}=\left\{n_{j_1},\ldots, n_{j_l} \right\}\subset {\rm supp}\,c$ and $\mathbb{A}\neq\emptyset$ then, using the relations $$\begin{aligned} & c=\left(n_1\;n_k \right)\cdots \left(n_1\;n_{j_i} \right)\cdots\left(n_1\; n_2 \right) \text{ and }\\ &\epsilon_{\{n_1\}}\cdot \left(n_1\;n_{j_i} \right)\cdot\epsilon_{\{n_1\}}=\epsilon_{\{n_{j_i}\}}\cdot \left(n_1\;n_{j_i} \right)\cdot\epsilon_{\{n_{j_i}\}} =\epsilon_{\{n_{j_i}\}}\cdot \epsilon_{\{n_1\}}, \end{aligned}$$ we obtain $f\left(c\cdot\epsilon_\mathbb{A} \right)= f\left( \left(n_1\;n_k \right)\cdots \left(n_1\;n_{j_1} \right)\cdots\left(n_1\; n_2 \right)\cdot\epsilon_{n_{j_1}}\cdot\epsilon_{\mathbb{A}}\right)$ $= f\left(\epsilon_{n_{j_1}}\cdot \left(n_1\;n_k \right)\right.$ $ \left.\cdots \left(n_1\;n_{j_1} \right)\cdots\left(n_1\; n_2 \right)\cdot\epsilon_{n_{j_1}}\cdot\epsilon_{\mathbb{A}}\right)$ $=f\left( \left(n_1\;n_k \right) \cdots\epsilon_{n_{j_1}}\cdot \left(n_1\;n_{j_1} \right)\cdot\epsilon_{n_{j_1}}\cdots\left(n_1\; n_2 \right)\cdot\epsilon_{\mathbb{A}}\right)$ $=f\left( \left(n_1\;n_k \right) \cdots\left(n_1\;n_{j_1+1} \right)\cdot\epsilon_{n_1}\cdot\epsilon_{n_{j_1}}\cdot\left(n_1\;n_{j_1-1} \right)\cdots\left(n_1\; n_2 \right)\cdot\epsilon_{\mathbb{A}}\right)$ $= f\left(\epsilon_{n_{j_1+1}}\cdot \left(n_1\;n_k \right) \right.$ $\left. \cdots\left(n_1\;n_{j_1+1} \right)\cdot\left(n_1\;n_{j_1-1} \right)\cdots\left(n_1\; n_2 \right)\cdot\epsilon_{\mathbb{A}}\right)$ $=f\left( \widetilde{c}\cdot\epsilon_{n_{j_1+1}}\cdot\epsilon_{\mathbb{A}}\right)$, where $\widetilde{c}=\left(n_1\;\cdots\; n_{j_1-1} \;n_{j_1+1}\;\cdots\;n_k \right)$. Applying these equalities $k$ times, we obtain $f\left(c\cdot\epsilon_\mathbb{A} \right)=f\left(\epsilon_{{\rm supp}\,c} \right)$. Therefore, the next corollary is the supplement to theorem \[Mult\_th\] and does not need of the proof already. \[Corollary\] There exists $\rho\in[0,1]$ such that for any quasicycle $q=c\cdot\epsilon_\mathbb{A}$, where $\mathbb{A}\neq\emptyset$, we have $f\left( q\right)=\rho^{\#({{\rm supp}\,c})}$. Next statement follows from theorem \[Mult\_th\] and [@Thoma]. \[restriction\_to\_symmetric\] The restriction of $f$ to the symmetric subgroup $\mathfrak{S}_\infty\subset R_\infty$ is the character of $\mathfrak{S}_\infty$. Denote by $\alpha=\left\{\alpha_1\geq\alpha_2\geq\ldots>0 \right\}$ and $\beta=\left\{\beta_1\geq\alpha_2\geq\ldots>0\right\}$ the corresponding [*Thoma parameters*]{}. Let $s$ be the permutation from $\mathfrak{S}_\infty$ and let $s=c_1\cdot c_2 \cdots c_k$ be its cycle decomposition, where the length of the cycle $c_j$ equal $l(c_j)>1$. Then $$\begin{aligned} f(c_j)=\sum\limits_i \alpha_i^{l(c_j)}+(-1)^{l(c_j)}\sum\limits_i\beta_i^{l(c_j)}\text{ \rm and } f(s)=\prod\limits_{j=1}^k f\left(c_j \right).\end{aligned}$$ Further, we denote this restriction by $\chi_{\alpha\beta}$. The main result of this paper is the following theorem \[main\_th\] Under the notations of Corollary \[Corollary\] and Proposition \[restriction\_to\_symmetric\], we have $\rho\in \alpha\cup 0$. The proof of the main theorem ============================= Let us recall first to the important statement from [@Ok1; @Ok2] that we will use below. \[OkUnkov\_operator\] For any $k$ the sequence $\left\{\pi\left(\left(k\;n \right) \right) \right\}_{n=1}^\infty$ converges in the weak operator topology to self-adjoint operator $\mathcal{O}_k\in\pi\left(\mathfrak{S}_\infty \right)^{\prime\prime}\subset\pi\left(R_\infty \right)^{\prime\prime}$. To prove, it is suffices to notice that $\left(\pi\left(\left(k\;n+1 \right) \right)\cdot\pi(r_1)\xi,\pi(r_2)\xi \right)$ $=\left(\pi\left(\left(k\;N \right) \right)\cdot\pi(r_1)\xi,\pi(r_2)\xi \right)$ for all $r_1, r_2\in R_\infty$, $N>n$ and apply (\[bicyclic\_1\]). Let $S(\mathcal{O}_k)$ be the spectrum of operator $\mathcal{O}_k$ and let $\mu$ denotes the spectral measure of operator $\mathcal{O}_k$, corresponding to vector $\xi$. Then the following hold: - [**1**]{}) the measure $\mu$ is discrete and its atoms can only to zero; - [**2**]{}) if $\;\mathcal{O}_k=\sum\limits_{\lambda\in S\left(\mathcal{O}_k \right)} \lambda E_k(\lambda)$ is the spectral decomposition of $\mathcal{O}_k$ then $\left(E_k(\lambda)\xi,\xi \right)=m(\lambda)\cdot|\lambda|$, where $m(\lambda)\in\mathbb{N}\cup0$; - [**3**]{}) if $\lambda\in S(\mathcal{O}_k)$ is positive (negative) then there exists some Thoma parameter (see Proposition \[restriction\_to\_symmetric\]) such that $\lambda$ $=\alpha_k$ $\left(\lambda=-\beta_k \right)$ and $m(\lambda)=\#\left\{k:\alpha_k=\lambda \right\}\;\;\left(\#\left\{k:-\beta_k=\lambda \right\} \right)$. \[commutativity-lemma\] The operators $\mathcal{O}_j$ and $\pi(\epsilon_{\{k\}})$ mutually commute. In the case $k\neq j$ lemma is obvious. By (\[bicyclic\_1\]), it is sufficient to show that $$\begin{aligned} \label{commutativity} \left(\pi(\epsilon_{\{k\}})\cdot\mathcal{O}_k\xi,\pi(r)\xi \right)=\left(\mathcal{O}_k\cdot\pi(\epsilon_{\{k\}})\xi,\pi(r)\xi \right)\;\text{ for all } \; r\in R_\infty.\end{aligned}$$ Fix the naturale number $N(r)$ such that $r\in N(r)$. For any sufficiently large number $N$ the next chain of the equalities holds: $$\begin{aligned} &\left(\pi(\epsilon_{\{k\}})\cdot\mathcal{O}_k\xi,\pi(r)\xi \right)\stackrel{\text{Lemma }\ref{OkUnkov_operator}}{=}\lim\limits_{n\to\infty}\left(\pi(\epsilon_{\{k\}})\cdot\pi\left(\left(k\;n\right) \right)\xi,\pi(r)\xi \right)\\ &\stackrel{N>{\rm max}\{k,N(r)\}}{=}\left(\pi(\epsilon_{\{k\}})\cdot\pi\left(\left(k\;N\right) \right)\xi,\pi(r)\xi \right)=\left(\pi(\epsilon_{\{k\}})\cdot\pi\left(\left(k\;N\right) \right)\cdot(\pi(r))^*\xi,\xi \right)\\ &\left(\pi\left(\left(k\;N\right) \right)\cdot\pi(\epsilon_{\{N\}})\cdot(\pi(r))^*\xi,\xi \right)\stackrel{N>N(r)}{=}\left(\pi\left(\left(k\;N\right) \right)\cdot (\pi(r))^*\cdot\pi(\epsilon_{\{N\}})\xi,\xi \right)\\ &=\left(\pi(\epsilon_{\{N\}})\cdot\pi\left(\left(k\;N\right) \right)\cdot (\pi(r))^*\xi,\xi \right)=\left(\pi(\epsilon_{\{N\}})\cdot\pi\left(\left(k\;N\right) \right) \xi,\pi(r)\xi \right)\\ &=\left(\pi\left(\left(k\;N\right) \right)\cdot\pi(\epsilon_{\{k\}}) \xi,\pi(r)\xi \right)=\left(\mathcal{O}_k\cdot\pi(\epsilon_{\{k\}})\xi,\pi(r)\xi \right).\end{aligned}$$ The equality (\[commutativity\]) is proved. Let $\mathfrak{A}_j$ be $w^*$-algebra, generated by the operators $\mathcal{O}_j$ and $\pi_j{\{\}}$. Then the following hold: - [i]{}) $\pi(\epsilon_{\{j\}})$ is a minimal projection in $\mathfrak{A}_j$; - [ii]{}) if $\lambda\leq 0$ then $E_j(\lambda)\cdot\pi(\epsilon_{\{j\}})=0$. To prove the property [i]{}), we notice that $$\begin{aligned} \label{relations_generators} \pi(\epsilon_{\{j\}})\cdot\pi\left(\left(\;n \right) \right)\cdot\pi(\epsilon_{\{j\}})=\pi(\epsilon_{\{j\}})\cdot\pi(\epsilon_{\{n\}}).\end{aligned}$$ Since $\pi$ is ${\rm II}_1$-factor representation, the limit of the sequence $\left\{ \pi(\epsilon_{\{n\}})\right\}$ exists in the weak operator topology. Namely, $$\begin{aligned} w^*{\text{-}}\lim\limits_{n\to\infty}\pi(\epsilon_{\{n\}})=\kappa\cdot I, \text{ where } \kappa=\left(\pi(\epsilon_{\{1\}})\xi,\xi \right).\end{aligned}$$ Hence, applying (\[relations\_generators\]), lemma \[OkUnkov\_operator\] and passing to the limit $n\to\infty$, we obtain $$\begin{aligned} \label{need_for_zero} \mathcal{O}_j\cdot\pi(\epsilon_{\{j\}})=\pi(\epsilon_{\{j\}})\cdot\mathcal{O}_j\cdot\pi(\epsilon_{\{j\}})=\kappa\cdot\pi(\epsilon_{\{j\}}).\end{aligned}$$ Therefore, $\mathcal{O}_j^m\cdot\pi(\epsilon_{\{j\}})=\kappa^m\cdot \pi(\epsilon_{\{j\}})$ for all $m\in\mathbb{N}$. Property [i]{}) is proved. We now come to the proof of [ii]{}). By property [i]{}) and lemma \[commutativity-lemma\], in the case, when $\pi(\epsilon_{\{j\}})\neq 0$, there exists only one spectral projection, for example $E_j(\hat{\lambda})$, such that $$\begin{aligned} E_j(\hat{\lambda})\cdot\pi(\epsilon_{\{j\}})= \pi(\epsilon_{\{j\}}) \text{ and } E(\lambda)\cdot\pi(\epsilon_{\{j\}})=0\; \text{ for all } \lambda\neq\hat{\lambda}.\end{aligned}$$ Hence, under the condition $\hat{\lambda}\neq0$, we obtain $$\begin{aligned} &\hat{\lambda}\left(\pi(\epsilon_{\{j\}})\xi,\xi \right)=\left(\mathcal{O}_j\cdot\pi(\epsilon_{\{j\}})\xi,\xi \right) \stackrel{\text{Lemma }\ref{OkUnkov_operator}}{=}\lim\limits_{n\to\infty}\left(\pi((\; n))\cdot \pi(\epsilon_{\{j\}})\xi,\xi\right)\\ &\stackrel{N\neq j}{=}\left(\pi((j\; N))\cdot \pi(\epsilon_{\{j\}})\xi,\xi \right)=\left(\pi(\epsilon_{\{j\}})\cdot\pi((j\; N))\cdot \pi(\epsilon_{\{j\}})\xi,\xi \right)\\ &=\left(\pi(\epsilon_{\{j\}})\cdot \pi(\epsilon_{\{N\}})\xi,\xi \right)\stackrel{\text{Òheorem \ref{Mult_th}}}{=}\left(\pi(\epsilon_{\{j\}})\xi,\xi \right)\left( \pi(\epsilon_{\{N\}})\xi,\xi \right)=\left(\pi(\epsilon_{\{j\}})\xi,\xi \right)^2.\end{aligned}$$ Since $\pi(\epsilon_{\{j\}})\neq 0$, then, by (\[bicyclic\_1\]), $\left(\pi(\epsilon_{\{j\}})\xi,\xi \right)\neq0$. Therefore, $$\begin{aligned} \label{nonzero} \hat{\lambda}=\left(\pi(\epsilon_{\{j\}})\xi,\xi \right)>0.\end{aligned}$$ If $\hat{\lambda}=0$, i.e. $\pi(\epsilon_{\{j\}})\leq E_k(0)$, then, using (\[need\_for\_zero\]), we obtain $\left(\pi(\epsilon_{\{j\}})\xi,\xi \right)=0$. Thus, by (\[bicyclic\_1\]), $\pi(\epsilon_{\{j\}})=0$. The next statement follows from preceding lemma and lemma \[commutativity-lemma\]. \[co\_main\] If $f\left(\epsilon_{\{k\}} \right)\neq0$ then there exists positive $\lambda\in S\left(\mathcal{O}_k \right)$ such that $\pi(\epsilon_{\{k\}})\leq E_k(\lambda)$ and $f\left(\epsilon_{\{k\}}\right)=\lambda$. The proof of Theorem \[main\_th\] {#proof_main_th} --------------------------------- By theorem \[Mult\_th\] and proposition \[restriction\_to\_symmetric\], it is sufficient to find the value of the character $f$ on quasicycle $q=c\cdot\epsilon_\mathbb{A}$, where $c=(1\;2\;\ldots\;k)$, $\mathbb{A}\subset{\rm supp}\,c$ and $\mathbb{A}\neq\emptyset$. Without loss of generality we can assume that $\pi(\epsilon_{\{k\}})\neq0$. Define the map $\vartheta_{\mathbb{A}}: \left\{1,2,\ldots,k \right\}\mapsto \left\{e,\epsilon_1 \right\}$, where $e$ is unit of $R_\infty$, by $$\begin{aligned} \vartheta_\mathbb{A}(j)=\left\{ \begin{array}{rl} \epsilon_{\{1\}},&\text{ if } j\in\mathbb{A}\\ e,&\text{ if } j\notin\mathbb{A}. \end{array}\right.\end{aligned}$$ Since $c=(1\;k)\cdot(1\;k-1)\cdots (1\;2)$, we have $$\begin{aligned} c\cdot\epsilon_{\mathbb{A}}=\vartheta_{\mathbb{A}}(k)\cdot(1\;k)\cdot\vartheta_{\mathbb{A}}(k-1)\cdot(1\;k-1)\cdots\vartheta_{\mathbb{A}}(2)\cdot(1\;2)\cdot\vartheta_{\mathbb{A}}(1).\end{aligned}$$ Therefore, for any collection $\left\{n_2, n_3,\ldots, n_k \right\}\subset\mathbb{N}$, where $n_2>k$, we obtain $$\begin{aligned} f\left(c\cdot\epsilon_{\mathbb{A}} \right)=\left(\pi\left(\vartheta_{\mathbb{A}}(k)\cdot(1\;k)\cdot\vartheta_{\mathbb{A}}(k-1)\cdot(1\;k-1)\cdots\vartheta_{\mathbb{A}}(2)\cdot(1\;2) \cdot\vartheta_{\mathbb{A}}(1)\right)\xi,\xi\right)\\ =\left(\pi\left(\vartheta_{\mathbb{A}}(k)\cdot(1\;n_k)\cdot\vartheta_{\mathbb{A}}(k-1)\cdot(1\;n_{k-1)}\cdots\vartheta_{\mathbb{A}}(2)\cdot(1\;n_2)\cdot\vartheta_{\mathbb{A}}(1) \right)\xi,\xi\right),\end{aligned}$$ Passing in series to the limits $n_k\to\infty, n_{k-1}\to\infty,\ldots,n_2\to\infty$, we come to the relation $$\begin{aligned} \begin{split} f\left(c\cdot\epsilon_{\mathbb{A}} \right)\stackrel{\text{Lemma \ref{OkUnkov_operator}}}{=}\left(\pi\left(\vartheta_{\mathbb{A}}(k) \right)\cdot\mathcal{O}_1\cdot \pi\left(\vartheta_{\mathbb{A}}(k-1) \right)\cdot\mathcal{O}_1\cdots \pi\left(\vartheta_{\mathbb{A}}(2) \right)\cdot\mathcal{O}_1\cdot\pi\left(\vartheta_{\mathbb{A}}(1) \right)\xi,\xi\right)\\ \stackrel{\text{Lemma \ref{commutativity-lemma} }}=\left\{ \begin{array}{rl} \left(\pi\left(\epsilon_{\{1\}} \right)\mathcal{O}_1^{k-1}\xi,\xi \right),&\text{ if } \mathbb{A}\neq \emptyset\\ \left(\mathcal{O}_1^{k-1}\xi,\xi \right),&\text{ if } \mathbb{A}= \emptyset. \end{array}\right. \stackrel{{\text{Corollary \ref{co_main}}}}{=}\left\{ \begin{array}{rl} \lambda^k,&\text{ if } \mathbb{A}\neq \emptyset\\ \chi_{\alpha\beta}(c),&\text{ if } \mathbb{A}= \emptyset. \end{array}\right. \end{split}\end{aligned}$$ Theorem \[main\_th\] is proved. The realisations of ${\rm II}_1$-factor-representations ======================================================= Our aim in this section is the construction of ${\rm II}_1$factor-representations for $R_\infty$. The parameters of the ${\rm II}_1$-factor-representations. {#parameters_of_repr} ---------------------------------------------------------- Let $B(\mathbb{\mathbf{H}})$ denote the set of all (bounded linear) operators acting on the complex Hilbert space $\mathbf{H}$. Let ${\rm Tr}$ be the ordinary[^1] trace on $B(\mathbb{\mathbf{H}})$. Fix the self-adjoint operator $A\in B(\mathbf{H})$ and the [*minimal*]{} orthogonal projection $\mathbf{q}\in B(\mathbf{H})$ such that $A\cdot\mathbf{q}=\mathbf{q}\cdot A$. Let ${\rm Ker}\, A=\left\{ u\in B(\mathbf{H}):Au=0\right\}$, and let $({\rm Ker} A)^\perp=\mathbf{H}\ominus {\rm Ker}\, A $. Denote by $E(\Delta)$ the spectral projection of operator $A$, corresponding to $\Delta\subset \mathbb{R}$. Suppose that occur the next conditions: - [(a)]{} ${\rm Tr} (|A|)\leq1$, - [(b)]{} if $\mathbf{q}\neq0$, then ${\rm Tr}(\mathbf{q})=1$; - [(c)]{} if ${\rm Tr} (|A|)=1$ then ${\rm Ker}\, A=0$; - [(d)]{} if ${\rm Tr} (|A|)<1$ then $\dim ({\rm Ker}\, A)=\infty$; - [(e)]{} $\mathbf{q}\cdot E([-1,0])=0$; Hilbert space $\mathcal{H}_A^\mathbf{q}$. {#hp} ----------------------------------------- Let $\mathbb{S}=\left\{ 1,2,\ldots, {\rm dim}\,\mathbf{H} \right\}$. Fix the matrix unit $\left\{ e_{kl}\right\}_{k,l\in\mathbb{S}}\subset B(\mathbf{H})$. Suppose for the convenience that $$\begin{aligned} Ae_{ll}=e_{ll}A \text{ for all } l.\end{aligned}$$ Let $\mathbb{S}_{reg}=\left\{ n_1,n_2,\ldots\right\}=\left\{ l:e_{ll}\mathbf{H}\subset {\rm Ker}\, A\right\}$ \[s\_regular\](see ([d]{})), where $n_k<n_{k+1}$. Define a state $\psi_k$ on $B\left(\mathbf{H} \right)$ as follows $$\begin{aligned} \label{psik} \psi_k\left(b \right)={\rm Tr}\left(b|A| \right)+\left(1-{\rm Tr}\left(|A| \right)\right) {\rm Tr}\left(b e_{n_kn_k}\right),\;\;b\in B\left(\mathbf{H} \right).\end{aligned}$$ Let $ _1\psi_k$ denote the product-state on $B\left(\mathbf{H}\right)^{\otimes k}$: $$\begin{aligned} \label{product_psik} _1\psi_k\left(b_1\otimes b_2\otimes \ldots\otimes b_k \right)=\prod\limits_{j=1}^k\psi_j\left(b_j \right).\end{aligned}$$ Now define inner product on $B \left( \mathbf{H}\right)^{\otimes k}$ by $$\begin{aligned} \label{inner_product_psik} \left( v,u\right)_k=\,_1\psi_k\left( u^*v \right).\end{aligned}$$ Let $\mathcal{H}_k$ denote the Hilbert space obtained by completing $B \left( \mathbf{H}\right)^{\otimes k}$ in above inner product norm. Now we consider the natural isometrical embedding $$\begin{aligned} v\ni\mathcal{H}_k\mapsto v\otimes {\rm I}\in\mathcal{H}_{k+1},\end{aligned}$$ and define Hilbert space $\mathcal{H}^\mathbf{q}_A$ as completing $\bigcup\limits_{k=1}^\infty \mathcal{H}_k$. The action of $R_\infty$ on $\mathcal{H}^\mathbf{q}_A$. {#action} -------------------------------------------------------- First, using the embedding\ $a\ni B\left(\mathbf{H} \right)^{\otimes k}\mapsto a\otimes{\rm I}\in B\left(\mathbf{H} \right)^{\otimes (k+1)}$, we identify $B\left(\mathbf{H} \right)^{\otimes k}$ with subalgebra $B\left(\mathbf{H} \right)^{\otimes k}\otimes\mathbb{C}\subset B\left(\mathbf{H} \right)^{\otimes (k+1)}$. Therefore, algebra $B\left(\mathbf{H}\right)^{\otimes\infty}=\bigcup\limits_{n=1}^\infty B\left(\mathbf{H} \right)^{\otimes n}$ is well defined. Now we construct the explicit embedding of $\mathfrak{S}_\infty$ into the unitary subgroup of $B\left(\mathbf{H}\right)^{\otimes\infty}$. For $a\in B\left(\mathbf{H}\right)$ put $a^{(k)}={\rm I}\otimes\cdots\otimes{\rm I}\otimes\underbrace{a}_{k}\otimes{\rm I}\otimes{\rm I}\cdots$. Let $E_{-}=E([-1,0[)$ and let $$\begin{aligned} U_{E_{-}}^{(k,\,k+1)}=({\rm I}-E_{-})^{(k)}({\rm I}-E_{-})^{(k+1)}+E_{-}^{(k)} ({\rm I}-E_{-})^{(k+1)}\\ + ({\rm I}-E_{-})^{(k)}E_{-}^{(k+1)}-E_{-}^{(k)}E_{-}^{(k+1)}.\end{aligned}$$ Define the unitary operator $T\left((k\;k+1)\right)\in B\left(\mathbf{H} \right)^{\otimes\infty}$ as follows $$\begin{aligned} \label{embeding_T} T\left((k\;k+1)\right)=U_{E_{-}}^{(k,\, k+1)}\sum\limits_{ij}e_{ij}^{(k)}e_{ji}^{(k+1)}.\end{aligned}$$ Put $T(\epsilon _{\{1\}})=\mathbf{q}^{(1)}$. Using the relation $(k\;k+m)=(k+m-1\;k+m)$ $\cdots(k $ $+1\;k+2)$ $\cdot(k\;k+1)\cdot(k+1\;k+2)\cdots(k+m-1\;k+m)$, we can to prove that $$\begin{aligned} \label{any_transposition} \begin{split} T\left((k\;k+m)\right)=U_{E_{-}}^{(k,\, k+m)}\sum\limits_{ij}e_{ij}^{(k)}e_{ji}^{(k+m)}, \text{ where }\\ U_{E_{-}}^{(k,\, k+m)}=({\rm I}-E_{-})^{(k)}({\rm I}-E_{-})^{(k+m)}-E_{-}^{(k)}E_{-}^{(k+m)}\\ +\left(({\rm I}-E_{-})^{(k)}E_{-}^{(k+m)}+E_{-}^{(k)}({\rm I}-E_{-})^{(k+m)}\right)\prod\limits_{j=k+1}^{k+m-1}\left({\rm I}-2E_- \right)^{(j)}. \end{split}\end{aligned}$$ A easy verification of the standard relations between $\left\{ T((k\;k+1))\right\}_{k\in\mathbb{N}}$ and $\mathbf{q}^{(1)}$ shows that $T$ extends by multiplicativity to the $\star$-homomorphism of $R_\infty$ to $B\left(\mathbf{H} \right)^{\otimes\infty}$. Left multiplication in $B\left(\mathbf{H}\right)^{\otimes\infty}$ defines $\star$-representation $\mathfrak{L}_A$ of $B\left(\mathbf{H}\right)^{\otimes\infty}$ by bounded operators on $\mathcal{H}^\mathbf{q}_A$. Put $\Pi_A(r)=\mathfrak{L}_A\left( T(r)\right)$, $r\in R_\infty$. Denote by $\pi_A^{(0)}$ the restriction of $\Pi_A$ to $\left[ \Pi_A\left( R_\infty\right)\xi _0\right]$, where $\xi _0$ is the vector from $\mathcal{H}^\mathbf{q}_A$ corresponding to the unit element of $B\left(\mathbf{H}\right)^{\otimes\infty}$. If $T\left( \epsilon_{\{1\}} \right)=0$ and $T(s)$ $(s\in\mathfrak{S}_\infty)$ is defined by (\[embeding\_T\]), then $\left\{ \pi_A^{(0)}\left( R_\infty \right) \right\}^{\prime\prime}=\left\{ \pi_A^{(0)}\left( \mathfrak{S}_\infty \right) \right\}^{\prime\prime}$ and the corresponding representation $\pi_A^{(0)}$ is ${\rm II}_1$-factor-representation of $R_\infty$. If $A=\mathbf{p}$, where $\mathbf{p}$ is one-dimensional projection, then $\left(\Pi_\mathbf{p}(s)\xi _0,\xi _0 \right)_{\mathcal{H}^\mathbf{q}_A}=1$ for all $s\in\mathfrak{S}_\infty$. Therefore, we obtain two corresponding representations: - [**1**]{}) $\pi_\mathbf{p}^{(0)}(s)=I$ for all $s\in\mathfrak{S}_\infty$, $\pi_\mathbf{p}^{(0)}(\epsilon_{\{k\}})=I$; - [**2**]{}) $\pi_\mathbf{p}^{(0)}(s)=I$ for all $s\in\mathfrak{S}_\infty$, $\pi_\mathbf{p}^{(0)}(\epsilon_{\{k\}})=0$. If $A=-\mathbf{p}$, then we have the unique representation: $\pi_{-\mathbf{p}}^{(0)}(s)=({\rm sign}\,s)\cdot I$ for all $s\in\mathfrak{S}_\infty$ and $\pi_\mathbf{p}^{(0)}(\epsilon_{\{k\}})=0$. The character formula --------------------- Let $f(r)=\left(\pi_A^{(0)}(r)\xi_0,\xi_0 \right)$, $r\in R_\infty$. Here we will find a formula for $f$. Next statement follows from (\[any\_transposition\]), by the ordinary calculation. \[Realisations\_okounkov\] Let $E_-\neq I$. Then $\lim\limits_{n\to\infty}\pi_A^{(0)}((k\;n))=\mathfrak{L}_A\left( A^{(k)} \right)$ in the weak operator topology. It follows from definition of $\pi_A^{(0)}$ that $f$ satisfies to theorem \[Mult\_th\]. Therefore, it is sufficient to find the value $f$ on quasicycle $q=c\epsilon_\mathbb{A}$, where $c=(1\;2\;\ldots\;k)$ and $\mathbb{A}\subset{\rm supp}\,c$. As in the proof of theorem \[main\_th\](section \[proof\_main\_th\], lemma \[Realisations\_okounkov\] gives $$\begin{aligned} f\left(c\cdot\epsilon_{\mathbb{A}}\right) = \left\{\begin{array}{rl} {\rm Tr}\left(|A|\cdot\mathbf{q}\cdot A^{k-1} \right),&\text{ if } \mathbb{A}\neq \emptyset\\ {\rm Tr}\left( |A|\cdot A^{k-1} \right) ,&\text{ if } \mathbb{A}= \emptyset. \end{array}\right.\end{aligned}$$ It follows from p. \[parameters\_of\_repr\] that there exists a spectral projection $E(\lambda)$ of operator $A$, where $\lambda>0$, with the property $\mathbf{q}\cdot E(\lambda)=\mathbf{q}$. Hence, using proposition \[restriction\_to\_symmetric\], we obtain $$\begin{aligned} f\left(c\cdot\epsilon_{\mathbb{A}}\right) = \left\{\begin{array}{rl} \lambda^k,&\text{ if } \mathbb{A}\neq \emptyset\\ \chi_{\alpha\beta}(c) ,&\text{ if } \mathbb{A}= \emptyset. \end{array}\right.\end{aligned}$$ A.L.T. Paterson, [*Groupoids, Inverse Semigroup, and their Operator Algebras*]{}, Progress in Mathematic (Boston, Mass), V. 170, Springer-Science, 1998, 274 pp. W. D. Munn, [*Matrix representations of semigroups*]{}, Proc. Cambridge Philos. Soc., [**51**]{}, 1955, 1-15. W. D. Munn, [*The characters of the symmetric inverse semigroup*]{}, Proc. Cambridge Philos. Soc., [**53**]{}, 1957, 13-18. L. Solomon, [*Representations of the rook monoid*]{}, J. Algebra 256 2002, 309–342. A. M. Vershik, P. P. Nikitin, Description of the characters and factor representations of the infinite symmetric inverse semigroup; Funct. Anal. Appl., 45:1 (2011), 13–24. C. Grood, [*A Specht Module Analog for the Rook monoid*]{}, The Electronic Journal of Combinatorics 9 (2002), $\#R2$. A.Okounkov, [*The Thoma theorem and representation of the infinite bisymmetric group*]{}, Funct. Anal. Appl. [**28**]{} (1994), no. 2, 100-107. A.Okounkov, [*On the representation of the infinite symmetric group*]{}, Journal of Mathematical Sciences October 1999, Volume 96, Issue 5, pp 3550-3589, arxiv RT-9803037. E.Thoma, [*Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen symmetrischen Gruppe*]{}, Math. Zeitschr. [**85**]{} (1964), no.1, 40-61. Adv. Stud. Contemp. Math., [**7**]{} (1990), 39-117. B. Verkin ILTPE of NASU - B.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine n.nessonov@gmail.com [^1]: ${\rm Tr}(\mathfrak{p})$=1 for any nonzero minimal projection $\mathfrak{p}\in B(\mathbb{\mathbf{H}})$.

--- abstract: 'We present a classification of non-hermitian random matrices based on implementing commuting discrete symmetries. It contains 43 classes. This generalizes the classification of hermitian random matrices due to Altland-Zirnbauer and it also extends the Ginibre ensembles of non-hermitian matrices [@engarde].' author: - Denis Bernard - André LeClair --- Random matrix theory originates from the work of Wigner and Dyson on random hamiltonians [@Dyson]. Since then it has been applied to a large variety of problems ranging from enumerative topology, combinatorics, to localization phenomena, fluctuating surfaces, integrable or chaotic systems, etc... Non-hermitian random matrices also have applications to interesting quantum problems such as open choatic scattering, dissipative quantum maps, non-hermitian localization, etc... See e.g. ref.[@revue] for an introduction. The aim of this short note is to extend the Dyson [@Dyson] and Altland-Zirnbauer [@AltZirn] classifications of hermitian random matrix ensembles to the non-hermitian ones. What are the rules? =================== As usual, random matrix ensembles are constructed by selecting classes of matrices with specified properties under discrete symmetries [@Dyson; @Mehta]. To define these ensembles we have to specify (i) what are the discrete symmetries, (ii) what are the equivalence relations among the matrices, and (iii) what are the probablility measures for each class. (i)\ Let $h$ denote a complex matrix. We demand that the transformations specifying random matrix classes are involutions — their actions are of order two. So we consider the following set of symmetries: &:&h = \_c c h\^T c\^[-1]{},c\^Tc\^[-1]{}= \[Csym\]\ [P sym.]{}&:&h = - p h p\^[-1]{}, p\^2=[**1**]{} \[Psym\]\ [Q sym.]{}&:&h = q h\^ q\^[-1]{}, q\^q\^[-1]{}=[**1**]{} \[Qsym\]\ [K [sym.]{}]{}&:&h = k h\^\* k\^[-1]{}, kk\^\*= \[Ksym\] $h^T$ denotes the transposed matrix of $h$, $h^*$ its complex conjugate and $h^{\dag}$ its hermitian conjugate. The factor $\ep_c$ is just a sign $\ep_c=\pm$. We could have introduced similar signs in the definitions of type $Q$ and type $K$ symmetries; however they can be removed by redefining $h \to ih$. We also demand that these transformations are implemented by [*unitary*]{} transformations: cc\^=[**1**]{}, pp\^=[**1**]{},  qq\^=[**1**]{}, kk\^=[**1**]{} In the case of hermitian matrices one refers to type $C$ symmetries as particle/hole symmetries or time reversal symmetries depending on whether $\ep_c=-$ or $\ep_c=+$ respectively. Matrices with type $P$ symmetry are said to be chiral. Both type $Q$ and type $K$ symmetries impose reality conditions on $h$ and they are redundant for hermitian matrices. (ii)\ We consider matrices up to [*unitary*]{} changes of basis, huhu\^ \[equiv\] In other words, matrices linked by unitary similarity transformations are said to be gauge equivalent. For the symmetries (\[Csym\]–\[Ksym\]), this gauge equivalence translates into: cucu\^T, pupu\^[-1]{}, ququ\^, kuku\^[\*-1]{} \[symgauge\] The classification relies heavily on this rule and on the assumed unitary implementations of the discrete symmetries. We shall only classify minimal classes, which by definition are those whose matrices do not commute with a fixed matrix. (iii)\ Since each of the classes we shall describe below is a subset of the space of complex matrices, the simplest probability measure $\mu(dh)$ one may choose is obtained by restriction of the gaussian one defined by (dh) = [N]{} ( - [Tr]{}hh\^ ) dh \[gauss\] with $\CN$ a normalization factor. It is invariant under the map (\[equiv\]). There is of course some degree of arbitrariness in formulating these rules, in particular concerning the choice of the gauge equivalence (\[equiv\]). It however originates on one hand by requiring the gaussian measure (\[gauss\]) be invariant, and one the another hand from considering auxiliary hermitian matrices $\cal H$ obtained by doubling the vector spaces on which the matrices $h$ are acting. These doubled matrices are defined by: = \[double\] They are always chiral as they anticommute with $\ga_5={\rm diag}(1,-1)$. Any similarity transformations $h\to uhu^{-1}$ are mapped into $\CH\to\CU\CH\CU^{\dag}$ with $\CU={\rm diag}(u,u^{\dag\,-1})$. So, demanding that these transformations also act by similarity on $\CH$ imposes $u$ to be unitary. On $\CH$, both type $P$ and $Q$ symmetries act as chiral transformations, $\CH\to -\CP\CH\CP^{-1}$ with $\CP={\rm diag}(P,P)$ and $\CH\to \CQ\CH\CQ^{-1}$ with $\CQ=\pmatrix{0&Q\cr Q&0\cr}$, and $\CH$ may be block diagonalized if $h$ is $Q$ or $P$ symmetric. Indeed, if $h$ is $Q$ symmetric then $\CQ$ and $\CH$ may be simultaneously diagonalized since they commute. If $h$ is $P$ symmetric, $\CH$ commutes with the product $\ga_5\CP$. Type $C$ and $K$ symmetries both act as particle/hole symmetries relating $\CH$ to its transpose $\CH^T$. The classification of the doubled hamiltonians $\CH$ thus reduces to that of chiral random matrices, cf. [@AltZirn]. However, the spectra of $h$ and $\CH$ may differ significantly so that we need a finer classification involving $h$ per se. Intrinsic definition of classes. ================================ To specify classes we demand that the matrices belonging to a given class be invariant under one or more of the symmetries (\[Csym\]–\[Ksym\]). It is important to bear in mind that when imposing two or more symmetries it is the group generated by these symmetries which is meaningful. Indeed, these groups may be presented in various ways depending on which generators one picks. For instance, if a matrix possesses both a type $P$ and a type $C$ symmetry, then it automatically has another type $C$ symmetry with $c'=pc$ and $\ep'_c=-\ep_c$. The intrinsic classification concerns the classification of the symmetry groups generated by the transformations (\[Csym\]–\[Ksym\]). We demand, as usual, that the transformations (\[Csym\]–\[Ksym\]) commute. For any pair of symmetries the commutativity conditions read: c=pcp\^T &;p\^\*=k\^[-1]{}pk&; q=pqp\^\[com1\]\ q\^T=c\^q\^[-1]{}c &;q\^\*=k\^[-1]{}qk\^[-1]{} &;k\^Tc\^[-1]{}kc\^\*= The signs $\pm$ are arbitrary; they shall correspond to different groups. This arises if no type $P$ and type $Q$ symmetry is imposed so that no reality condition is specified and $h$ is simply a complex matrix. We may then impose either a type $P$ or a type $C$ symmetry or both. Not all groups generated by a type $P$ and a type $C$ symmetry are distinct since, as mentioned above, the product of these symmetries is another type $C$ symmetry but with an opposite sign $\ep_c$. The list of inequivalent symmetry groups, together with the inequivalent choices of the sign $\ep_c$, is the following: [ccc]{} Generators & Discrete symmetry group & Number\ & Defining relations & of classes\ \ No sym & no condition & 1\ $P$ sym & $p^2=1$ & 1\ $C$ sym & $c^T=\pm c,\ \ep_c$ & 4\ $P$, $C$ sym & $p^2=1,\ c^T=\pm c,\ pcp^T=c$ & 2\ $P$, $C$ sym & $p^2=1,\ c^T=\pm \ep_c \, c,\ pcp^T=-c$ & 2\  &  \ If the sign $\ep_c$ does not appear as an entry it means that the value of this sign is irrelevent — opposite values correspond to identical groups. The sign factors $\pm$ written explicitly are relevant — meaning that e.g. the groups generated by a type $C$ symmetry with $c^T=c$ or $c^T=-c$ are inequivalent. The equivalences among the defining relations for groups generated a type $P$ and a type $C$ symmetry are the following: ------------------------------------------------- --------- ------------------------------------------------- $(p^2=1,c^T=\pm c, c = pcp^T)_{\ep_c}$ $\cong$ $(p^2=1,c^T=\pm c, c = pcp^T)_{-\ep_c}$ $(p^2=1,c^T=\pm \ep_c\, c, c =- pcp^T)_{\ep_c}$ $\cong$ $(p^2=1,c^T=\pm\ep_c\, c, c =- pcp^T)_{-\ep_c}$ ------------------------------------------------- --------- ------------------------------------------------- In the above table we anticipate the numbers of classes which we shall describe in the following section. They depend on the numbers of inequivalent representations of each set of defining relations. Considering discrete groups generated by more symmetries of type $P$ or $C$ does not lead to new minimal classes. For instance, if the group is generated by two type $P$ symmetries, then their product commutes with $h$ and thus they do not define a minimal class. Similarly, suppose that we impose two type $C$ symmetries with sign $\ep_{c1}$ and $\ep_{c2}$. If $\ep_{c1}\ep_{c2}=-$, their product makes a type $P$ symmetry and the group they generate is among the ones listed above. If $\ep_{c1}\ep_{c2}=+$, their product commutes with $h$ and they do not specify a minimal class. More generally, considering more combinations of type $P$ and type $C$ symmetries does not lead to new minimal classes as in such cases one may always define fixed matrices commuting with $h$. This arises if we impose at least a type $Q$ or a type $K$ symmetry, but we may simultaneously also impose symmetries of other types. Again, there could be different but equivalent presentations of the same discrete group as not all of these symmetries are independent. For instance, a product of a type $P$ symmetry with a type $C$, $Q$ or $K$ symmetry is again a type $C$, $Q$ or $K$. The list of inequivalent groups generated by two or three of these symmetries is the following: [ccc]{} Generators & Discrete symmetry groups & Number\ & Defining relations & of classes\ \ $Q$ sym & $ q=q^{\dag} $ & 2\ $K$ sym & $ kk^* = \pm 1 $ & 2\ $P$, $Q$ sym & $ p^2=1,\ q^2=1,\ q=\pm pqp^{\dag} $& 2\ $P$, $K$ sym & $ p^2=1,\ kk^* = \pm 1,\ kp^*=pk $& 2\ $P$, $K$ sym & $ p^2=1,\ kk^* = 1,\ kp^*=-pk $& 1\ $Q$, $C$ sym & $q=q^{\dag},\ c^T=\pm c,\ q^T=c^{\dag}q^{-1}c,\ \ep_c$& 8\ $Q$, $C$ sym & $q=q^{\dag},\ c^T=\pm c,\ q^T=-c^{\dag}q^{-1}c,\ \ep_c$& 4\ $P$, $Q$, $C$ sym & $p^2=1,\ q=q^{\dag},\ c^T=\pm c,\ \ep_c$&\    & $c=\ep_{cp}\, pcp^T,\ q=\ep_{pq}\, pqp^{\dag},\ q^T=\ep_{cq}\,c^{\dag}q^{-1}c$& 12\   &  \ As above, the explicitly mentioned signs $\pm$ correspond to inequivalent groups. The groups with defining relations $( p^2=1,\ kk^* =1,\ kp^*=-pk)$ or $( p^2=1,\ kk^* =-1,\ kp^*=-pk)$ are equivalent. The groups generated either by a type $Q$ and a type $K$, or by a type $C$ and a type $K$ symmetries are included in this list because the symmetries of type $C$, $Q$ or $K$ are linked by the fact the product of two of them produces a symmetry of the third type. The last cases, quoted in the last line of the above list, are made of groups generated by three symmetries, one of a type $P$ and two of type either $C$, $Q$ or $K$. Their defining relations depend on the choices of the signs $\ep_{cp}$, $\ep_{pq}$ and $\ep_{cq}$. The equivalences between these choices are the following: ------------------------------------------------------------- --------- -------------------------------------------------------------- $(c^T=\pm c; \ep_{cp}=\ep_{pq}=\ep_{cq}=+)_{\ep_c}$ $\cong$ $(c^T=\pm c; \ep_{cp}=\ep_{pq}=\ep_{cq}=+)_{-\ep_c}$ $(c^T=\pm c; \ep_{cp}=\ep_{pq}=-\ep_{cq}=+)_{\ep_c}$ $\cong$ $(c^T=\pm c; \ep_{cp}=\ep_{pq}=-\ep_{cq}=+)_{-\ep_c}$ $(c^T=\pm c; \ep_{cp}=-\ep_{pq}=\ep_{cq}=+)_{\ep_c}$ $\cong$ $(c^T=\pm c; -\ep_{cp}=\ep_{pq}=\ep_{cq}=-)_{-\ep_c}$ $(c^T=\pm\ep_c\, c; -\ep_{cp}=\ep_{pq}=\ep_{cq}=+)_{\ep_c}$ $\cong$ $(c^T=\pm\ep_c\, c; \ep_{cp}=-\ep_{pq}=\ep_{cq}=-)_{-\ep_c}$ $(c^T=\pm\ep_c\, c; \ep_{cp}=\ep_{pq}=-\ep_{cq}=-)_{\ep_c}$ $\cong$ $ (c^T=\pm\ep_c\, c; \ep_{cp}=\ep_{pq}=\ep_{cq}=-)_{-\ep_c}$ ------------------------------------------------------------- --------- -------------------------------------------------------------- Considering groups generated by more symmetries does not lead to new minimal classes since in such cases one may construct matrices commuting with $h$. Explicit realizations of the classes. ===================================== Having determined the inequivalent groups of commuting discrete symmetries, the second step consists in finding all inequivalent representations of the defining relations of those groups. Due to the rules we choose, in particular the second one, eq.(\[equiv\]), we only consider representations in which all symmetries are unitarily implemented and which are not unitarily equivalent. We shall list all these representations, adding an index $\ep_c$ to recall when the action (\[Csym\]) of the corresponding discrete group depends on $\ep_c$. If we impose no symmetry at all, the class is simply the set of complex matrices. If we impose only a type $P$ symmetry. The matrix $p$ is unitary and square to the identity, so that it is diagonalizable with eigenvalues $\pm 1$. The solution $p=1$ is trivial as it implies $h=0$. Assuming for simplicity that the numbers of $+1$ and $-1$ eigenvalues are equal, we may choose a basis diagonalizing $p$: ( p = \_z1); \[p1\] Here and below, $\sigma_z,\ \sigma_x$ and $\sigma_y$ denote the standard Pauli matrices. If we only impose a type $C$ symmetry, $c$ is unitary and either symmetric and antisymmetric. As is well known [@Dyson], up to an appropriate gauge choice it may be presented into one of the following forms: ( c = 1)\_[\_c]{};(c=i\_y1)\_[\_c]{}; \[c1\] Let us now impose a type $P$ and a type $C$ symmetry. In the basis diagonalizing $p$ with $p = \sigma_z\otimes1$, the commutativity relation $pcp^T=\pm c$ means that either $p$ and $c$ commute or anticommute. If they commute, then $c$ is block diagonal in this basis so that $c=1\otimes 1$ or $c=1\otimes i\sigma_y$ depending whether it is symmetric or antisymmetric. If they anticommute, $c$ is block off-diagonal so that, modulo unitary change of basis, it may be reduced to $c=\sigma_x\otimes 1$ or $c=i\sigma_y\otimes 1$. However, as explained in the previous intrinsic classification, these two cases correspond the same symmetry group but with opposite signs $\ep_c$. So a set of inequivalent representations is: && (p=\_z1, c=11); (p=\_z11, c=1i\_y1) ;\ &&                     (p=\_z1, c=\_x1)\_[\_c]{} ; \[pc1\] Altogether there are $10$ classes of non-hermitian random matrices without reality conditions. They are of course parallel to the $10$ classes of hermitian random matrices [@AltZirn]. Imposing only a type $Q$ symmetry, we have $q=q^{\dag}$ and $qq^{\dag}=1$ since, by choice, we implement the symmetry unitarily. So $q$ is diagonalizable with eigenvalues $\pm 1$. The solution $q=1$ is non-trivial as it simply means that $h$ is hermitian. When $q$ possesses both $+1$ and $-1$ eigenvalues we assume that they are in equal numbers, hence: (q=1);(q=\_z1) ; \[q1\] Imposing only a type $K$ symmetry, we have $kk^*=\pm1$, $kk^{\dag}=1$, and their classification is similar to that of type $C$ symmetries. There are two cases: (k=1);(k=i\_y1); \[k1\] When imposing both type $P$ and $Q$ symmetries, $p$ and $q$ both square to the identity and either commute and anticommute. In the gauge in which $p=\sigma_z\otimes1$ the possible $q$ are $1\otimes1$, $\sigma_z\otimes1$ or $\sigma_x\otimes 1$, up to unitary similarity transformations preserving the form of $p$. However, the two first possibilities generate identical groups, thus the inequivalent representations are: (p=\_z1, q=11);(p=\_z1, q=\_x1) ; Let us next impose a type $P$ and a type $K$ symmetry. With the unitarity property of $p$ and $k$, the commutativity relations between these symmetries are similar to those between type $P$ and $C$ symmetries. Thus, the classification of these representations may be borrowed from eq.(\[pc1\]) and we have: && (p=\_z1, k=11); (p=\_z11, k=1i\_y1) ;\ &&                     (p=\_z1, k=\_x1) ; \[pk1\] When imposing simultaneously a type $Q$ and a type $C$ symmetry, their product is a symmetry of type $K$. The commutativity constraints between type $Q$ and $C$ symmetries are solved in the same way as the commutativity conditions for type $P$ and $C$ symmetries; the only difference being that $q=1$ is now a non-trivial solution, contrary to $p=1$. The inequivalent representations for $q$ and $c$ are: (q=1, c=1)\_[\_c]{} &;& (q=11, c=i\_y1)\_[\_c]{} ; \[qc1\]\ (q=\_z1, c=11)\_[\_c]{} &;& (q=\_z1, c=1i\_y)\_[\_c]{}\ (q =\_z1, c=i\_y1)\_[\_c]{} &;& (q=\_z1, c=\_x1)\_[\_c]{}; Finally, let us impose a type $P$ symmetry together with two among the three types of symmetries $Q$, $C$ and $K$. The commutativity constraints are solved by first choosing a gauge in which $p=\sigma_z\otimes1$. The list of inequivalent solutions is: (p=\_z1, & q =11,  &c=11 [or]{} c=1i\_y);\ (p=\_z1, & q =11,  &c=\_x1)\_[\_c]{} ;\[pcq1\]\ (p=\_z1, & q =\_x1,  &c=11 [or]{} c=1i\_y \ &   & [or]{} c=\_x1 [or]{}  c=\_xi\_y)\_[\_c]{};Here, each choice of $c$ corresponds to a different class. Altogether, eqs.(\[p1\]–\[pcq1\]) give $43$ classes of random non-hermitian matrices. As for the Ginibre ensembles [@Gini] — which correspond to the classes without symmetry or with type $K$ symmetry with $k=1$ or $k=i\sigma_y\otimes 1$ — we expect that in the large matrix size limit the density of states in each class covers a bounded domain of the complex plane with the topology of a disc. [&lt;widest bib entry&gt;]{} This note does not describe the seminar presented by one of the authors (D.B.) at the conference these proceedings are meant for; the content of the seminar may be found in ref.[@marche]. Rather these notes present, with more details, results obtained later in ref.[@Dirac] as a byproduct of the classification of random Dirac fermions in two dimensions. F. Dyson, J. Math. Phys. [**3**]{} (1962) 140; M. Mehta, [*Random matrices*]{}, Academic Press, Boston, 1991. A. Altland and M. Zirnbauer, Phys. Rev. [**B 55**]{} (1997) 1142; M. Zirnbauer, J. Math. Phys. [**37**]{} (1996) 4986; J. Ginibre, J. Math. Phys. [**6**]{} (1965) 440; Y.V. Fyodorov and H.J. Sommers, J. Math. Phys. [**38**]{} (1997) 1918, and references therein; M. Bauer, D. Bernard and J.-M. Luck, J. Phys. A [**34**]{} (2001) 2659; D. Bernard and A. LeClair, “A classification of 2d random Dirac fermions”, arXiv:cond-mat/0109552.

--- abstract: | Identification schemes are interactive protocols typically involving two parties, a *prover*, who wants to provide evidence of his or her identity and a *verifier*, who checks the provided evidence and decide whether it comes or not from the intended prover. In this paper, we comment on a recent proposal for quantum identity authentication from Zawadzki [@Zawadzki19], and give a concrete attack upholding theoretical impossibility results from Lo [@Lo97] and Buhrman et al. [@Buhrman12]. More precisely, we show that using a simple strategy an adversary may indeed obtain non-negligible information on the shared identification secret. While the security of a quantum identity authentication scheme is not formally defined in [@Zawadzki19], it is clear that such a definition should somehow imply that an external entity may gain no information on the shared identification scheme (even if he actively participates injecting messages in a protocol execution, which is not assumed in our attack strategy). author: - 'Carlos E. González-Guillén[^1]' - 'María Isabel González Vasco[^2]' - 'Floyd Johnson[^3]' - 'Ángel L. Pérez del Pozo[^4]' bibliography: - 'QIA.bib' title: Concerning Quantum Identity Authentication Without Entanglement --- Introduction ============ One of the major goals of cryptography is authentication in different flavours, namely, providing guarantees that certain interaction is actually involving certain parties from a designated presumed set of users. In the two party scenario, cryptographic constructions towards this goal are called *identity authentication schemes*, and have been extensively studied in classical cryptography. The advent of quantum computers spells the possible end for many of these protocols however. Since Wiesner proposed using quantum mechanics in cryptography in the 1970’s multiple directions using this concept have undergone serious research. One major role quantum mechanics has played in cryptography is the development of quantum key distribution (QKD) where two parties can securely share a one time pad using quantum mechanics, for example the seminal protocol BB84 [@BB84]. One drawback most of these protocols share is the need for authentication, which is traditionally done over an authenticated classical channel. Classically, there are different ways of defining so-called *identification schemes*, for mutual authentication of peers, mainly depending on whether the involved parties share some secret information (such as a password) or should rely on different (often certified) keys provided by a trusted third party. In the quantum scenario, different identification protocols have been introduced following the first approach, e.g., assuming that two parties may obtain authentication evidence from the common knowledge of a shared secret. These kind of constructions, often called *quantum identity authentication schemes* (or just *quantum identification schemes*), are thus closely related to protocols for *quantum equality tests* and *quantum private comparison*. All these constructions are concrete examples of two-party computations with asymmetric output, i.e. allowing only one of the two parties involved to learn the result of a computation on two inputs. Without posing restrictions on an adversary it was shown by Lo in [@Lo97] and and Buhrman et al. in [@Buhrman12] that these constructions are impossible, even in a quantum setting. As a consequence, constructions for generic unrestricted adversaries in the quantum setting are doomed to failure. All in all, the necessity for authentication in QKD has led to many authors considering approaches which are strictly quantum in nature, such as those in [@Penghao16; @Zeng00; @Huang11] which are based off entanglement or more recently [@Zawadzki19; @Hong17] which do not rely on entanglement. These are known as *quantum identity authentication* (QIA) protocols. For protocols such as BB84 that do not rely on entanglement it would be more appealing to not rely on entanglement for entity authentication purposes. [*Our Contribution.*]{} Recently, an original work about authentication without entanglement by Hong et. al. in [@Hong17] was improved by Zawadzki using tools from classical cryptography in [@Zawadzki19]. We start this contribution by summarizing in section \[sec:impossibility\] the impossibility results from Lo [@Lo97] and Buhrman et al. [@Buhrman12], concerning generic quantum two party protocols. Further, we present and discuss the Zawadzki protocol in section \[sec:zawadzki\_protocol\] and show how it succumbs under a simple attack, which we outline in section \[sec:attack\]. Our attack evidences the practical implications of the proven impossibility of identification schemes as conceived in Zawadki’s design, and thus we stress that fundamental changes in the original proposal, beyond preventing our attack, would be needed in order to derive a secure identification scheme. Quantum Equality Tests are Impossible {#sec:impossibility} ===================================== A *one sided equality test* is a cryptographic protocol in which one party, Alice, convinces another, Bob, that they share a common key by revealing nothing to either party except equality (or inequality) to Bob. Formally we define a key space $K$ and a function $F:K^2\to \{0,1\}$ which checks for equality. Let $i\in K$ be Alice’s key and $j\in K$ be Bob’s key. The goals of a one sided equality test are as follows: 1\) $F(i,j)=1$ if and only if $i=j$. 2\) Alice learns nothing about $j$ nor about $F(i,j)$. 3\) Bob learns $F(i,j)$ with certainty. If $F(i,j)=0$ then Bob learns nothing about $i$ except $i\neq j$. The above is a specific case of a one-sided two-party secure computation protocol as described in [@Lo97]. In this work, a very general result is proven indicating that any protocol realising a one-sided two party secure computation task is impossible, even in a quantum setting. In particular, Lo shows in [@Lo97] that if a protocol satisfies 1) and 2) then Bob can know the output of $F(i,j)$ for any $j$. Furthermore, the one sided equality test with some small relaxations on points 1) and 3) is also proven impossible. Hence, any one-sided QIA protocol which validates identities using equality tests by use of quantum mechanics is impossible without imposing restrictions on an adversary. Note that the above argument says nothing about protocols with built in adversarial assumptions such as those presented in [@Damgard14; @Bouman13]. Further, note that many of the QIA schemes end up with a round where Bob accepts or rejects, which makes Alice aware of the success or failure of the protocol. Indeed, those schemes can be straightforwardly turned into one-sided equality tests by suppressing Bob’s final message announcing the result. Hence, they are clearly insecure against a dishonest Bob. However, note that if any such protocol can be modified so that Alice may obtain information on the identification output at some point before the last protocol round, it is unclear how Lo’s impossibility result would apply. However, if they are built upon equality tests we can get impossibility from another well know result by Buhrman el al.[@Buhrman12]. Certainly, two-sided QIA schemes, in which both Alice and Bob learn the result of the protocol, are a particular case of two-sided two-party computations. It is shown in [@Buhrman12] that a correct quantum protocol for a classical two-sided two-party computation that is secure against one of the parties is completely insecure against the other. For equality tests, if one of the parties, say Alice, learns nothing else than $F(i,j)$, the other party, Bob, will indeed be able to compute $F(i,j)$ for all possible inputs $j$. Thus, any two-sided QIA protocol which validates identities using equality tests is also impossible without imposing further restrictions on the adversary. QIA without Entanglement {#sec:zawadzki_protocol} ======================== Here we will outline the protocol proposed in [@Zawadzki19] with some minor modifications, discussed afterword. Suppose Alice and Bob have keys $k_a$ and $k_b$ respectively. Bob wishes to verify that $k_b=k_a$ without leaking any information about $k_b$ or $k_a$. Bob randomly generates a nonce $r$ from a designated domain and generates a universal hash function $H:\{0,1\}^N\to \{0,1\}^{2d}$. This hash function may be chosen by Bob or sampled at random, in the below description we sample from a space of universal hash functions with image $\{0,1\}^{2d}$ called $\mathbb{H}$. Bob sends Alice $r$ and $H$. Alice then calculates the value $h_a=H(r||k_a)$. Alice then acts on pairs in $h_a$ with an embedding function $Q:\{0,1\}^2\to \CC ^2$. This function $Q$ uses the first of the two binary values to determine the measurement basis (horizontal/vertical or diagonal/antidiagonal) and the second to determine the specific qubit in $\{|0\rangle, |1\rangle , |+\rangle, |-\rangle\}$. For example, $Q(0,0)=|0\rangle$ and $Q(1,0)=|+\rangle$. Applying $Q$ to the pairs of bits in $h_a$ Alice prepares and sends $d$ qubits to Bob over the quantum channel. Upon reception, Bob computes $h_b=H(r||k_b)$. Note that if $k_a=k_b$ then $h_a=h_b$. Using the first bit of each pair Bob measures the quantum states and insures he obtains the correct qubit according to the second bit of the pair. If Bob measures something that disagrees with the even bits of $h_b$ then Bob rejects Alice’s challenge. If after measuring all qubits Bob has not yet rejected Alice’s challenge then he accepts her challenge. Changes made to the protocol are as follows: (1) Bob generates $r$ and $H$, this is done to thwart a simple attack discussed later; (2) the hash function changes between trials, this has no impact on the security of the protocol due to the public nature of the hash in both instances; and finally (3) here we assume for simplicity that Alice and Bob obtain the same nonce $r$ with certainty, using classical error correction techniques one can be relatively certain both parties obtain the same nonce. See below for a schematic overview of the protocol. $\quad$\ Figure 1. The protocol presented in [@Zawadzki19] The reason we force Bob to generate the randomness instead of Alice is that an adversary with unbound quantum memory may impersonate Bob but not make a measurement. Suppose an adversary does not know the key but requests Alice to identify herself. If Alice generates and sends $r,H$ with the string of states $|\varphi _i \rangle$ then the adversary may record $r,H$ and hold in memory, but not measure, the qubits. At a later time an honest participant may ask the adversary to identify themselves, in this case the adversary may send $r,H$ and the qubits in memory. Thus, the adversary correctly forges an authentication. Note that as we have presented the algorithm an adversary may still make this impersonation by waiting between Alice and Bob then passing the information between the two. The difference is as long as Bob generates the nonce then this attack must only be done while Alice and Bob are both online, whereas if Alice generates and sends the nonce then an adversary may hold the states for as long as is technologically feasible. The proposed protocol is claimed to be leakage resistant when considering an adversary measuring in a random basis. The reasoning behind this is that unless an adversary, Eve, correctly guesses the correct basis for each round, she will obtain different values for at least one of the bits of the hash. Now suppose an adversary is capable of computing preimages of hash functions through brute force with unbounded classical computational power or through dictionary attacks with unbounded classical memory. In this case it is unlikely that there will exist a $k_e\in K$ such that $H(r||k_e)$ matches what Eve measured. In the event there does exist such a $k_e$ then with overwhelming probability $k_e\neq k_a=k_b$ and Eve will not be able to falsify authentication of Alice or Bob. Unfortunately, the proposed protocol is claimed to be exactly a two-sided equality test with possible, though unlikely, relaxation of $F(i,j)=1$ if and only if $i=j$ (in this case $i$ is $k_a$ and $j$ is $k_b$). We know such a protocol has necessary leakage and due to the non-interactive nature of Bob we know that $k_b$ has no leakage, thus we know there must exist some leakage on $k_a$. Although Eve may not be able to determine any exact bit of $k_a$ she may drastically reduce the number of possible options for $k_a$ and hence construct a proper subset of $K$ such that the true value for $k_a$ is contained in this subset. An attack exemplifying this phenomenon is described in the next section. A Key Space Reduction Attack on QIA without Entanglement (our contribution) {#sec:attack} =========================================================================== Before discussing the specific attack, let $B$ be a set of orthogonal bases in ${\mathbb{C}}^2$ and consider the following fact. If a quantum state is prepared in a basis $b\in B$ with value $v\in \{0,1\}$, then an adversary may always remove one possible combination of $b$ and $v$ with a single measurement. Upon measuring in basis $b'\in B$ an adversary obtains $v'\in \{0,1\}$. The adversary is then certain the original pair $(b,v)$ was not $(b',1\bigoplus v')$, as when measured in the basis $b$ the qubit prepared by $b$ and $v$ will yield $v$ with certainty. Note that the adversary cannot say with certainty how the qubit was prepared, but they can always remove one possible option. Suppose now that instead of sampling at random for $b$ and $v$, the qubit is prepared using a private key $k\in K$ and a set of public parameters $p$, namely $b=b(k,p)$ and $v=v(k,p)$. An adversary once again measures in basis $b'\in B$ (chosen or taken at random) to obtain $v'\in \{0,1\}$, they may then determine a basis/value pair in which the qubit was not prepared. Because the adversary is assumed to be computationally unbounded they may then compute $b(\hat{k},p)$ and $v(\hat{k},p)$ for all $\hat{k}\in K$. Whenever these computations output the impossible pair $k',v'$ the adversary becomes aware that $\hat{k}\neq k$, hence reducing the key space. The extent to which the key space is reduced depends on the number of basis in $B$. If the distribution of basis choices in $B$ is low entropy the attack may be accomplished as described while if $B$ is high entropy then a probabilistic version decreases the space of likely keys. The assumption that the adversary is computationally unbounded may be lifted if $k$ is low entropy (for he can then indeed test all possible values for $k$ — given there are only a polynomial set of candidates), however assuming a computationally bounded adversary immediately removes unconditional security as an end goal. Let us now apply this key space reduction to the QIA protocol proposed in [@Zawadzki19], in this case the private key is $k$ and the public parameters are $r$ and $H$. Suppose an Eve has no a priori knowledge of the key except its existence in $K$. After receiving $r$ and $H$ over the classical channel she measures all qubits $|\varphi _i \rangle$ received from Alice in the horizontal/vertical basis and records the outputs as $M$. In the case where Eve is utilizing man-in-the-middle she is done, if she is impersonating Bob she accepts or rejects the protocol. After the protocol finishes the adversary may then compute $h_{\hat{k}}=H(r||\hat{k})$ for all $\hat{k}\in K$. Suppose the first qubit Eve measured in $M$ was $|0\rangle$. She now examines the first two bits of each $h_{\hat{k}}$, those that begin 00, 10, or 11 are all possible of obtaining the qubit $|0\rangle$ after measurement. The first of these three tuples will yield $|0\rangle$ with certainty and the later two with a probability of 0.5. The final tuple 01 however is not possible as that would imply that the qubit started in the state $|1\rangle$ and measured in $|0\rangle$. Thus, Eve knows that any $\hat{k}$ such that $h_{\hat{k}}$ begins 01 is not the key. The hash function is assumed to be independent and identically distributed so this removes approximately $\frac{1}{4}$ of all possible keys. Repeat this process for all qubits. After completion of all hash and check operations the adversary has obtained a subset of the key space which contains the key, hence causing information leakage. Specifically, the adversary knows the key is in subset $S$ defined by $$S=\{s\in K: h_{s_{2i}}=M_{i} \text{ and } h_{s_{2i-1}}=0\ \forall i\leq d\}.$$ Note that the true key $k\in S$ and $|S|\approx (\frac{3}{4})^d |K|$. This attack may be repeated using other choice of bases (i.e.- not always selecting the horizontal/vertical basis) and utilizing the same approach these different bases will likely yield different subsets of $K$. The key is in the intersection of all these subsets, decreasing the possible key space further. Note that after applying this attack the advantage of adversary may be negligible yet if $(\frac{3}{4})^d |K|$ is still not negligible in the security parameter. Parameters are not listed in [@Zawadzki19] however it does not seem unreasonable that $d$ is sufficiently large compared to $|K|$, otherwise a false positive for authentication is more likely. Other QIA protocols =================== It is worth pointing out that the attack described in section \[sec:attack\] also applies to the protocol by Hong et al. [@Hong17], which Zawadzki [@Zawadzki19] modifies. In more detail, the protocol in [@Hong17] is similar to Zawadzki’s, but do not use a hash function. Instead, whenever Alice transmits the qubits sequentially and, before sending each qubit, she randomly decides if she is going to use *security mode* or *authentication mode*. In the first case, she sends a decoy state while in the second one, a qubit encoding two bits of the authentication string is sent, similarly to [@Zawadzki19]. After Bob’s reception, Alice announces which mode she just has used. Therefore an adversary using the same strategy described in our attack in section \[sec:attack\] and collecting the information obtained whenever Alice announces authentication mode, will be able to shrink the size of the key space in the same way we have previously stated. On the other hand, other quantum identification protocols proposed in the literature are not vulnerable to our attack neither contradict the impossibility result mentioned in section \[sec:impossibility\]. For instance, some of them [@Penghao16; @Zeng00; @Yang13] are aided by the presence of a trusted third party, therefore not being real two-party protocols. Another type of protocols, such as [@Mihara02; @Shi01; @Zhang00], make use of an entangled quantum state shared between both parties. In [@Mihara02] the users, in addition, share a bitstring used as a password; both parties measures their part of the entangled state to produce a one time key that one of the users XORs with the password and sends the result to the other who checks for consistency. The downside of this approach is that, to repeat the identification process, the parties need to be provided again with new entangled states. In [@Shi01; @Zhang00] the users do not share any classical secret, they just use the entangled state to identify themselves. Conclusion ========== The protocol given by Zawadzki in [@Zawadzki19] may be secure against hash preimage attacks when attempting to find an exact match, however when considering impossible results from quantum measurements we see some hashed key values are not possible. Proverbially, the forest may be secure but each of the trees reveals enough information to reconstruct the possible forests. By eliminating approximately one quarter of the key options from each qubit we see that by measuring all the individual qubits in a random basis does in fact reveal a great deal about the key. This attack has no concern on quantum memory though relies heavily on classical computation power. Hence, unlike [@Damgard14; @Bouman13] where the authors consider a bounded quantum storage model, the only way to make this protocol secure without greatly changing its construction is to constrict an adversaries computational power. The attack proposed here is general in the sense of QIA protocols in the prepare and measure setup, thus any future protocol of this type must consider possible key space reduction attacks. Regardless of the method it is known that any identification protocol which poses no bounds on the adversary will inevitably fail due to results of Lo and Buhrman et al. For this reason we advise that any future attempts at identification schemes consider, and clearly communicate, their assumptions and objectives. [**[Acknowledgements:]{}**]{} This research was sponsored in part by the NATO Science for Peace and Security Programme under grant G5448, in part by Spanish MINECO under grants MTM2016-77213-R and MTM2017-88385-P, and in part by Programa Propio de I+D+i of the Universidad Politécnica de Madrid. [^1]: carlos.gguillen@upm.es [^2]: mariaisabel.vasco@urjc.es [^3]: johnsonf2017@fau.edu [^4]: angel.perez@urjc.es

--- abstract: 'We study the ground states of the single- and two-qubit asymmetric Rabi models, in which the qubit-oscillator coupling strengths for the counterrotating-wave and corotating-wave interactions are unequal. We take the transformation method to obtain the approximately analytical ground states for both models and numerically verify its validity for a wide range of parameters under the near-resonance condition. We find that the ground-state energy in either the single- or two-qubit asymmetric Rabi model has an approximately quadratic dependence on the coupling strengths stemming from different contributions of the counterrotating-wave and corotating-wave interactions. For both models, we show that the ground-state energy is mainly contributed by the counterrotating-wave interaction. Interestingly, for the two-qubit asymmetric Rabi model, we find that, with the increase of the coupling strength in the counterrotating-wave or corotating-wave interaction, the two-qubit entanglement first reaches its maximum then drops to zero. Furthermore, the maximum of the two-qubit entanglement in the two-qubit asymmetric Rabi model can be much larger than that in the two-qubit symmetric Rabi model.' author: - 'Li-Tuo Shen' - 'Zhen-Biao Yang' - Mei Lu - 'Rong-Xin Chen' - 'Huai-Zhi Wu' title: Ground state of the asymmetric Rabi model in the ultrastrong coupling regime --- Introduction ============ The Rabi model [@PR-49-324-1936], describing the interaction between a two-level system and a quantized harmonic oscillator, is a fundamental model in quantum optics. For the cavity quantum electrodynamics (QED) experiments, the qubit-oscillator coupling strength of the Rabi model is far smaller than the oscillator’s frequency and the corotating-wave approximation (RWA) works well, bringing in the ubiquitous Jaynes-Cummins model [@IEEE-51-89-1963; @JMO-40-1195-1993; @PRL-87-037902-2001; @PRA-71-013817-2005]. With recent experiment progresses in Rabi models [@PT-58-42-2005; @Science-326-108-2009; @PR-492-1-2010; @RPP-74-104401-2011; @Nature-474-589-2011; @RMP-84-1-2012; @RMP-85-623-2013; @arxiv1308-6253-2014] in the ultrastrong coupling regime [@PRB-78-180502-2008; @PRB-79-201303-2009; @Nature-458-178-2009; @Nature-6-772-2010; @PRL-105-237001-2010; @PRL-105-196402-2010; @PRL-106-196405-2011; @Science-335-1323-2012; @PRL-108-163601-2012; @PRB-86-045408-2012; @NatureCommun-4-1420-2013], in which the qubit-oscillator coupling strength becomes a considerable fraction of the oscillator’s or qubit’s frequency, the RWA breaks down but relatively complex quantum dynamics arises, bringing about many fascinating quantum phenomena [@NJP-13-073002-2011; @PRL-109-193602-2012; @PRA-81-042311-2010; @PRA-87-013826-2013; @PRA-59-4589-1999; @PRA-62-033807-2000; @PRB-72-195410-2005; @PRA-74-033811-2006; @PRA-77-053808-2008; @PRA-82-022119-2010; @PRL-107-190402-2011; @PRL-108-180401-2012; @PLA-376-349-2012]. Explicitly analytic solution to the Rabi model beyond the RWA is hard to obtain due to the non-integrability in its infinite-dimensional Hilbert space. Since it is difficult to capture the physics through numerical solution [@JPA-29-4035-1996; @EPL-96-14003-2011], various approximately analytical methods for obtaining the ground states of the symmetric Rabi models (SRM) have been tried [@RPB-40-11326-1989; @PRB-42-6704-1990; @PRL-99-173601-2007; @EPL-86-54003-2009; @PRA-80-033846-2009; @PRL-105-263603-2010; @PRA-82-025802-2010; @EPJD-66-1-2012; @PRA-86-015803-2012; @PRA-85-043815-2012; @PRA-86-023822-2012; @EPJB-38-559-2004; @PRB-75-054302-2007; @EPJD-59-473-2010; @arXiv-1303-3367v2-2013; @arXiv-1305-1226-2013; @PRA-87-022124-2013; @PRA-86-014303-2012; @arXiv-1305-6782-2013]. Especially, Braak [@PRL-107-100401-2011] used the method based on the $Z_{2}$ symmetry to analytically determine the spectrum of the single-qubit Rabi model, which was dependent on the composite transcendental function defined through its power series but failed to derive the concrete form of the system’s ground state. In Ref. [@PRA-81-042311-2010], Ashhab *et al.* applied the method of adiabatic approximation to treat two extreme situations to obtain the eigenstates and eigenenergies in the single-qubit SRM, i.e., the situation with a high-frequency oscillator or a high-frequency qubit. Ashhab [@PRA-87-013826-2013] used different order parameters to identify the phase regions of the single-qubit SRM and found that the phase-transition-like behavior appeared when the oscillator’s frequency was much lower than the qubit’s frequency. Lee and Law [@arXiv-1303-3367v2-2013] used the transformation method to seek the approximately analytical ground state of the two-qubit SRM in the near-resonance regime, and found that the two-qubit entanglement drops as the coupling strength further increased after it reached its maximum. Previous studies consider the ground state of the SRM, i.e., the qubit-oscillator coupling strengths of the counterrotating-wave and corotating-wave interactions are equal. In this paper, we study the asymmetic Rabi models (ASRM), i.e., the coupling strengths for the counterrotating-wave and corotating-wave interactions are unequal, which helps to gain deep insight into the fundamentally physical property of such models. Different from Refs. [@PRA-81-042311-2010; @PRA-87-013826-2013], we here use the transformation method to obtain the ground state of the single-qubit ASRM under the near-resonance situation, where the oscillator’s frequency approximates the qubit’s frequency. Differ further from Ref. [@arXiv-1303-3367v2-2013], our investigation for the two-qubit ASRM intuitively identifies the collective contribution to its ground-state entanglement caused by the corotating-wave and counterrotating-wave interactions. We investigate the single- and two-qubit ASRMs and show that their approximately analytical ground states agree well with the exactly numerical solutions for a wide range of parameters under the near-resonance situation, and the ground-state energy has an approximately quadratic dependence on the coupling strengths stemming from contributions of the counterrotating-wave and corotating-wave interactions. Besides, we show that the ground-state energy is mainly contributed by the counterrotating-wave interaction in both models. For the two-qubit ASRM, we obtain the approximately analytical negativity. Interestingly, for the two-qubit ASRM, we find that, with the increase of the coupling strength in the counterrotating-wave or corotating-wave interaction, the two-qubit entanglement first reaches its maximum then drops to zero. The advantages of our result are the collective contributions to the ground state of the ASRM caused by the corotating-wave interaction and counterrotating-wave interaction can be determined approximately, and the contribution of the counterrotating-wave interaction on the ground state energy is larger than that of the corotating-wave interaction. We find that the maximal two-qubit entanglement of the ASRM is larger than that in the case of SRM. However, the transformation method here is applicable to the ASRM only under the near-resonant regime, where the oscillator’s frequency approximates the qubit’s frequency. When the corotating-wave and counterrotating-wave coupling constants are large enough in the ASRM, the result obtained by the transformation method has a big error compared with that obtained by the exactly numerical method. Such an investigation can also be generalized to the complex cases of three- and more-qubit ASRM. Note that the ASRM can be realized by using two unbalanced Raman channels between two atomic ground states induced by a cavity mode and two classical fields in theory [@PRA-75-013804-2007]. The single-qubit ASRM ===================== Transformed ground state ------------------------ The Hamiltonian of the single-qubit ASRM is [@PRA-8-1440-1973]: (assume $\hbar=1$ for simplicity hereafter) $$\begin{aligned} \label{1} H_{1}&=&\frac{1}{2}w_{a}\sigma_{z}+w_{b}b^{\dagger}b\cr&& +\frac{\lambda_{1}}{2}(b^{\dagger}\sigma_{-}+b\sigma_{+}) +\frac{\lambda_{2}}{2}(b^{\dagger}\sigma_{+}+b\sigma_{-}),\end{aligned}$$ where $w_{a}$ is the qubit’s frequency. $\sigma_{z}$ and $\sigma_{\pm}$ are the Pauli matrices, describing the qubit’s energy operator and the spin-flip operators, respectively. We assume that $|\downarrow\rangle_{A}$ and $|\uparrow\rangle_{A}$ are the eigenstates of $\sigma_{z}$, i.e., $\sigma_{z}$ $|\downarrow\rangle_{A}$ $=$ $-|\downarrow\rangle_{A}$ and $\sigma_{z}$ $|\uparrow\rangle_{A}$ $=$ $|\uparrow\rangle_{A}$. $b^{\dagger}$ ($b$) is the creation (annihilation) operator of the harmonic oscillator with the frequency $w_{b}$. The qubit-oscillator coupling strengths of the corotating-wave interaction $(b^{\dagger}\sigma_{-}+b\sigma_{+})$ and the counterrotating-wave interaction $(b^{\dagger}\sigma_{+}+b\sigma_{-})$ are denoted by $\lambda_{1}$ and $\lambda_{2}$, respectively. However, when $\lambda_{1}$ $\neq$ $\lambda_{2}$ (here $\lambda_{1}$, $\lambda_{2}$, $w_{a}$ $\neq0$), to our knowledge, there is still no analytical solution to the ground state of the single-qubit ASRM. Our task in this paper is to determine the ground-state energy $E_g$ and the ground-state vector $|\phi_g\rangle$ for the single- (Section II) or two-qubit (Section III) ASRM, where $H_{1}|\phi_g\rangle$ $=$ $E_{g}|\phi_g\rangle$. In this paper, the subscripts $A$ and $F$ denote the vectors of the atomic state and field state, respectively. To deal with the counterrotating-wave terms in Eq. (\[1\]), we apply a unitary transformation to the Hamiltonian $H_{1}$ [@EPJD-59-473-2010; @PRB-75-054302-2007; @PRA-82-022119-2010]: $$\begin{aligned} \label{3} H_{1}^{'}&=&e^{S_{1}}H_{1}e^{-S_{1}},\end{aligned}$$ with $$\begin{aligned} \label{4} S_{1}=\xi_{1}(b^{\dagger}-b)\sigma_{x},\end{aligned}$$ where $\xi_{1}$ is a variable to be determined later. Then the transformed Hamiltonian $H_{1}^{'}$ can be decomposed into three parts: $$\begin{aligned} \label{5} H_{1}^{'}&=&H_{1}^{a}+H_{1}^{b}+H_{1}^{c},\end{aligned}$$ with $$\begin{aligned} \label{6-7-8} H_{1}^{a}&=&\frac{1}{2}\big[w_{a}\eta_{1}-(\lambda_{1}-\lambda_{2})\xi_{1}\eta_{1}\big]\sigma_{z}\cr\cr&&+ \big[w_{b}-(\lambda_{1}-\lambda_{2})\xi_{1}\eta_{1}\sigma_{z}\big]b^{\dagger}b\cr\cr&& +w_{b}\xi_{1}^{2}-\frac{1}{2}(\lambda_{1}+\lambda_{2})\xi_{1},\\ H_{1}^{b}&=&\big[\frac{1}{4}(\lambda_{1}+\lambda_{2})-w_{b}\xi_{1}\big](b^{\dagger}+b)\sigma_{x}\cr\cr&& -i\big[\frac{1}{4}(\lambda_{1}-\lambda_{2})\eta_{1}+w_{a}\xi_{1}\eta_{1}\big](b^{\dagger}-b)\sigma_{y},\\ H_{1}^{c}&=&\frac{1}{2}w_{a}\sigma_{z}\bigg\{\cosh\big[2\xi_{1}(b^{\dagger}-b)\big]-\eta_{1} \bigg\}\cr&& -\frac{i}{2}w_{a}\sigma_{y}\bigg\{\sinh\big[2\xi_{1}(b^{\dagger}-b)\big]-2\xi_{1}\eta_{1}(b^{\dagger}-b) \bigg\}\cr&& -\frac{i}{4}(\lambda_{1}-\lambda_{2})(b^{\dagger}-b)\sigma_{y}\bigg\{\cosh\big[2\xi_{1}(b^{\dagger}-b)\big]\cr&&-\eta_{1} \bigg\}+\frac{1}{4}(\lambda_{1}-\lambda_{2})(b^{\dagger}-b)\sigma_{z}\bigg\{\sinh\big[2\xi_{1}(b^{\dagger} \cr&&-b)\big]-2\xi_{1}\eta_{1}(b^{\dagger}-b) \bigg\}+O(b^{\dagger2},b^{2}),\end{aligned}$$ where $\eta_{1}$$=_{F}$$\langle 0|\cosh[2\xi_{1}(b^{\dagger}-b)]|0\rangle_{F}$ $=$ $e^{-2\xi_{1}^2}$ and $O(b^{\dagger2},b^{2})=\frac{1}{2}(\lambda_{1}-\lambda_{2})\xi_{1}\eta_{1}(b^{\dagger2}+b^2)\sigma_{z}$. The terms $\cosh[2\xi_{1}(b^{\dagger}-b)]$ and $\sinh[2\xi_{1}(b^{\dagger}-b)]$ in $H_{1}^{c}$ have the dominating expansions [@EPJD-59-473-2010]: $$\begin{aligned} \label{9-10} \cosh[2\xi_{1}(b^{\dagger}-b)]&\simeq&\eta_{1}+O(\xi_{1}^2),\\ \sinh[2\xi_{1}(b^{\dagger}-b)]&\simeq&2\xi_{1}\eta_{1}(b^{\dagger}-b)+O(\xi_{1}^3),\end{aligned}$$ where $O(b^{\dagger2},b^{2})$, $O(\xi_{1}^2)$ and $O(\xi_{1}^3)$ are higher-order terms of $b^{\dagger}$ and $b$, which represent the double- and three-photon transition processes and can be neglected as an approximation when $\xi_1$ and $|\lambda_1\pm\lambda_2|$ are much smaller than the frequency sum $w_a+w_b$ where $w_a\approx w_b$. Thus, $H_{1}^{'}\simeq H_{1}^{a}+H_{1}^{b}$. When the parameter $\xi_{1}$ is chosen such that it satisfies the condition: $$\begin{aligned} \label{11} e^{2\xi_{1}^2}\big[(\lambda_{1}+\lambda_{2})-4w_{b}\xi_{1}\big]=(\lambda_{1}-\lambda_{2})+4w_{a}\xi_{1},\end{aligned}$$ the qubit and the oscillator are coupled in the following form: $$\begin{aligned} \label{12} H_{1}^{b}&=&\frac{1}{2}\big[(\lambda_{1}+\lambda_{2})-4w_{b}\xi_{1}\big]\times\big(b^{\dagger} \sigma_{-}+b\sigma_{+}\big).\end{aligned}$$ Note that $H_{1}^{b}$ in Eq. (\[12\]) contains no counterrotating-wave interactions in which the qubit excitation (deexcitation) is accompanied by the emission (absorption) of a photon. Therefore, the transformed Hamiltonian $H_{1}^{'}$ is exactly solvable when we eliminate the counterrotating-wave terms by choosing $\xi_{1}$ to satisfy Eq. (\[11\]) and by neglecting higher-order transition processes which are presented by terms $O(b^{\dagger2},b^{2})$, $O(\xi_{1}^2)$ and $O(\xi_{1}^3)$. It is easy to show that the eigenvector $|\downarrow\rangle_A|0\rangle_F$ is the ground-state vector of the transformed Hamiltonian $H_{1}^{'}$, with $|0\rangle_{F}$ being the vacuum state of the harmonic oscillator, and the corresponding eigenenergy $E_{g1}$ is: $$\begin{aligned} \label{13} E_{g1}=\xi_1^2w_{b}-\frac{1}{2}(\lambda_1+\lambda_2)\xi_1-\frac{1}{2}\eta_1[w_a-\xi_1(\lambda_1-\lambda_2)].\cr&&\end{aligned}$$ We see that when $\lambda_{1}=\lambda_{2}$, $E_{g1}$ reduces to the transformed ground-state energy derived in Ref. [@EPJD-59-473-2010]. Therefore, the ground state of the original Hamiltonian (\[1\]) can be approximately constructed: $$\begin{aligned} \label{14} |\phi_{g1}\rangle&=&e^{-S_{1}}|\downarrow\rangle_{A}|0\rangle_{F}\cr &=&\frac{1}{\sqrt{2}}(|\psi_{A}^{+}\rangle|-\xi_1\rangle_F-|\psi_{A}^{-}\rangle|\xi_1\rangle_F),\end{aligned}$$ with $|\xi_{1}\rangle_{F}$ and $|-\xi_{1}\rangle_{F}$ being the coherent states of the oscillator with the amplitudes $\xi_{1}$ and $-\xi_{1}$. $|\psi_{A}^{+}\rangle=(|\uparrow\rangle_A+|\downarrow\rangle_A)/\sqrt{2}$ and $|\psi_{A}^{-}\rangle=(|\uparrow\rangle_A-|\downarrow\rangle_A)/\sqrt{2}$ are the eigenstates of $\sigma_x$. ![(Color online) The ground-state energy for the single-qubit ASRM obtained by the transformation method $E_0=E_{g1}$ (red grid) and by the numerical solution $E_0=E_{g}$ (blue grid) versus the coupling strengths $\lambda_{1}$ and $\lambda_{2}$: (a) $w_{b}=0.8w_{a}$; (b) $w_{b}=w_{a}$; (c) $w_{b}=1.2w_{a}$. The energy deviation $\Delta E_{g1}=E_{g1}-E_g$ versus $\lambda_{1}$ and $\lambda_{2}$: (d) $w_{b}=0.8w_{a}$; (e) $w_{b}=w_{a}$; (f) $w_{b}=1.2w_{a}$. []{data-label="Fig.1."}](fig1.eps){width="0.9\columnwidth"} The value of $\xi_{1}$ is obtained by numerically solving the nonlinear equation (\[11\]). $\xi_{1}$ has an approximately linear dependence on the counterrotating-wave coupling strength by neglecting high-order terms of the field mode as: $$\begin{aligned} \label{29} \xi_{1}&\simeq&\frac{\lambda_2}{2(w_a+w_b)}.\end{aligned}$$ In Fig. 1, we compare the ground-state energy obtained by the transformation method and that by the numerical solution. Especially, we find that the ground-state energy obtained by the transformation method coincides very well with the exactly numerical solution when $|\lambda_1-\lambda_2|\leq0.15w_a$. Therefore, when $\lambda_{1},\lambda_{2}\leq w_{a}$, the transformed ground-state energy $E_{g1}$ approximates: $$\begin{aligned} \label{15} E_{g1}\simeq-\frac{1}{2}w_a-\frac{\lambda_2^2}{4(w_a+w_b)}+\frac{\lambda_2^3(\lambda_1-\lambda_2)}{8(w_a+w_b)^3},\end{aligned}$$ which shows that the ground-state energy has an approximately quadratic dependence on the coupling strength by neglecting high-order terms of the field mode for the small factor $|\lambda_1-\lambda_2|$ and is mainly contributed by the counterrotating-wave interaction. This result differs further from that of the SRM [@EPJD-59-473-2010]. Considering the fidelity $F_{1}$ for the ground state $|\phi_{g1}\rangle$, where $F_{1}=\langle\phi_{g1}|\phi_{g}\rangle$ and $|\phi_g\rangle$ is the ground state obtained through numerical solutions [@arXiv-1303-3367v2-2013], we plot $F_1$ as a function of the coupling strengths $\lambda_{1}$ and $\lambda_{2}$ under different detunings in Fig. 2. The result shows that the fidelity is higher than $99.9\%$ when $\lambda_1\leq0.5w_{a}$ and $\lambda_2\leq0.5w_{a}$. Furthermore, the fidelity under the positive-detuning case ($w_{b}-w_{a}>0$) decreases slowest among all the cases in Fig. 2 (a) - (c) when $\lambda_1$ and $\lambda_2$ increase. ![(Color online) The fidelity $F_{1}$ of the ground state for the single-qubit ASRM obtained by the transformation method versus the coupling strengths $\lambda_{1}$ and $\lambda_{2}$: (a) $w_{b}=0.8w_{a}$; (b) $w_{b}=w_{a}$; (c) $w_{b}=1.2w_{a}$. []{data-label="Fig.2."}](fig2.eps){width="1\columnwidth"} Ground-state entanglement ------------------------- In this section, we focus on the entanglement between the qubit and the oscillator in the ground state of the single-qubit ASRM. Since the ground state is a pure state, we take the von Neumann entropy as an entanglement measure. If a pure state of a composite system $XY$ is given by the density matrix $\rho_{XY}$, the entropy of the subsystem $X$ is defined as: $$\begin{aligned} \label{SS} S_{\rho_{X}} = -Tr(\rho_{X}log_2\rho_{X}),\end{aligned}$$ where $\rho_{X}=Tr_{Y}(\rho_{XY})$ is the reduced density matrix for the subsystem $X$ by tracing out the freedom degree of the subsystem $Y$. Note that $S_{\rho_{X}}$ measures the entanglement between the subsystems $X$ and $Y$ of the system, which has a maximum value of $log_{2}K$ in a $K$-dimensional Hilbert space. In the standard basis $\{|\uparrow\rangle_A,|\downarrow\rangle_A \}$, the reduced density matrix of the qubit is $\rho_{A}=Tr_{F}(|\Phi_{G}\rangle\langle\Phi_{G}|)$, where $|\Phi_{G}\rangle$ is the exactly numerical ground state of the single-qubit ASRM. The entropy of the qubit $S_{\rho_{A}}$ = $-Tr(\rho_{A}log_2\rho_{A})$ is numerically plotted in Fig. 3, which shows that the entanglement between the qubit and the oscillator increases from as $\lambda_1$ and $\lambda_2$ increase from zero to values close to $w_a$ and $w_b$. ![(Color online) The degree of entanglement $S_{\rho_A}$ for the qubit in the ground state of the single-qubit ASRM obtained by the numerical simulation versus the coupling strengths $\lambda_{1}$ and $\lambda_{2}$: (a) $w_{b}=0.8w_{a}$; (b) $w_{b}=w_{a}$; (c) $w_{b}=1.2w_{a}$. []{data-label="Fig.3."}](fig3.eps){width="1\columnwidth"} The two-qubit ASRM ================== Transformed ground state ------------------------ The Hamiltonian of the two-qubit ASRM is [@PRA-8-1440-1973]: $$\begin{aligned} \label{17} H_{}&=&w_{a}J_{z}+w_{b}b^{\dagger}b+g_{1}(b^{\dagger}J_{-}+bJ_{+})\cr\cr&& +g_{2}(b^{\dagger}J_{+}+bJ_{-}),\end{aligned}$$ where $w_{a}$ is the frequency of each qubit. $J_{l}\{ l=x,y,z,\pm\}$ describes the collective qubit operator of a spin-$1$ system. $b^{\dagger}$ ($b$) is the creation (annihilation) operator of the harmonic oscillator with the frequency $w_{b}$. The qubit-oscillator coupling strengths of the corotating-wave and counterrotating-wave interactions are $g_{1}$ and $g_{2}$, respectively. We denote the eigenstates of $J_{z}$ by $|-1\rangle_{A}$, $|0\rangle_{A}$, and $|1\rangle_{A}$, i.e., $J_{z}|m\rangle_{A}=m|m\rangle_{A}$ ($m=0,\pm1$). $|0\rangle_{F}$ is the vacuum state of the harmonic oscillator, and $|\alpha\rangle_{F}$ denotes the coherent-state field with the amplitude $\alpha$. When a rotation around the $y$ axis is performed, the Hamiltonian of the two-qubit ASRM can be written as : $$\begin{aligned} \label{18} H_{2}&=&w_{a}J_{x}+w_{b}b^{\dagger}b+(g_{1}+g_{2})(b^{\dagger}+b)J_{z}\cr\cr&& +i(g_{1}-g_{2})(b^{\dagger}-b)J_{y}.\end{aligned}$$ To transform the Hamiltonian $H_{2}$ into a mathematical form without counterrotating-wave terms, we apply a unitary transformation to $H_{2}$: $$\begin{aligned} \label{19} H_{2}^{'}&=&e^{S_{2}}H_{2}e^{-S_{2}},\end{aligned}$$ with $$\begin{aligned} \label{20} S_{2}&=&\xi_{2}(b^{\dagger}-b)J_{z},\end{aligned}$$ where $\xi_{2}$ is a variable to be determined. Therefore, the transformed Hamiltonian $H_{2}^{'}$ is decomposed into three parts: $$\begin{aligned} \label{21} H_{2}^{'}&=&H_{2}^{a}+H_{2}^{b}+H_{2}^{c},\end{aligned}$$ with $$\begin{aligned} \label{22-23-24} H_{2}^{a}&=&w_{b}b^{\dagger}b+\bigg[ w_{a}\eta_{2}-(g_{1}-g_{2})\eta_{2}\xi_{2}\bigg]J_{x}\cr&& +\bigg[w_{b}\xi_{2}^{2}-2\xi_{2}(g_{1}+g_{2}) \bigg]J_{z}^{2},\\ H_{2}^{b}&=&\bigg[ (g_{1}+g_{2})-w_{b}\xi_{2}\bigg](b^{\dagger}+b)J_{z}\cr&&+ i\bigg[ w_{b}\eta_{2}\xi_{2}+(g_{1}-g_{2})\eta_{2}\bigg](b^{\dagger}-b)J_{y},\\ H_{2}^{c}&=&w_{a}J_{x}\bigg\{ \cosh[\xi_{2}(b^{\dagger}-b)]-\eta_{2}\bigg\} \cr&&+ iw_{a}J_{y}\bigg\{ \sinh[\xi_{2}(b^{\dagger}-b)]-\eta_{2}\xi_{2}(b^{\dagger}-b) \bigg\}\cr&&+ (g_{1}-g_{2})(b^{\dagger}-b)J_{x}\bigg\{ \sinh[\xi_{2}(b^{\dagger}-b)]\cr&&-\eta_{2}\xi_{2}(b^{\dagger}-b) \bigg\}+i(g_{1}-g_{2})(b^{\dagger}-b)J_{y}\cr&&\times \bigg\{ \cosh[\xi_{2}(b^{\dagger}-b)]-\eta_{2}\bigg\}+ O(b^{\dagger2},b^{2}),\end{aligned}$$ where $\eta_{2}=$ $_{F}\langle 0|\cosh[\xi_{2}(b^{\dagger}-b)]|0\rangle_{F}$ $=$ $e^{-\xi_{2}^{2}/2}$ and $O(b^{\dagger2},b^{2})=(g_{1}-g_{2})\eta_{2}\xi_{2}J_{x} (b^{\dagger2}-2b^{\dagger}b-b^{2})$. As shown in the single-qubit ASRM, when $\xi_2$ and $|g_1\pm g_2|$ are much smaller than the frequency sum $w_a+w_b$ where $w_a\approx w_b$, $H_{2}^{c}$ can be neglected, thus $H_{2}^{'}\simeq H_{2}^{a}+H_{2}^{b}$. Compared with $H_{1}^{a}$ in the single-qubit ASRM of Sec. II, the main difference is the presence of the $J_{z}^{2}$ operator term in $H_{2}^{a}$, but in the single-qubit ASRM the corresponding term $\sigma_z^2=1$ is just a constant. Therefore, $H_{2}^{a}$ here represents a renormalized three-level system in which we need to diagonalize $H_{2}^{a}$ to remove counterrotating-wave terms. The eigenvalues $\nu_{k}$ ($k=1,2,3$) and eigenstates $|\varphi_{k}\rangle_{A}$ of the Hamiltonian $H_{2}^{''}=H_{2}^{a}-w_{b}b^{\dagger}b$ are: $$\begin{aligned} \label{25} \nu_{1}&=&\frac{A}{2}-\frac{1}{2}\sqrt{A^{2}+8B^2}, \cr |\varphi_{1}\rangle_{A}&=&\frac{1}{N_{1}}\bigg\{|-1\rangle_{A}-\frac{(A+\sqrt{A^2+8B^2})}{2B}|0\rangle_{A}+|1\rangle_{A}\bigg\},\cr\cr \nu_{2}&=&A, \cr\cr |\varphi_{2}\rangle_{A}&=&\frac{1}{N_{2}}\bigg\{-|-1\rangle_{A}+|1\rangle_{A}\bigg\},\cr\cr \nu_{3}&=&\frac{A}{2}+\frac{1}{2}\sqrt{A^{2}+8B^2}, \cr\cr |\varphi_{3}\rangle_{A}&=&\frac{1}{N_{3}}\bigg\{|-1\rangle_{A}-\frac{(A-\sqrt{A^2+8B^2})}{2B}|0\rangle_{A}+|1\rangle_{A}\bigg\},\cr&&\end{aligned}$$ with $$\begin{aligned} \label{26} A&=&w_{b}\xi_{2}^2-2\xi_{2}(g_{1}+g_{2}), \cr B&=&\frac{1}{\sqrt{2}}[w_{a}\eta_{2}-\eta_{2}\xi_{2}(g_{1}-g_{2})],\end{aligned}$$ where $N_{k}$ is the normalization factor of the eigenvector $|\varphi_{k}\rangle_{A}$. Here the eigenvalues are arranged in the decreasing order: $\nu_{1}<\nu_{2}<\nu_{3}$. Then $H_{2}^{'}$ can be expanded in terms of the renormalized eigenvectors: $$\begin{aligned} \label{27} H_{2}^{'}&\simeq&\sum_{k=1}^{3}\nu_{k}|\varphi_{k}\rangle_{A}\langle \varphi_{k}|+\bigg[ (D_{1}b+D_{2}b^{\dagger})|\varphi_{1}\rangle_{A}\langle \varphi_{2}|\cr&&+(D_{3}b+D_{4}b^{\dagger})|\varphi_{2}\rangle_{A}\langle \varphi_{3}|+H.c.\bigg]+w_{b}b^{\dagger}b,\end{aligned}$$ where $D_{x}\ (x=1,2,3,4)$ is the coefficient depending on the variable $\xi_{2}$. After transforming the Hamiltonian $H_{2}$ into $H_{2}^{'}$, we can eliminate counterrotating-wave terms describing the coupling between the lowest two eigenstates by setting: $$\begin{aligned} \label{28} D_{1}&=&\eta_{2}\bigg[ w_{a}\xi_{2}+(g_{1}-g_{2})\bigg]\bigg(A+\sqrt{A^2+8B^2}\bigg)\cr&&-2\sqrt{2}B\bigg[ (g_{1}+g_{2})-w_{b}\xi_{2}\bigg]=0.\end{aligned}$$ The value of $\xi_{2}$ is obtained by numerically solving the nonlinear equation (\[28\]). We find that when $g_{1}\leq0.5w_{a}$ and $g_{2}\leq0.5w_{a}$, $\xi_{2}$ has an approximately linear dependence on the coupling strengths: $$\begin{aligned} \label{29} \xi_{2}&\simeq&\frac{(w_{b}-w_{a})g_{1}+(w_{b}+w_{a})g_{2}}{w_{b}^2+w_{a}^2}.\end{aligned}$$ ![(Color online) The ground-state energy for the two-qubit ASRM obtained by the transformation method $E_0=E_{g2}$ (red grid) and the numerical solution $E_0=E_{g}$ (blue grid) versus the coupling strengths $g_{1}$ and $g_{2}$: (a) $w_{b}=0.8w_{a}$; (b) $w_{b}=w_{a}$; (c) $w_{b}=1.2w_{a}$, where $E_{g2}$ is plotted by using $\nu_{1}$ in Eq. (\[25\]). The energy deviation $\Delta E_{g2}=E_{g2}-E_g$ versus $g_{1}$ and $g_{2}$: (d) $w_{b}=0.8w_{a}$; (e) $w_{b}=w_{a}$; (f) $w_{b}=1.2w_{a}$. []{data-label="Fig.4."}](fig4.eps){width="1\columnwidth"} In Fig. 4, we compare the ground-state energy obtained by the transformation method and that obtained by the numerical solution. We find that when $g_{1}\leq0.25w_{a}$ and $g_{2}\leq0.25w_{a}$, the ground-state energy obtained by the transformation method coincides very well with the exact value even for $|g_{1}-g_{2}|=0.24w_{a}$. Therefore, when $g_{1}\leq0.5w_{a}$ and $g_{2}\leq0.5w_{a}$, $|\varphi_{1}\rangle_{A}|0\rangle_{F}$ is expected to be the approximately analytical ground state of the transformed Hamiltonian, and the ground state $|\phi_{g}\rangle$ of the two-qubit ASRM can be expressed by the transformed ground state $|\phi_{g2}\rangle$: $$\begin{aligned} \label{30} |\phi_{g2}\rangle&=&e^{-S_{2}}|\varphi_{1}\rangle_{A}|0\rangle_{F} \cr&&=\frac{1}{N_{1}}\big( |-1\rangle_{A}|\xi_{2}\rangle_{F}-\frac{\nu_{3}}{B}|0\rangle_{A}|0\rangle_{F}+|1\rangle_{A}|-\xi_{2}\rangle_{F}\big),\cr&&\end{aligned}$$ and the ground-state energy $E_{g2}$ is: $$\begin{aligned} \label{31} E_{g2}&\simeq&\nu_{1}\simeq -w_{a}-\frac{(g_{1}+g_{2})g_{2}}{w_{a}w_{b}},\end{aligned}$$ which directly shows that $E_{g2}$ has an approximately quadratic dependence on the qubit-oscillator coupling strengths by neglecting high-order terms of the field mode. This result differs further from that in the two-qubit SRM [@arXiv-1303-3367v2-2013]. ![(Color online) The fidelity $F_{2}$ of the ground state for the two-qubit ASRM obtained by the transformation method versus the coupling strengths $g_{1}$ and $g_{2}$: (a) $w_{b}=0.8w_{a}$; (b) $w_{b}=w_{a}$; (c) $w_{b}=1.2w_{a}$. []{data-label="Fig.5."}](fig5.eps){width="1\columnwidth"} The fidelity $F_{2}$ of the ground state as a function of the qubit-oscillator coupling strengths $g_{1}$ and $g_{2}$ under different detunings is plotted in Fig. 5. The result shows that $F_{2}$ keeps higher than $99.9\%$ when $g_{1}\leq0.25w_{a}$ and $g_{2}\leq0.25w_{a}$, which coincides with the behavior of the transformed ground-state energy shown in Fig. 5. Ground-state entanglement ------------------------- We also examine the ground-state entanglement of the two-qubit ASRM by taking into account both the transformation method and the exactly numerical treatment. Negativity is taken to quantify the entanglement for two qubits, which is defined as [@PRA-65-032314-2002]: $$\begin{aligned} \label{32} M_{\rho_A}&=&\frac{\|\rho_A^{T}\|-1}{2},\end{aligned}$$ where $\rho_A^{T}$ is the partially transposed matrix of the two-qubit reduced density matrix $\rho_A$, with $\rho_A=Tr_{F}(\rho_{AF})$ and $\rho_{AF}=|\phi_{g}\rangle\langle \phi_{g}|$, and $\|\rho_A^{T}\|$ is the trace norm of $\rho_A^{T}$. Thus, $M_{\rho_A}$ alternatively equals the absolute value for the sum of the negative eigenvalues of $\rho_A^{T}$. For the transformed ground state $|\phi_{g2}\rangle$ in Eq. (\[30\]), the partially transposed matrix of the reduced density operator for the two qubits in the qubit basis $\Gamma_{q}$ $=$ $\{$ $|\uparrow_{1}\rangle|\uparrow_{2}\rangle, |\uparrow_{1}\rangle|\downarrow_{2}\rangle, |\downarrow_{1}\rangle|\uparrow_{2}\rangle, |\downarrow_{1}\rangle|\downarrow_{2}\rangle$ $\}$, where $|\uparrow_{l}\rangle$ and $|\downarrow_{l}\rangle$ ($l=1,2$) correspond to the excited and ground states of the $l$th qubit respectively, is obtained as follows: ![(Color online) The negativity $M_{\rho_A}$ of two qubits in the ground state of the two-qubit ASRM versus the coupling strengths $g_{1}$ and $g_{2}$: (a) $w_{b}=0.8w_{a}$; (b) $w_{b}=w_{a}$; (c) $w_{b}=1.2w_{a}$. The results obtained by the transformation method $M_{\rho_A}=M_{\rho_A}^{t}$ and the numerical simulation $M_{\rho_A}=M_{\rho_A}^{n}$ are represented by the red grid and the blue grid, respectively. The deviation in the two-qubit negativity $\Delta M=M_{\rho_A}^{n}-M_{\rho_A}^{t}$ obtained by transformation method: (d) $w_{b}=0.8w_{a}$; (e) $w_{b}=w_{a}$; (f) $w_{b}=1.2w_{a}$. []{data-label="Fig.6."}](fig6.eps){width="0.95\columnwidth"} $$\begin{aligned} \label{33} \rho_A^{T}&=&\frac{1}{(2+\beta^2)} \left(\begin{array}{cccc} 1 & \frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}} & \frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}} & \frac{\beta^2}{2} \\ \frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}} & \frac{\beta^2}{2} & e^{-2\alpha^2} & \frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}}\\ \frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}} & e^{-2\alpha^2} & \frac{\beta^2}{2} & \frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}} \\ \frac{\beta^2}{2} & \frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}} & \frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}} & 1 \end{array} \right),\cr&&\end{aligned}$$ where $\alpha=\xi_{2}$ and $\beta=-\frac{\nu_{3}}{B}$. With Eq. (\[33\]), we can calculate the negative $M_{\rho_A}$: $$\begin{aligned} \label{34} M_{\rho_A}&=&\max \bigg\{ \frac{2e^{-2\xi_{2}^2}-(\frac{\nu_{3}}{B})^2}{2[2+(\frac{\nu_{3}}{B})^2]}, 0 \bigg\}.\end{aligned}$$ When $g_1\leq0.25w_{a}$ and $g_2\leq0.25w_{a}$, $M_{\rho_A}$ approximates: $$\begin{aligned} \label{35} M_{\rho_A}&\simeq& \frac{w_{b}\big[(1-\frac{1}{\sqrt{2}})^2g_{2}^{2}+g_1g_2\big]}{4w_{a}(w_{a}+w_{b})^2}.\end{aligned}$$ From Eq. (\[35\]), we see that the two-qubit entanglement increases with $g_2^2$ and $g_1g_2$. The two-qubit negativity as a function of the qubit-oscillator coupling strengths $g_{1}$ and $g_{2}$ under different detunings has been plotted in Fig. 6 (a) - (c), and the corresponding deviation from the numerical simulation is plotted in Fig. 6 (d) - (f). For $0<g_1\leq0.25w_a$ and $0<g_2\leq0.25w_a$, the two-qubit negativity has a linear dependence on $g_1$ for the fixed $g_2$ and a quadratic dependence on $g_2$ for the fixed $g_1$; For $0<g_1\leq0.25w_a$ and $0.25w_a<g_2<0.5w_a$, the negativity keeps close to zero; However, for $0.25w_a<g_1<0.5w_a$ and $0<g_2<0.5w_a$, the negative has a similar dependence on $g_1$ and $g_2$ with the case of $0<g_1\leq0.25w_a$ and $0<g_2\leq0.25w_a$. We find that when $g_1\leq0.25w_{a}$ and $g_2\leq0.25w_{a}$ the deviation in the negativity is close to zero, meaning the ground state obtained by the transformation method agrees well with the exact one. This directly shows that the two-qubit entanglement is caused by the counterrotating-wave interaction in the Hamiltonian. Interestingly, after the negativity has reached its maximum, it will monotonically decrease when $g_{1}$ or $g_{2}$ further increases. Furthermore, the maximum of the two-qubit entanglement in the two-qubit ASRM is far larger than that in the two-qubit SRM, and the two-qubit entanglement mainly appears when the coupling strength of the corotating-wave interaction is bigger than that of the counterrotating-wave interaction, which is because the contribution to the two-qubit entanglement from the counterrotating-wave interaction is larger than that from the corotating-wave interaction in Eq. (\[35\]). As seen from Fig. 7, when $g_{1}>1.11w_{a}$ or $g_{2}>0.88w_{a}$ at $w_{b}=w_{a}$, $M_{\rho_A}$ decreases to zero and never increases again, and the maximum negativity is about $0.10$ which is only $3.5\times10^{-2}$ in the two-qubit SRM [@arXiv-1303-3367v2-2013]. ![(Color online) The negativity $M_{\rho_A}$ of two qubits in the ground state of the two-qubit ASRM obtained by the numerical simulation versus the coupling strengths $g_{1}$ and $g_{2}$ when $w_{b}=w_{a}$. []{data-label="Fig.7."}](fig7.eps){width="0.5\columnwidth"} In Fig. 8, we numerically plot the entropy $S_{\rho_A}$ of two qubits versus the coupling strengths $g_1$ and $g_2$ in the ground state of the two-qubit ASRM, where $S_{\rho_A}$ $=$ $-Tr(\rho_{A}log_2\rho_{A})$. The result shows that the entanglement between the qubit and the oscillator increases from as $g_1$ and $g_2$ increase from zero to values close to $w_a$ and $w_b$. ![(Color online) The degree of entanglement $S_{\rho_A}$ for the qubits in the ground state of the two-qubit ASRM obtained by numerical simulations versus the coupling strengths $g_{1}$ and $g_{2}$: (a) $w_{b}=0.8w_{a}$; (b) $w_{b}=w_{a}$; (c) $w_{b}=1.2w_{a}$. []{data-label="Fig.8."}](fig8.eps){width="1\columnwidth"} Conclusion ========== In conclusion, we have used the transformation method to obtain the approximately analytical ground states of the single- and two-qubit ASRMs, and shown that the transformed results coincided well with those obtained by numerical simulations for a wide range of parameters under the near-resonance condition. We find that the ground-state energy in either the single- or two-qubit ASRM has an approximately quadratic dependence on the qubit-oscillator coupling strengths, and the contribution of the counterrotating-wave interaction on the ground state energy is larger than that of the corotating-wave interaction. Interestingly, we also find that the two-qubit entanglement of the two-qubit ASRM decreases to zero and never increases again as long as the qubit-oscillator coupling strengths are large enough. 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--- abstract: 'We consider the $\Lambda N\to NN$ weak transition, responsible for a large fraction of the non-mesonic weak decay of hypernuclei. We follow on the previously derived effective field theory and compute the next-to-leading one-loop corrections. Explicit expressions for all diagrams are provided, which result in contributions to all relevant partial waves.' author: - 'A. Pérez-Obiol' - 'D. R. Entem' - 'B. Juliá-Díaz' - 'A. Parreño' title: 'One-loop contributions in the EFT for the $\Lambda N \to NN$ transition' --- Introduction ============ One of the major challenges in nuclear physics is to understand the interactions among hadrons from first principles. For more than twenty years, many research groups have directed their efforts to develop Effective Field Theories (EFT), working with the idea of separating the nuclear force in long-range and short-range components. The underlying premise was that low-energy processes, as the ones encountered in nuclear physics, should not be affected by the specific details of the high-energy physics. The typical energies associated to nuclear phenomena suggest that the appropriate degrees of freedom are nucleons and pions (or the ground state baryon and pseudo scalar octets for processes involving strangeness), interacting derivatively as it is dictated by the effective chiral Lagrangian. The nuclear interaction is characterized by the presence of very different scales, going from the values of the masses of the light pseudo-scalar bosons to the ones of the ground-state octet baryons. The EFT formalism makes use of this separation of scales to construct an expansion of the Lagrangian in terms of a parameter built up from ratios of these scales. For example, in the study of the low-energy nucleon-nucleon interaction, a clear separation of scales is seen between the external momentum of the interacting nucleons, a soft scale which typically takes values up to the pion mass, and a hard scale corresponding to the nucleon mass. While the long-range part of this interaction is governed by the light scale through the pion-exchange mechanism, short-range forces are accounted for by zero-range contact operators, organized according to an increasing number of derivatives. These contact terms, which respect chiral symmetry, have values which are not constrained by the chiral Lagrangian, and therefore, their relative strength (encapsulated in the size of the low-energy coefficients, LECs) has to be obtained from a fit to nuclear observables. The large amount of experimental data for the interaction among pions and nucleons has made possible to perform successful EFT calculations of the strong nucleon-nucleon interaction up to fourth order in the momentum expansion (${\cal O}(p^4)$), at next-to-next-to-next-to-leading order (N$^3$LO) in the heavy-baryon formalism [@Epelbaum:2012vx; @entem]. In the weak sector, the study of nucleon-nucleon Parity Violation (PV) with an Effective Field Theory at leading order has been undertaken in Ref. [@nucPV05], where the authors discuss existing and possible few-body measurements that can help in constraining the relevant (five) low-energy constants at order $p$ in the momentum expansion and the ones associated with dynamical pions. In the strange sector, the experimental situation is less favorable due to the short life-time of hyperons, unstable against the weak interaction. This fact complicates the extraction of information regarding the strong interaction among baryons in free space away from the nucleonic sector. Nevertheless, SU(3) extensions of the EFT for nucleons and pions have been developed at leading order (LO) [@SW96; @KDT01; @H02; @BBPS05] and next-to-leading (NLO) order [@PHM06]. In the present work we consider the weak four-body $\Lambda N \to NN$ interaction, which is accessible experimentally by looking at the decay of $\Lambda-$hypernuclei, bound systems composed by nucleons and one $\Lambda$ hyperon. These aggregates decay weakly through mesonic ($\Lambda \to N \pi$) and non-mesonic ($\Lambda N \to NN$) modes, the former being suppressed for mass numbers of the order or larger than 5, due to the Pauli blocking effect acting on the outgoing nucleon. In contrast to the weak NN PV interaction, which is masked by the much stronger Parity Conserving (PC) strong NN signal, the weak $|\Delta S|=1 \, \Lambda N$ interaction has the advantage of presenting a change of flavor as a signature, favoring its detection in the presence of the strong interaction. The first studies of the weak $\Lambda N$ interaction using a lowest order effective theory were presented in Refs. [@Jun; @PBH05; @PPJ11] . These works included the exchange of the lighter pseudoscalar mesons while parametrizing the short-range part of the interaction with contact terms at order ${\cal O}(q^0)$, where $q$ denotes the momentum exchanged between the interacting baryons. While the results of Ref. [@PPJ11] show that it is possible to reproduce the hypernuclear decay data with the lowest order effective Lagrangian, the stability of the momentum expansion has to be checked by including the next order in the EFT. If an effective field theory can be built for the weak $\Lambda N \to NN$ transition, the values for the LECs of the theory, which encode the high-energy components of the interaction, should vary within a reasonable and natural range when one includes higher orders in the calculation. Compared to the LO calculation, which involves two LECs, the unknown baryon-baryon-kaon vertices and the pseudoscalar cut-off parameter in the form-factor, the NLO calculation introduces additional unknowns. Namely, the parameters associated to the new contact terms (three when one neglects the small value of the momentum of the initial particles, a nucleon and a hyperon bound in the hypernucleus, in front of the momentum of the two outgoing nucleons) and the couplings appearing in the two-pion exchange diagrams. Therefore, in order to constrain the EFT at NLO, one needs to collect enough data, either through the accurate measure of hypernuclear decay observables, or through the measure of the inverse reaction in free space, $n p \to \Lambda p$. Unfortunately, the small values of the cross-sections for the weak strangeness production mechanism, of the order of $10^{-12}$ mb [@Haidenbauer1995; @Parreno1998; @Inoue2001], has prevented, for the time being, its consideration as part of the experimental data set, despite the effort invested in extracting different polarization observables for this process [@Kishimoto2000; @Ajimura2001]. At present, quantitative experimental information on the $|\Delta S|=1$ weak interaction in the baryonic sector comes from the measure of the total and partial decay rates of hypernuclei, and an asymmetry in the number of protons detected parallel and antiparallel to the polarization axis, which comes from the interference between the PC and PV weak amplitudes. Since observables from one hypernucleus to another can be related through hypernuclear structure coefficients, one has to be careful in selecting the data that can be used in the EFT calculation. For example, while one may indeed expect measurements from different p-shell hypernuclei, say, A=12 and 16, to provide with the same constraint, the situation is different when including data from s-shell hypernuclei like A=5. For the latter, the initial $\Lambda N$ pair can only be in a relative s-state, while for the former, relative p-states are allowed as well. In this paper we present the analytic expressions to be included at next-to-leading order in the effective theory for the weak $\Lambda N$ interaction. These expressions have been derived by considering four-fermion contact terms with a derivative operator insertion together with the two-pion exchange mechanism. The paper is organized as follows. In Section II we introduce the Lagrangians and the power counting scheme we use to calculate the relevant Feynman diagrams. In Sections \[ss:loc\] and \[ss:nloc\] we present the LO and NLO potentials for the $\Lambda N\rightarrow NN$ transition, and a comparison between both contributions is performed in Section \[sec:bc\]. We conclude and summarize in Section \[sec:conclusions\]. Interaction Lagrangians and counting scheme {#sec2} =========================================== The non-mesonic weak decay of the $\Lambda$ involves both the strong and electroweak interactions. The $\Lambda$ decay is mediated by the presence of a nucleon which in the simplest meson-exchange picture, exchanges a meson, e.g. $\pi$, $K$, with the $\Lambda$. Thus, computing the transition requires the knowledge of the strong and weak Lagrangians involving all the hadrons entering in the process. In this section we describe the strong and weak Lagrangians entering at leading order (LO) and next-to-leading order (NLO) in the $\Lambda N\to NN$ interaction. ![Weak vertices for the $\Lambda N\pi$, $\Sigma N\pi$ and $NNK$ stemming from the Lagrangians in Eq. (\[eq:weakl\]). The weak vertex is represented by a solid black circle. \[vf2\]](nnpw "fig:") ![Weak vertices for the $\Lambda N\pi$, $\Sigma N\pi$ and $NNK$ stemming from the Lagrangians in Eq. (\[eq:weakl\]). The weak vertex is represented by a solid black circle. \[vf2\]](nspw "fig:") ![Weak vertices for the $\Lambda N\pi$, $\Sigma N\pi$ and $NNK$ stemming from the Lagrangians in Eq. (\[eq:weakl\]). The weak vertex is represented by a solid black circle. \[vf2\]](nnkw "fig:") The weak interaction between the $\Sigma$, $\Lambda$ and $N$ baryons and the pseudoscalar $\pi$ and $K$ mesons is described by the phenomenological Lagrangians: $$\begin{aligned} \label{eq:weakl} {\mathcal{L}}_{\Lambda N\pi}^w=&-iG_Fm_\pi^2\overline{\Psi}_N(A+B\gamma^5) {\vec{\tau}}\cdot\vec{\pi}\Psi_\Lambda \\\nonumber {\mathcal{L}}_{\Sigma N\pi}^w=& -iG_Fm_\pi^2\overline{\Psi}_N(\vec{A}_{\Sigma_i}+\vec{B}_{\Sigma_i}\gamma^5) \cdot\vec{\pi}\Psi_{\Sigma_i}\,, \\\nonumber {\cal L}^{w}_{NN K} =& -iG_Fm_\pi^2 \, \left[ \, \overline{\psi}_{N} \left( ^0_1 \right) \,\,( C_{K}^{PV} + C_{K}^{PC} \gamma_5) \,\,(\phi^{K})^\dagger \psi_{N} \right. \\ \nonumber & \left. + \, \overline{\psi}_{N} \psi_{N} \,\,( D_{K}^{PV} + D_{K}^{PC} \gamma_5 ) \,\,(\phi^{K})^\dagger \,\, \left( ^0_1 \right) \right] \ ,\end{aligned}$$ where $G_Fm_\pi^2=2.21\times10^{-7}$ is the weak Fermi coupling constant, $\gamma$ are the Dirac matrices and $\tau$ the Pauli matrices. The index $i$ appearing in the $\Sigma$ field refers to the different isospurion states for the $\Sigma$ hyperon: $$\Psi_{\Sigma\frac12}= \left(\begin{array}{c}-\sqrt{\frac23}\Sigma_+\\\frac{1}{\sqrt3}\Sigma_0\end{array}\right)\,, ~~ \Psi_{\Sigma\frac32}= \left(\begin{array}{c}0\\-\frac{1}{\sqrt3}\Sigma_+\\\sqrt{\frac23}\Sigma_0\\\Sigma_-\, \end{array}\right)\,.$$ The PV and PC structures, $\vec{A}_{\Sigma_i}$ and $\vec{B}_{\Sigma_i}$ contain the corresponding weak coupling constants together with the isospin operators $\tau^a$ for $\frac12\to\frac12$ transitions and $T^a$ for $\frac12\to\frac32$ transitions. The weak couplings $A=1.05$, $B=-7.15$, $A_{\Sigma\frac12}=-0.59$, $A_{\Sigma\frac32}=2.00$, $B_{\Sigma\frac12}=-15.68$, and $B_{\Sigma\frac32}=-0.26$ [@DF96] are fixed to reproduce the experimental data of the corresponding hyperon decays, while the ones involving kaons, $C_K^{PC}=-18.9$, $D_K^{PC}=6.63$, $C_{K}^{PV}=0.76$ and $D_K^{PV}=2.09$, are derived using SU(3) symmetry. ![Weak vertices corresponding to the $\Lambda N\pi\pi$, and $\Lambda N$ interactions. The solid black circle represents the weak vertex. The corresponding Lagrangians are given in Eq. (\[eq:weakl2\]). \[vf3\]](nnppw "fig:") ![Weak vertices corresponding to the $\Lambda N\pi\pi$, and $\Lambda N$ interactions. The solid black circle represents the weak vertex. The corresponding Lagrangians are given in Eq. (\[eq:weakl2\]). \[vf3\]](nlw "fig:") The other two weak vertices entering at the considered order (Fig. \[vf3\]) are obtained from the weak SU(3) chiral Lagrangian, $$\begin{aligned} {\mathcal{L}}^w_{\Lambda N\pi\pi}=& G_Fm_\pi^2\frac{h_{2\pi}}{f_\pi^2}(\vec{\pi}\cdot\vec{\pi}) \overline{\Psi}\Psi_\Lambda\,, \label{eq:weakl2}\\ {\mathcal{L}}^w_{\Lambda N} =& iG_Fm_\pi^2 h_{\Lambda N} \overline{\Psi}\Psi_\Lambda \nonumber\,,\end{aligned}$$ with $ h_{2\pi}=(D+3F)/(8\sqrt6 G_Fm_\pi^2)=-10.13\text{ MeV} $ and $ h_{\Lambda N}=-(D+3F)/(\sqrt6 G_F m_\pi^2)=81.02\text{ MeV} \,. $ $D$ and $F$ are the couplings parametrizing the weak chiral SU(3) Lagrangian, and can be fitted through the pole model to the experimentally known hyperon decays. In that case, one finds that when s-wave amplitudes are correctly reproduced, p-wave amplitude predictions disagree with the experiment [@donoghue]. [cc]{} ![Strong vertices for the $NN\pi$, $NN\pi\pi$, $\Lambda \Sigma\pi$ and $\Lambda N K$ which arise from the Lagrangians in Eq. (\[eq:str\]). \[vf1\]](nnps "fig:")& ![Strong vertices for the $NN\pi$, $NN\pi\pi$, $\Lambda \Sigma\pi$ and $\Lambda N K$ which arise from the Lagrangians in Eq. (\[eq:str\]). \[vf1\]](nnpps "fig:")\ \ ![Strong vertices for the $NN\pi$, $NN\pi\pi$, $\Lambda \Sigma\pi$ and $\Lambda N K$ which arise from the Lagrangians in Eq. (\[eq:str\]). \[vf1\]](lspw "fig:")& ![Strong vertices for the $NN\pi$, $NN\pi\pi$, $\Lambda \Sigma\pi$ and $\Lambda N K$ which arise from the Lagrangians in Eq. (\[eq:str\]). \[vf1\]](lnks "fig:") The strong vertices for the interaction between our baryonic and mesonic degrees of freedom are obtained from the strong SU(3) chiral Lagrangian [@donoghue], $$\begin{aligned} {\mathcal{L}}^s_{NN\pi}=&-\frac{g_A}{2f_\pi}\overline{\Psi}\gamma^\mu\gamma_5 {\vec{\tau}}\Psi\cdot\partial_\mu\vec{\pi}\,,\nonumber\\ {\mathcal{L}}^s_{NN\pi\pi} =&-\frac{1}{4f_\pi^2}\overline{\Psi}\gamma^\mu {\vec{\tau}}\cdot(\vec{\pi}\times\partial_\mu\vec{\pi})\Psi\,,\nonumber\\ {\mathcal{L}}^s_{\Lambda\Sigma\pi}=&-\frac{D_s}{\sqrt{3}}\,\overline{\Psi}_\Lambda\gamma^\mu\gamma_5 \Psi_\Sigma\cdot\partial_\mu\vec{\pi} \,,\label{eq:str} \\\nonumber {\mathcal{L}}^{s}_{\Lambda N K} =& \, \frac{D_s+3F_s}{2\sqrt3f_\pi} \, \overline{\Psi}_{N} \gamma^\mu\gamma_5 \,\partial_\mu\phi_{K} \Psi_\Lambda \,,\end{aligned}$$ where we have taken the convention which gives us $\Psi_\Sigma\cdot\vec{\pi}=\Psi_{\Sigma_+}\pi_-+\Psi_{\Sigma_-}\pi_++\Psi_{\Sigma_0}\pi_0$, and we consider, $g_A=1.290$, $f_\pi=92.4$ MeV, $D_s=0.822$, and $F_s=0.468$. These strong coupling constants are taken from $NN$ interaction models such as the Jülich [@JB] or Nijmegen [@nij99] potentials. The four interaction vertices corresponding to these Lagrangians are depicted in Fig. \[vf1\]. Once the interaction Lagrangians involving the relevant degrees of freedom have been presented, we need to define the power counting scheme which allows us to organize the different contributions to the full amplitude. Power counting scheme {#ss:cs} --------------------- The amplitude for the $\Lambda N\to NN$ transition is built as the sum of a medium and long-range one meson exchanges (i.e. $\pi$ and $K$), the contribution from the two-pion exchanges, and the contribution of the contact interactions up to ${\cal O} (q^2/M^2)$ as described below. The order at which the different terms enter in the perturbative expansion of the amplitudes is given by the so-called Weinberg power counting scheme [@W9091]. In our calculations we will employ the heavy baryon formalism [@jm]. This technique introduces a perturbative expansion in the baryon masses appearing in the Lagrangians, so that this new large scale does not disrupt the well-defined Weinberg power counting. It is worth noting that, in the heavy baryon formalism, terms of the type $\overline{\Psi}_B\gamma^5\Psi_B$ are subleading in front of terms like $\overline{\Psi}_B\Psi_B$, since they show up at one order higher in the heavy baryon expansion. In our calculation, we choose to keep both terms in our Lagrangians of Eqs. (\[eq:weakl\]) because the experimental values for the couplings $B_\Lambda$ and $B_\Sigma$ are much larger than $A_\Lambda$ and $A_\Sigma$. For example, $A_\Lambda=1.05$ and $B_\Lambda=-7.15$ [@donoghue]. Our calculation is characterized by the presence of different octet baryons in the relevant Feynman diagrams, contributing in both, the spinors and propagators. The spinors for the incoming $\Lambda$ and $N$ with masses $M_\Lambda$ and $M_N$, energies $E_p^{\Lambda}$ and $E_p^N$, and momenta ${\vec{p}}$ and $-{\vec{p}}$ are $$\begin{aligned} \nonumber u_1(E_p^\Lambda,{\vec{p}}\,)= \sqrt{\frac{E_p^\Lambda+M_\Lambda}{2M_\Lambda}} \left(\begin{array}{c} 1\\\nonumber \frac{{\vec{\sigma}_1}\cdot{\vec{p}}}{E_p^\Lambda+M_\Lambda} \end{array}\right)\,, \\\\\nonumber u_2(E_p^N,-{\vec{p}}\,)= \sqrt{\frac{E_p^N+M_N}{2M_N}} \left(\begin{array}{c} 1\\ -\frac{{\vec{\sigma}_2}\cdot{\vec{p}}}{E_p^N+M_N} \end{array}\right)\,,\end{aligned}$$ and for the outgoing nucleons with momenta ${\vec{p}\,'}$ and $-{\vec{p}\,'}$, and energy $E'\equiv\frac12\left(E_p^\Lambda+E_p^N\right)$, $$\begin{aligned} \nonumber \bar{u}_1(E',{\vec{p}\,'})= \sqrt{\frac{E'+M_N}{2M_N}} \left(\begin{array}{cc} 1& -\frac{{\vec{\sigma}_1}\cdot{\vec{p}\,'}}{E'+M_N} \end{array}\right)\,, \\\\\nonumber \bar{u}_2(E',-{\vec{p}\,'})= \sqrt{\frac{E'+M_N}{2M_N}} \left(\begin{array}{cc} 1& ñ\frac{{\vec{\sigma}_2}\cdot{\vec{p}\,'}}{E'+M_N} \end{array}\right)\,.\end{aligned}$$ The relativistic propagator of a baryon with mass $M_B$ and momentum $p$ reads $$\frac{i}{\cancel{p}-M_B+i\epsilon} =\frac{i(\cancel{p}+M_B)}{p^2-M_B^2+i\epsilon} \,.$$ Making the heavy baryon expansion with these spinors and propagators introduces mass differences ($M_\Lambda-M_N$, $M_\Sigma-M_\Lambda$) in the baryonic propagators. A reasonable approach would be to consider these mass differences of order ${\cal O}\left({\vec{q}}^{\,2}/\Lambda^2\right)$ ($M_B={\overline{M}}+{\cal O}\left({\vec{q}}^{\,2}/\Lambda^2\right)$), and thus they would not enter in the loop diagrams. We have chosen to leave the physical masses in both the initial and final spinors and also in the intermediate propagators; i.e. we consider the mass differences as another scale in the heavy baryon expansion. The corresponding SU(3) symmetric limit is also given at the end of section \[ss:tped\], and can be easily obtained from our expressions by setting the mass differences, which we explicitly retain, to zero. The procedure we follow to compute the different Feynman diagrams entering the transition amplitude is the following: first we write down the relativistic expressions for each diagram, and then afterwards, we perform the heavy baryon expansion. In the next sections we will describe the LO and NLO contributions to the process $\Lambda N\to NN$, following the scheme presented here. The explicit expressions and details of the calculations are given in the Appendices. Leading order Contributions {#ss:loc} =========================== ------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------- ![One-pion and one-kaon exchange contributions to the transition.\[fig:loc\]](ope "fig:") ![One-pion and one-kaon exchange contributions to the transition.\[fig:loc\]](oke "fig:") ![One-pion and one-kaon exchange contributions to the transition.\[fig:loc\]](contactw0 "fig:") ------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------- For completeness, we rewrite here the LO EFT already presented in Ref. [@PPJ11], and then build the NLO contributions in the next section. At tree level, the transition potential $\Lambda N\to NN$ involves the LO contact terms, and $\pi$ and $K$ exchanges, as depicted in Fig. \[fig:loc\]. First, the contact interaction can be written as the most general Lorentz invariant potential with no derivatives. The four-fermion (4P) interaction in momentum space at leading order (in units of $G_F$) is $$\begin{aligned} V_{4P} ({\vec q} \, ) &=& C_0^0 + C_0^1 \; {\vec \sigma}_1 {\vec \sigma}_2 \,,\label{eq:vlo}\end{aligned}$$ where $C_0^0$ and $C_0^1$ are low energy constants which need to be fitted by direct comparison to experimental data. In Ref. [@PPJ11] we presented several sets of values which were to a large extent compatible with the scarce data on hypernuclear decay. The potentials for the one pion and one kaon exchanges, as functions of transferred momentum ${\vec{q}}\equiv{\vec{p}\,'}-{\vec{p}}$, read, respectively [@PRB97] $$\begin{aligned} {V_{\pi}} ({\vec q}\,) =& \nonumber -\frac{G_F m_\pi^2g_{NN\pi}}{2 M_N} \left( A_\pi - \frac{B_\pi}{2 {\overline{M}}}{\vec \sigma}_1 \, {\vec q} \, \right) \frac{{\vec \sigma}_2 \, {\vec q}\,}{-q_0^2+{\vec q}^{\; 2}+m_\pi^2} \, \\&\times {\vec{\tau}_1}\cdot{\vec{\tau}_2}{\rm ,} \label{eq:pion}\\ {V_{K}} ({\vec q}\,) =& -\frac{G_F m_\pi^2g_{\Lambda N K}}{2{\overline{M}}} \left( \hat{A} - \frac{\hat{B}}{2 M_N}{\vec \sigma}_1 \, {\vec q} \, \right) \nonumber\\&\times\frac{{\vec \sigma}_2 \, {\vec q}\,} {-q_0^2+{\vec q}^{\; 2}+m_K^2} \, {\rm ,} \label{eq:kaon}\end{aligned}$$ where $m_\pi=138$ MeV and $m_K=495$ MeV, $q_0\equiv \frac12(M_\Lambda-M_N)$, $g_{NN\pi}\equiv \frac{g_A M_N}{f_\pi}$, $g_{\Lambda N K}\equiv-\frac{D_s+3F_s}{2\sqrt3f_\pi}$, ${\overline{M}}\equiv\frac12(M_N+M_\Lambda)$, and $$\begin{aligned} {\hat A} &=\left( \frac{ C^{PV}_{K}}{2} + D^{PV}_{K} + \frac{ C^{PV}_{K}}{2} {\vec \tau}_1 {\vec \tau}_2 \,\right), \\ {\hat B}&= \left( \frac{ C^{PC}_{K}}{2} + D^{PC}_{K} + \frac{ C^{PC}_{K}}{2} {\vec \tau}_1 \, {\vec \tau}_2 \right) \,.\end{aligned}$$ Next-to-leading order contributions {#ss:nloc} =================================== Order Parity Structures ------- -------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 0 PC $1$, ${\vec{\sigma}_1}\cdot{\vec{\sigma}_2}$ ${\vec{\sigma}_1}\cdot{\vec{q}}$, ${\vec{\sigma}_1}\cdot{\vec{p}}$, ${\vec{\sigma}_2}\cdot{\vec{q}}$, ${\vec{\sigma}_2}\cdot{\vec{p}}$, $({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}$, $({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{p}}$, ${\vec{q}}^2$, ${\vec{p}}^2$, $({\vec{\sigma}_1}\cdot{\vec{\sigma}_2}){\vec{q}}^2$, $({\vec{\sigma}_1}\cdot{\vec{\sigma}_2}){\vec{p}}^2$, $({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})$, $({\vec{\sigma}_1}\cdot{\vec{p}})({\vec{\sigma}_2}\cdot{\vec{p}})$, $({\vec{\sigma}_1}+{\vec{\sigma}_2})\cdot({\vec{q}}\times{\vec{p}})$ ${\vec{q}}\cdot{\vec{p}}$, $({\vec{\sigma}_1}\cdot{\vec{\sigma}_2}){\vec{q}}\cdot{\vec{p}}$, $({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{p}})$, $({\vec{\sigma}_1}\cdot{\vec{p}})({\vec{\sigma}_2}\cdot{\vec{q}})$, $({\vec{\sigma}_1}-{\vec{\sigma}_2})\cdot({\vec{q}}\times{\vec{p}})$ : All possible PC and PV NLO operational structures connecting the initial and final spin and angular momentum states. There are a total of 18. \[tab:contacts\] The NLO contribution to the weak decay process, $\Lambda N\to NN$, includes contact interactions with one and two derivative operators, caramel diagrams and two-pion-exchange diagrams. NLO contact potential {#sec:nlocontact} --------------------- In principle the NLO contact potential should include, in the center of mass, structures involving both the initial (${\vec{p}}\,$) and final (${\vec{p}\,'}$) momenta, or independent linear combinations, e.g. ${\vec{q}}\equiv{\vec{p}\,'}-{\vec{p}}$ and ${\vec{p}}$. Table \[tab:contacts\] lists all these possible structures. At NLO there are 18 LECs —6 PV ones at order ${\cal O}\left(q/M\right)$, 7 PC ones at order ${\cal O}\left(q^2/M^2\right)$ and 5 PV ones at order ${\cal O}\left(q^2/M^2\right)$—, which must be fitted to experiment. This is not feasible with current experimental data on hypernuclear decay. A reasonable way to reduce the number of LECs and render the fitting procedure more tractable is to note that the pionless weak decay mechanism we are interested in takes place inside a bound hypernucleus. Thus, one can consider that in the $\Lambda N\rightarrow NN$ transition potential the initial baryons have a fairly small momentum. Moreover, the final nucleons gain an extra momentum from the surplus mass of the $\Lambda$ ($M_\Lambda-M_N=116$ MeV), which in most cases allow to consider, ${\vec{p}\,'}\gg{\vec{p}}$. In this case, one may approximate ${\vec{q}}\simeq{\vec{p}\,'}$ and ${\vec{p}}=0$. Within this approximation, the NLO part of the contact potential reads (in units of $G_F$): $$\begin{aligned} V_{4P} ({\vec q} \, ) &= C_1^0 \; \displaystyle\frac{{\vec \sigma}_1{\vec q}}{2 M_N} \label{eq:vnlo} + \,C_1^1 \; \displaystyle\frac{{\vec \sigma}_2{\vec q}}{2 M_N} + {\im} \, C_1^2 \; \displaystyle \frac{({\vec \sigma}_1 \times {\vec\sigma}_2)\;{\vec q}}{2 M_N} \\ &+ C_2^0 \; \displaystyle\frac{{\vec \sigma}_1{\vec q} \;{\vec\sigma}_2{\vec q}}{4 M_N^2}+ C_2^1 \; \displaystyle\frac{{\vec \sigma}_1 {\vec \sigma}_2 \; {\vec q}^{\; 2}} {4 M_N^2}+ C_2^2 \; \displaystyle\frac{{\vec q}^{\,2}}{4 M_N^2} \,. \nonumber\end{aligned}$$ ----------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------- -- ![ Caramel diagrams contributing to the process at NLO. The solid circle represents the weak vertex. \[fig:caramels\]](caramel1 "fig:") ![ Caramel diagrams contributing to the process at NLO. The solid circle represents the weak vertex. \[fig:caramels\]](caramel2 "fig:") ![ Caramel diagrams contributing to the process at NLO. The solid circle represents the weak vertex. \[fig:caramels\]](caramel3 "fig:") (a) (b) (c) ----------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------- -- Using strong and weak LO contact interactions and two baryonic propagators one can also build three diagrams that enter at NLO. These caramel-like diagrams are shown in Fig. \[fig:caramels\]. They only differ in the position of the strong and weak vertices and in the mass of upper-leg baryonic propagator. In order to write a general expression for the three caramel diagrams we label the mass of the upper-leg propagating baryon $M_\alpha$ ($M_a=M_N$, $M_b=M_\Lambda$ and $M_c=M_\Sigma$) and the corresponding strong and weak contact vertices $C_{S(s)}^\alpha+C_{T(s)}^\alpha{\vec{\sigma}_1}\cdot{\vec{\sigma}_2}$ and $C_{S(w)}^\alpha+C_{T(w)}^\alpha{\vec{\sigma}_1}\cdot{\vec{\sigma}_2}$, where $\alpha=a,b,c$ corresponds to the labels of Fig. \[fig:caramels\]. It is also convenient to define $M_\alpha=M_N+\Delta_\alpha$. In the heavy baryon formalism these diagrams only contribute with an imaginary part of the form $$\begin{aligned} V_\alpha&=i\frac{G_Fm_\pi^2}{16\pi M_N} (C_{S(s)}^{\alpha}+C_{T(s)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\, \\&\times\nonumber (C_{S(w)}^{\alpha}+C_{T(w)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\, \\&\times\nonumber \sqrt{(\Delta_b-\Delta_\alpha)(\frac12(\Delta_b+\Delta_\alpha)+M_N)+{\vec{p}}^2}\,.\end{aligned}$$ Few more details are given in App. \[sec:caramels\]. One pion corrections to the LO contact interactions, shown in Fig. \[fig:contact.corrections\], also enter at NLO. The net contribution of these diagrams is to shift the coefficients of the LO contact terms with functions dependent on $m_\pi$, $M_\Lambda-M_N$ and $M_\Sigma-M_N$. ![Corrections to the LO contact interactions. The contributions of all these diagrams can be accounted for by an adequate shift of the coefficients of the LO contact terms.[]{data-label="fig:contact.corrections"}](contacttots "fig:")\ ![Corrections to the LO contact interactions. The contributions of all these diagrams can be accounted for by an adequate shift of the coefficients of the LO contact terms.[]{data-label="fig:contact.corrections"}](contacttots2 "fig:") Two-pion-exchange diagrams {#ss:tped} -------------------------- The two-pion-exchange contributions are organized according to the different topologies — balls, triangles, and boxes—, such that most of the integration techniques are shared by each class of diagrams. There are two types of ball diagrams, of which only one gives a non-zero contribution, depicted in Fig. \[fig:ball\]. In addition, there are four triangle diagrams, shown in Fig. \[fig:triangle\], and two box and crossed box diagrams, shown in Fig. \[fig:box\]. The topologies contain, respectively, zero, one, and two baryonic propagators, which may correspond to $N$ or $\Sigma$ baryons. All the diagrams contain two relativistic propagators from the 2$-\pi$ exchange. -------------------------------------------------------------------------------------------------------------------------------- ![The ball diagram contributing to the process at NLO. The solid circle represents the weak vertex.\[fig:ball\]](ball2 "fig:") (a) -------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Triangle diagrams which contribute to the process at NLO. The solid circle represents the weak interaction vertex.\[fig:triangle\]](uptriangle1 "fig:") ![Triangle diagrams which contribute to the process at NLO. The solid circle represents the weak interaction vertex.\[fig:triangle\]](uptriangle2 "fig:") ![Triangle diagrams which contribute to the process at NLO. The solid circle represents the weak interaction vertex.\[fig:triangle\]](downtriangle1 "fig:") ![Triangle diagrams which contribute to the process at NLO. The solid circle represents the weak interaction vertex.\[fig:triangle\]](downtriangle2 "fig:") (b) (c) (d) (e) ----------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ![Box diagrams which contribute to the process at NLO. The solid circle stands for the weak interaction vertex.\[fig:box\]](box1 "fig:") ![Box diagrams which contribute to the process at NLO. The solid circle stands for the weak interaction vertex.\[fig:box\]](box2 "fig:") ![Box diagrams which contribute to the process at NLO. The solid circle stands for the weak interaction vertex.\[fig:box\]](box3 "fig:") ![Box diagrams which contribute to the process at NLO. The solid circle stands for the weak interaction vertex.\[fig:box\]](box4 "fig:") (f) (g) (h) (i) ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ The technical details of the evaluation of the Feynman diagrams for the ball, triangle and box diagrams are given in the App. \[sec:balls\], \[sec:triangles\], and \[sec:boxs\] respectively. The main technique used is to introduce a number of master integrals, which appear in different diagrams, and which reduce the mathematical complexity of the problem (see App. \[sec:mi\]). Once they are defined, we derive a number of relations between the master integrals, which can in most cases be easily checked. Full details are provided to ensure the future use of these expressions. Using the labels defined in Figs. \[fig:ball\], \[fig:triangle\] and \[fig:box\] we organize the contributions of all the $2-\pi$ exchange diagrams in Eq. (\[eq:tots\]). The corresponding coefficients in terms of the coupling constants, baryon and meson masses, and momenta can be read off from the full expressions given in the Appendices \[sec:balls\], \[sec:triangles\] and \[sec:boxs\]. $$\begin{aligned} \label{eq:tots} V_a=&c_{a1}\,{\vec{\tau}_1}\cdot{\vec{\tau}_2}\\\nonumber V_b=&c_{b1} \\\nonumber V_c=&c_{c1}\,{\vec{\tau}_1}\cdot{\vec{\tau}_2}\\\nonumber V_d=& \left[c_{d1}+c_{d2}\,{\vec{\sigma}_1}\cdot{\vec{q}}+c_{d3}\,({\vec{q}}\cdot{\vec{p}}) +c_{d4}\,{\vec{\sigma}_1}\cdot({\vec{q}}\times{\vec{p}})\right] ({\vec{\tau}_1}\cdot{\vec{\tau}_2}) \\\nonumber V_e=&(c_{e1}+c_{e2}{\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\tau}_1}\cdot{\vec{\tau}_2})\end{aligned}$$ $$\begin{aligned} \label{eq:tots2} V_f=& \Big[ c_{f1} +c_{f2}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2}+c_{f3}{\vec{\sigma}_1}\cdot{\vec{q}}+c_{f4}({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\nonumber\\ +&c_{f5}({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}}) +c_{f6}({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{p}}) \\+&c_{f7}{\vec{\sigma}_1}\cdot({\vec{p}}\times{\vec{q}})\nonumber +c_{f8}{\vec{\sigma}_2}\cdot({\vec{p}}\times{\vec{q}})\Big] (c_{f1}'+c_{f2}'\,{\vec{\tau}_1}\cdot{\vec{\tau}_2}) \\\nonumber V_g=& \Big[ c_{g1} +c_{g2}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2}+c_{g3}({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}}) \Big] \\\times&\nonumber (c_{g1}'+c_{g2}'\,{\vec{\tau}_1}\cdot{\vec{\tau}_2}) \\+&\nonumber \Big[ c_{g4}{\vec{\sigma}_1}\cdot{\vec{q}}+c_{g5}({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\Big] (c_{g1}''+c_{g2}''\,{\vec{\tau}_1}\cdot{\vec{\tau}_2}) \\\nonumber V_h=& \Big[ c_{h1} +c_{h2}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2}+c_{h3}{\vec{\sigma}_1}\cdot{\vec{q}}+c_{h4}({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\\+&c_{h5}({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})\nonumber +c_{h6}({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{p}}) \\+&c_{h7}{\vec{\sigma}_1}\cdot({\vec{p}}\times{\vec{q}})\nonumber +c_{h8}{\vec{\sigma}_2}\cdot({\vec{p}}\times{\vec{q}})\Big] (c_{h1}'+c_{h2}'\,{\vec{\tau}_1}\cdot{\vec{\tau}_2}) \\\nonumber V_i=& \Big[ c_{i1} +c_{i2}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2}+c_{i3}({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}}) \Big] \\\times&\nonumber (c_{i1}'+c_{i2}'\,{\vec{\tau}_1}\cdot{\vec{\tau}_2}) \\+&\nonumber \Big[ c_{i4}{\vec{\sigma}_1}\cdot{\vec{q}}+c_{i5}({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\Big] (c_{i1}''+c_{i2}''\,{\vec{\tau}_1}\cdot{\vec{\tau}_2})\,.\end{aligned}$$ Considering the SU(3) limit where all the baryon masses are considered to take the same value ($q_0=q_0'=0$) the expressions above become much more simple. Defining $$\begin{aligned} At(q)\equiv&\frac{1}{2q}\arctan\left(\frac{q}{2m_\pi}\right) \\ L(q)\equiv&\frac{\sqrt{4m_\pi^2+q^2}}{q}\ln\left(\frac{\sqrt{4m_\pi^2+q^2}+q}{2m_\pi}\right), \\q\equiv &\sqrt{{\vec{q}}^{\,2}},\end{aligned}$$ and extracting the baryonic poles and the polynomial terms, one obtains, $$\begin{aligned} V_a=&\label{eq:va} -\frac{h_{\Lambda N}}{192\pi^2f_\pi^4(M_\Lambda-M_N)} (4m_\pi^2+q^2)L(q)({\vec{\tau}_1}\cdot{\vec{\tau}_2}) \\ V_b=& \frac{3g_A^2h_{2\pi}}{32\pi f_\pi^4}(2m_\pi^2+q^2)At(q) \\\label{eq:vc} V_c=& -\frac{g_A^2h_{\Lambda N}}{384\pi^2 f_\pi^4(M_\Lambda-M_N)}(8m_\pi^2+5q^2)L(q)({\vec{\tau}_1}\cdot{\vec{\tau}_2}) \\ V_d=&\nonumber \frac{g_A}{64\pi^2 f_\pi^3 M_N} L(q)({\vec{\tau}_1}\cdot{\vec{\tau}_2}) \left( -2Bm_\pi^2-B{\vec{q}}^2+B({\vec{q}}\cdot{\vec{p}}) \right.\\&\left.+6A M_N ({\vec{\sigma}_1}\cdot{\vec{q}})-3iB {\vec{\sigma}_1}\cdot({\vec{q}}\times{\vec{p}}) \right) \\ V_e=& \frac{\sqrt{3}D_s}{384\pi^2 f_\pi^3 M_N} L(q) \left( B_{\Sigma1}(4m_\pi^2+3{\vec{q}}^2)-4A_{\Sigma1} M_N({\vec{\sigma}_1}\cdot{\vec{q}}) \right)\,,\end{aligned}$$ $$\begin{aligned} V_f=&\nonumber \frac{g_A^3}{512\pi^2 f_\pi^3 M_N(4m_\pi^2+{\vec{q}}^2)}L(q) (-3+2{\vec{\tau}_1}\cdot{\vec{\tau}_2}) \\&\nonumber \times\left[ \frac{1}{6}B(448m_\pi^4+4m_\pi^2(-24{\vec{q}}\cdot{\vec{p}}+47{\vec{q}}^2)+25{\vec{q}}^4 \right.\\&\left.\nonumber -36{\vec{q}}^2({\vec{q}}\cdot{\vec{p}})) +4iB(4m_\pi^2+{\vec{q}}^2){\vec{\sigma}_2}\cdot({\vec{q}}\times{\vec{p}}) \right.\\&\left.\nonumber -4A M_N(8m_\pi^2+3{\vec{q}}^2){\vec{\sigma}_1}\cdot{\vec{q}}\right.\\&\left.\nonumber +2iB(8m_\pi^2+3{\vec{q}}^2){\vec{\sigma}_1}\cdot({\vec{q}}\times{\vec{p}}) \right.\\&\left.\nonumber +4B(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{p}}) \right.\\&\left.\nonumber -4B(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}}) \right.\\&\left.\nonumber -4B(4m_\pi^2+{\vec{q}}^2)({\vec{q}}\cdot{\vec{p}}-{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{\sigma}_2}) \right.\\&\left. -8iAM_N(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\right] \\\nonumber\\ V_g=&\nonumber \frac{D_s g_A^2}{256\sqrt{3}\pi^2 f_\pi^3 M_N(4m_\pi^2+{\vec{q}}^2)}L(q) \\&\nonumber \times\left[ -\frac{1}{6}B_{\Sigma2}(448m_\pi^4+188m_\pi^2{\vec{q}}^2+25{\vec{q}}^4) \right.\\&\nonumber\left. +4A_{\Sigma2}M_N(8m_\pi^2+3{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}}) \right.\\&\nonumber\left. +4B_{\Sigma2}(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}}) \right.\\&\left.\nonumber -4B_{\Sigma2}(4m_\pi^2+{\vec{q}}^2){\vec{q}}^2({\vec{\sigma}_1}\cdot{\vec{\sigma}_2}) \right.\\&\left. -8iA_{\Sigma2}M_N(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\right]\,,\end{aligned}$$ $$\begin{aligned} V_h=&\nonumber \frac{g_A^3}{512\pi^2 f_\pi^3 M_N(4m_\pi^2+{\vec{q}}^2)}L(q) (3+2{\vec{\tau}_1}\cdot{\vec{\tau}_2}) \\&\nonumber \times\left[ \frac{1}{6}B(448m_\pi^4+4m_\pi^2(-24{\vec{q}}\cdot{\vec{p}}+47{\vec{q}}^2)+25{\vec{q}}^4 \right.\\&\nonumber\left. -36{\vec{q}}^2({\vec{q}}\cdot{\vec{p}})) -4iB(4m_\pi^2+{\vec{q}}^2){\vec{\sigma}_2}\cdot({\vec{q}}\times{\vec{p}}) \right.\\&\nonumber\left. -4A M_N(8m_\pi^2+3{\vec{q}}^2){\vec{\sigma}_1}\cdot{\vec{q}}\right.\\&\nonumber\left. -2iB(8m_\pi^2+3{\vec{q}}^2){\vec{\sigma}_1}\cdot({\vec{q}}\times{\vec{p}}) \right.\\&\nonumber\left. +4B(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{p}}) \right.\\&\nonumber\left. -4B(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}}) \right.\\&\nonumber\left. -4B(4m_\pi^2+{\vec{q}}^2)({\vec{q}}\cdot{\vec{p}}-{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{\sigma}_2}) \right.\\&\left. +8iAM_N(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\right] \\\nonumber\\ V_i=&\nonumber \frac{D_s g_A^2}{256\sqrt{3}\pi^2 f_\pi^3 M_N(4m_\pi^2+{\vec{q}}^2)}L(q) \\&\nonumber \times\left[ \frac{1}{6}B_{\Sigma3}(448m_\pi^4+188m_\pi^2{\vec{q}}^2+25{\vec{q}}^4) \right.\\&\nonumber\left. +A_{\Sigma3}M_N(8m_\pi^2+3{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}}) \right.\\&\nonumber\left. +4B_{\Sigma3}(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}}) \right.\\&\nonumber\left. -4B_{\Sigma3}(4m_\pi^2+{\vec{q}}^2){\vec{q}}^2({\vec{\sigma}_1}\cdot{\vec{\sigma}_2}) \right.\\&\left. +4iA_{\Sigma3}M_N(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\right]\,.\end{aligned}$$ The isospin part for the potentials that contain $\Sigma$ propagators ($V_e$, $V_g$, $V_i$) is taken into account by making the replacements: $$\begin{aligned} A_{\Sigma1}\to&\frac{2}{3}\left(\sqrt3 A_{\Sigma\frac12}+ A_{\Sigma\frac32}\right) \vec{\tau_1}\cdot\vec{\tau_2} \\ B_{\Sigma1}\to& \frac{2}{3}\left(\sqrt3 B_{\Sigma\frac12}+ B_{\Sigma\frac32}\right) \vec{\tau_1}\cdot\vec{\tau_2}\,,\end{aligned}$$ $$\begin{aligned} A_{\Sigma2}\to& -\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32} +\frac23(\sqrt3A_{\Sigma\frac12}+A_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\\ B_{\Sigma2}\to& -\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32} +\frac23(\sqrt3B_{\Sigma\frac12}+B_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\,.\end{aligned}$$ $$\begin{aligned} A_{\Sigma3}\to& -\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32} -\frac23(\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\\ B_{\Sigma3}\to& -\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32} -\frac23(\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\,.\end{aligned}$$ Note that Eqs. (\[eq:va\]) and (\[eq:vc\]) only have physical meaning away from the SU(3) limit. Brief comparison of LO and NLO contributions {#sec:bc} ============================================ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(UP) Medium-Long range part of the potentials for the one-pion-exchange, one-kaon-exchange, ball diagram and triangle diagrams. (DOWN) Medium-Long range part of the potentials for the one-pion-exchange, one-kaon-exchange, box and crossed box diagrams. \[fig:pot\]](pottriangles "fig:")![(UP) Medium-Long range part of the potentials for the one-pion-exchange, one-kaon-exchange, ball diagram and triangle diagrams. (DOWN) Medium-Long range part of the potentials for the one-pion-exchange, one-kaon-exchange, box and crossed box diagrams. \[fig:pot\]](potboxes "fig:") -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(UP) Medium-Long range part of the potentials in the SU(3) limit for the one-pion-exchange, one-kaon-exchange, ball diagram and triangle diagrams. (DOWN) Medium-Long range part of the potentials in the SU(3) limit for the one-pion-exchange, one-kaon-exchange, box and crossed box diagrams. \[fig:pot0\]](pottriangles0 "fig:")![(UP) Medium-Long range part of the potentials in the SU(3) limit for the one-pion-exchange, one-kaon-exchange, ball diagram and triangle diagrams. (DOWN) Medium-Long range part of the potentials in the SU(3) limit for the one-pion-exchange, one-kaon-exchange, box and crossed box diagrams. \[fig:pot0\]](potboxes0 "fig:") ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- In Eqs. (\[eq:tots\]) and (\[eq:tots2\]) we provide the explicit momentum and spin structures arising from the different Feynman diagrams. Some features can be easily read off from the different terms. First, the ball (a) and first two triangle diagrams (b,c) only contribute to the parity conserving part of the transition potential. Most other diagrams have a non-trivial contribution, involving all allowed momenta and spin structures. To provide a sample of the contribution of the different diagrams to the full amplitude, we consider one particular transition, $^3 S_1\rightarrow ^3S_1$. In particular, we compare the $\pi$ and $K$ exchanges with the ball, triangle and box diagrams for the $\Lambda n\rightarrow nn$ interaction. Since the transition is parity conserving, none of the parity violating structures of Table \[tab:contacts\] contribute. For structures of the type $({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})$ we have that $$\begin{aligned} ({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})=\frac{{\vec{q}}^{\,2}}{3}({\vec{\sigma}_1}\cdot{\vec{\sigma}_2})+\frac{{\vec{q}}^2}{3}\hat{S}_{12}(\hat{q}),\end{aligned}$$ where the tensor operator $\hat{S}_{12}(\hat{q})$ changes two units of angular momentum and does not contribute to this transition. The potential, therefore, depends only on the modulus of the momentum (or ${\vec{q}}^{\,2}$). To obtain the potential in position space we Fourier-transform the expressions for the one-meson-exchange contributions, Eqs. \[eq:pion\] and \[eq:kaon\], and the loop expressions in the appendices \[sec:balls\], \[sec:triangles\] and \[sec:boxs\]. More explicitly, $$\begin{aligned} \tilde{V}(r)= {\cal F}\left[V({\vec{q}}^{\,2}) F({\vec{q}}^{\,2})\right]&\equiv \int_{-\infty}^{\infty}\frac{d^3q}{(2\pi)^3} e^{i{\vec{q}}\cdot{\vec{r}}}V({\vec{q}}^2)F({\vec{q}}^{\,2}) {}\end{aligned}$$ with $q\equiv|{\vec{q}}|$ and $r\equiv|{\vec{r}}|$ and where we have included a form factor in order to regularize the potential. Following the formalism developed in Ref. [@PRB97] we use a monopole form factor for the meson exchange contribution at each vertex, while the $2-\pi$ terms use a Gaussian form of the type $F({\vec{q}}^2)\equiv e^{-{\vec{q}}^{\,4}/\Lambda^4}$. The expressions for each loop have been calculated using dimensional regularization and are shown in the appendices $B$, $C$ and $D$. They are written in terms of the couplings appearing in Sec. \[sec2\] and of the master integrals appearing in App. \[sec:mi\]. $\eta$ is the regularization parameter that appears when integrating in $D\equiv4-\eta$ dimensions. The modified minimal subtraction scheme ($\overline{MS}$) has been used—we have expanded in powers of $\eta$ the expressions for the different loop contributions and then subtracted the term $R\equiv-\frac{2}{\eta}+\gamma-1-\ln\left(4\pi\right)$—. In Fig. \[fig:pot\], we show the respective contributions to the potential in position space. The contribution from the different $2-\pi$ exchange potentials are seen to be sizable at all distances. In particular, the box (f, g, h) and triangle (d) diagrams give larger contributions than the pion in the medium and long-range. The ball diagram (a) and the triangles (c), (e), (h) and (i) are attractive while all the others are repulsive. Notice that diagrams (d), (f) and (h) contribute with an imaginary part. This is characteristic of diagrams with a $\Lambda N\pi$ vertex, which may be on shell since $M_\Lambda>M_N+m_\pi$. This imaginary part is taking into account the amplitude for the possible $\Lambda N\rightarrow NN\pi$ transition. We stress that the imaginary part of the box diagram (f) that comes from the baryonic pole has been extracted, so no iterated part is considered in Fig. \[fig:pot\]. Fig. \[fig:pot0\], shows the same potentials but taking $q_0=q_0'=0$. All diagrams seem to have a smaller contribution when the baryon mass differences are neglected. The attractive and repulsive character of the different potentials does not change except for the second box diagram and the second crossed box diagram, which turn to be attractive and repulsive, respectively, when taking the SU(3) limit. Conclusions {#sec:conclusions} =========== The weak decay of hypernuclei is dominated for large enough number of nucleons by the non-mesonic weak decay modes. In these modes, the bound $\Lambda$ particle decays in the presence of nucleons by means of a process which involves weak and strong interaction vertices describing the production and absorption of mesons. The relevant, experimentally known, partial and total decay rates of hypernuclei, are successfully described by meson-exchange models and also by a lowest-order effective field theory description of the weak $\Lambda N\to NN$ process, when appropriate nuclear wave functions are used for the initial and final nuclear systems. Nevertheless, the stability of the EFT approach which has to be tested by looking at higher orders in the theory, could not be analyzed yet, mainly because of the very scarce world-database for such observables, a situation which should be improved in the near future. In this article we have presented the one-loop contribution to the previously obtained LO EFT for the weak $\Delta S=1$ $\Lambda N$ transition. As expected, the structure of the transition amplitude is considerably more involved than the corresponding LO amplitude and contains more low-energy coefficients which ought to be fitted to data. In the present formal work we have solely presented the calculation of the amplitude terms and have not attempted to make any comparison to experimental data, therefore, no fit in order to extract the new unknowns has been performed. The different structures which appear in the obtained transition amplitude, involving spin, isospin and orbital degrees of freedom, produce sizable contributions to all relevant partial waves. To illustrate this fact, we have presented the potential in $r$ space corresponding to the different Feynman diagrams for the $^3S_1- ^3S_1$ partial wave. Box and cross-box diagrams are found to produce substantial contributions at distances of the order of 1 fm, larger than the ones corresponding to the one-pion-exchange and one-kaon-exchange mechanisms. In view of this result, it would be interesting to see if one-loop contributions play an equivalent role in other partial wave transitions, testing possible cancellations or enhancements that would leave the results for the decay rates either unchanged or modified. A complete analysis of the higher order terms would require a larger set of independent hypernuclear decay measurements and a more accurate measure of some observables, specially those related to the parity violating asymmetry for s-shell and p-shell hypernuclei. Moreover, it would be desirable to arrange for alternative experiments focused to obtain information on the weak $\Delta S=1$ interaction. A step in this direction was taken more than ten years ago by experimental groups at RCNP in Osaka (Japan) \[15,16\], by looking at the weak strangeness production reaction $np \to \Lambda p$. Unfortunately, the small value for the cross-section for this process precluded the compilation of new data. We think that it is important to foster new experimental avenues of approaching the weak interaction among baryons in the strange sector, and even try to recover the Osaka experiment within the research plan of the new experimental facilities devoted to the study of strange systems. To ease the use of the obtained EFT amplitudes, we have provided with the explicit analytic expressions for all diagrams which will in future work be implemented in the calculation of hypernuclear decay observables. We thank J. Soto, J. Tarrús, J. Haidenbauer and A. Nogga for the helpful comments and discussions. This work is partly supported by grants FPA2010-21750-C02-02 and FIS2011-24154 from MICINN, 283286 from European Community-Research Infrastructure Integrating Activity ‘Study of Strongly Interacting Matter’, CSD2007-00042 from Spanish Ingenio-Consolider 2010 Program CPAN, and 2009SGR-1289 from Generalitat de Catalunya. A.P-O. acknowledges support by the APIF Ph.D. program of the University of Barcelona. B.J.D. is supported by the Ramon y Cajal program. Caramel diagrams {#sec:caramels} ================ ![First caramel-type Feynman diagram \[caramel1\]](caramel1g) ![Second caramel-type Feynman diagram \[caramel2\]](caramel2g) ![Third caramel-type Feynman diagram \[caramel3\]](caramel3g) Using the same notation that is described in section \[sec:nlocontact\] we write a general expression for the three caramel diagrams that depends on the label $\alpha=a,b,c$, which corresponds, respectively, to the masses and vertices of Figs. \[caramel1\], \[caramel2\], and \[caramel3\]. The relativistic expression for our caramel diagrams is, $$\begin{aligned} V_\alpha&=iG_Fm_\pi^2 (C_{S(s)}^{\alpha}+C_{T(s)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\, (C_{S(w)}^{\alpha}+C_{T(w)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\, \\&\times {\int\frac{d^4l}{(2\pi)^4}}\frac{1}{(E_p-l_0)^2-{\vec{l}}^2-M_N^2+i\epsilon} \\&\times\frac{1}{(E_p^\Lambda+l_0)^2-{\vec{l}}^2-M_\alpha^2}\end{aligned}$$ In order to not miss the relativistic pole we must first integrate the temporal part ($l_0$) before heavy-baryon expand the expression. Proceeding in this manner one obtains a purely imaginary part (the real is suppressed in the heavy baryon expansion). $$\begin{aligned} V_\alpha&=-\frac{G_Fm_\pi^2}{4M_N} (C_{S(s)}^{\alpha}+C_{T(s)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\, (C_{S(w)}^{\alpha}+C_{T(w)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\, \\&\times {\int\frac{d^3l}{(2\pi)^3}}\, \frac{1}{(\Delta_b-\Delta_\alpha)(\frac12(\Delta_b+\Delta_\alpha)+M_N)+{\vec{p}}^2-{\vec{l}}^2} \\&=i\frac{G_Fm_\pi^2}{16\pi M_N} (C_{S(s)}^{\alpha}+C_{T(s)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\, (C_{S(w)}^{\alpha}+C_{T(w)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\, \\&\times \sqrt{ (\Delta_b-\Delta_\alpha)(\frac12(\Delta_b+\Delta_\alpha)+M_N)+{\vec{p}}^2}\end{aligned}$$ Ball diagrams {#sec:balls} ============= In our calculation we have two different kind of ball diagrams depending on the position of the weak vertex, although only one of them actually contributes. Their contribution can be written in terms of the $B$ integrals defined in Appendix  \[sec:mi\]. Here and in the following sections we first write the relativistic amplitude using $V=i \ M$ and then the corresponding heavy baryon expression. ![Kinematical variables of the first kind of ball-diagram.\[fball1\]](ball11) For the first type of ball diagram, depicted in Fig. \[fball1\], we obtain the following contribution, $$\begin{aligned} V_{\text{ball 1}}=& \frac{G_Fm_\pi^2 h_{2\pi}}{4f_\pi^4} \delta_{ab}\ \epsilon^{abc}\tau^c \nonumber\\ &\times {\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon} \frac{1}{(l-q)^2-m_\pi^2+i\epsilon}\nonumber \\&\times {\overline{u}}_1({\overline{E}},{\vec{p}\,'}) u_1(E_p^\Lambda,{\vec{p}})\nonumber\\ &\times {\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'}) \gamma_\mu(q^\mu-2l^\mu) u_2(E_p,-{\vec{p}}) \\=&0\,,\end{aligned}$$ which is shown to vanish due to the isospin factor, $\delta_{ab}\epsilon^{abc}\tau^c=0$. ![Kinematical variables of the second kind of ball-diagram.\[fball2g\]](ball2g) The amplitude corresponding to the diagram in Fig. \[fball2g\] reads, $$\begin{aligned} V_a&=&-i \frac{G_Fm_\pi^2h_{\Lambda N}}{8f_\pi^4} ({\vec{\tau}_1}\cdot{\vec{\tau}_2}) \nonumber\\ &\times&{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon} \,\frac{1}{(l+q)^2-m_\pi^2+i\epsilon} \nonumber\\ &\times& \frac{(2l^\mu+q^\mu)(q^\nu+2l^\nu)}{k_N^2-M_N^2+i\epsilon} \nonumber\\ &\times& {\overline{u}}_1({\overline{E}},{\vec{p}\,'}) \gamma_\mu ({\cancel{k}_N}+M_N) u_1(E_p^\Lambda,{\vec{p}}) \nonumber\\ &\times& {\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'}) \gamma_\nu u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using heavy baryon expansion, $$\begin{aligned} V_a &=&\ \frac{G_Fm_\pi^2h_{\Lambda N}}{8\Delta Mf_\pi^4} ({\vec{\tau}_1}\cdot{\vec{\tau}_2}) (4 { B}_{20} +4q_0 { B}_{10} +q_0^2 { B}) \,,\nonumber\\\end{aligned}$$ where we have used the master integrals with $q_0=-\frac{M_\Lambda-M_N}{2}$ and ${\vec{q}}={\vec{p}\,'}-{\vec{p}}$. Triangle diagrams {#sec:triangles} ================= Two up triangles and two down triangles contribute to the interaction. The final expressions are written in terms of the integrals $I$ defined in Appendix \[sec:mi\]. The amplitude for the first up triangle, depicted in Fig. \[uptri\], is ![Up triangle diagram contributing at NLO. \[uptri\]](uptriangle112) $$\begin{aligned} V_b=&-i\frac38\frac{G_F m_\pi^2h_{2\pi}g_A^2}{M_N f_\pi^4} {\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon} \nonumber\\\times&\, \frac{1}{(l+q)^2-m_\pi^2+i\epsilon}\, \frac{(l^\mu+q^\mu)l^\nu}{k_N^2-M_N^2+i\epsilon} \\\times&\nonumber\, \,{\overline{u}}_1({\overline{E}}_p,{\vec{p}\,'}){\overline{u}}_1(E_p^\Lambda,{\vec{p}}) \\\times&\nonumber\, {\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\mu\gamma_5({\cancel{k}_N}+M_N)\gamma_\nu\gamma_5 u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using heavy baryon expansion, $$\begin{aligned} V_b =\frac34\frac{G_F m_\pi^2h_{2\pi}g_A^2}{f_\pi^4} \left[ (3-\eta)I_{22}+{\vec{q}}^2I_{23}+{\vec{q}}^2I_{11} \right]\,,\end{aligned}$$ where, we have used the master integrals with $q_0=\frac{M_\Lambda-M_N}{2}$, $q_0'=0$ and ${\vec{q}}={\vec{p}\,'}-{\vec{p}}$. ![Second up triangle contribution at NLO.[]{data-label="uptri2"}](uptriangle2g) For the second up triangle, depicted in Fig. \[uptri2\], the relativistic amplitude is $$\begin{aligned} V_c=& -i\frac{G_Fm_\pi^2h_{\Lambda N} g_A^2}{8f_\pi^4(r_N^2-M_N^2)} {\vec{\tau}_1}\cdot{\vec{\tau}_2}{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon} \nonumber\\\times&\, \frac{1}{(l+q)^2-m_\pi^2+i\epsilon}\, \frac{(2l^\rho+q^\rho)(l^\mu+q^\mu)l^\nu}{k_N^2-M_N^2+i\epsilon} \\\times&\nonumber\, {\overline{u}}_1({\overline{E}},{\vec{p}\,'})\gamma_\rho({\cancel{k}_N}'+M_N) u_1(E_p^\Lambda,{\vec{p}}) \\\times&\nonumber\, {\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\mu\gamma_5({\cancel{k}_N}+M_N) \gamma_\nu\gamma_5 u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using heavy baryon expansion, $$\begin{aligned} V_c=& \frac{G_Fm_\pi^2h_{\Lambda N} g_A^2}{8\Delta M f_\pi^4} {\vec{\tau}_1}\cdot{\vec{\tau}_2}\left[ 2(3-\eta)I_{32}+2{\vec{q}}^2I_{33}+2{\vec{q}}^2I_{21} \nonumber\right.\\+&\left. (3-\eta)q_0I_{22}+q_0{\vec{q}}^2I_{23}+q_0{\vec{q}}^2I_{11} \right]\,,\end{aligned}$$ where, we have used the master integrals with $q_0=\frac{M_\Lambda-M_N}{2}$, $q_0'=0$ and ${\vec{q}}={\vec{p}\,'}-{\vec{p}}$. ![“Down”-triangle contribution at NLO.\[downtri\]](downtriangle1g) The amplitude for the first down triangle (Fig. \[downtri\]) is $$\begin{aligned} V_d=& i\frac{G_Fm_\pi^2g_A}{4 f_\pi^3} (\vec{\tau}_1\cdot\vec{\tau}_2) {\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon} \\\times&\nonumber\, \frac{1}{(l+q)^2-m_\pi^2+i\epsilon} \frac{(l^\nu+q^\nu)(2l^\mu+q^\mu)}{k_N^2-M_N^2+i\epsilon} \\\times&\nonumber\, {\overline{u}}_1({\overline{E}},{\vec{p}\,'}) \gamma_{\nu}\gamma_5 ({\cancel{k}_N}+M_N) (A+B\gamma_5) u_1(E_p^\Lambda,{\vec{p}}) \\\times&\,\nonumber {\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\mu u_2(E_p,-{\vec{p}})\,,\end{aligned}$$ with the heavy baryon expansion, it reduces to, $$\begin{aligned} V_d=& -\frac{G_Fm_\pi^2g_A}{8M_N f_\pi^3} (\vec{\tau}_1\cdot\vec{\tau}_2) \Big[ B(2I_{30}+7q_0I_{20}+7q_0^2I_{10} \nonumber\\+& 2q_0^3I -2(3-\eta)I_{32}-(3-\eta)q_0I_{22}) \\-&\nonumber B(2I_{21}+q_0I_{11}+2I_{33}+q_0I_{23}){\vec{q}}^2 \\-&\nonumber B(2I_{10}+2I_{21}+q_0 I+q_0I_{11})({\vec{q}}\cdot{\vec{p}}) \\+&\nonumber 2A\,M_N(2I_{21} +q_0I_{11}-2I_{10}-q_0I){\vec{\sigma}_1}\cdot{\vec{q}}\\+&\nonumber iB(-2I_{21}-q_0I_{11}+2I_{10}+q_0I){\vec{\sigma}_1}({\vec{q}}\times{\vec{p}}) \Big]\,.\end{aligned}$$ We have used the master integrals with $q_0=-\frac{M_\Lambda-M_N}{2}$, $q_0'=-M_\Lambda+M_N$ and ${\vec{q}}={\vec{p}}'-{\vec{p}}$. ![Second type of down-triangle involving the intermediate exchange of a $\Sigma$.[]{data-label="downtri2"}](downtriangle2g) The second type of down-triangle diagram involves the intermediate exchange of the $\Sigma$ (Fig. \[downtri2\]). Its amplitude is $$\begin{aligned} V_e=& \frac{G_Fm_\pi^2D_s}{4\sqrt{3}f_\pi^3} {\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon} \nonumber\\\times&\nonumber\, \frac{1}{(l+q)^2-m_\pi^2+i\epsilon}\, \frac{(2l^\mu+q^\mu)l^\nu}{k_N^2-M_\Sigma^2+i\epsilon} \\\times&\nonumber\, {\overline{u}}_1({\overline{E}},{\vec{p}\,'})(A_{\Sigma}+B_{\Sigma}\gamma_5)({\cancel{k}_N}+M_\Sigma) \gamma_{\nu}\gamma_5 u_1(E_p^\Lambda,{\vec{p}}) \\\times&\nonumber\, {\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\mu u_2(E_p,-{\vec{p}}) \,.\end{aligned}$$ Using the heavy baryon expansion $$\begin{aligned} V_e=& -\frac{G_Fm_\pi^2D_s}{8\sqrt{3}M_N f_\pi^3} \Big[ B_{\Sigma}\Big(-2I_{30}+(-5q_0-2\Delta M_\Sigma)I_{20} \\+&\nonumber 2(3-\eta)I_{32}+2{\vec{q}}^2I_{33}+2{\vec{q}}^2I_{21}+{\vec{q}}^2I_{21} \\+&\nonumber q_0(-2q_0-\Delta M_\Sigma)I_{10} +(3-\eta)q_0I_{22}+q_0{\vec{q}}^2I_{23}+q_0{\vec{q}}^2I_{11}\Big) \\-&\nonumber 2A_{\Sigma} M_N(2I_{21} +q_0I_{11})({\vec{\sigma}_1}\cdot{\vec{q}}) \Big]\,.\end{aligned}$$ The isospin is taken into account by replacing every $A_\Sigma$ and $B_\Sigma$ by $$\begin{aligned} \frac{2}{3}\left(\sqrt3 A_{\Sigma\frac12}+ A_{\Sigma\frac32}\right) \vec{\tau_1}\cdot\vec{\tau_2}, ~~~ \frac{2}{3}\left(\sqrt3 B_{\Sigma\frac12}+ B_{\Sigma\frac32}\right) \vec{\tau_1}\cdot\vec{\tau_2}\,,\end{aligned}$$ where, we have used the master integrals with $q_0=-\frac{M_\Lambda-M_N}{2}$, $q_0'=M_\Sigma-M_\Lambda$ and ${\vec{q}}={\vec{p}}'-{\vec{p}}$. Box diagrams {#sec:boxs} ============ We have two kind of direct box diagrams and two cross-box ones. Direct box diagrams usually present a pinch singularity. This is because the poles appearing in the baryonic propagators get infinitesimally close to one another. In our integrals the denominators appearing in the baryonic propagators also contain terms proportional to $M_\Lambda-M_N$ and $M_\Sigma-M_\Lambda$, and this avoids the singularity. The integrals entering in the expression of the amplitudes are the $J$ and $K$ defined in Appendix \[sec:mi\]. The amplitude for the first type of box diagram (Fig. \[box1\]) is ![Box diagram contributing at NLO.[]{data-label="box1"}](box1g) $$\begin{aligned} V_f=& i\frac{G_Fm_\pi^2g_A^3}{8f_\pi^3} (3-2{\vec{\tau}_1}\cdot{\vec{\tau}_2}) {\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon} \nonumber\\\times&\nonumber\, \frac{1}{(l+q)^2-m_\pi^2+i\epsilon}\, \frac{1}{k_N^2-M_N^2+i\epsilon}a \\\times&\nonumber\, \frac{(l^\rho+q^\rho)(l^\nu+q^\nu)l^\mu}{r_N^2-M_N^2+i\epsilon} \\\times&\nonumber\, {\overline{u}}_1({\overline{E}},{\vec{p}\,'}) \gamma_\rho\gamma_5 ({\cancel{k}_N}+M_N) (A+B\gamma_5) u_1(E_p^\Lambda,{\vec{p}}) \\\times&\nonumber\, {\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'}) \gamma_\nu\gamma_5 ({\cancel{r}_N}+M_N) \gamma_\mu\gamma_5 u_2(E_p,-{\vec{p}}) \,.\end{aligned}$$ Using the heavy baryon expansion, $$\begin{aligned} V_f&=-\frac{G_Fm_\pi^2g_A^3}{32M_N f_\pi^3} (3-2{\vec{\tau}_1}\cdot{\vec{\tau}_2})\Bigg[ -4A M_N \left(4K_{22} + K_{11} {\vec{q}}^2 \nonumber\right.\\+&\left.\nonumber 2 K_{23} {\vec{q}}^2+K_{35} {\vec{q}}^2+(5-\eta) K_{34}\right){\vec{\sigma}_1}\cdot {\vec{q}}\nonumber\\-&\nonumber 2B K_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot {\vec{q}}) + 2B K_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot {\vec{p}}) \nonumber\\-&\nonumber 4i A M_N K_{22} \left({\vec{\sigma}_1}\times{\vec{\sigma}_2}\right)\cdot{\vec{q}}- 2 B \left({\vec{p}}\cdot {\vec{q}}-{\vec{q}}^2\right) K_{22} {\vec{\sigma}_1}\cdot {\vec{\sigma}_2}\nonumber\\+&\nonumber 2 i B \left(K_{11} {\vec{q}}^2+2 K_{23} {\vec{q}}^2+K_{35} {\vec{q}}^2 \nonumber\right.\\+&\left.\nonumber (4-\eta) K_{22}+(5-\eta) K_{34}\right) {\vec{\sigma}_1}\cdot\left({\vec{p}}\times {\vec{q}}\right) +2 i B K_{22} {\vec{\sigma}_2}\cdot\left({\vec{p}}\times {\vec{q}}\right) \nonumber\\-&\nonumber 2 B \Big(K_{11}{\vec{q}}^2 \left({\vec{p}}\cdot {\vec{q}}+2 q_0{}^2\right)+ K_{23}( 2{\vec{p}}\cdot {\vec{q}}{\vec{q}}^2+2q_0^2{\vec{q}}^2+{\vec{q}}^4) \nonumber\\+&\nonumber K_{35}( {\vec{p}}\cdot {\vec{q}}{\vec{q}}^2+2{\vec{q}}^4) +K_{22}((4-\eta) {\vec{p}}\cdot{\vec{q}}+{\vec{q}}^2+(6-2\eta)q_0^2) \nonumber\\+&\nonumber (5-\eta)K_{34}({\vec{p}}\cdot {\vec{q}}+2{\vec{q}}^2) +K_{48} {\vec{q}}^4+K_{21} {\vec{q}}^2 q_0+K_{33} {\vec{q}}^2 q_0 \nonumber\\-&\nonumber K_{31} {\vec{q}}^2 -K_{43} {\vec{q}}^2 +2(5-\eta) K_{47} {\vec{q}}^2 +(3-\eta) K_{32} q_0 \nonumber\\-&\nonumber (3-\eta) K_{42}+(15-8\eta) K_{46}\Big) \Bigg]\,,\end{aligned}$$ where we have used the master integrals with $q_0=-\frac{M_\Lambda-M_N}{2}$, $q_0'=M_N-M_\Lambda$, and ${\vec{q}}={\vec{p}}'-{\vec{p}}$. ![Second box-type Feynman diagram. \[box2\]](box3g) The second box diagram (Fig. \[box2\]), which involves a $\Sigma$ propagator, contributes with $$\begin{aligned} V_g=& -i\frac{G_Fm_\pi^2g_A^2D_s}{4\sqrt{3} f_\pi^3} {\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon} \\\times&\nonumber\, \frac{1}{(l+q)^2-m_\pi^2+i\epsilon}\, \frac{1}{k_N^2-M_\Sigma^2+i\epsilon} \\\times&\nonumber\, \frac{l^\rho(l^\nu+q^\nu)l^\mu}{r_N^2-M_N^2+i\epsilon} \\\times&\nonumber\, {\overline{u}}_1({\overline{E}},{\vec{p}\,'})(A_\Sigma+B_{\Sigma}\gamma_5)({\cancel{k}_N}+M_N)\gamma_\rho\gamma_5 u_1(E_p^\Lambda,{\vec{p}}) \\\times&\nonumber\, {\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\nu\gamma_5({\cancel{r}_N}+M_N)\gamma_\mu\gamma_5 u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using the heavy baryon expansion $$\begin{aligned} V_g=& \frac{G_Fm_\pi^2g_A^2D_s}{16\sqrt{3}M_N f_\pi^3} \Big[ -2B_{\Sigma} K_{22} {\vec{q}}^2 {\vec{\sigma}_1}\cdot {\vec{\sigma}_2}\nonumber\\-&\nonumber\, 4A_{\Sigma} K_{22} M_N i\left({\vec{\sigma}_1}\times {\vec{\sigma}_2}\right){\vec{q}}\\-&\nonumber\, 4 A_{\Sigma} M_N\left({\vec{q}}^2 K_{23}+5 K_{34}+{\vec{q}}^2 K_{35}+K_{22}\right){\vec{\sigma}_1}\cdot {\vec{q}}\\+&\nonumber\, 2B_{\Sigma} K_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot {\vec{q}}) + 2 B_{\Sigma} \left({\vec{q}}^2 K_{22}+{\vec{q}}^4 K_{23} \right.\\-&\nonumber\left.\, {\vec{q}}^2 K_{31}+(3-\eta) (\Delta M-\Delta M_\Sigma) K_{32} \right.\\+&\nonumber\left.\, {\vec{q}}^2 (\Delta M -\Delta M_\Sigma)K_{33}+2(5-\eta) {\vec{q}}^2 K_{34} +2 {\vec{q}}^4 K_{35} \right.\\-&\nonumber\left.\, (3-\eta) K_{42}-{\vec{q}}^2 K_{43}+(15-8\eta) K_{46} \right.\\+&\nonumber\left.\, 2(5-\eta) {\vec{q}}^2 K_{47} + {\vec{q}}^4K_{48} +{\vec{q}}^2 K_{21} \left(\text{$\Delta $M}-\text{$\Delta $M}_{\Sigma }\right) \right) \Big]\,.\end{aligned}$$ To take into account the isospin we must replace every $A_\Sigma$ and $B_\Sigma$ by $$\begin{aligned} A\to& -\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32} +\frac23(\sqrt3A_{\Sigma\frac12}+A_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\\ B\to& -\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32} +\frac23(\sqrt3B_{\Sigma\frac12}+B_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\,.\end{aligned}$$ We have used the master integrals with $q_0=-\frac{M_\Lambda-M_N}{2}$, $q_0'=M_\Sigma-M_\Lambda$, and ${\vec{q}}={\vec{p}}'-{\vec{p}}$. ![Crossed-box diagram contributing at NLO.[]{data-label="box2g"}](box2g) The second crossed box diagram (Fig. \[box2g\]) includes a $\Sigma$-propagator and contributes to the potential with $$\begin{aligned} V_h=& i\frac{G_Fm_\pi^2g_A^3}{8f_\pi^3} (3+2{\vec{\tau}_1}\cdot{\vec{\tau}_2}) {\int\frac{d^4l}{(2\pi)^4}}\frac{1}{(l+q)^2-m_\pi^2+i\epsilon} \nonumber\\\times&\nonumber\, \frac{1}{l^2-m_\pi^2+i\epsilon}\, \frac{1}{r_N^2-M_N^2+i\epsilon}\, \frac{(l^\rho)(l^\nu+q^\nu)(l^\mu)}{k_N^2-M_N^2+i\epsilon} \\\times&\nonumber\, {\overline{u}}_1({\overline{E}},{\vec{p}\,'}) \gamma_\rho\gamma_5({\cancel{k}_N}+M_N)(A+B\gamma_5) u_1(E_p^\Lambda,{\vec{p}}) \\\times&\nonumber\, {\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\nu\gamma_5({\cancel{r}_N}+M_N)\gamma_\mu\gamma_5 u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using heavy baryon expansion and the master integrals of Sec. \[sec:mi\], and redefining ${\vec{q}}\equiv{\vec{p}\,'}-{\vec{p}}$, $$\begin{aligned} V_h=& -\frac{G_Fm_\pi^2g_A^3} {32M_N f_\pi^3} (3+2{\vec{\tau}_1}\cdot{\vec{\tau}_2})\Big[ -2 iB J_{22}{\vec{\sigma}_2}\left({\vec{p}}\times {\vec{q}}\right) \nonumber\\+& 2B J_{22} \left(-{\vec{p}}\cdot {\vec{q}}+{\vec{q}}^2\right) {\vec{\sigma}_1}\cdot {\vec{\sigma}_2}\\+&\nonumber 2 i B\left(J_{22}+{\vec{q}}^2 J_{23}+(5+\eta) J_{34}+{\vec{q}}^2 J_{35}\right) {\vec{\sigma}_1}\cdot\left({\vec{p}}\times {\vec{q}}\right) \\+&\nonumber 4i A J_{22} M_N\left({\vec{\sigma}_1}\times {\vec{\sigma}_2}\right){\vec{q}}\\+&\nonumber 4 AM_N \left({\vec{q}}^2 J_{23}+5 J_{34}+{\vec{q}}^2 J_{35}+J_{22}\right){\vec{\sigma}_1}\cdot{\vec{q}}\\+&\nonumber 2BJ_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot {\vec{p}}) -2BJ_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}}) \\-&\nonumber 2B\left({\vec{q}}^2 q_0 J_{21}+\left(-{\vec{p}}\cdot {\vec{q}}+{\vec{q}}^2\right) J_{22}+(-{\vec{p}}\cdot {\vec{q}}{\vec{q}}^2 +{\vec{q}}^4) J_{23} \right.\\-&\left.\nonumber {\vec{q}}^2 J_{31}+(3-\eta) q_0 J_{32}+{\vec{q}}^2 q_0 J_{33} \right.\\+&\left.\nonumber (5-\eta)(-{\vec{p}}\cdot {\vec{q}}+2 {\vec{q}}^2 )J_{34} + (-{\vec{p}}\cdot {\vec{q}}{\vec{q}}^2 +2 {\vec{q}}^4 )J_{35} \right.\\-&\left.\nonumber (3-\eta) J_{42} -{\vec{q}}^2 J_{43}+(15-8\eta)J_{46} + 2(5-\eta) {\vec{q}}^2 J_{47} \right.\\+&\left.\nonumber {\vec{q}}^4 J_{48}\right) \Big]\,.\end{aligned}$$ We have used the master integrals with $q_0=\frac{M_\Lambda-M_N}{2}$, $q_0'=-\frac{M_\Lambda-M_N}{2}$, and ${\vec{q}}={\vec{p}}'-{\vec{p}}$. ![Second crossed-box-type Feynman diagram \[xbox2\]](box4g) The amplitude for the crossed-box diagram with a $\Sigma$ propagator is $$\begin{aligned} V_i=& -i\frac{G_Fm_\pi^2g_A^2D_s}{16\sqrt{3}M_N^2f_\pi^3} {\int\frac{d^4l}{(2\pi)^4}}\frac{1}{(l+q)^2-m_\pi^2+i\epsilon} \nonumber\\\times&\, \frac{1}{l^2-m_\pi^2+i\epsilon}\, \frac{1}{r_N^2-M_N^2+i\epsilon} \\\times&\nonumber\, \frac{(l^\rho+{\vec{q}}^\rho)(l^\nu+q^\nu)(l^\mu)}{k_N^2-M_\Sigma^2+i\epsilon} \\\times&\nonumber\, {\overline{u}}_1({\overline{E}},{\vec{p}\,'})(A_\Sigma+B_\Sigma\gamma_5)({\cancel{k}_N}+M_\Sigma)\gamma_\rho\gamma_5 u_1(E_p^\Lambda,{\vec{p}}) \\\times&\nonumber\, {\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\nu\gamma_5({\cancel{r}_N}+M_N)\gamma_\mu\gamma_5 u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using heavy baryon expansion and the master integrals of Sec. \[sec:mi\], and redefining ${\vec{q}}\equiv{\vec{p}\,'}-{\vec{p}}$, $$\begin{aligned} V_i=& \frac{G_Fm_\pi^2g_A^2D_s}{16\sqrt{3}M_N f_\pi^3} \Big[ 2B_\Sigma J_{22}{\vec{q}}^2 {\vec{\sigma}_1}\cdot {\vec{\sigma}_2}\\-&\nonumber 2iA_\Sigma J_{22} M_N \left({\vec{\sigma}_1}\times {\vec{\sigma}_2}\text{)$\cdot$}{\vec{q}}\right. \\-&\nonumber A_\Sigma M_N \left({\vec{q}}^2 J_{11}+2 {\vec{q}}^2 J_{23}+5 J_{34}+{\vec{q}}^2 J_{35}+4J_{22}\right){\vec{\sigma}_1}\cdot{\vec{q}}\\-&\nonumber 2B_\Sigma J_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot {\vec{q}}) \\+&\nonumber 2B_\Sigma \left( ({\vec{q}}^2-(3-\eta)q_0(q_0+\Delta M_\Sigma) J_{22} \right.\\+&\left.\nonumber ({\vec{q}}^4-{\vec{q}}^2q_0^2-{\vec{q}}^2q_0\Delta M_\Sigma)J_{23} -{\vec{q}}^2 J_{31} \right.\\-&\left.\nonumber (3-\eta)(2q_0+\Delta M_\Sigma)J_{32} -(2 {\vec{q}}^2 q_0 +{\vec{q}}^2\Delta M_\Sigma)J_{33} \right.\\+&\left.\nonumber 2(5-\eta) {\vec{q}}^2 J_{34} +2 {\vec{q}}^4 J_{35}-(3-\eta) J_{42}-{\vec{q}}^2 J_{43} \right.\\+&\left.\nonumber (15-8\eta)J_{46}+ 2(5-\eta) {\vec{q}}^2 J_{47}+{\vec{q}}^4 J_{48} \right.\\-&\left.\nonumber {\vec{q}}^2 q_0 J_{11} \left(q_0+\text{$\Delta $M}_{\Sigma }\right)-{\vec{q}}^2 J_{21} \left(2 q_0+\text{$\Delta $M}_{\Sigma }\right)\right) \Big]\,.\end{aligned}$$ To take into account the isospin we must replace every $A_\Sigma$ and $B_\Sigma$ by $$\begin{aligned} A_\Sigma\to& -\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32} -\frac23(\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\\ B_\Sigma\to& -\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32} -\frac23(\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\,.\end{aligned}$$ We have used the master integrals with $q_0=\frac{M_\Lambda-M_N}{2}$, $q_0'=M_\Sigma-M_\Lambda+\frac{M_\Lambda-M_N}{2}$, and ${\vec{q}}={\vec{p}}'-{\vec{p}}$. Master integrals {#sec:mi} ================ Definitions ----------- We need the following integrals in order to calculate the Feynman diagrams. The $B$’s, $I$’s, $J$’s and $K$’s appear, respectively, in the ball, triangle, box and crossed box diagrams: $$B_{;\mu;\mu\nu} \equiv\, \frac{1}{i}{\int\frac{d^4l}{(2\pi)^4}}\frac{1 }{l^2-m^2+i\epsilon}\, \frac{(1;l_\mu;l_\mu l_\nu)}{(l+q)^2-m^2+i\epsilon} \,,$$ $$\begin{aligned} I_{;\mu;\mu\nu;\mu\nu\rho} \equiv\,& \frac{1}{i} {\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m^2+i\epsilon}\, \frac{1}{(l+q)^2-m^2+i\epsilon}\, \nonumber\\&\nonumber \times\frac{1}{-l_0-q_0'+i\epsilon} (1;{l_{\mu}};{l_{\mu}}{l_{\nu}};{l_{\mu}}{l_{\nu}}{l_{\rho}}) \,,\end{aligned}$$ $$\begin{aligned} J_{;\mu;\mu\nu;\mu\nu\rho} \equiv\,& \frac1i{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m^2+i\epsilon}\, \frac{1}{(l+q)^2-m^2+i\epsilon}\, \\&\times \frac{1}{-l_0-q_0'+i\epsilon} \,\frac{(1;{l_{\mu}};{l_{\mu}}{l_{\nu}};{l_{\mu}}{l_{\nu}}{l_{\rho}})}{-l_0+i\epsilon} \,,\end{aligned}$$ $$\begin{aligned} K_{;\mu;\mu\nu;\mu\nu\rho} \equiv\,& \frac1i{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m^2+i\epsilon}\, \frac{1}{(l+q)^2-m^2+i\epsilon}\, \\&\times \frac{1}{-l_0-q_0'+i\epsilon} \frac{(1;{l_{\mu}};{l_{\mu}}{l_{\nu}};{l_{\mu}}{l_{\nu}}{l_{\rho}})}{l_0+i\epsilon} \,.\end{aligned}$$ The strategy is to calculate explicitly the integrals with no subindex (no integrated momenta in the numerators), and then relate the others to simpler integrals. To do so we also need to explicitly calculate the following integrals: $$A(m)\equiv\, \frac{1}{i} {\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m^2+i\epsilon} \,,$$ $$\begin{aligned} A_{;\mu;\mu\nu}(q,q')\equiv\,& \frac{1}{i}{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{(l+q)^2-m^2+i\epsilon} \\&\times \frac{1}{-l_0-q_0'+i\epsilon}(1;l_\mu;l_\mu l_\nu) \,,\end{aligned}$$ $$\begin{aligned} C_{;\mu;\mu\nu;\mu\nu\rho}(q_0,q_0')\equiv\,& \frac{1}{i}{\int\frac{d^4l}{(2\pi)^4}}\, \frac{1}{(l+q)^2-m^2+i\epsilon}\, \\&\times \frac{1}{-l_0-q_0'+i\epsilon} \frac{(1;l_\mu;l_\mu l_\nu;l_\mu l_\nu l_\rho)}{-l_0+i\epsilon} \,,\end{aligned}$$ $$\begin{aligned} D_{;\mu;\mu\nu;\mu\nu\rho}(q_0,q_0')\equiv\,& \frac{1}{i}{\int\frac{d^4l}{(2\pi)^4}}\, \frac{1}{(l+q)^2-m^2+i\epsilon} \\&\times \frac{1}{-l_0-q_0'+i\epsilon} \frac{(1;l_\mu;l_\mu l_\nu;l_\mu l_\nu l_\rho)}{l_0+i\epsilon} \,.\end{aligned}$$ The integrals can be divided depending on their subindexes being temporal or spatial. We show explicitly all the cases for the integrals $J$. The same definitions are used for all the other integrals. Therefore, to know any other integral one needs to replace in Eq. (\[eq:many\]) $J$ by $A$, $B$, $I$, etc. $$\begin{aligned} J_\mu\equiv\,& \delta_{\mu0}J_{10}+\delta_{\mu i}J_{11}{\vec{q}}_i \label{eq:many}\\\nonumber\\ J_{\mu\nu}\equiv\,& \delta_{\mu0}\delta_{\nu0}J_{20} +(\delta_{\mu0}\delta_{\nu i} +\delta_{\mu i}\delta_{\nu 0})J_{21}{\vec{q}}_i \nonumber\\& +\delta_{\mu i}\delta_{\nu j}(J_{22}\delta_{ij} +J_{23}{\vec{q}}_i{\vec{q}}_j) \nonumber\\\nonumber\\ J_{\mu\nu\rho}\equiv\,& \delta_{\mu0}\delta_{\nu0}\delta_{\rho0}J_{30} +\delta\delta\delta_{\{\mu\nu\rho 00i\}}{\vec{q}}_iJ_{31} \nonumber\\& +\delta\delta\delta_{\{\mu\nu\rho 0ij\}} (\delta_{ij}J_{32}+{\vec{q}}_i{\vec{q}}_jJ_{33}) \nonumber\\& +\delta_{\mu i}\delta_{\nu j}\delta_{\rho k} (\delta{\vec{q}}_{\{ijk\}}J_{34}+{\vec{q}}_i{\vec{q}}_j{\vec{q}}_kJ_{35}) \nonumber\\\nonumber\\ J_{\mu\nu\rho\sigma}\equiv\,& \delta_{\mu0}\delta_{\nu0}\delta_{\rho0}\delta_{\sigma0}J_{40} +\delta\delta\delta\delta_{\{\mu\nu\rho\sigma000i\}}{\vec{q}}_iJ_{41} \nonumber\\& +\delta\delta\delta\delta_{\{\mu\nu\rho\sigma00ij\}} (\delta_{ij}J_{42}+{\vec{q}}_i{\vec{q}}_jJ_{43}) \nonumber\\& +\delta\delta\delta\delta_{\{\mu\nu\rho\sigma0ijk\}} (\delta{\vec{q}}_{\{ijk\}}J_{44}+{\vec{q}}_i{\vec{q}}_j{\vec{q}}_kJ_{45}) \nonumber\\& +\delta_{\mu i}\delta_{\nu j}\delta_{\rho k}\delta_{\sigma l} (\delta\delta_{\{ijkl\}}J_{46} +\delta{\vec{q}}{\vec{q}}_{\{ijkl\}}J_{47} \nonumber\\& +{\vec{q}}_i{\vec{q}}_j{\vec{q}}_k{\vec{q}}_lJ_{48})\,. \nonumber\end{aligned}$$ All coefficients $J_{10}$, $J_{11}$, etc. have been written explicitly as functions of $I$, $J$, $K$, which can be integrated numerically, and the other simpler functions. The following definitions have been employed: $$\begin{aligned} \delta{\vec{q}}_{\{ijk\}}=&\, \delta_{ij}{\vec{q}}_k +\delta_{ik}{\vec{q}}_j +\delta_{jk}{\vec{q}}_i\,, \\ \delta{\vec{q}}{\vec{q}}_{\{ijkl\}}=&\, \delta_{ij}{\vec{q}}_k{\vec{q}}_l +\delta_{ik}{\vec{q}}_j{\vec{q}}_l +\delta_{il}{\vec{q}}_j{\vec{q}}_k \nonumber\\ & +\delta_{jk}{\vec{q}}_i{\vec{q}}_l +\delta_{jl}{\vec{q}}_i{\vec{q}}_k +\delta_{kl}{\vec{q}}_i{\vec{q}}_j\,, \\ \delta\delta_{\{ijkl\}}=&\, \delta_{ij}\delta_{kl} +\delta_{ik}\delta_{jl} +\delta_{il}\delta_{jk}\,.\end{aligned}$$ The other quantities, $\delta\delta\delta_{\{\mu\nu\rho00i\}}$, $\delta\delta\delta_{\{\mu\nu\rho0ij\}}$, etc, are not meant to be contracted with the indexes $i$, $j$, and $k$ appearing in the rest of the expressions. They only indicate how many of the indexes $\mu$, $\nu$, $\rho$, and $\sigma$ must be temporal and how many spatial. It does not matter the order in which $0$, $i$, $j$, and $k$ are assigned to $\mu$, $\nu$, $\rho$, and $\sigma$, since all the integrals $J_{\mu\nu}$, $J_{\mu\nu\rho}$, etc, are symmetric with respect to these indexes. For example $$J_{00i}=J_{0i0}=J_{i00}={\vec{q}}_i J_{31} \,.$$ Results for the master integrals -------------------------------- We have regularized the master integrals via dimensional regularization, where the integrals depend on the momentum dimension $D_\eta$, or more specifically, on the parameter $\eta$, defined through $D_\eta=4-\eta$, and on the renormalization scale $\mu$, for which we have taken $\mu=m_\pi$. In the following we use, $$\begin{aligned} R=&-\frac{2}{\eta}-1+\gamma-\log(4\pi) \,, \\ q_0''=&q_0'-q_0 \,.\end{aligned}$$ The integrals $A(m)$, $A(q_0,q_0')$ and $B(q_0,|{\vec{q}}|)$ appear, for example, in [@scherer02]. We have checked that both results coincide. It is important to maintain the $-i\epsilon$ prescription, otherwise the integrals may give a wrong result. We take it into account by replacing $q_0'\to q_0'-i\epsilon$ when evaluating the integrals. ### $A(m), A(q_0,q'_0)$ and $B(q_0,{\vec{q}})$ We have, $$A(m)= -\frac{1}{8\pi^2}m^2\left(\frac12R+\log\left(\frac{m}{\mu}\right)\right) \,.$$ $$\begin{aligned} A(q_0,q_0')\equiv -\frac{q_0''}{8\pi^2} \left[ \pi\frac{\sqrt{m^2-q_0''^2}}{q_0''} +1-R-2\log\left(\frac{m}{\mu}\right) \right. & \nonumber\\\left. -\frac{2 \sqrt{{q_0''}^2(m^2-{q_0''}^2)} \tan ^{-1}\left(\frac{\sqrt{{q_0''}^2}}{\sqrt{m^2-{q_0''}^2 }}\right)}{{q_0''}^2} \right] &\end{aligned}$$ $$\begin{aligned} B(q_0,{\vec{q}})&= -\frac{1}{16\pi^2}\left[-1+R+2\log\left(\frac{m}{\mu}\right)+2L(|q|)\right]\end{aligned}$$ with $$\begin{aligned} L(|q|)\equiv &\frac{w}{|q|}\log\left(\frac{w+|q|}{2m}\right) \,,\end{aligned}$$ $w\equiv \sqrt{4m^2+|q|^2}$, $|q|\equiv \sqrt{{\vec{q}}^2-q_0^2}$, and $q^2\equiv q_0^2-{\vec{q}}^2\le0$. ### $C(q_0,q_0')$ and $D(q_0,q_0')$ $$\begin{aligned} C(q_0,q_0')\equiv& -\frac{1}{16\pi^2}{\int_0^1dx}{\int_{0}^1dy}\Bigg[ 3y^{-\frac12}(1-y) \\& \left[ -\frac43 -\frac12(R-1+\log(4)) -\frac12\log\left(\frac{s_{xy}}{4\mu^2}\right) \right] \\&+y^{-\frac12}(1-y)(m^2+q_0''r_0') s_{xy}^{-1} \\& -\pi(q_0''+r_0')s_x^{-\frac12} \Bigg] \,,\end{aligned}$$ with $s_x=m^2-q_0^2+x(q_0^2-q_0''^2)$, $s_{xy}=m^2+(1-y)(-q_0^2+x(q_0^2-q_0''^2))$. $$\begin{aligned} D(q_0,q_0')=& -C(q_0,q_0')+\frac{1}{q_0'}\frac{1}{4\pi}\sqrt{m^2-q_0^2} \,.\end{aligned}$$ ### $I(q_0,|{\vec{q}}|,q_0')$ $$\begin{aligned} I(q_0,q,q_0')&= -\frac{1}{8\pi^2}\int_0^1dx\int_0^1dy \left[ \frac{\pi}{2}\frac{1}{\sqrt{s_x}} \right.\\&\left. -\frac34y^{-\frac12}(1-y){C_q'}\frac{1}{s_{xy}} +\frac12y^{\frac12}(1-y){C_q'}^3\frac{1}{s_{xy}^2} \right]\end{aligned}$$ with $C_q'=-q_0(1-x)+q_0'$, $s_x\equiv -q^2x(1-x)-\left(q_0'-q_0+q_0x\right)^2+m_\pi^2$, and $s_{xy}\equiv -q^2x(1-x)-\left(q_0'-q_0+q_0x\right)^2(1-y)+m_\pi^2$. ### $J(q_0,|{\vec{q}}|,q_0')$ and $K(q_0,|{\vec{q}}|,q_0')$ $$\begin{aligned} J{}=& -\frac{1}{8\pi^2}\int_0^1dx{\int_{0}^1dy}\,y(1-y) \Bigg\{ \left(-{C_q'}^3-{C_q'}^2{C_q}\right.\\&\left. -{C_q'}{C_q}^2-{C_q}^3 +2 s_x ({C_q'}+{C_q})\right)\frac{3\pi}{8s_{xy}^{\frac52}} \\& +({C_q'}+{C_q})\frac{\pi}{8s_{xy}^{\frac32}} + \frac{105}{16}{\int_0^1dz}\,z^3\sqrt{1-z} \Big[ -\frac{3}{s_{xyz}} \\& +\left({C_q'}^2-5 {C_q'}{C_q}+{C_q}^2-9s_x\right)\frac{2}{7s_{xyz}^2} \\&+ \left(-9s_x^2+2s_x\left({C_q'}^2-5 {C_q'}{C_q}+{C_q}^2\right)+ \right.\\&\left. 3 {C_q'}^3{C_q}+{C_q'}^2{C_q}^2+3 {C_q'}{C_q}^3\right)\frac{8}{35s_{xyz}^3} \\&+\left(-3s_x^3+s_x^2\left({C_q'}^2-5 {C_q'}{C_q}+{C_q}^2\right) \right.\\&\left. +s_x\left(3 {C_q'}^3{C_q}+{C_q'}^2{C_q}^2+3 {C_q'}{C_q}^3\right) \right.\\&\left. -{C_q'}^3 {C_q}^3\right)\frac{16}{35s_{xyz}^4} \Big] \Bigg\}\end{aligned}$$ with $C_{q}\equiv -q_0(1-x)$, $C_q'\equiv -q_0(1-x)+q_0'$, $s_x\equiv -q^2x(1-x)+m_\pi^2$, $s_{xy}\equiv s_x-{C_q}^2+y({C_q}^2-{C_q'}^2)$, and $ s_{xyz}\equiv s_x+z\cdot y({C_q}^2-{C_q'}^2)-z{C_q}^2$. $$\begin{aligned} K&= -J +\frac{1}{8\pi q_0'}\int_0^1 dx \frac{1}{\sqrt{m^2+(1-x)({\vec{q}}^2x-q_0^2)}} \,.\end{aligned}$$ Results for the master integrals with $q_0=q_0'=0$ -------------------------------------------------- $$\begin{aligned} A(m)&= -\frac{1}{8\pi^2}m^2\left(\frac12R+\log\left(\frac{m}{\mu}\right)\right) \\ A(0,0)&=-\frac{m}{8\pi} \\ B(0,{\vec{q}})&= -\frac{1}{16\pi^2}\left[-1+R+2\log\left(\frac{m}{\mu}\right)+2L(q)\right] \\ C(0,0)&=-\frac{1}{4\pi^2} \left( -\frac{R}{2}-\frac12 -\log(\frac{m}{\mu})\right) \\ I(0,{\vec{q}},0)&= -\frac{1}{4\pi}At(q) \\ J(0,{\vec{q}},0)&=\frac{1}{2\pi^2{\vec{q}}^2}L(q),\end{aligned}$$ where $L(q)$ and $At(q)$ are defined with $$\begin{aligned} At(q)\equiv&\frac{1}{2q}\arctan\left(\frac{q}{2m_\pi}\right) \\ L(q)\equiv&\frac{\sqrt{4m_\pi^2+q^2}}{q}\log\left(\frac{\sqrt{4m_\pi^2+q^2}+q}{2m_\pi}\right)\,.\end{aligned}$$ Relations between master integrals ---------------------------------- ### $A_\mu(q_0,q_0')$ $$\begin{aligned} A_{10}&=-A(m)-q_0'A \\ A_{11}&=-A\end{aligned}$$ ### $A_{\mu\nu}(q,q')$ $$\begin{aligned} A_{20}&= \left[ (q_0+q_0')A(m)+ {q_0'}^2A\right] \\A_{21}&= A(m)+q_0'A \\A_{22}&= \frac{1}{D_\eta-1}\left[ q_0''A(m)+({q_0''}^2-m^2)A\right] \\A_{23}&=A\end{aligned}$$ ### $B_\mu(q)$ $$\begin{aligned} B_{10}&=-\frac{q_0}{2}B \\ B_{11}&=-\frac12B\end{aligned}$$ ### $B_{\mu\nu}(q)$ $$\begin{aligned} B_{20}&= \frac{1}{2(D_\eta-1)q^2}\Bigg[ (q^2+q_0^2(D_\eta-2))A(m) \\& -\left(2{\vec{q}}^2m^2+\frac12q^2(q^2-D_\eta q_0^2)\right)B \Bigg] \\\\ B_{21}&= \frac{q_0}{2(D_\eta -1)q^2}\left[ (D_\eta -2)A(m)+\left(\frac{ D_\eta}{2}q^2-2m^2\right)B\right] \\\\ B_{22}&= -\frac{1}{2(D_\eta -1)}\left[A(m)+\left(2m^2-\frac{q^2}{2}\right)B\right] \\\\ B_{23}&= \frac{1}{2(D_\eta -1)q^2}\left[ (D_\eta -2)A(m)+\left(\frac{D_\eta}{2}q^2-2m^2\right)B \right]\end{aligned}$$ ### $C_\mu(q_0,q_0')$ $$\begin{aligned} C_{10}&=-A \\ C_{11}&=-C\end{aligned}$$ ### $C_{\mu\nu}(q_0,q_0')$ $$\begin{aligned} C_{20}&=-A_{10} \\ C_{21}&\equiv -A_{11} \\ C_{22}&=\frac{1}{D_\eta-1}(C_{20}+2q_0C_{10}+(q_0^2-m^2)C) \\ C_{23}&=C\end{aligned}$$ ### $C_{\mu\nu\rho}(q_0,q_0')$ $$\begin{aligned} C_{30}&=-A_{20} \\ C_{31}&=-A_{21} \\ C_{32}&=-A_{22} \\ C_{33}&=-A_{23} \\ C_{34}&\equiv-C_{22} \\ C_{35}&=-6C_{11}-3C_{23}-4C\end{aligned}$$ ### $D_\mu(q_0,q_0')$ $$\begin{aligned} D_{10}&=A \\ D_{11}&=-D\end{aligned}$$ ### $D_{\mu\nu}(q_0,q_0')$ $$\begin{aligned} D_{20}&\equiv A_{10} \\ D_{21}&\equiv A_{11} \\ D_{22}&=\frac{1}{D_\eta-1}(D_{20}+2q_0D_{10}+(q_0^2-m^2)D) \\ D_{23}&=D\end{aligned}$$ ### $D_{\mu\nu\rho}(q_0,q_0')$ $$\begin{aligned} D_{30}&\equiv A_{20} \\ D_{31}&\equiv A_{21} \\ D_{32}&\equiv A_{20} \\ D_{33}&\equiv A_{21} \\ D_{34}&\equiv -D_{22} \\ D_{35}&\equiv-6D_{11}-3D_{23}-4D\end{aligned}$$ ### $I_\mu$ $$\begin{aligned} I_{10}&= -B-q_0'I \\ I_{11}&= \frac{1}{2{\vec{q}}^2} \left[-A(0,q_0',r_0)+A -2q_0B+(q_0^2-{\vec{q}}^2-2q_0q_0')I \right]\end{aligned}$$ ### $I_{\mu\nu}$ $$\begin{aligned} I_{20}&=-B_{10}-q_0'I_{10} \\ I_{21}&=-B_{11}-q_0'I_{11} \\ I_{22}&=\frac{1}{(D_\eta -2){\vec{q}}^{\,2}} \left[-I_{({\vec{l}}\cdot{\vec{q}})^2}+{\vec{q}}^2I_{({\vec{l}}^2)}\right] \\ I_{23}&=\frac{1}{(D_\eta -2){\vec{q}}^{\,4}} \left[(D_\eta -1)I_{({\vec{l}}\cdot{\vec{q}})^2}-{\vec{q}}^2I_{({\vec{l}}^2)}\right]\end{aligned}$$ $$\begin{aligned} I_{({\vec{l}}^2)}&= -A(q,q')-m^2I_0-B_{10}-q_0'I_{10} \\ I_{({\vec{l}}\cdot{\vec{q}})^2}&= \frac12{\vec{q}}^2 \left[ A_{11}(q,q') -2q_0B_{11}+(q^2-2q_0q_0')I_{11} \right]\end{aligned}$$ ### $I_{\mu\nu\rho}$ $$\begin{aligned} I_{30}&=-B_{20}-q_0'I_{20} \\\\ I_{31}&=-B_{21}-q_0'I_{21} \\\\ I_{32}&= -B_{22}-q_0'I_{22} \\\\ I_{33}&= -B_{23}-q_0'I_{23} \\\\ I_{34}&= \frac{-I_{({\vec{l}}\cdot{\vec{q}})^3}+{\vec{q}}^2I_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}} {{\vec{q}}^4(D_\eta -2)} \\\\ I_{35}&= \frac{(D_\eta +1)I_{({\vec{l}}\cdot{\vec{q}})^3}-3{\vec{q}}^2I_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}} {{\vec{q}}^6(D_\eta -2)}\end{aligned}$$ $$\begin{aligned} I_{({\vec{l}}\cdot{\vec{q}})^3}&= \frac12{\vec{q}}^2 \left[-A_{22}(0,q_0')-{\vec{q}}^2A_{23}(0,q_0')-{\vec{q}}^2A(0,q_0') \right.\\&-2{\vec{q}}^2 A_{11}(0,q_0') A_{22}+{\vec{q}}^2A_{23}+q^2I_{22}+q^2{\vec{q}}^2I_{23} \\&\left. -2q_0B_{22}-2q_0{\vec{q}}^2B_{23} -2q_0q_0'I_{22} -2q_0q_0'{\vec{q}}^2I_{23} \right]\end{aligned}$$ $$\begin{aligned} I_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}&= {\vec{q}}^2\left( -A_{11}-m^2I_{11}-B_{21}-q_0'I_{21} \right)\end{aligned}$$ ### $J_\mu$ $$\begin{aligned} J_{10}&\equiv-I \\ J_{11}&\equiv \frac{1}{2{\vec{q}}^2} \left[ -C(0,q_0')+C -2q_0I+q^2J\right]\end{aligned}$$ ### $J_{\mu\nu}$ $$\begin{aligned} J_{20}&\equiv -I_{10} \\ J_{21}&\equiv -I_{11} \\ J_{22}&\equiv\frac{1}{(D_\eta -2){\vec{q}}^{\,2}} \left[-J_{({\vec{l}}\cdot{\vec{q}})^2}+{\vec{q}}^2J_{({\vec{l}}^2)}\right] \\ J_{23}&\equiv\frac{1}{(D_\eta -2){\vec{q}}^{\,4}} \left[(D_\eta -1)J_{({\vec{l}}\cdot{\vec{q}})^2}-{\vec{q}}^2J_{({\vec{l}}^2)}\right]\end{aligned}$$ $$\begin{aligned} J_{({\vec{l}}^2)}&= -C-m^2J-I_{10} \\\\ J_{({\vec{l}}\cdot{\vec{q}})^2}&= \frac12 \left[ C_{11}+q^2J_{11} -2q_0I_{11} \right]{\vec{q}}^2\end{aligned}$$ ### $J_{\mu\nu\rho}$ $$\begin{aligned} J_{30}&\equiv-I_{20} \\ J_{31}&\equiv-I_{21} \\ J_{32}&\equiv-I_{22} \\ J_{33}&\equiv-I_{23} \\ J_{34}&\equiv \frac{-J_{({\vec{l}}\cdot{\vec{q}})^3}+{\vec{q}}^2J_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}} {{\vec{q}}^4(D_\eta -2)} \\ J_{35}&\equiv \frac{(D_\eta +1)J_{({\vec{l}}\cdot{\vec{q}})^3}-3{\vec{q}}^2J_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}} {{\vec{q}}^6(D_\eta -2)}\end{aligned}$$ $$\begin{aligned} J_{({\vec{l}}\cdot{\vec{q}})^3}&= \frac{{\vec{q}}^2}{2}\left[ -C_{20}(0,q_0') -{\vec{q}}^2C_{21}(0,q_0') -{\vec{q}}^2C(0,q_0') \right.\\&\left. -2{\vec{q}}^2C_{11}(0,q_0') +C_{20}+C_{21}{\vec{q}}^2 \right.\\&\left. +q^2(J_{22}+J_{23}{\vec{q}}^2) -2q_0(I_{22}+I_{23}{\vec{q}}^2) \right] \\ J_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}&= -{\vec{q}}^2\left[C_{11} +m^2J_{11} +I_{21} \right]\end{aligned}$$ ### $J_{\mu\nu\rho\sigma}$ $$\begin{aligned} J_{40}&\equiv -I_{30} \\ J_{41}&\equiv -I_{31} \\ J_{42}&\equiv -I_{32} \\ J_{43}&\equiv-I_{33} \\ J_{44}&\equiv -I_{34} \\ J_{45}&\equiv -I_{35} \\ J_{46}&= 2\,\frac{-J_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}+{\vec{q}}^2J_{{\vec{l}}^4}} {{\vec{q}}^2(D-2)(2D+3)} \\J_{47}&= \frac{-(2D+3)J_{({\vec{l}}\cdot{\vec{q}})^4}+2(2+D){\vec{q}}^2J_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}-{\vec{q}}^4J_{{\vec{l}}^4}} {{\vec{q}}^6(D-2)(2D+3)} \\J_{48}&= \frac{(D+4)J_{({\vec{l}}\cdot{\vec{q}})^4}-6{\vec{q}}^2J_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}} {{\vec{q}}^8(D-2)}\end{aligned}$$ $$\begin{aligned} J_{({\vec{l}}\cdot{\vec{q}})^4}&= \frac{{\vec{q}}^4}{2}\left[ 3C_{34}+{\vec{q}}^2C_{35} \right.\\&\left. +q^2(3J_{34}+{\vec{q}}^2J_{35}) \right.\\&\left. -2q_0(3I_{34}+{\vec{q}}^2I_{35}) \right] \\ J_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}&= -{\vec{q}}^2\Big[C_{22}+{\vec{q}}^2C_{23} +m^2(J_{22}+J_{23}{\vec{q}}^2) +I_{32} \\& +{\vec{q}}^2I_{33} \Big] \\ J_{{\vec{l}}^4}&= -(C_{22}(D_\eta -1)+C_{23}{\vec{q}}^2) \\& -m^2(J_{22}(D_\eta -1)+J_{23}{\vec{q}}^2) -(I_{32}(D_\eta -1)+I_{33}{\vec{q}}^2)\end{aligned}$$ ### $K_\mu$ $$\begin{aligned} K_{10}&=I \\ K_{11} &\equiv \frac{1}{2{\vec{q}}^2} \left[ -D(0,q_0')+D+q^2K +2q_0I \right]\end{aligned}$$ ### $K_{\mu\nu}$ For the first two cases we apply the following tricks, $$\begin{aligned} K_{20}&\equiv I_{10} \\ K_{21}&\equiv I_{11} \\ K_{22}&\equiv\frac{1}{(D_\eta -2){\vec{q}}^{\,2}} \left[-K_{({\vec{l}}\cdot{\vec{q}})^2}+{\vec{q}}^2K_{({\vec{l}}^2)}\right] \\ K_{23}&\equiv \frac{1}{(D_\eta -2){\vec{q}}^{\,4}} \left[(D_\eta -1)K_{({\vec{l}}\cdot{\vec{q}})^2}-{\vec{q}}^2K_{({\vec{l}}^2)}\right]\end{aligned}$$ Giving the following results, $$\begin{aligned} K_{({\vec{l}}^2)}&=-D-m^2K+I_{10}-r_0K_{10} \\ K_{({\vec{l}}\cdot{\vec{q}})^2}&= \frac12 \left[ D_{11}+q^2K_{11} +2q_0I_{11} \right]{\vec{q}}^2\end{aligned}$$ ### $K_{\mu\nu\rho}$ $$\begin{aligned} K_{30}&\equiv I_{20} \\ K_{31}&\equiv I_{21} \\ K_{32}&\equiv I_{22} \\ K_{33}&\equiv I_{23} \\ K_{34}&= \frac{-K_{({\vec{l}}\cdot{\vec{q}})^3}+{\vec{q}}^2K_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}} {{\vec{q}}^4(D_\eta -2)} \\ K_{35}&\equiv \frac{(D_\eta +1)K_{({\vec{l}}\cdot{\vec{q}})^3}-3{\vec{q}}^2K_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}} {{\vec{q}}^6(D_\eta -2)}\end{aligned}$$ $$\begin{aligned} K_{({\vec{l}}\cdot{\vec{q}})^3}&= \frac{{\vec{q}}^2}{2}\left[ -D_{22}(0,q_0') -{\vec{q}}^2D_{23}(0,q_0') -{\vec{q}}^2D(0,q_0') \right.\\&\left. -2{\vec{q}}^2D_{11}(0,q_0') +D_{22} +{\vec{q}}^2D_{23} \right.\\&\left. +q^2(K_{22}+K_{23}{\vec{q}}^2) +2q_0(I_{22}+I_{23}{\vec{q}}^2) \right] \\ K_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}&= -{\vec{q}}^2\left[D_{11} +m^2K_{11} -I_{21} +r_0K_{21}\right]\end{aligned}$$ ### $K_{\mu\nu\rho\sigma}$ $$\begin{aligned} K_{40}&\equiv I_{30} \\ K_{41}&\equiv I_{31} \\ K_{42}&\equiv I_{32} \\ K_{43}&\equiv I_{33} \\ K_{44}&\equiv I_{34} \\ K_{45}&\equiv I_{35} \\ k_{46}&= 2\,\frac{-K_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}+{\vec{q}}^2K_{{\vec{l}}^4}} {{\vec{q}}^2(D-2)(2D+3)} \\K_{47}&= \frac{-(2D+3)K_{({\vec{l}}\cdot{\vec{q}})^4}+2(2+D){\vec{q}}^2K_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}-{\vec{q}}^4K_{{\vec{l}}^4}} {{\vec{q}}^6(D-2)(2D+3)} \\K_{48}&= \frac{(D+4)K_{({\vec{l}}\cdot{\vec{q}})^4}-6{\vec{q}}^2K_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}} {{\vec{q}}^8(D-2)}\end{aligned}$$ $$\begin{aligned} K_{({\vec{l}}\cdot{\vec{q}})^4}&= \frac12\left[ 2D_{10}{\vec{q}}^4+{\vec{q}}^4D \right.\\&\left. +q^2(K_{22}{\vec{q}}^2+K_{23}{\vec{q}}^4) \right.\\&\left. +2q_0(I_{22}{\vec{q}}^2+ I_{23}{\vec{q}}^4) \right] \\ K_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}&= -\Big[D_{22}+D_{23}{\vec{q}}^2 +m^2(K_{22}+K_{23}{\vec{q}}^2) -I_{32} \\& -I_{33}{\vec{q}}^2 \Big]{\vec{q}}^2 \\ K_{{\vec{l}}^4}&= -(D_{22}(D_\eta-1)+D_{23}{\vec{q}}^2) \\& -m^2(K_{22}(D_\eta-1)+K_{23}{\vec{q}}^2) \\& +(I_{32}(D_\eta-1)+I_{33}{\vec{q}}^2)\end{aligned}$$ [99]{} D. R. Entem and R. Machleidt, 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--- abstract: 'In the paper by means of Fourier transform method and similarity method we solve the Dirichlet problem for a multidimensional equation wich is a generalization of the Tricomi, Gellerstedt and Keldysh equations in the half-space, in which equation have elliptic type, with the boundary condition on the boundary hyperplane where equation degenerates.The solution is presented in the form of an integral with a simple kernel which is an approximation to the identity and self-similar solution of Tricomi-Keldysh type equation . In particular, this formula contains a Poisson’s formula, which gives the solution of the Dirichlet problem for the Laplace equation for the half-space. If the given boundary value is a generalized function of slow growth, the solution of the Dirichlet problem can be written as a convolution of this function with the kernel (if a convolution exists).' author: - '**Oleg D. Algazin**' date: Bauman Moscow State Technical University title: 'Exact solution to the Dirichlet problem for degenerating on the boundary elliptic equation of Tricomi -Keldysh type in the half-space' --- MSC2010: 35Q99, 35J25, 35J70 **Keywords**: Fourier transform, Tricomi equation, Dirichlet problem, approximation to the identity, self-similar solution, the similarity method, the generalized functions of slow growth. Introduction {#introduction .unnumbered} ============ In the paper is considered the multidimensional elliptic equation in the half-space $$y^m\Delta_xu+u_{yy}=0,~~ y>0,~~ m>-2,\eqno{(\textup{T})}$$ where $x=(x_1,x_2,\dots,x_n)\in\mathbb{R}^n, u=u(x,y)$  is a function of variabels $(x,y)\in\mathbb{R}^{n+1}$, $$\Delta_x=\frac{\partial^2 }{\partial x_1^2}+\dots+\frac{\partial^2 }{\partial x_n^2}$$ is the Laplace operator on the variable $x$. 1. If $n=1, m=1$ we obtain the Tricomi equation $$yu_{xx}+u_{yy}=0.$$ 2. If $n=1, m>0$ we obtain the Gellerstedt equation $$y^mu_{xx}+u_{yy}=0,~~m>0.$$ 3. If $n=1, m<0$ the equation (T) can be written as $$u_{xx}+y^{-m}u_{yy}=0,~~0<-m<2,$$ it is a special case of the Keldych equation [@Kel]. These equations are used in transonic gas dynamics [@Ber], and in mathematical models of cold plasma [@Otw]. 4. If $m=0$ we obtain the Laplace equation $$\Delta u(x,y)=0.$$ A bounded (as $y\to\infty$) solution of the Dirichlet problem for the Laplace equation for the half-space $$\Delta u(x,y)=0,~~x\in\mathbb{R}^n,~~y>0,$$ $$u(x,0)=\psi(x),~~x\in\mathbb{R}^n,$$ is given by the Poisson integral [@Bit],[@Ste] $$u(x,y)=\frac{\Gamma((n+1)/2)}{\pi^{(n+1)/2}}\int_{\mathbb{R}^n}\frac{y\psi(t)}{(|x-t|^2+y^2)^{(n+1)/2}}dt.$$ A similar formula is derived by us in this paper for the solution of the Dirichlet problem for an Tricomi-Keldysh type equation (T) by means of the Fourier transform to the variables in the boundary hyperplane $y=0$ in the case $m=1$, and by the similarity method in the case of $m>-2$. In this formula, in particular, is contained the Poisson’s integral formula ($m=0$) , which can also be obtained using the Fourier transform. For the case $ -2 <m <0$ this formula was earlier obtained by L.S.Parasyuk by Fourier transform [@Par]. In the case of $m> 0$ in the calculation of multidimensional Fourier transformations there are great difficulties (except the case $m = 1$, wich we consider in section 2). Therefore, we apply the similarity method. With it, in section 3, we find a self-similar solution of the equation of Tricomi -Keldysh type for any $m> -2$, which is an approximation to the identity in the space of integrable functions. Solution of the Dirichlet problem is represented as a convolution of the self-similar solution of equation of Tricomi-Keldysh type with boundary function (if thise convolution exists). The general properties of the approximation to the identity implies that in the case of bouded piecewise continuous boundary function this convolution is written in the form of an integral and gives the classical solution of the Dirichlet problem, i.e. the boundary values of integral coincide with the boundary function at all points of continuity. In the case of the boundary function, which is a generalized function of slow growth, the convolution gives a generalized solution of the Dirichlet problem, i.e. weakly converges in the space of generalized functions of slow growth to a given boundary generalized function. In particular, the kernel is a solution of the Dirichlet problem, where the boundary function is the Dirac delta function. We note that by the similarity method was obtained the fundamental solutions for the Tricomi operator ($m = 1$) in the works of J.Barros-Neto and I.M. Gelfand [@Gel1],[@Gel2],[@Gel3] and by Fourier transform method ($m = 1$) in the work of J.Barros-Neto and F.Cardoso [@Car]. Earlier, using the Fourier transform method we solved in our joint works the Dirichlet and Dirichlet-Neumann problem for the Laplace and Poisson equations in a multidimensional infinite layer [@Alg1],[@Alg2]. Notations and statement of the problem ====================================== We introduce the following notations: $$x=(x_1+\dots+x_n)\in\mathbb{R}^n,~~(x,y)=(x_1+\dots+x_n,y)\in\mathbb{R}^{n+1},~~y\in\mathbb{R},$$ $$|x|=\sqrt{x_1^2+\dots+x_n^2},~~xt=x_1t_1+\dots+x_nt_n,~~dx=dx_1\dots dx_n,$$ $$F(t)=\mathscr{F}[f](t)=\int_{\mathbb{R}^n}f(x)e^{ixt} dx-$$ Fourier transform of an integrable function $f(x)$ . If integrable in $x$ function $f(x,y)$ depends on the variables $x$ and $y$, then its Fourier transform with respect to $x$ will be denoted $$\mathscr{F}_x[f](t,y)=\int_{\mathbb{R}^n}f(x,y)e^{ixt} dx.$$ Similarly we define the inverse Fourier transform of an integrable function $F(t)$ $$f(x)=\mathscr{F}^{-1}[F](x)=\frac{1}{(2\pi)^n}\int_{\mathbb{R}^n}F(t)e^{ixt} dt$$ and an integrable in $t$ function $F(t,y)$ $$\mathscr{F}_t^{-1}[F](x,y)=\frac{1}{(2\pi)^n}\int_{\mathbb{R}^n}F(t,y)e^{ixt} dt.$$ The definition of the Fourier transform of generalized functions of slow growth, see.[@Vla]. Consider the Dirichlet problem for the Tricomi-Keldysh type equation: $$y^m\Delta_xu+u_{yy}=0,~~x\in\mathbb{R}^n,~~ y>0,~~ m>-2,\eqno{(1.1)}$$ $$u(x,0)=\psi(x),~~x\in\mathbb{R}^n,\eqno{(1.2)}$$ $$u(x,y)~\text{is bounded as}~y\to\infty.\eqno{(1.3)}$$ Solution the Dirichlet problem for an equation of Tricomi type in the case $m=1$ by means the Fourier transform method ====================================================================================================================== $$y\Delta_xu+u_{yy}=0,~~x\in\mathbb{R}^n,~~ y>0,\eqno{(2.1)}$$ $$u(x,0)=\psi(x),~~x\in\mathbb{R}^n,\eqno{(2.2)}$$ $$u(x,y)~\text{is bounded as}~y\to\infty.\eqno{(2.3)}$$ Applying the Fourier transform with respect to x to the equation (2.1) and denoting $$U(t,y)=\mathscr{F}_x[u](t,y),~~\Psi(t)=\mathscr{F}[\psi](t).$$ we get the boundary value problem for ordinary differential equation with the parameter $t\in\mathbb{R}^n$: $$-y|t|^2U(t,y)+U_{yy}(t,y)=0,$$ $$U(t,0)=\Psi(t),~~U(t,y)~\text{is bounded as}~y\to\infty.$$ This is Airy equation , its general solution is written in terms of the Airy functions $$U(t,y)=c_1(t)\textup{Ai}(|t|^{2/3}y)+c_2(t)\textup{Bi}(|t|^{2/3}y).$$ Because the $$\lim_{y\to\infty}\textup{Bi}(|t|^{2/3}y)=\infty~\text{and}~\textup{Ai}(0)=\frac{1}{3^{2/3}\Gamma(2/3)},$$ then, taking into account the boundary conditions, we obtain the solution of the boundary problem $$U(t,y)=3^{2/3}\Gamma(2/3)\Psi(t)\textup{Ai}(|t|^{2/3}y).$$ Applying the inverse Fourier transform, we obtain the solution of the original Dirichlet problem for the equation of Tricomi type (2.1) - (2.3) in the form of a convolution (if this convolution exists) $$u(x,y)=\psi(x)*k_{n1}(x,y),\eqno{(2.4)}$$ where the kernel $$k_{n1}(x,y)=3^{2/3}\Gamma(2/3)\mathscr{F}_t^{-1}[\textup{Ai}(|t|^{2/3}y)](x,y).$$ Let $|x|=r,~|t|=\rho,~\sigma_{n-1}$ - the area of the unit sphere in $\mathbb{R}^n$. To calculate the inverse Fourier transform we pass to spherical coordinates taking into account that for positive values of the argument the Airy function is expressed via the Macdonald function $$\textup{Ai}(\rho^{2/3}y)=\frac{1}{\pi\sqrt{3}}\rho^{1/3}\sqrt{y}K_{1/3}(2/3y^{3/2}\rho),~~y>0.$$ We have $$k_{n1}(x,y)=\frac{3^{2/3}\Gamma(2/3)}{(2\pi)^n}\int_{\mathbb{R}^n}\textup{Ai}(|t|^{2/3}y)e^{-ixt}dt=$$ $$=3^{2/3}\Gamma(2/3)\frac{\sigma_{n-1}}{(2\pi)^n}\int_0^{\infty}\textup{Ai}(\rho^{2/3}y)\rho^{n/2}d\rho\int_{0}^{\pi}e^{-ir\rho\cos \theta}\sin^{n-2} \theta d\theta=$$ $$=\frac{3^{2/3}\Gamma(2/3)}{(2\pi)^{n/2}r^{n/2-1}}\int_0^{\infty}\textup{Ai}(\rho^{2/3}y)\rho^{n/2}J_{n/2-1}(r\rho) d\rho=$$ $$=\frac{3^{2/3}\Gamma(2/3)\sqrt{y}}{(2\pi)^{n/2}r^{n/2-1}\pi\sqrt{3}}\int_0^{\infty}K_{1/3}(2/3y^{3/2}\rho)\rho^{1/3+n/2}J_{n/2-1}(r\rho) d\rho,$$ where $J_{n/2-1}(r\rho)$ – is Bessel function of the 1st kind of order $\nu=n/2-1$. This formula is valid for $n=1$, it is easy to check. The last integral is expressed in terms of the hypergeometric function $F$ ([@Rys] p. 684, the formula 6.576.3): $$\int_0^{\infty}K_{1/3}(2/3y^{3/2}\rho)\rho^{1/3+n/2}J_{n/2-1}(r\rho) d\rho=$$ $$=\frac{3^{n+1/3}\Gamma(n/2+1/3)}{2^{n/2+1}y^{3n/2+1/2}}F\left(\frac{n}{2}+\frac{1}{3},\frac{n}{2};\frac{n}{2};-\frac{9r^2}{4y^3}\right)=$$ $$=\frac{3^{n+1/3}\Gamma(n/2+1/3)}{2^{n/2+1}y^{3n/2+1/2}}\left(1+\frac{9r^2}{4y^3}\right)^{-n/2-1/3}.$$ Finally, we obtain an expression for the kernel $$k_{n1}(x,y)=C_{n1}^*\frac{y}{(4y^3+9|x|^2)^{n/2+1/3}},~~x\in\mathbb{R}^n,~~y>0,$$ $$C_{n1}^*=\frac{3^{n+1/2}\Gamma(2/3)\Gamma(n/2+1/3)}{2^{1/3}\pi^{n/2+1}}.$$ The received kernel has the following properties for $y>0$: $$\begin{aligned} 1)~&k_{n1}(x,y)>0,\\ 2)~&\int_{\mathbb{R}^n}k_{n1}(x,y)dx=1,\\ 3)~&\forall\delta>0, \lim_{y\to+0} \sup_{|x|\ge\delta} k_{n1}(x,y)=0.\end{aligned}$$ Property 1) is obvious. Property 2) follows from the fact that the Fourier transform of $k_{n1} (x,y)$ there are $3^{2/3}\Gamma(2/3)\textup{Ai}(|t|^{2/3}y)$, $$\int_{\mathbb{R}^n} k_{n1}(x,y)e^{ixt}dx=3^{2/3}\Gamma(2/3)\textup{Ai}(|t|^{2/3}y).$$ Putting $t=0$ , we obtain $$\int_{\mathbb{R}^n} k_{n1}(x,y)dx=3^{2/3}\Gamma(2/3)\textup{Ai}(0)=1.$$ Property 3) follows from the fact that $k_{n1} (x,y)$ decreases monotonically as a function of $|x|$ . These properties mean that $k_{n1} (x,y)$ is *the approximation to the identity* or $\delta$–shaped system of functions of $x$ (with parameter $y$ ), for $y\to+0 , k_{n1} (x,y)$ weakly converges to $\delta$–function $\delta(x)$. If $\psi(x)$ is a bounded piecewise continuous function then convolution (2.4) exists and is recorded as an integral $$u(x,y)=C_{n1}^*\int_{\mathbb{R}^n}\frac{\psi(x)y}{(4y^3+9|x-t|^2)^{n/2+1/3}}dt.$$ From the fact that the kernel of integral is the approximation to the identity follows the equality $$\lim_{y\to+0} u(x,y)=\psi(x)$$ at the points of continuity of $\psi(x)$, which means that the integral is a classical solution of the Dirichlet problem. For generalized functions of slow growth $\psi(x)\in\mathscr{S}'(\mathbb{R}^n)$ , for which a convolution exists, the function $$u(x,y)=\psi(x)*k_{n1}(x,y)$$ is a generalized solution of the Dirichlet problem: $$\lim_{y\to+0} u(x,y)=\psi(x)~~ \text{in}~~ \mathscr{S}'.$$ For example, if $\psi(x)=\delta(x)$, then the solution of the Dirichlet problem is the kernel of the integral $$u(x,y)=\delta(x)*k_{n1}(x,y)=k_{n1}(x,y),$$ $$\lim_{y\to+0} k_{n1}(x,y)=\delta(x)~~\text{in}~~\mathscr{S}'.$$ If $\psi(x)\in L^p(\mathbb{R}^n),~1 \le p \le \infty$, then from the properties of the approximation to the identity \[5\] follows that $$\lim_{y\to+0} u(x,y)=\psi(x)$$ for almost every $x$, and if $p<\infty$ , then $u(x,y)$ converges to $\psi(x)$ in the norm of $L^p(\mathbb{R}^n)$, as $y\to+0$. For the case $n=1$ the integral, which gives the solution of the Dirichlet problem for the Tricomi equation (2.1) - (2.3), has the form $$u(x,y)=\frac{3^{3/2}\Gamma(2/3)\Gamma(5/6)}{\pi^{3/2}2^{1/3}}\int_{-\infty}^{\infty}\frac{y\psi(t)}{(4y^3+9(x-t)^2)^{5/6}}dt.$$ The kernel of the integral representation of the solution of the Dirichlet problem for multidimensional Tricomi equation, which is an approximation to the identity, represented in the form $$k_{n1}(x,y)=C_{n1}^*\frac{y}{(4y^3+9|x|^2)^{n/2+1/3}}=\frac{1}{y^{n3/2}}\varphi\left(\frac{|x|}{y^{3/2}}\right)$$ where $$\varphi(r)=\frac{C_{n1}^*}{(4+9r^2)^{n/2+1/3}}.$$ That is, the kernel $k_{n1} (x,y)$ is the *self-similar solution* of the Tricomi equation, which can be found by the *similarity method* ([@Bar],[@Pol], Ch. 3). We use this method to solve the Dirichlet problem for the equation of Tricomi-Keldysh type. Solution the Dirichlet problem for the equation of Tricomi-Keldysh type in the case $m>-2$ by the similarity method =================================================================================================================== We will search the kernel of the integral representation of the solution of the Dirichlet problem (1.1) - (1.3), which is an approximation to the identity, in the form of a self-similar solution of the equation of Tricomi-Keldysh type (1.1) $$u(x,y)=\frac{1}{y^{\alpha}}\varphi\left(\frac{r}{y^{\beta}}\right),~~\alpha>0,~~\beta>0,\eqno{(3.1)}$$ where $r=|x|$, that is, we are looking for a spherically symmetric solution, depending only on $|x|=r$. The equation of Tricomi-Keldysh type (1.1) for a spherically symmetric function takes the form $$y^m\left(u_{rr}+\frac{n-1}{r}u_r\right)+u_{yy}=0,~~y>0,~~m>-2,\eqno{(3.2)}$$ To determine the constants $\alpha$ and $\beta$ we will do in equation (3.2) the change of variables $$u=C^l \bar u,~~r=C^k \bar r,~~y=C \bar y,~~(C>0),$$ and require that this equation is transferred into itself. We obtain the equation in the new variables $$C^{m+l-2k}\left(\bar u_{\bar r\bar r}+\frac{n-1}{\bar r}\bar u_{\bar r}\right)+C^{l-2}\bar u_{\bar y\bar y}=0.$$ In order to this equation coincides with the equation (3.2), we set $$m+l-2k=l-2.$$ Hence $$k=\frac{m+2}{2} ,~~l~\text{is any}.$$ In the new variables, the self-similar solution should have the same form (3.1) $$\bar u=\frac{1}{\bar y^{\alpha}}\varphi\left(\frac{\bar r}{\bar y^{\beta}}\right).$$ Returning to the old variables, we obtain $$u=\frac{C^{\alpha+l}}{y^{\alpha}}\varphi\left(\frac{C^{-k+\beta}r}{y^{\beta}}\right),$$ in order to coincide this expression with the (3.1), we set $$\beta=k=\frac{m+2}{2},~\alpha=-l, \text{that is, any. We take}~ \alpha=kn=n\frac{m+2}{2}.$$ From the condition $\alpha>0, \beta>0$ , it follows that $m>-2$. And so, we seek a solution of equation (3.2) in the form $$u=\frac{1}{y^{kn}}\varphi\left(\frac{r}{y^k}\right),~~k=\frac{m+2}{2},~~m>-2.\eqno{(3.3)}$$ Substituting the function (3.3) in the equation (3.2), we obtain the equation $$(1+k^2r^2)\varphi''(r)+\left(\frac{n-1}{r}+2k^2nr+k^2r+kr\right)\varphi'(r)+kn(kn+1)\varphi(r)=0.$$ By doing in this equation the change of variable $$1+k^2r^2=\xi,$$ we obtain for the function $\bar \varphi(\xi)$ hypergeometric equation $$\xi(1-\xi)\bar \varphi''(\xi)+\left(\frac{n}{2}+1+\frac{1}{2k}-\xi\left(n+1+\frac{1}{2k}\right)\right)\bar \varphi'(\xi)-\left(\frac{n^2}{4}+\frac{n}{4k}\right)\bar \varphi(\xi)=0.$$ We write it in the form $$\xi(1-\xi)\bar \varphi''(\xi)+\left(c-\xi(a+b+1)\right)\bar \varphi'(\xi)-ab\bar \varphi(\xi)=0,\eqno{(3.4)}$$ where $$c=\frac{n}{2}+1+\frac{1}{2k},~~a=\frac{n}{2},~~b=\frac{n}{2}+\frac{1}{2k}.$$ The general solution of the hypergeometric equation (3.4) has the form $$\begin{aligned} \bar \varphi(\xi)=C_1F(a,b;c;\xi)+C_2\xi^{1-c}F(b-c+1,a-c+1;2-c;\xi)=\\ =C_1F\left(\frac{n}{2},\frac{n}{2}+\frac{1}{2k};\frac{n}{2}+1+\frac{1}{2k};\xi\right)+C_2\xi^{-n/2-1/2k}F\left(0,-\frac{1}{2k};1+\frac{n}{2}+\frac{1}{2k};\xi\right)=\\ =C_1F\left(\frac{n}{2},\frac{n}{2}+\frac{1}{m+2};\frac{n}{2}+1+\frac{1}{m+2};\xi\right)+C_2\xi^{-n/2-1/(m+2)},\end{aligned}$$ where $F(a,b;c;\xi)$ is the hypergeometric function. We will take a particular solution (denoting constant $C_2$ through $C_{nm}$) $$\bar \varphi(\xi)=\frac{C_{nm}}{\xi^{n/2+1/(m+2)}}.$$ Returning to the old variables, we obtain $$\varphi(r)=\frac{C_{nm}}{\left(1+\left(\frac{m+2}{2}\right)^2r^2\right)^{n/2+1/(m+2)}}.$$ We choose a such constant $C_{nm}$ that the integral of $\varphi(|x|)$ over the entire space $\mathbb{R}^n$ is equal to unity. Passing to the spherical coordinates and denoting $\sigma_{n-1}$ area of the unit sphere in $\mathbb{R}^n$, we will have $$C_{nm}^{-1}=\int_{\mathbb{R}^n}\varphi(|x|)dx=\sigma_{n-1}\int_0^{\infty}\frac{r^{n-1}dr}{\left(1+\left(\frac{m+2}{2}\right)^2r^2\right)^{n/2+1/(m+2)}}=$$ $$=\frac{\sigma_{n-1}}{2\left(\frac{m+2}{2}\right)^n}\int_0^{\infty}\frac{t^{n/2-1}}{(1+t)^{n/2+1/(m+2)}}dt=\frac{\sigma_{n-1}2^{n-1}}{(m+2)^n}\frac{\Gamma\left(\frac{n}{2}\right)\Gamma\left(\frac{1}{m+2}\right)}{\Gamma\left(\frac{n}{2}+\frac{1}{m+2}\right)}.$$ We made the change of variable $\left(\left(\frac{m+2}{2}\right)r\right)^2=t$, and used the formula $$\int_0^{\infty}\frac{t^{a-1}}{(1+t)^{b+c}}dt=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)},~a>0,b>0,$$ where $\Gamma$ is Euler gamma function. Thus $$C_{nm}=\frac{(m+2)^n\Gamma\left(\frac{n}{2}+\frac{1}{m+2}\right)}{\pi^{n/2}2^n\Gamma\left(\frac{1}{m+2}\right)}.\eqno{(3.5)}$$ Self-similar solution of the equation of Tricomi-Keldysh type, which is obtained by the formula (3.3), we denote $$k_{nm}(x,y)=\frac{1}{y^{n(m+2)/2}}\varphi\left(\frac{|x|}{y^{(m+2)/2}}\right)=$$ $$=\frac{C_{nm}y}{\left(y^{m+2}+\left(\frac{m+2}{2}\right)^2\right)^{n/2+1/(m+2)}},~~m>-2,\eqno{(3.6)}$$ where $C_{nm}$ determined by the formula (3.5). If $m=0$, we obtain the Poisson kernel $$k_{n0}(x,y)=\frac{C_{n0}y}{(y^2+|x|^2)^{(n+1)/2}},$$ where $$C_{n0}=\frac{\Gamma\left((n+1)/2\right)}{\pi^{(n+1)/2}}.$$ If $m=1$ we obtain the kernel of the integral representation of the solution of the Dirichlet problem (2.1) - (2.3), which was found in the previous section by Fourier transform $$k_{n1}(x,y)=\frac{C_{n1}y}{\left(y^3+\frac{9}{4}|x|^2\right)^{n/2+1/3}}=\frac{2^{n+2/3}C_{n1}y}{(4y^3+9|x|^2)^{n/2+1/3}},$$ where $$2^{n+2/3}C_{n1}=\frac{2^{2/3}3^n\Gamma(n/2+1/3)}{\pi^{n/2}\Gamma(1/3)}=\frac{3^{n+1/2}\Gamma(2/3)\Gamma(n/2+1/3)}{2^{1/3}\pi^{n/2+1}}=C_{n1}^*.$$ We show that the function $k_{nm} (x,y)$, defined by (3.6), is an approximation to the identity in the space of integrable on $\mathbb{R}^n$ functions, that is, has the following properties for $y>0$: $$\begin{aligned} 1)~&k_{nm}(x,y)>0,\\ 2)~&\int_{\mathbb{R}^n}k_{nm}(x,y)dx=1,\\ 3)~&\forall\delta>0, \lim_{y\to+0} \sup_{|x|\ge\delta} k_{nm}(x,y)=0.\end{aligned}$$ Properties 1) and 3) are proved like in section 2 for the case $m=1$. Let us prove property 2). We have $$\int_{\mathbb{R}^n}k_{nm}(x,y) dx=\int_{\mathbb{R}^n}\frac{1}{y^{n(m+2)/2}}\varphi\left(\frac{|x|}{y^{(m+2)/2}}\right)dx=\int_{\mathbb{R}^n}\varphi(|t|)dt=1.$$ Therefore, the solution to the Dirichlet problem of the equation of Tricomi-Keldysh type can be written as a convolution of the boundary function $\psi(x)$ with the kernel $k_{nm} (x,y)$ (if a convolution exists) $$u(x,y)=\psi(x)*k_{nm}(x,y).$$ If $\psi(x)$ is a bounded piecewise continuous function, the convolution exists and is recorded as an integral $$u(x,y)=C_{nm}\int_{\mathbb{R}^n}\frac{\psi(t)y}{(y^{m+2}+((m+2)/2)^2|x|^2)^{n/2+1/(m+2)}}dt.$$ At points of continuity $\psi(x)$ $$\lim_{y\to+0}u(x,y)=\psi(x).$$ That is, the integral is the classical solution of the Dirichlet problem. For generalized functions of slow growth $\psi(x)\in\mathscr{S}'(\mathbb{R}^n)$ , for which a convolution exists, the function $$u(x,y)=\psi(x)*k_{nm}(x,y)$$ is a generalized solution of the Dirichlet problem: $$\lim_{y\to+0} u(x,y)=\psi(x)~~ \text{in}~~ \mathscr{S}'.$$ For example, if $\psi(x)=\delta(x)$, then the solution of the Dirichlet problem is the kernel of the integral $$u(x,y)=\delta(x)*k_{nm}(x,y)=k_{nm}(x,y),$$ $$\lim_{y\to+0} k_{nm}(x,y)=\delta(x)~~\text{in}~~\mathscr{S}'.$$ If $\psi(x)\in L^p(\mathbb{R}^n),~1 \le p \le \infty$, then from the properties of the approximation to the identity \[5\] follows that $$\lim_{y\to+0} u(x,y)=\psi(x)$$ for almost every $x$, and if $p<\infty$ , then $u(x,y)$ converges to $\psi(x)$ in the norm of $L^p(\mathbb{R}^n)$, as $y\to+0$. **Example.** For $n=1,m=-1$, we have the Dirichlet problem for the Keldysh equation $$\begin{gathered} u_{xx}+yu_{yy}=0,~~-\infty<x<\infty,~~y>0,\\ u(x,0)=\psi(x),~~~-\infty<x<\infty,\\ u(x,y)~\text{is bounded as}~ y\to\infty.\end{gathered}$$ If $\psi(x)$ is a bounded piecewise continuous function, the solution to this problem is given by the integral $$u(x,y)=2\int_{-\infty}^{\infty}\frac{y\psi(t)dt}{(4y+(x-t)^2)^{3/2}}.$$ Take $\psi(x)=\left\{\begin{aligned}a,~x<0\\b,~x>0\end{aligned}\right.$, then $$u(x,y)=2ay\int_{-\infty}^0\frac{dt}{(4y+(x-t)^2)^{3/2}}+2by\int_0^{\infty}\frac{dt}{(4y+(x-t)^2)^{3/2}}=$$ $$=\frac{a+b}{2}+\frac{b-a}{2}\frac{x}{\sqrt{4y+x^2}}.\eqno{(3.7)}$$ It is easy to verify that the function (3.7) 1. satisfies the Keldysh equation $u_{xx}+yu_{yy}=0,~y>0$, 2. is bounded, $|u(x,y)|\le\max (|a|,|b|),~y>0$, 3. satisfies the boundary condition, $$\lim_{y\to+0}u(x,y)=\frac{a+b}{2}+\frac{b-a}{2}\frac{x}{|x|}=\left\{\begin{aligned}a,~x<0\\b,~x>0\end{aligned}\right.=\psi(x).$$ At the point of discontinuity $$\lim_{y\to+0}u(0,y)=\frac{a+b}{2}.$$ Conclusion {#conclusion .unnumbered} =========== In the paper is considered an elliptic equation in a multidimensional half-space $$y^m\Delta_x+u_{yy}=0,~~x\in\mathbb{R}^n,~~ y>0,~~ m>-2,\eqno{(\textup{T})}$$ which is a generalization of the equations of Tricomi ($n=1,m=1$), Gellerstedt ($n=1,m>0$), Keldysh ($n = 1,-2 <m <0$) and Laplace ($n\ge1,m = 0$). We are looking for the solution of this equation, which is bounded when $y\to\infty$ and satisfies on the boundary of the half-space Dirichlet boundary condition $$u(x,0)=\psi(x),~~x\in\mathbb{R}^n.$$ Is found an exact solution of the Dirichlet problem in the form of an integral that generalizes the well-known Poisson formula giving the solution of the Dirichlet problem for the Laplace equation. The kernel of the integral representation of the solution of the Dirichlet problem for the case $m = 1$ is found by the Fourier transform method, and for the general case of $m> -2$ by the similarity method in form a self-similar solution of equation (T). It is shown that this kernel is an approximation to the identity. The general properties of the approximation to the identity implies that the integral is a classical solution of the Dirichlet problem if the boundary function is bounded and piecewise continuous, and if the boundary function is a generalized function of slow growth, and exists a convolution of the function with the kernel, the convolution is a generalized solution of the Dirichlet problem . That is, a solution for $y\to + 0$ tends to the boundary function in $\mathscr{S}'(\mathbb{R}^n)$ in the sense of theory of generalized functions . [99]{} Keldysh M.V. On some cases of degenerate elliptic equations on the boundary of a domain ,Doklady Acad. Nauk USSR. Vol. 77 (1951), 181-183. Bers L. Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Surveys Appl. Math. 3, Wiley, New York, 1958. Otway T.H. Dirichlet Problem for Elliptic-Hyperbolic Equations of Keldych Type. Springer-Verlag, Berlin, Heidelberg, 2012. Bitsadze A.V. Equations of mathematical physics, Moscow, Mir Publishers, 1980. Stein I. Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. Parasyuk L.S. Boundary problems for elliptic differential equations that degenerate on the boundary of domain, Ukrainska akademia drukarstva. Naukovi zapysky. 1961, No.13, 65-75.(in Ukrainian) Barros-Neto J., Gelfand I.M. Fundamental solutions for the Tricomi Operator , Duke Math. J. 98 (3) (1999), 465–483. Barros-Neto J., Gelfand I.M. Fundamental solutions for the Tricomi operator, II,Duke Math. J. 2002. 111 (3), 561–584. Barros-Neto J., Gelfand I.M. Fundamental solutions for the Tricomi operator, III,Duke Math. J. 2005. 128 (1), 119–140. Barros-Neto J., Cardoso F. Bessel Integrals and Fundamental Solutions for a Generalized Tricomi Operator, Journal of Functional Analysis, 183 (2001), 472-497. DOI:10.1006/jfan.2001.3749. Algazin O.D., Kopaev A.V. Solution to the Mixed Boundary-Value Problem for Laplace Equation in Multidimensional Infinite Layer, Herald of the BMSTU, Series “Natural Sciences” (2015), No.1, 3-13. DOI:10.18698/1812-3368-2015-1-3-13. Algazin O.D., Kopaev A.V. Solution of the Dirichlet Problem for the Poisson’s equation in a Multidimensional Infinite Layer, Mathematics and Mathematical Modelling of the Bauman MSTU (2015), No.4, 41-53 (in Russian). DOI: 10.7463/mathm.0415.0812943. Vladimirov V.S. Generalized Functions in Mathematical Physics, Moscow, Nauka Publ., 1979 (in Russian). Ryshik I.M., Gradstein I.S. Tables of Integrals, Series, and Products, Academic Press, New York, 2007. Barenblatt G.I. Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge University Press, 2002. Polyanin A.D., Zaitsev V.F., Zhurov A.I. Methods for solving nonlinear equations of mathematical physics and mechanics, Moscow, Fizmatlit Publ., 2005 (in Russian).

--- abstract: | Internet-scale distributed systems often replicate data within and across data centers to provide low latency and high availability despite node and network failures. Replicas are required to accept updates without coordination with each other, and the updates are then propagated asynchronously. This brings the issue of conflict resolution among concurrent updates, which is often challenging and error-prone. The Conflict-free Replicated Data Type (CRDT) framework provides a principled approach to address this challenge. This work focuses on a special type of CRDT, namely the Conflict-free Replicated Data Collection (CRDC), e.g. list and queue. The CRDC can have complex and compound data items, which are organized in structures of rich semantics. Complex CRDCs can greatly ease the development of upper-layer applications, but also makes the conflict resolution notoriously difficult. This explains why existing CRDC designs are tricky, and hard to be generalized to other data types. A design framework is in great need to guide the systematic design of new CRDCs. To address the challenges above, we propose the Remove-Win Design Framework. The remove-win strategy for conflict resolution is simple but powerful. The remove operation just wipes out the data item, no matter how complex the value is. The user of the CRDC only needs to specify conflict resolution for non-remove operations. This resolution is destructed to three basic cases and are left as open terms in the CRDC design skeleton. Stubs containing user-specified conflict resolution logics are plugged into the skeleton to obtain concrete CRDC designs. We demonstrate the effectiveness of our design framework via a case study of designing a conflict-free replicated priority queue. Performance measurements also show the efficiency of the design derived from our design framework. author: - | Yuqi Zhang, Yu Huang, Hengfeng Wei, Jian Lu\ \ \ bibliography: - 'rwf.bib' title: 'Remove-Win: a Design Framework for Conflict-free Replicated Data Collections' --- \#1

David G. CHARLTON$^{a}$, Guido MONTAGNA$^{b,c}$,\ Oreste NICROSINI$^{c,b}$ and Fulvio PICCININI$^{c}$ [*$^a$Royal Society University Research Fellow, School of Physics and Space Research, University of Birmingham, Birmingham B15 2TT, UK*]{}\ [*$^b$Dipartimento di Fisica Nucleare e Teorica, Università di Pavia, via A. Bassi n. 6 - 27100 PAVIA - ITALY*]{}\ [*$^c$ INFN, Sezione di Pavia, via A. Bassi n. 6 - 27100 PAVIA - ITALY*]{} Program classification: 11.1\ [The Monte Carlo program [WWGENPV]{}, designed for computing distributions and generating events for four-fermion production in $e^+ e^- $ collisions, is described. The new version, 2.0, includes the full set of the electroweak (EW) tree-level matrix elements for double- and single-$W$ production, initial- and final-state photonic radiation including $p_T / p_L$ effects in the Structure Function formalism, all the relevant non-QED corrections (Coulomb correction, naive QCD, leading EW corrections). An hadronisation interface to [JETSET]{} is also provided. The program can be used in a three-fold way: as a Monte Carlo integrator for weighted events, providing predictions for several observables relevant for $W$ physics; as an adaptive integrator, giving predictions for cross sections, energy and invariant mass losses with high numerical precision; as an event generator for unweighted events, both at partonic and hadronic level. In all the branches, the code can provide accurate and fast results. ]{} [*Program obtainable from:*]{} CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) [*Reference to original program:*]{} [WWGENPV]{}; [*Cat. no.:*]{} ACNT; [*Ref. in CPC:* ]{} [**90**]{} (1995) 141 [*Authors of original program:*]{} Guido Montagna, Oreste Nicrosini and Fulvio Piccinini [*The new version supersedes the original program*]{} [*Computer for which the new version is designed:*]{} DEC ALPHA 3000, HP 9000/700 series; [*Installation:*]{} INFN, Sezione di Pavia, via A. Bassi 6, 27100 Pavia, Italy [*Keywords:*]{} $e^+ e^-$ collisions, LEP, $W$-mass measurement, radiative corrections, QED corrections, QCD corrections, Minimal Standard Model, four-fermion final states, electron structure functions, Monte Carlo integration/simulation, hadronisation. The precise measurement of the $W$-boson mass $M_W$ constitutes a primary task of the forthcoming experiments at the high energy electron–positron collider LEP2 ($2 M_W \leq \sqrt{s} \leq 210$ GeV). A meaningful comparison between theory and experiment requires an accurate description of the fully exclusive processes $e^+ e^- \to 4f$, including the main effects of radiative corrections, with the final goal of providing predictions for the distributions measured by the experiments. Same as in the original program, as far as weighted event integration and unweighted event generation are concerned. Adaptive Monte Carlo integration for high numerical precision purposes is added. Optional hadronic interface in the generation branch is supplied. The most promising methods for measuring the $W$-boson mass at LEP2 are the so called “threshold” and “direct reconstruction” methods \[5\]. For the first one, a precise evaluation of the threshold cross section is required. For the second one, a precise description of the invariant-mass shape of the hadronic system in semileptonic decays is mandatory. In order to meet these requirements, the previous version of the program has been improved by extending the class of the tree-level EW diagrams taken into account, by including $p_T / p_L $ effects both in initial- and final-state QED radiation, by supplying an hadronic interface in the generation branch. While the semileptonic decay channels are complete at the level of the Born approximation EW diagrams ([CC11/CC20]{} diagrams), neutral current backgrounds are neglected in the fully hadronic and leptonic decay channels. QED radiation is treated at the leading logarithmic level. Due to the absence of a complete ${\cal O} (\alpha)$/${\cal O} (\alpha_s)$ diagrammatic calculation, the most relevant EW and QCD corrections are effectively incorporated according to the recipe given in \[6\]. No anomalous coupling effects are at present taken into account. As adaptive integrator, the code provides cross section and energy and invariant-mass losses with a relative accuracy of about 1% in 8 min on HP 9000/735. As integrator of weighted events, the code produces about $10^5$ events/min on the same system. The generation of a sample of $10^3$ hadronised unweighted events requires about 8 min on the same system. Subroutines from the library of mathematical subprograms [NAGLIB]{} \[3\] for the numerical integrations are used in the program, when the adaptive integration branch is selected. \[1\] CERN Program Library, CN Division, CERN, Geneva. \[2\] NAG Fortran Library Manual Mark 16 (Numerical Algorithms Group, Oxford, 1991). \[3\] T. Sjöstrand, Comp. Phys. Commun. [**82**]{} (1994) 74; Lund University Report LU TP 95-20 (1995). \[4\] F. James, Comput. Phys. Commun. 79 (1994) 111. \[5\] [*Physics at LEP2*]{}, CERN Report 96-01, Theoretical Physics and Particle Physics Experiments Divisions, G. Altarelli, T. Sjöstrand and F. Zwirner, eds., Vols. 1 and 2, Geneva, 19 February 1996. \[6\] W. Beenakker, F. Berends et al., “WW cross-sections and distributions”, in \[5\], Vol. 1, pag. 79. Introduction ============ The precise measurement of the $W$-boson mass $M_W$ constitutes a primary task of the forthcoming experiments at the high energy electron–positron collider LEP2 ($2 M_W \leq \sqrt{s} \leq 210$ GeV). A meaningful comparison between theory and experiment requires an accurate description of the fully exclusive processes $e^+ e^- \to 4f$, including the main effects of radiative corrections, with the final goal of providing predictions for the distributions measured by the experiments. A large effort in the direction of developing tools dedicated to the investigation of this item has been spent within the Workshop “Physics at LEP2”, held at CERN during 1995. Such an effort has led to the development of several independent four-fermion codes, both semianalytical and Monte Carlo, extensively documented in [@wweg]. [WWGENPV]{} is one of these codes, and the aim of the present paper is to describe in some detail the developments performed with respect to the original version [@cpcww], where a description of the formalism adopted and the physical ideas behind it can be found. As discussed in [@wmass], the most promising methods for measuring the $W$-boson mass at LEP2 are the so called “threshold” and “direct reconstruction” methods. For the first one, a precise evaluation of the threshold cross section is required. For the second one, a precise description of the invariant-mass shape of the hadronic system in semileptonic and hadronic decays is mandatory. In order to meet these requirements, the previous version of the program has been improved, both from the technical and physical point of view. On the technical side, in addition to the “weighted event integration” and “unweighted event generation” branches, the present version can also be run as an “adaptive Monte Carlo” integrator, in order to obtain high numerical precision results for cross sections and other relevant observables. In the “weighted event integration” branch, a “canonical” output can be selected, in which several observables are processed in parallel together with their most relevant moments [@wweg]. Moreover, the program offers the possibility of generating events according to a specific flavour quantum number assignment for the final-state fermions, or of generating “mixed samples”, namely a fully leptonic, fully hadronic or semileptonic sample. On the physical side, the class of tree-level EW diagrams taken into account has been extended to include all the single resonant diagrams ([CC11/CC20]{}), in such a way that all the charged current processes are covered. Motivated by the physical relevance of keeping under control the effects of the transverse degrees of freedom of photonic radiation, both for the $W$ mass measurement and for the detection of anomalous couplings, the contribution of QED radiation has been fully developed in the leading logarithmic approximation, going beyond the initial-state, strictly collinear approximation, to include $p_T / p_L $ effects both for initial- and final-state photons. Last, an hadronic interface to [JETSET]{} in the generation branch has been added. In the present version the neutral current backgrounds are neglected in the fully hadronic and leptonic decay channels, but this is not a severe limitation of the program since, at least in the LEP2 energy range, these backgrounds can be suppressed by means of proper invariant mass cuts. On the other hand, the semileptonic decay channels are complete at the level of the Born approximation EW diagrams ([CC11/CC20]{} diagrams), and this feature allows to treat at best those channels that are expected to be the most promising for the direct reconstruction of the $W$ mass, because free of systematics such as the “color reconnection” and the “Bose-Einstein correlation” problems. In the development of the code, particular attention has been paid to the possibility of obtaining precise results in relatively short CPU time. As shown in [@wweg], [WWGENPV]{} is one of the most precise four fermion Monte Carlo’s from the numerical point of view. This feature allows the use of the code also for fitting purposes. The most important new features =============================== In the following we list and briefly describe the most important technical and physical developments implemented in the new version of [WWGENPV]{}. TECHNICAL IMPROVEMENTS The present version of the program consists of three branches, two of them already present in the original version but upgraded in some respect, the third one completely new. - Unweighted event generation branch. This branch, meant for simulation purposes, has been improved by supplying an option for an hadronisation interface (see more details later on). - Weighted event integration branch. This branch, intended for computation only, includes as a new feature an option for selecting a “canonical” output containing predictions for several observables and their most relevant moments together with a Monte Carlo estimate of the errors. According to the strategy adopted in [@wweg], the first four Chebyshev/power moments of the following quantities are computed: the production angle of the $W^+$ with respect to the positron beam ([TNCTHW, N=1,2,3,4]{}), the production angle $\vartheta_{d}$ of the down fermion with respect to the positron beam ([TNCTHD]{}), the decay angle $\vartheta_{d}^*$ of the down fermion with respect to the direction of the decaying $W^-$ measured in its rest frame ([TNCTHSR]{}), the energy $E_{d}$ of the down fermion normalized to the beam energy ([XDN]{}), the sum of the energies of all radiated photons ([XGN]{}) normalized to the beam energy, the lost and visible photon energies normalized to the beam energy ([XGNL]{} and [XGNV]{}), respectively, and, finally $$\begin{aligned} <x_m> \, = \, {1 \over \sigma} \, \int \left( { { \sqrt{s_+} + \sqrt{s_-} -2 M_W} \over {2 E_b} } \right) \, d \sigma\end{aligned}$$ where $s_+$ and $s_-$ are the invariant masses of the $W^+$ and $W^-$ decay products, respectively ([XMN]{}). - Adaptive integration branch. This new branch is intended for computation only, but offers high precision performances. On top of the importance sampling, an adaptive Monte Carlo integration algorithm is used. The code returns the value of the cross section together with a Monte Carlo estimate of the error. Moreover, if QED corrections are taken into account, also the average energy and invariant mass losses are printed. The program must be linked to NAG library for the Monte Carlo adaptive routine. Full consistency between non-adaptive and adaptive integrations has been explicitly proven. In each of these three branches the user is asked to specify the four-fermion final state which is required. The final states at present available are those containing [CC03]{} diagrams as a subset. Their list, as appears when running the code, is the following: PURELY LEPTONIC PROCESSES [0] ---> E+ NU_E E- BAR NU_E [1] ---> E+ NU_E MU- BAR NU_MU [2] ---> E- BAR NU_E MU+ NU_MU [3] ---> E+ NU_E TAU- BAR NU_TAU [4] ---> E- BAR NU_E TAU+ NU_TAU [5] ---> MU+ NU_MU MU- BAR NU_MU [6] ---> MU+ NU_MU TAU- BAR NU_TAU [7] ---> MU- BAR NU_MU TAU+ NU_TAU [8] ---> TAU+ NU_TAU TAU- BAR NU_TAU SEMILEPTONIC PROCESSES [9] ---> E+ NU_E D BAR U [10] ---> E- BAR NU_E BAR D U [11] ---> E+ NU_E S BAR C [12] ---> E- BAR NU_E BAR S C [13] ---> MU+ NU_MU D BAR U [14] ---> MU- BAR NU_MU BAR D U [15] ---> MU+ NU_MU S BAR C [16] ---> MU- BAR NU_MU BAR S C [17] ---> TAU+ NU_TAU D BAR U [18] ---> TAU- BAR NU_TAU BAR D U [19] ---> TAU+ NU_TAU S BAR C [20] ---> TAU- BAR NU_TAU BAR S C HADRONIC PROCESSES [21] ---> D BAR U BAR D U [22] ---> D BAR U BAR S C [23] ---> S BAR C BAR D U [24] ---> S BAR C BAR S C MIXED SAMPLES [25] ---> LEPTONIC SAMPLE [26] ---> SEMILEPTONIC SAMPLE [27] ---> HADRONIC SAMPLE It is worth noting that in the weighted event integration and unweighted event generation branches, besides the possibility of selecting a specific four-fermion final state, an option is present for considering three realistic mixed samples corresponding to the fully leptonic, fully hadronic or semileptonic decay channel, respectively. When the generation of a mixed sample is required, as a a first step the cross section for each contributing channel is calculated; as a second step, the unweighted events are generated for each contributing channel with a frequency given by the weight of that particular channel with respect to the total. In the generation branch, if the hadronisation interface is not enabled, an $n$-tuple is created with the following structure: 'X_1','X_2','EB' ! x_{1,2} represent the ! energy fractions of incoming e^- and ! e^+ after ISR; EB is the beam energy; 'Q1X','Q1Y','Q1Z','Q1LB' ! x,y,z components of the momentum ! of particle 1 and the particle label ! according to PDG; the final-state ! fermions are assumed to be massless; 'Q2X','Q2Y','Q2Z','Q2LB' ! as above, particle 2 'Q3X','Q3Y','Q3Z','Q3LB' ! " " " 3 'Q4X','Q4Y','Q4Z','Q4LB' ! " " " 4 'AK1X','AK1Y','AK1Z' ! x,y,z components of the momentum of ! the photon from particle 1; ! they are 0 if no FSR has been chosen ! and/or if particle 1 is a neutrino; 'AK2X','AK2Y','AK2Z' ! as above, particle 2 'AK3X','AK3Y','AK3Z' ! " " " 3 'AK4X','AK4Y','AK4Z' ! " " " 4 'AKEX','AKEY','AKEZ' ! x,y,z components of the momentum of ! the photon from the initial-state ! electron; they are 0 if no ISR has ! been chosen; 'AKPX','AKPY','AKPZ' ! as above, initial-state positron; If the hadronisation interface is enabled, fully hadronised events are instead made available to a user routine in the [/HEPEVT/]{} format (see below). PHYSICAL IMPROVEMENTS\ The main theoretical developments with respect to the original version concern the inclusion of additional matrix elements to the tree-level kernel and a more sophisticated treatment of the photonic radiation, beyond the initial-state, strictly collinear approximation. Moreover, an hadronisation interface to [ JETSET]{} has been also provided. [*Tree-level EW four-fermion diagrams*]{} – In addition to double-resonant charged-current diagrams [CC03]{} already present in the previous version, the matrix element includes also the single-resonant charged-current diagrams [CC11]{} for $\mu $ and $\tau$’s in the final state, and [CC20]{} for final states containing electrons. This allows a complete treatment at the level of four-fermion EW diagrams of the semileptonic sample, which appears the most promising and cleanest for the direct mass reconstruction method due to the absence of potentially large “interconnection” effects [@wmass]. Concerning [CC20]{} diagrams, the importance sampling technique has been extended to take care of the peaking behaviour of the matrix element when small momenta of the virtual photon are involved. As a consequence of the fact that the tree-level matrix-element is computed in the massless limit, a cut on the minimum electron (positron) scattering angle must be imposed. The inclusion of such a cut eliminates the problems connected with gauge-invariance in the case of [CC20]{} processes, for which the present version does not include any so-called “reparation” scheme [@bhf; @wwcd]. Anyway, when for instance a set of “canonical cuts” [@wweg] is used, the numerical relevance of such gauge-invariance restoring schemes has been shown [@bhf; @wwcd] to be negligible compared with the expected experimental accuracy. [*Photonic corrections*]{} – As far as photonic effects are concerned, the original version, as stated above, included only leading logarithmic initial-state corrections in the collinear approximation within the SF formalism. The treatment has been extended in a two-fold way: the contribution of final-state radiation has been included and the $p_T / p_L$ effects have been implemented both for initial- and final-state radiation. The inclusion of the transverse degrees of freedom has been achieved by generating the fractional energy $x_{\gamma}$ of the radiated photons by means of resummed electron structure functions $D(x; s)$ [@sf] ($x_{\gamma} = 1 - x$) and the angles using an angular factor inspired by the pole behaviour $1 / (p \cdot k)$ for each charged emitting fermion. This allows to incorporate leading QED radiative corrections originating from infrared and collinear singularities, taking into account at the same time the dominant kinematic effects due to non-strictly collinear photon emission, in such a way that the universal factorized photonic spectrum is recovered. According to this procedure, the leading logarithmic corrections from initial- and final-state radiation are isolated as a gauge-invariant subset of the full calculation (not yet available) of the electromagnetic corrections to $e^+ e^- \to 4f$. Due to the inclusion of $p_T$-carrying photons at the level of initial-state radiation, the Lorentz boost allowing the reconstruction of the hard-scattering event from the c.m. system to the laboratory one has been generalized to keep under control the $p_T$ effects on the beam particles. Final state radiation and $p_T / p_L$ effects are not taken into account in the adaptive integration branch. [*Non-QED corrections*]{} – Coulomb correction is treated as in the original version on double-resonant [CC03]{} diagrams. QCD corrections are implemented in the present version in the naive form according to the recipe described in [@wweg; @wwcd]. The treatment of the leading EW contributions is unchanged with respect to the original version. [*Hadronisation*]{} – Final-state quarks issuing from the electroweak 4-fermion scattering are not experimentally observable. An hadronisation interface is provided to the [JETSET]{} package [@jetset] to allow events to be extrapolated to the hadron level, for example for input to a detector simulation program. Specifically, the 4-fermion event structure is converted to the [/HEPEVT/]{} convention, then [JETSET]{} is called to simulate QCD partonic evolution (via routines [LUJOIN]{} and [LUSHOW]{}) and hadronisation (routine [LUEXEC]{}). In making this conversion, masses must be added to the outgoing fermions, considered massless in the hard scattering process. This is done by rescaling the fermion momenta by a single scale factor, keeping the flight directions fixed in the rest frame of the four fermion system. In the QCD evolution phase, strings join quarks coming from the same W decay. The virtuality scale of the QCD evolution is taken to be the invariant mass-squared of each evolving fermion pair. No colour reconnection is included by default, although it could be implemented by appropriate modification of routine [WWGJIF]{} if required. Bose - Einstein correlations are neglected in the present version. The resulting event structure is then made available to the user in the [/HEPEVT/]{} common block via a routine [WWUSER]{} for further analysis, such as writing out for later input to a detector simulation program. The [WWUSER]{} routine is also called at program initialisation time to allow the user to set any non-standard [JETSET]{} program options, for example, and at termination time to allow any necessary clean-up. A dummy [WWUSER]{} routine is supplied with the program. The only [JETSET]{} option which is changed by [WWGENPV]{} from its default value controls emission of gluons and photons by final-state partons[^1], turning off final-state photon emission simulation from [JETSET]{} if activated in [WWGENPV]{}, to avoid double counting.\ All the new features of the program can be switched on/off by means of separate flags, as described in the following. Input ===== Here we give a short explanation of the input parameters and flags required when running the program. **** OGEN(CHARACTER*1) It controls the use of the program as a Monte Carlo event generator of unweighted events ([OGEN = G]{}) or as a Monte Carlo/adaptive integrator for weighted events ([OGEN = I]{}). **** RS(REAL*8) The centre-of-mass energy (in GeV). **** OFAST(CHARACTER*1) It selects ([OFAST = Y]{}) the adaptive integration branch, when [OGEN = I]{}. When this choice is done, the required relative accuracy of the numerical integration has to be supplied by means of the [REAL\*8]{} variable [EPS]{}. **** NHITWMAX(INTEGER) Required by the Monte Carlo integration branch. It is the maximum number of calls for the Monte Carlo loop. **** NHITMAX(INTEGER) Required by the event-generation branch. It is the maximum number of hits for the hit-or-miss procedure. **** IQED(INTEGER) This flag allows the user to switch on/off the contribution of the initial-state radiation. If [IQED = 0]{} the distributions are computed in lowest-order approximation, while for [IQED = 1]{} the QED corrections are included in the calculation. **** OPT(CHARACTER*1) This flag controls the inclusion of $p_T / p_L$ effects for the initial-state radiation. It is ignored in the adaptive integration branch where only initial-state strictly collinear radiation is allowed. **** OFS(CHARACTER*1) It is the option for including final-state radiation. It is assumed that final-state radiation can be switched on only if initial-state radiation including $p_T / p_L$ effects is on, in which case final-state radiation includes $p_T / p_L$ effects as well. Ignored in the adaptive integration branch. **** ODIS(CHARACTER*1) Required by the integration branch. It selects the kind of experimental distribution. For [ODIS = T]{} the program computes the total cross section (in pb) of the process; for [ODIS = W]{} the value of the invariant-mass distribution $d \sigma / d M$ of the system $d \bar u$ ([IWCH = 1]{}) or of the system ${\bar d} u$ ([IWCH = 2]{}) is returned (in pb/GeV). **** OWIDTH(CHARACTER*1) It allows a different choice of the value of the $W$-width. [OWIDTH = Y]{} means that the tree-level Standard Model formula for the $W$-width is used; [OWIDTH = N]{} requires that the $W$-width is supplied by the user in GeV. **** NSCH(INTEGER) The value of [NSCH]{} allows the user to choose the calculational scheme for the weak mixing angle and the gauge coupling. Three choices are available. If [NSCH=1]{}, the input parameters used are $G_F, M_W, M_Z$ and the calculation is performed at tree level. If [NSCH = 2]{} or [3]{}, the input parameters used are $\alpha(Q^2), G_F, M_W$ or $\alpha(Q^2), G_F, M_Z$, respectively, and the calculation is performed using the QED coupling constant at a proper scale $Q^2$, which is requested as further input. The recommended choice is [NSCH = 2]{}, consistently with [@wweg]. **** OCOUL(CHARACTER*1) This flag allows the user to switch on/off the contribution of the Coulomb correction. Unchanged with respect to the old version of the program. **** OQCD(CHARACTER*1) This flag allows the user to switch on/off the contribution of the naive QCD correction. **** ICHANNEL(INTEGER) A channel corresponding to a specific flavour quantum number assignment can be chosen. **** ANGLMIN(REAL*8) The minimum electron (positron) scattering angle (deg.) in the laboratory frame. It is ignored when [CC20]{} graphs are not selected. **** SRES(CHARACTER*1) Option for switching on/off single-resonant diagrams ([CC11]{}). **** OCC20(CHARACTER*1) Option for switching on/off single-resonant diagrams when electrons (positrons) occur in the final state ([CC20]{}). **** OOUT(CHARACTER*1) Option for “canonical” output containing results for several observables and their most important moments. It is active only in the Monte Carlo integration branch. **** OHAD(CHARACTER*1) Option for switching on/off hadronisation interface in the unweighted event generation branch. Test run output =============== The typical new calculations that can be performed with the updated version of the program are illustrated in the following examples. An example of adaptive integration is provided. The process considered is $e^+ e^- \to e^+ \nu_e d \bar u$ ([CC20]{}). The output gives the cross section, together with the energy and invariant-mass losses from initial-state radiation. “Canonical” cuts are imposed as in [@wweg]. The input card is as follows: OGEN = I RS = 190.D0 OFAST = Y EPS = 1.D-2 IQED = 1 OPT = N OFS = N ODIS = T OWIDTH = Y NSCH = 2 ALPHM1 = 128.07D0 OCOUL = N OQCD = N ICHANNEL = 9 ANGLMIN = 10.D0 SRES = Y OCC20 = Y OOUT = N An example of weighted event integration is provided. Here the process considered is $e^+ e^- \to \mu^+ \nu_{\mu} d \bar u$ ([CC11]{}). “Canonical” cuts are imposed as before. The “canonical” output is provided. The input card differs from the previous one as follows: RS = 175.D0 OFAST = N NHITWMAX = 100000 OPT = Y OFS = Y OCOUL = Y OQCD = Y ICHANNEL = 13 OCC20 = N OOUT = Y An example of unweighted event generation including hadronisation is provided. A sample of 100 events corresponding to the full semileptonic channel is generated. The detailed list of an hadronised event is given. The input card differs from the first one as follows: OGEN = G RS = 175.D0 NHITMAX = 100 OPT = Y OFS = Y OCOUL = Y OQCD = Y ICHANNEL = 26 ANGLMIN = 5.D0 OHAD = Y Conclusions =========== The program [WWGENPV 2.0]{} has been described. In its present version it allows the treatment of all the four-fermion reactions including the [CC03]{} class of diagrams as a subset. This means that all the semileptonic channels are complete from the tree-level diagrams point of view, whereas fully leptonic and fully hadronic channels are treated in the CC approximation. Since the most promising channels for the $W$ mass reconstruction are the semileptonic ones, the present version of the code allows a precise analysis of such data. Moreover, NC backgrounds can be suppressed by proper invariant mass cuts, so that the code is also usable for fully hadronic and leptonic events analysis, with no substantial loss of reliability. Initial- and final-state QED radiation is taken into account within the SF formalism, including finite $p_T / p_L$ effects in the leading logarithmic approximation. The Coulomb correction is included for the [CC03 ]{} graphs. Naive QCD and leading EW corrections are implemented as well. An hadronic interface to [JETSET]{} is also provided. The code as it stands is a valuable tool for the analysis of LEP2 data, with particular emphasis to the threshold and direct reconstruction methods for the measurement of the $W$-boson mass. Speed and high numerical accuracy allow the use of the program also for fitting purposes. The code is supported. Future releases of [WWGENPV]{} will include: - an interface to the code [HIGGSPV]{} [@wweg; @egdp] in order to treat all the possible four-fermion processes in the massless limit, including Higgs-boson signals; - implementation of anomalous couplings; - implementation of CKM effects; - the extension of the hadronic interface to [HERWIG]{} [@herwig]. [99]{} D. Bardin, R. Kleiss et al., “Event generators for WW physics”, in [*Physics at LEP2*]{}, CERN [**96-01**]{}, Theoretical Physics and Particle Physics Experiments Divisions, G. Altarelli, T. Sjöstrand and F. Zwirner, eds., Vol. 2, pag. 3, Geneva, 19 February 1996. G. Montagna, O. Nicrosini and F. Piccinini, Comput. Phys. Commun. [**90**]{} (1995) 141. Z. Kunszt, W.J. Stirling et al., “Determination of the mass of the $W$ boson”, in [*Physics at LEP2*]{}, CERN [**96-01**]{}, Theoretical Physics and Particle Physics Experiments Divisions, G. Altarelli, T. Sjöstrand and F. Zwirner, eds., Vol. 1, pag. 141, Geneva, 19 February 1996. E. N. Argyres et al., Phys. Lett. [**B358**]{} (1995) 339. W. Beenakker, F. Berends et al., “WW cross-sections and distributions”, in [*Physics at LEP2*]{}, CERN [**96-01**]{}, Theoretical Physics and Particle Physics Experiments Divisions, G. Altarelli, T. Sjöstrand and F. Zwirner, eds., Vol. 1, pag. 79. T. Sjöstrand, Comp. Phys. Commun. [**82**]{} (1994) 74; Lund University Report LU TP 95-20 (1995). M.L. Mangano, G. Ridolfi et al., “Event Generators for Discovery Physics”, in [*Physics at LEP2*]{}, CERN [**96-01**]{}, Theoretical Physics and Particle Physics Experiments Divisions, G. Altarelli, T. Sjöstrand and F. Zwirner, eds., Vol.2, pag. 299. G. Marchesini, B. R. Webber et al., Comput. Phys. Commun. [**67**]{} (1992) 465. HADRONISATION VIA JETSET WWGJIF: Initialising WWGENPV to JETSET interface PHOTON FSR WILL BE HANDLED BY WWGENPV: TURNING OFF PHOTON FSR FROM JETSET 1 SQRT(S) = 175.0 GEV M_Z = 91.1888 GEV G_Z = 2.4974 GEV M_W = 80.23 GEV G_W = 2.08823657515165 GEV STH2 = .2310309124510679 GVE = -1.409737267299689E-02 GAE = -.185794027211796 GWF = .230409927395451 GWWZ = -.571479454308384 Event listing (summary) I particle/jet KS KF orig p_x p_y p_z E m 1 !e+! 21 -11 0 .000 .000 -87.500 87.500 .001 2 !e-! 21 11 0 .000 .000 87.500 87.500 .001 3 !e+! 21 -11 1 .000 .000 -87.500 87.500 .001 4 !e-! 21 11 2 .000 .000 87.500 87.500 .001 5 gamma 1 22 1 .000 .000 .000 .000 .000 6 nu_e 1 12 0 45.589 -8.504 -7.296 46.945 .000 7 e+ 1 -11 0 -32.518 1.009 -24.213 40.555 .001 8 gamma 1 22 7 -.063 -.003 -.054 .083 .000 9 (c~) A 11 -4 0 -6.482 5.087 -14.629 16.844 1.350 10 (g) V 11 21 0 -.912 -.166 -1.330 1.621 .000 11 (g) A 11 21 0 -1.717 2.148 .149 2.754 .000 12 (g) V 11 21 0 -.404 -.150 -.153 .457 .000 13 (g) A 11 21 0 .524 .257 -.485 .759 .000 14 (g) V 11 21 0 .568 5.813 -3.909 7.028 .000 15 (g) A 11 21 0 -.833 .123 -.634 1.054 .000 16 (g) V 11 21 0 -1.616 -.218 .984 1.904 .000 17 (g) A 11 21 0 -.893 -.594 -.345 1.127 .000 18 (g) V 11 21 0 -.064 -1.133 .124 1.141 .000 19 (u) A 11 2 0 -1.275 -.657 4.823 5.032 .006 20 (u~) V 11 -2 0 -1.276 .758 3.201 3.528 .006 21 (g) A 11 21 0 .733 -.103 2.158 2.281 .000 22 (g) V 11 21 0 .691 -.234 2.326 2.438 .000 23 (g) A 11 21 0 .410 -1.788 16.927 17.027 .000 24 (s) V 11 3 0 -.461 -1.646 22.354 22.420 .199 25 (string) 11 92 9 -13.104 10.511 -15.404 39.722 32.532 26 (D~0) 11 -421 25 -6.057 4.073 -13.996 15.894 1.865 27 (pi0) 11 111 25 -1.112 1.280 -.302 1.728 .135 28 (rho-) 11 -213 25 -1.158 1.158 -2.172 2.803 .676 29 (omega) 11 223 25 -.792 .558 .401 1.303 .774 30 (Sigma*~+) 11 -3114 25 .123 2.038 -1.332 2.787 1.351 31 (K*-) 11 -323 25 .319 1.963 -1.639 2.729 .896 32 (Delta+) 11 2214 25 -.155 1.330 -1.314 2.263 1.266 33 (rho-) 11 -213 25 -1.026 .221 .345 1.292 .670 34 (pi0) 11 111 25 -.256 .036 -.546 .619 .135 35 (rho0) 11 113 25 -.488 -.118 .538 1.103 .822 36 pi+ 1 211 25 -.420 .013 -.275 .521 .140 37 pi- 1 -211 25 -.612 -.620 .002 .882 .140 38 (rho+) 11 213 25 -.474 -1.082 1.048 1.662 .520 39 (rho0) 11 113 25 -.997 -.339 3.839 4.136 1.122 40 (string) 11 92 20 .096 -3.014 46.967 47.695 7.733 41 p~- 1 -2212 40 -.233 .669 3.532 3.723 .938 42 p+ 1 2212 40 .259 -.320 2.633 2.826 .938 43 (Delta~--) 11 -2224 40 -.208 -.282 7.298 7.431 1.357 44 pi+ 1 211 40 .568 -.518 3.830 3.909 .140 45 (Lambda0) 11 3122 40 -.290 -2.562 29.673 29.806 1.116 46 (K*+) 11 323 26 -3.673 2.525 -9.054 10.129 .875 47 (rho-) 11 -213 26 -2.384 1.548 -4.942 5.765 .856 48 gamma 1 22 27 -1.055 1.186 -.310 1.617 .000 49 gamma 1 22 27 -.058 .094 .008 .111 .000 50 pi- 1 -211 28 -.837 1.095 -1.923 2.370 .140 51 (pi0) 11 111 28 -.321 .063 -.249 .433 .135 52 pi+ 1 211 29 -.610 .314 .197 .727 .140 53 pi- 1 -211 29 -.236 .248 .217 .429 .140 54 (pi0) 11 111 29 .055 -.005 -.014 .146 .135 55 (Lambda~0) 11 -3122 30 .085 1.705 -.918 2.237 1.116 56 pi+ 1 211 30 .038 .333 -.414 .551 .140 57 (K~0) 11 -311 31 .462 1.760 -1.507 2.414 .498 58 pi- 1 -211 31 -.143 .203 -.133 .315 .140 59 n0 1 2112 32 .123 .910 -1.024 1.665 .940 60 pi+ 1 211 32 -.277 .420 -.290 .598 .140 61 pi- 1 -211 33 .005 .082 .159 .227 .140 62 (pi0) 11 111 33 -1.031 .138 .186 1.065 .135 63 gamma 1 22 34 -.201 -.017 -.301 .362 .000 64 gamma 1 22 34 -.055 .053 -.245 .257 .000 65 pi- 1 -211 35 -.071 -.134 .622 .655 .140 66 pi+ 1 211 35 -.417 .016 -.084 .448 .140 67 pi+ 1 211 38 .032 -.277 .386 .497 .140 68 (pi0) 11 111 38 -.506 -.804 .661 1.166 .135 69 pi- 1 -211 39 -.186 .113 .094 .275 .140 70 pi+ 1 211 39 -.811 -.452 3.745 3.861 .140 71 p~- 1 -2212 43 -.135 -.544 5.493 5.600 .938 72 pi- 1 -211 43 -.073 .262 1.805 1.831 .140 73 n0 1 2112 45 -.262 -2.385 26.596 26.721 .940 74 (pi0) 11 111 45 -.028 -.177 3.077 3.085 .135 75 (K0) 11 311 46 -2.419 1.634 -6.536 7.176 .498 76 pi+ 1 211 46 -1.253 .891 -2.517 2.953 .140 77 pi- 1 -211 47 -.404 .604 -1.633 1.792 .140 78 (pi0) 11 111 47 -1.980 .945 -3.310 3.973 .135 79 gamma 1 22 51 -.223 .051 -.091 .246 .000 80 gamma 1 22 51 -.098 .012 -.158 .187 .000 81 gamma 1 22 54 .048 -.066 .004 .082 .000 82 gamma 1 22 54 .006 .062 -.017 .064 .000 83 p~- 1 -2212 55 .064 1.274 -.756 1.755 .938 84 pi+ 1 211 55 .021 .431 -.161 .482 .140 85 (K_S0) 11 310 57 .462 1.760 -1.507 2.414 .498 86 gamma 1 22 62 -.111 .044 -.011 .120 .000 87 gamma 1 22 62 -.920 .094 .197 .945 .000 88 gamma 1 22 68 -.034 -.065 .012 .074 .000 89 gamma 1 22 68 -.472 -.740 .649 1.091 .000 90 gamma 1 22 74 -.035 -.088 2.449 2.451 .000 91 gamma 1 22 74 .007 -.089 .628 .634 .000 92 (K_S0) 11 310 75 -2.419 1.634 -6.536 7.176 .498 93 gamma 1 22 78 -1.877 .868 -3.149 3.768 .000 94 gamma 1 22 78 -.103 .077 -.160 .205 .000 95 pi- 1 -211 85 .270 .383 -.379 .618 .140 96 pi+ 1 211 85 .192 1.377 -1.128 1.796 .140 97 pi+ 1 211 92 -.563 .463 -2.034 2.165 .140 98 pi- 1 -211 92 -1.857 1.171 -4.502 5.011 .140 sum: .00 .000 .000 .000 175.000 175.000 NHIT = 100 XSECT FOR UNWEIGHTED EVENTS EFF = 1.887112905965163E-03 XSECT = 6.79360646147459 +- .6787193283232224 (PB) NBIAS/NMAX = .0 USER TERMINATION A TOTAL OF 100 EVENTS WERE GENERATED THE TOTAL CROSS-SECTION WAS: 6.793606281280517E-09 MB [^1]: parameter [MSTJ(41)]{}

--- author: - '[Nikolai Nadirashvili[^1],.4 cm Serge Vlăduţ[^2] ]{}' title: '[Singular Solutions of Hessian Elliptic Equations in Five Dimensions]{}' --- § Ø Introduction ============ In this paper we study a class of fully nonlinear second-order elliptic equations of the form $$F(D^2u)=0\leqno(1.1)$$ defined in a domain of ${ \R}^n$. Here $D^2u$ denotes the Hessian of the function $u$. We assume that $F$ is a Lipschitz function defined on the space $ S^2({ \R}^n)$ of ${n\times n}$ symmetric matrices satisfying the uniform ellipticity condition, i.e. there exists a constant $C=C(F)\ge 1$ (called an [*ellipticity constant*]{}) such that $$C^{-1}||N||\le F(M+N)-F(M) \le C||N||\; \leqno(1.2)$$ for any non-negative definite symmetric matrix $N$; if $F\in C^1(S^2({ \R}^n))$ then this condition is equivalent to $$\frac{1}{ C'}|\xi|^2\le F_{u_{ij}}\xi_i\xi_j\le C' |\xi |^2\;, \forall\xi\in { \R}^n\;.\leqno(1.2')$$ Here, $u_{ij}$ denotes the partial derivative $\pt^2 u/\pt x_i\pt x_j$. A function $u$ is called a [*classical*]{} solution of (1) if $u\in C^2(\Om)$ and $u$ satisfies (1.1). Actually, any classical solution of (1.1) is a smooth ($C^{\alpha +3}$) solution, provided that $F$ is a smooth $(C^\alpha )$ function of its arguments. For a matrix $S \in S^2({ \R}^n)$ we denote by $\lambda(S)=\{ \lambda_i : \lambda_1\leq...\leq\lambda_n\} \in { \R}^n$ the (ordered) set of eigenvalues of the matrix $S$. Equation (1.1) is called a Hessian equation (\[T1\],\[T2\] cf. \[CNS\]) if the function $F(S)$ depends only on the eigenvalues $\lambda(S)$ of the matrix $S$, i.e., if $$F(S)=f(\lambda(S)),$$ for some function $f$ on ${ \R}^n$ invariant under permutations of the coordinates. In other words the equation (1.1) is called Hessian if it is invariant under the action of the group $O(n)$ on $S^2({ \R}^n)$: $$\forall O\in O(n),\; F({^t O}\cdot S\cdot O)=F(S) \;.\leqno(1.3)$$ The Hessian invariance relation (1.3) implies the following: \(a) $F$ is a smooth (real-analytic) function of its arguments if and only if $f$ is a smooth (real-analytic) function. \(b) Inequalities (1.2) are equivalent to the inequalities $${\mu\over C_0} \leq { f ( \lambda_i+\mu)-f ( \lambda_i) } \leq C_0 \mu, \; \forall \mu\ge 0,$$ $\forall i=1,...,n$, for some positive constant $C_0$. \(c) $F$ is a concave function if and only if $f$ is concave. Well known examples of the Hessian equations are Laplace, Monge-Ampère, Bellman, Isaacs and Special Lagrangian equations. Bellman and Isaacs equations appear in the theory of controlled diffusion processes, see \[F\]. Both are fully nonlinear uniformly elliptic equations of the form (1.1). The Bellman equation is concave in $D^2u \in S^2({ \R}^n)$ variables. However, Isaacs operators are, in general, neither concave nor convex. In a simple homogeneous form the Isaacs equation can be written as follows: $$F(D^2u)=\sup_b \inf_a L_{ab}u =0, \leqno (1.4)$$ where $L_{ab}$ is a family of linear uniformly elliptic operators of type $$L= \sum a_{ij} {\partial^2 \over \partial x_i \partial x_j } \leqno (1.5)$$ with an ellipticity constant $C>0$ which depends on two parameters $a,b$. Consider the Dirichlet problem $$\label{dir}\begin{cases} F(D^2u, Du, u, x)=0 &\text{in}\; \Om \cr \quad \quad u=\vph & \text{on}\; \pt\Om\;,\cr\end{cases}$$ where $\Omega \subset {\R}^n$ is a bounded domain with a smooth boundary $\partial \Omega$ and $\vph$ is a continuous function on $\pt\Om$. We are interested in the problem of existence and regularity of solutions to the Dirichlet problem (1.6) for Hessian equations and Isaacs equation. The problem (1.6) has always a unique viscosity (weak) solution for fully nonlinear elliptic equations (not necessarily Hessian equations). The viscosity solutions satisfy the equation (1.1) in a weak sense, and the best known interior regularity (\[C\],\[CC\],\[T3\]) for them is $C^{1+\epsilon }$ for some $\epsilon > 0$. For more details see \[CC\], \[CIL\]. Until recently it remained unclear whether non-smooth viscosity solutions exist. In the recent papers \[NV1\], \[NV2\], \[NV3\], \[NV4\] the authors first proved the existence of non-classical viscosity solutions to a fully nonlinear elliptic equation, and then of singular solutions to Hessian uniformly elliptic equation in all dimensions beginning from 12. Those papers use the functions $$w_{12,\delta}(x)={P_{12}(x)\over |x|^{\delta }},\;w_{24,\delta}(x)= {P_{24} (x)\over |x|^{\delta }},\: \delta\in [1,2[,$$ with $P_{12}(x),P_{24}(x)$ being cubic forms as follows: $$P_{12}(x)=Re (q_1q_2q_3),\; x=(q_1,q_2,q_3)\in {\H}^3={ \R}^{12},$$ $ {\H}$ being Hamiltonian quaternions, $$P_{24}(x)={Re((o_1\cdot o_2)\cdot o_3)}={Re(o_1\cdot(o_2\cdot o_3))},\; x=(o_1,o_2,o_3)\in {\O}^3={\R}^{24}$$ $\O$ being the algebra of Caley octonions. Finally, the paper \[NTV\] gives a construction of non-smooth viscosity solution in 5 dimensions which is order 2 homogeneous, also for Hessian equations, the function $$w_5(x)={P_{5} (x)\over |x|},\;$$ being such solution for the Cartan minimal cubic $$P_{5}(x)=x_1^3+\frac{3 x_1}2\left(z_1^2 + z_2^2-2 z_3^2-2x_2^2\right)+\frac{3\sqrt 3}2\left(x_2z_1^2-x_2z_2^2 + 2z_1z_2z_3\right)$$ in 5 dimensions. However, the methods of \[NTV\] does not work for the function $w_{5,\delta}(x)=P_{5} (x)/ |x|^{\delta }, \;\delta>1$, and thus does not give singular (i.e. not in $C^{1,1}$) viscosity solutions to fully nonlinear equations in 5 dimensions. In the present paper we fill the gap and prove [*The function $$w_{5,\delta}(x)=P_{5} (x)/ |x|^{1+\delta }, \;\delta\in [0,1[$$ is a viscosity solution to a uniformly elliptic Hessian equation $(1.1)$ with a smooth functional $F$ in a unit ball $B\subset {\R}^{5}$ for the isoparametric Cartan cubic form $$P_{5}(x)=x_1^3+\frac{3 x_1}2\left(z_1^2 + z_2^2-2 z_3^2-2x_2^2\right)+\frac{3\sqrt 3}2\left(x_2z_1^2-x_2z_2^2 + 2z_1z_2z_3\right)$$ with $x=(x_1,x_2,z_1,z_2,z_3)$.*]{} In particular one gets the optimality of the interior $C^{1,\alpha}$-regularity of viscosity solutions to fully nonlinear equations in dimensions 5 and more; note also that all previous constructions give only Lipschitz Hessian functional $F$. Let us recall that in the paper \[NV5\] it is proven that there is no order 2 homogenous solutions to elliptic equations in 4 dimensions which suggests strongly that in 4 (and less) dimensions there is no homogenous non-classical solutions to uniformly elliptic equations. As in \[NV3\] we get also that $w_{5,\delta}(x), \;\delta\in [0,1[$ is a viscosity solution to a uniformly elliptic Isaacs equation: [**Corollary 1.2.**]{} [*The function $$w_{5,\delta}(x)=P_{5} (x)/ |x|^{1+\delta }, \;\delta\in [0,1[$$ is a viscosity solution to a uniformly elliptic Isaacs equation $(1.4)$ in a unit ball $B\subset { \R}^{5}$.*]{} The rest of the paper is organized as follows: in Section 2 we recall some necessary preliminary results and we prove our main results in Section 3. The proof in Section 3 extensively uses MAPLE but is completely rigorous. Preliminary results ==================== Let $w=w_n$ be an odd homogeneous function of order $2-\delta,\: 0\leq \delta <1$, defined on a unit ball $B =B_1\subset { \R}^n$ and smooth in $B \setminus\{0\}$. Then the Hessian of $w$ is homogeneous of order $-\delta$. Define the map $$\Lambda :B \longrightarrow \lambda (S) \in { \R}^n\; .$$ $\lambda(S)=\{ \lambda_i : \lambda_1\leq...\leq\lambda_n\} \in { \R}^n$ being the (ordered) set of eigenvalues of the matrix $S=D^2w$. Denote $\Sigma_n$ the permutation group of $\{ 1,...,n\}$. For any $\sigma \in \Sigma_n$, let $T_{\sigma}$ be the linear transformation of ${ \R}^n$ given by $x_i \mapsto x_{\sigma(i)}, \; i=1,...,n.$ Let $a,b\in B$. Denote by $\mu_1(a,b)\leq ...\leq \mu_n(a,b)$ the eigenvalues of $(D^2w(a)-D^2w(b))$. [**Lemma 2.1.**]{} [*Assume that for a smooth function $g: U \longrightarrow { \R}$ where the domain $U$ contains $$M:=\bigcup_{\sigma \in \Sigma_{n} }\ T_{\sigma }\Lambda (B)\subset { \R}^n$$ one has $$g_{|M}=0.$$ Assume also the condition $$\min_{i=1,\ldots,5}\inf_{x\in M}\left\{\frac{\partial g}{\partial \lambda_i}(\lambda )\right\}>0.\leqno (2.1)$$ Assume further that for any $a,b\in B$ either $\mu_1(a,b)= ...= \mu_n(a,b)=0$ or $$1/C\leq -\frac{\mu_1(a,b)}{\mu_n(a,b)}\leq C,\leqno (2.2)$$ where $C$ is a positive constant (may be, depending on $M,g$ but not on $a,b$). If $\delta> 0$ we assume additionally that $w$ changes sign in $B$. Then $w$ is a viscosity solution in $B$ of a uniformly elliptic Hessian equation $(1.1)$ with a smooth $F$. Function $w$ is as well a solution to a uniformly elliptic Isaacs equation.*]{} [*Proof.*]{} Denote for any $\theta>0$ by $K_{\theta }\subset { \R}^n$ the cone $\{\lambda \in { \R}^n,\, \lambda_i /|\lambda | >\theta\},\, $ and let $K^*_{\theta }$ be its dual cone. Let $x,y$ be orthogonal coordinates in ${ \R}^n$ such that $x=\lambda_1+...+\lambda_n$ and $y$ be the orthogonal complement of $x$. Denote by $p$ the orthogonal projection of ${ \R}^n$ on subspace $y$. Denote $$\Gamma =\{ g=0 \} \subset U,$$ $$G=p(\Gamma ),$$ $$m=p(M).$$ From (2.1), (2.2) it follows that the surface $\Gamma $ is a graph of a smooth function $h$ defined on $G$. By $k_{\theta}$ we denote the function on $y$ which graph is the surface $\partial K^*_{\theta }$. We define the function $H(y)$ by $$H(y) = \inf_{z\in G}\{ h(z)+k_{\theta }(y-z)\}.$$ We fix a sufficiently small $\theta >0$. Then from (2.1), (2.2) it follows that $H=h$ on $G$. Denote by $J$ the graph of $H$. It is easy to show, see similar argument in \[NV1\], \[NV3\], that for any $a,b\in J,\, a\neq b$, $$1/C\leq -\min_i(a_i-b_i)/\max_i(a_i-b_i)\leq C.\leqno (2.3)$$ Let $E$ be a smooth function in ${ \R}^{n-1}$ with the support in a unit ball and with the integral being equal to $1$. Denote $E_c(y)=c^{-n+1}E(y/c)$, $c>0$. Set $$H_c=H\ast E_c.$$ Then $H_c$ will be a smooth function such that any two points $a,b$ on its graph will satisfy (2.3). Moreover $H_c \rightarrow H$ in $C({ \R}^n)$ as $c$ goes to $0$, and $H_c \rightarrow h$ in $C^{\infty }$ on compact subdomains of $G$. Thus for a sufficiently small $c>0$ we can easily modify function $H_c$ to a function $\tilde H$ such that $\tilde H$ will coincide with $h$ in a neighborhood of $m$, coincide with $H$ in the complement of $G$ and the points on the graph of $\tilde H$ will still satisfy (2.3) possibly with a larger constant $C$. Define the function $F$ in ${ \R}^n$ by $$F=x-\tilde H(y).$$ Then $w$ is a solution in ${ \R}^n\setminus \{ 0\}$ of a uniformly elliptic Hessian equation (1.1) with such defined nonlinearity $F$. As in \[NV3\], \[NV4\] it follows that $w$ is a viscosity solution of (1.1) in the whole space ${ \R}^n$. In \[NV3\] we have shown that the equation (1.1) for the function $w$ can be rewritten in the form of the Isaacs equation. The lemma is proved. We will apply this result to the function $w_{5,\delta}(x)=P_{5} (x)/ |x|^{1+\delta } $. Let then recall some facts from \[NTV\] about the Cartan cubic form $P_5(x)$. [**Lemma 2.2.**]{} [*The form $P_{5}(x)$ admits a three-dimensional automorphism group.*]{} Indeed, one easily verifies that the orthogonal trasformations $$A_1(\phi):= \left(% \begin{array}{ccccc} {3\cos(\phi)^2-1\over 2}& { \sqrt 3\sin(\phi)^2\over 2}&0&0&{ \sqrt 3\sin(2\phi)\over 2} \\ { \sqrt 3\sin(\phi)^2\over 2}&{1+\cos(\phi)^2\over 2}&0&\;0&{-\sin(2\phi)\over 2} \\ 0& 0&\cos(\phi)&\sin(\phi)&0 \\ 0&0&-\sin(\phi)&\cos(\phi)&0\\ {- \sqrt 3\sin(2\phi)\over 2}&{\sin(2\phi)\over 2} &0&0&\cos(2\phi)\\ \end{array}% \right)$$ $$A_2(\psi):= \left(% \begin{array}{ccccc} 1& 0& 0& 0& 0\\ 0&\cos(2\psi)& 0& -\sin(2\psi)& 0\\ 0&0& \cos(\psi)&0&-\sin(\psi)\\ 0&\sin(2\psi)& 0&\cos(2\psi)& 0\\ 0&0&\sin(\psi)& 0& \cos(\psi)\\ \end{array}% \right)$$ $$A_3(\theta):= \left(% \begin{array}{ccccc} {3\cos(\theta)^2-1\over 2}& {-\sqrt 3\sin(\theta)^2\over 2}&0& 0& {-\sqrt 3\sin(2\theta)\over 2} \\ {-\sqrt 3\sin(\theta)^2\over 2}& {1+\cos(\theta)^2\over 2}& 0& 0& {-\sin(2\theta)\over 2}\\ 0&0&\cos(\theta)&-\sin(\theta)&0\\ 0& 0& \sin(\theta)&\cos(\theta)& 0\\ {\sqrt 3\sin(2\theta)\over 2}& {\sin(2\theta)\over 2}& 0& 0&\cos(2\theta)\\ \end{array}% \right)$$ do not change the value of $P_5(x)$. [**Lemma 2.3.**]{} *Let $G_P$ be subgroup of $SO(5)$ generated by* {$A_1(\phi),A_2(\psi), A_3(\theta):\:(\phi,\psi,\theta)\in { \R}^3\}.$ Then the orbit $G_PS^1$ of the circle $$S^1=\{(\cos(\chi),0,\sin(\chi),0,0):\chi\in { \R}\}\subset S^4$$ under the natural action of $G_P$ is the whole $S^4 . $ We need also the following two simple algebraic results ( \[NV3, Lemmas 2.2 and 4.1\]): [**Lemma 2.4.**]{} [*Let $ A,B$ be two real symmetric matrices with the eigenvalues $ \lambda_1\ge\lambda_2\ge\ldots\ge\lambda_{n} $ and $ \lambda'_1\ge\lambda'_2\ge\ldots\ge\lambda'_{n} $ respectively. Then for the eigenvalues $ \Lambda_1\ge\Lambda_2\ge\ldots\ge\Lambda_{n} $ of the matrix $A-B$ we have*]{} $$\Lambda_1\ge\max_{i=1,\cdots, n}(\lambda_i-\lambda'_i), \;\;\Lambda_n\le\min_{i=1,\cdots, n}(\lambda_i-\lambda'_i).$$ [**Lemma 2.5.** ]{}[*Let $\delta\in[0,1),\; W,\;\bar W\in { \R}$ with $|W|\le \frac{1}{3\sqrt 3},|\bar W|\le \frac{1}{3\sqrt 3}$ and let $ \mu_1(\delta)\ge \mu_2(\delta)\ge \mu_3(\delta)$ $($resp., $\bar\mu_1(\delta)\ge\bar\mu_2(\delta)\ge\bar\mu_3(\delta)\;)$ be the roots of the polynomial $$T^3+3W(1+\delta) T^2+(3W^2(1+\delta)^2-1)T+W(1-\delta) +W^3(1+\delta)^3$$ $($resp. of the polynomial $$T^3+3\bar W (1+\delta) T^2+(3 \bar W^2(1+\delta)^2-1)T+\bar W(1-\delta)+ \bar W^3(1+\delta)^3\; ).$$ Then for any $K>0$ verifying $ |K-1|+|\bar W-W|\neq 0$ one has $${1-\delta\over 5+\delta}=:\rho\le { \mu_+(K)\over -\mu_-(K)}\le {1\over\rho}= {5+\delta\over 1-\delta}$$ where $$\mu_-(K):= \min\{\mu_1(\delta)-K\bar\mu_1(\delta),\; \mu_2(\delta)-K\bar\mu_2(\delta),\; \mu_3(\delta)-K\bar\mu_3(\delta)\} ,$$ $$\mu_+(K):= \max\{\mu_1(\delta)-K\bar\mu_1(\delta),\; \mu_2(\delta)-K\bar\mu_2(\delta),\; \mu_3(\delta)-K\bar\mu_3(\delta)\}\;.$$* ]{} Proofs ====== Let $w_{5,\delta}=P_5/|x|^{1+\delta}, \: \delta\in [0,1[$. By Lemma 2.1 it is sufficient to prove the existence of a smooth function $g$ verifying the conditions (2.1) and (2.2). We beging with calculating the eigenvalues of $D^2w_{5,\delta}(x)$. More precisely, we need [**Lemma 3.1.**]{} *Let $x\in S^4$, and let $x\in G_P(p,0,q,0,0)$ with $p^2+q^2=1$. Then $$Spec(D^2w_{5,\delta}(x))= \{\mu_{1,\delta},\mu_{2,\delta},\mu_{3,\delta},\mu_{4,\delta},\mu_{5,\delta} \}$$ for $$\mu_{1,\delta}= \frac{p(p^2\delta+6-3\delta)}{2},$$ $$\mu_{2,\delta}= \frac{p(p^2\delta-3-3\delta)+3\sqrt{12-3p^2}}{2},$$ $$\mu_{3,\delta}=\frac{p(p^2\delta-3-3\delta)-3\sqrt{12-3p^2}}{2},$$ $$\mu_{4,\delta}= -\frac{p\delta(6-\delta)(3-p^2)+\sqrt{D(p,\delta)}}{4},$$ $$\mu_{5,\delta}= -\frac{p\delta(6-\delta)(3-p^2)-\sqrt{D(p,\delta)}}{4} ,$$ and $$D(p,\delta):=(6-\delta)(4-\delta)(2-\delta)\delta(p^2-3)^2p^2+144(\delta-2)^2>0.$$* The characteristic polynomial $F(S)$ of $D^2w$ is given by $$F(S)=S^5+a_{1,\delta} S^4+a_{2,\delta}S^3+a_{3,\delta}S^2+a_{4,\delta} S+a_{5,\delta}$$ for $$a_{1,\delta}= \frac{(\delta+1)(\delta-8)b}{2},$$ $$a_{2,\delta}= \frac{(\delta+1)(21\delta+13-4\delta^2)b^2}{4}+9(2\delta-\delta^2-4),$$ $$a_{3,\delta}= \frac{(6\delta^2-31\delta-1)(\delta+1)^2b^3}{8}+ \frac{27(4\delta-2\delta^2+5+\delta^3)}{2},$$ $$a_{4,\delta}= \frac{(2\delta-1)(5-\delta)(\delta+1)^2b^4}{8}+ \frac{9(\delta-1)(\delta^2-2\delta+9)}{2},$$ $$a_{5,\delta}= \frac{b(1-\delta)\left(b^2(\delta+1)^3+ 108(1-\delta)\right)\left(b^2(\delta+1)(\delta-5)+36(\delta-1)\right)}{32},$$ $$where \quad b:=p(p^2-3).$$ Note that the spectrum in this lemma is unordered one. [*Proof of Lemma 3.1.*]{} Since $w_{5,\delta}$ is invariant under $G_P$, we can suppose that $x=(p,0,q,0,0)$. Then $w_{5,\delta}(x)=\frac{p(3-p^2)}{2}$ and we get by a brute force calculation: $$D^2w_{5,\delta}(x):= \left(% \begin{array}{cc} M_{1,\delta}&0\\ 0&M_{2,\delta} \\ \end{array}% \right)$$ being a block matrix with $$M_{1,\delta}:= \frac{ 1}{2}\left(% \begin{array}{ccc} m_{1,1}& {m_{1,2}}& {m_{1,3}} \\ {m_{1,2}}& {m_{2,2}}& {m_{2,3}}\\ {m_{1,3}}& {m_{2,3}}& { m_{3,3}} \\ \end{array}% \right), \;$$ $m_{1,1}:=-(\delta+2)\delta p^5+(\delta+3)\delta p^3+(12-9\delta)p,$ $m_{1,2}:= 3\sqrt 3 p(p^2-1)\delta,$ $ m_{1,3}:= -q\left((\delta+2)\delta p^4+3\delta(1-\delta)p^2+3\delta-6)\right)$ $ m_{2,2}:=\delta p^3-3(\delta+4)p,$ $ m_{2,3}:=3\sqrt 3q(\delta p^2+2-\delta), $ $m_{3,3}:=(\delta+2)\delta p^5+(5-4\delta)\delta p^3-3(\delta-1)(2-\delta)p,$ $$M_{2,\delta}:=\frac{ 1}{2} \left(% \begin{array}{cc} { \delta p^3+3(2-\delta)p}& 6\sqrt 3 q \\ 6 \sqrt 3 q & { \delta p^3-3(4+\delta)p}\\ \end{array}% \right)$$ which gives for the characteristic polynomial $F(S) =F_1(S)\cdot F_2(S)\cdot F_3(S)$ where $$F_1(S):=S-\frac{p(p^2\delta+6-3\delta)}{2};$$ $$F_2(S)=S^2+\frac{\delta p (p^2-3) (\delta-6) S}{2}+\frac{(2-\delta)\left( (\delta-6)\delta p^6+6(6-\delta)\delta^2 p^4+9(\delta^2-6\delta) p^2+36(\delta-2)\right)}{4};$$ $$F_3(S) := S^2+(3 +3\delta-\delta p^2) p S+ \frac{ (p^2-3) (\delta^2 p^4-3\delta^2 p^2-6\delta p^2+36)}{4};$$ and the spectrum. Developing $F(S)$ we get the last formulas. This follows immediately from Lemma 3.1 and a simple calculation since $$\Delta(w)=-a_{1,\delta},\;S_2(w)= a_{2,\delta},\: S_4(w)= a_{4,\delta},\:\det(D^2w)=-a_{5,\delta}.$$ Let then determine the ordered spectrum $\{\lambda_1,\lambda_2,\ldots,\lambda_5 \}, \; $ $\linebreak\lambda_1\ge\lambda_2\ge\ldots\ge\lambda_5 $ of $D^2w$. *Let $\lambda_1\ge\lambda_2\ge\ldots\ge\lambda_5$ be the eigenvalues of $D^2w_{5,\delta}(x).$ Then* $$\lambda_1= \mu_{2,\delta}, \quad \lambda_5= \mu_{3,\delta},\quad\quad\quad\quad$$ $$\lambda_2 = \begin{cases} \mu_{4,\delta} & \text{for}\; p\in [-1,p_0(\delta)],\cr \mu_{1,\delta} & \text{for}\; p\in [p_0(\delta),1], \end{cases}$$ $$\lambda_3 = \begin{cases} \mu_{5,\delta} & \text{for}\; p\in [-1,-p_0(\delta)],\cr \mu_{1,\delta} & \text{for}\; p\in [-p_0(\delta),p_0(\delta)], \cr \mu_{4,\delta} & \text{for}\; p\in [ p_0(\delta),1], \end{cases}$$ $$\lambda_4 = \begin{cases} \mu_{1,\delta} & \text{for}\; p\in [-1,-p_0(\delta)],\cr\mu_{5,\delta} & \text{for}\; p\in [-p_0(\delta),1],\cr \end{cases}$$ where $$p_0(\delta):=\frac{3^{1/4}\sqrt{1-\delta}}{(3+2\delta-\delta^2)^{1/4}}=\frac{3^{1/4}\sqrt{ \varepsilon}}{(4 - \varepsilon^2)^{1/4}}\in ]0,1].$$ [*Proof.*]{} The inequalities $\mu_{2,\delta}(p)\ge\mu_{1,\delta}(p)\ge\mu_{3,\delta}(p)$ are obvious since $\mu_{2,\delta}(p)$ and $\mu_{3,\delta}(p)$ are decreasing in $p$, $\mu_{1,\delta}(p)$ is increasing in $p$, $\mu_{3,\delta}(-1)=\mu_{1,\delta}(-1),$ $\mu_{2,\delta}(1)=\mu_{1,\delta}(1).$ The resultant $$R(\delta,p)= Res(F_2, F_3)= 144(p-1)^2(p+1)^2\left(r_8p^8-r_6p^6+ r_4p^4-r_2p^2 +r_0 \right)$$ where $$r_8=(\varepsilon^2-4)^2,r_6=12 (\varepsilon^2-4)^2,r_4=3 (4-\varepsilon^2)(72-17\varepsilon^2 ),$$ $$r_2=108 (\varepsilon^2-4)^2 ,r_0=144(3-\varepsilon^2)^2$$ is strictly positive for $(\varepsilon,p)\in ]0,1[\times ]-1,1[$. Indeed, let $$r:=\frac{ R}{144(p-1)^2(p+1)^2}=r_8p^8-r_6p^6+ r_4p^4-r_2p^2 +r_0$$ then $$d:=\frac{\partial r}{4\varepsilon\partial \varepsilon}=(\varepsilon^2-4)p^8 +12p^6(4-\varepsilon^2)+3(17\varepsilon^2-70)p^4+ 108(4-\varepsilon^2)p^2 +144(\varepsilon^2-3)<0$$ for $(\varepsilon,p)\in ]0,1[\times [0,1[$ since $$\frac{\partial d}{ 4p\partial p}=(4-\varepsilon^2)(-2p^6+18p^4-51p^2+54)-6p^2\ge (4-\varepsilon^2)\cdot 19-6\ge 51,$$ and for $p=1$ one has $d=-166+76\varepsilon^2\le -90.$ For $\delta=0, \varepsilon=1$ we get $$R(\delta,p)\ge R(1,p)=9(1-p^2)(4-p^2))(p^4-7p^2+16)$$ which proves the positivity. Using then the inequalities $$\mu_{2,\delta}(-1)=\mu_{4,\delta}(-1)>\mu_{5,\delta}(-1)>\mu_{3,\delta}(-1),$$ $$\mu_{2,\delta}(1)>\mu_{4,\delta}(1)>\mu_{5,\delta}(-1)=\mu_{3,\delta}(-1)$$ and the postivity of the resultant we get $$\mu_{2,\delta}(p)\ge\mu_{4,\delta}(p)\ge\mu_{5,\delta}(p)\ge\mu_{3,\delta}(p)$$ for any $p\in[-1,1].$ Calculatig then $$R_1(\delta,p)= Res(F_1, F_3)=12(p^2-3) \left( p^4(\varepsilon^2-4)+3\varepsilon^2\right)$$ and taking into account the equalities $$\mu_{4,\delta}(p_0(\delta))=\mu_{1,\delta}(p_0(\delta)),\;\mu_{5,\delta}(-p_0(\delta))=\mu_{1,\delta}(-p_0(\delta))$$ we get the result. Note the oddness property of the spectrum: $$\lambda_{1,\delta}(-p)=-\lambda_{5,\delta}(p),\;\lambda_{2,\delta}(-p)=-\lambda_{4,\delta}(p),\;\lambda_{3,\delta}(-p)=-\lambda_{3,\delta}(p).$$ Let us now verify the second condition (2.2) of Lemma 2.1, namely the uniform hyperbolicity of $M_{\delta}(a,b,O)$. [*Proof.*]{} The proof depends on the value of $$k:=p_0(\delta):=\frac{3^{1/4}\sqrt{1-\delta}}{(3+2\delta-\delta^2)^{1/4}}=\frac{3^{1/4}\sqrt{ \varepsilon}}{(4 - \varepsilon^2)^{1/4}}.$$ Note that $C=\frac{1000(\delta+1)(3-\delta)}{3(1-\delta)^2}= \frac{1000}{k^4}$. We shall give the proof for $k\in ]0,\frac{1}{2}]$, the proof for $k\in [\frac{1}{2},1]$ is similar, simpler and uses $C=10^4$. Suppose that the conclusion does not hold, that is for some $a\neq b $ and some $O\in {\hbox {O}}({5} )$ one has $$M_\delta(a,b,O):=M_\delta(a)- {^tO}\cdot M_\delta(b)\cdot O\neq 0,$$ but $${1\over C}> - {\Lambda_1\over \Lambda_{5}}\;\text{\rm or} - {\Lambda_1\over \Lambda_{5}} >C .$$ We can suppose without loss that $|b|\le 1=|a|\in \S^4_1.$ Let $\overline{b}:=b/|b|\in \S^4_1,$ $W:=W(a),\overline W:=W(\overline b), K:= |b| ^{-1-\delta}.$ Note that since for any harmonic cubic polynomial $Q(x)$ on ${\R}^n$ and any $a\in \S_1^{n-1}\subset{\R}^n$ one has $$Tr\left (D^2\left ({Q(x)\over |x|^{1+\delta}}\right )(a)\right)= (\delta^2-2\delta-3-n) Q(a),$$ we get $Tr\left(M_{\delta}(a,b,O)\right)=(2\delta+8-\delta^2)(K\overline W-W),\; P_5 $ being harmonic. Let us prove the claim for $(K\overline W-W)\ge 0$, the proof for $(K\overline W-W)\le 0$ being the same while permuting $a$ with $b$ and $\Lambda_{1}$ with $\Lambda_{5}$. Since $$Tr\left(M_{\delta}(a,b,O)\right)=(2\delta+8-\delta^2)(K\overline W -W)\ge 0,$$ we get $ 4\Lambda_{1}+\Lambda_{5}\ge 0$ and $-\Lambda_{5}/\Lambda_{1}\le 4$. Therefore, we have only to rule out the inequality $ {1\over C}> - {\Lambda_1\over\Lambda_{5}}$ i.e. $ -\Lambda_{5}> C {\Lambda_1}.$ Recall that $$W=\frac{3p-p^3}{2},\overline W=\frac{3\overline p-\overline p^3}{2}$$ for some $p, \overline p\in [-1,1].$ We have then 3 possibilities: 1). $p,\overline p \in [-k,k]$; 2). $p \in [-k,k], \overline p \notin [-k,k]$; 3). $p,\overline p \notin [-k,k]$. In the cases 1) and 3) applying Lemma 2.4 we get $\Lambda_1\ge \mu_+(K)$, $\Lambda_{5}\le \mu_-(K)$ in the notation of Lemma 2.5 which permits to finish the proof as in Proposition 4.1 of \[NV3\]. We thus have to treat the (more difficult) case 2). Lemma 2.4 together with the inequality $ -\Lambda_{5}> C {\Lambda_1}$ gives $$-\min_{i=1,\cdots,5}\{K\lambda_{i,\delta}(\overline p )-\lambda_{i,\delta}( p )\}>C\max_{i=1,\cdots,5}\{K\lambda_{i,\delta}(\overline p )-\lambda_{i,\delta}( p )\}.$$ Thank to the oddness of the spectrum we suppose without loss that $\overline p >k$. Recall that then by Lemma 3.2 one has $$\lambda_{1,\delta}(\overline p )=\mu_{2,\delta}(\overline p ),\lambda_{1,\delta}( p )=\mu_{2,\delta}(p ), \;\lambda_{2,\delta}(\overline p )=\mu_{1,\delta}(\overline p ),\lambda_{1,\delta}( p )=\mu_{4,\delta}(p ),$$ $$\lambda_{3,\delta}(\overline p )=\mu_{4,\delta}(\overline p ),\lambda_{3,\delta}( p )=\mu_{1,\delta}(p ), \;\lambda_{4,\delta}(\overline p )=\mu_{5,\delta}(\overline p ),\;\lambda_{4,\delta}( p )=\mu_{5,\delta}(p ),$$ $$\lambda_{5,\delta}(\overline p )=\mu_{3,\delta}(\overline p ),\;\lambda_{5,\delta}( p )=\mu_{3,\delta}(p ).$$ We have then 2 possibilities for $p:$ 2a). $p \in [-k,0]$; 2b). $p \in ]0,k]$. Let $p \in [-k,0]$, then $\mu_{1,\delta}(p )\le 0$ and thus $$C\max_{i=1,\cdots,5}\{K\lambda_{i,\delta}(\overline p )-\lambda_{i,\delta}( p )\}\ge CK\lambda_{3,\delta}(\overline p ) =CK\mu_{4,\delta}(\overline p )\ge CK\mu_{4,\delta}(p_0(\delta) )=$$ $$=CK\mu_{1,\delta}(k) = CK(k^3(\sqrt {k^4+3}-k^2+3)/\sqrt {k^4+3})\ge 2CKk^{3}$$ since one verifies that the function $\mu_{4,\delta}(p)$ is increasing on $[k,1].$ On the other hand, $$|\min_{i=1,\cdots,5}\{K\lambda_{i,\delta}(\overline p )-\lambda_{i,\delta}( p )\}|\le K\max_{i=1,\cdots,5, p}|\{\lambda_{i,\delta}( p )\}|+\max_{i=1,\cdots,5, p}|\{\lambda_{i,\delta}( p )\}|\le 8(K+1).$$ Therefore one gets $8(K+1)\ge 2CKk^{3}$ which clearly is a contradiction for, say, $K\ge 1/4$. For $0<K\le 1/4$ we get $$C\max_{i=1,\cdots,5}\{K\lambda_{i,\delta}(\overline p )-\lambda_{i,\delta}( p )\}\ge C(K\lambda_{5,\delta}(\overline p )-\lambda_{5,\delta}( p ))\ge C(K(-8)-(-5))\ge 3C$$ which can not be less than $ 8(K+1)\le 10$. Let finally $p \in ]0,k]$. We consider then 2 possibilities for $K$: $(i)$ $K \le 20/31=(1.55)^{-1} $, $(ii)$ $K>20/31=(1.55)^{-1}$. In the case $(i)$ one has $$C\max_{i=1,\cdots,5}\{K\lambda_{i,\delta}(\overline p )-\lambda_{i,\delta}( p )\}\ge C(K\lambda_{5,\delta}(\overline p )-\lambda_{5,\delta}( p ))\ge$$ $$\ge C(K\mu_{3,\delta}(1)-\mu_{3,\delta}( 0 ))\ge C(3\sqrt 3+20(\varepsilon-8)/31)>C/30>8(K+1)$$ since $\lambda_{5,\delta}( p )=\mu_{3,\delta}(p)$ is decreasing, $\mu_{3,\delta}(0)=-3\sqrt 3,\mu_{3,\delta}(1)=\varepsilon-8.$ We suppose then $K>20/31=(1.55)^{-1}$. Then if $p\le 3k/4$ one has $$C\max_{i=1,\cdots,5}\{K\lambda_{i,\delta}(\overline p )-\lambda_{i,\delta}( p )\}\ge CK(\lambda_{3,\delta}(\overline p )-\lambda_{3,\delta}( p )/K) \ge CK(\mu_{4,\delta}(\overline p)-\mu_{1,\delta}( p )/K)\ge$$ $$\ge CK(\mu_{4,\delta}(k)-\mu_{1,\delta}( 3k/4 )/K)=CK(\mu_{1,\delta}(k)-\mu_{1,\delta}( 3k/4 )/K)>$$ $$> CK\mu_{1,\delta}\left( \frac{3k}{4}\right)\left(\frac{\mu_{4,\delta}(k))}{\mu_{1,\delta}\left( \frac{3k}{4}\right)}-\frac{31}{20}\right)\ge$$ $$\ge CK\frac{\mu_{1,\delta}\left( \frac{3k}{4}\right)}{ 100} \ge CK \frac{2k^3}{100 }>20K > 8(K+1),$$ contradiction for $k\in [0,1/2]$ since $\frac{\mu_{4,\delta}(k))}{\mu_{1,\delta}\left( \frac{3k}{4}\right)}>1.56,\mu_{1,\delta}\left( \frac{3k}{4}\right)>2k^3$ there. Thus we can suppose that $p\in ]\frac{3k}{4}, k] .$ One notes then that $\mu_{4,\delta}(p)\le \mu_{4,\delta}(k)$ for$p\in [\frac{k}{4},1].$ This permits to rule out the case $\overline p\ge \frac{3k}{2}$. Indeed, one has in this case $$C\max_{i=1,\cdots,5}\{K\lambda_{i,\delta}(\overline p )-\lambda_{i,\delta}( p )\}\ge CK(\lambda_{2,\delta}(\overline p )-\lambda_{2,\delta}( p )/K) \ge CK(\mu_{1,\delta}(\overline p)-\mu_{4,\delta}( p )/K)\ge$$ $$\ge CK(\mu_{1,\delta}(3k/2)-\mu_{1,\delta}( k)/K)=CK(\mu_{1,\delta}(3k/2)-\mu_{1,\delta}( k)/K),$$ and one gets a contradiction as above since $$\frac{\mu_{1,\delta}\left( \frac{3k}{2}\right)}{\mu_{1,\delta}(k)} >2$$ for $k\in [0,1/2]$. The last case to rule out is thus $ K\ge 20/31, p\in [\frac{3k}{4},k], \overline p \in [k,\frac{3k}{2}]$. Let then $$\alpha:=k-p\in [0,\frac{k}{4}] \subset [0,\frac{1}{8}],\;\beta:=\overline p -k\in [0,\frac{k}{2}]\subset [0,\frac{1}{4}],\; a:=\max \{\alpha,\beta\}.$$ It is easy to verify that on $\left[\frac{3k}{4},\frac{3k}{2}\right]$ one has the following inequalities: $$\frac{\partial \mu_{1,\delta}(p)}{\partial p}\ge 3k^2, \; \frac{11k^3 }{4}\ge \mu_{1,\delta}(k)\ge \frac{5k^3 }{2}\:;$$ $$\frac{\partial \mu_{2,\delta}(p)}{\partial p}\ge -5, \;4k-5 \ge\mu_{2,\delta}(k)\ge 4k-\frac{11}{2} \:;$$ $$\frac{\partial \mu_{3,\delta}(p)}{\partial p}\ge -\frac{ 9}{2}, \; -5-3k\ge\mu_{3,\delta}(k)\ge -\frac{11+7k }{2}\:;$$ $$\frac{\partial \mu_{4,\delta}(p)}{\partial p}\ge -\frac{k }{29}, \; \frac{11k^3 }{4}\ge \mu_{4,\delta}(k)\ge \frac{5k^3 }{2}\:;$$ $$\frac{\partial \mu_{5,\delta}(p)}{\partial p}\ge 10k^2-12, \; -10k\ge\mu_{5,\delta}(k)\ge -12k \:.$$ Let then $ K\in \left[\frac{20}{31},1\right].$ Therefore, $$C\max_{i=1,\cdots,5}\{K\lambda_{i,\delta}(\overline p )-\lambda_{i,\delta}( p )\}\ge$$ $$\ge C\max \{K\mu_{1,\delta}(k+\alpha )-\mu_{1,\delta}( k-\beta), K\mu_{3,\delta}(k+\alpha ) -\mu_{3,\delta}( k-\beta)\}\ge\max\left\{ M_1,M_2\right\}=$$ $$=C\max\left\{3(K-1)k^3a+3(K+1)k^2, (1-K) (5+3k)a-\frac{ (11+7k ) (K+1)}{10}\right\}$$ for linear forms $M_1,M_2$ in $K$. Note that the minimal value of $\max\{ M_1,M_2\}$ as a function of $K$ equals (recall that our $C=1000/k^4$): $$\frac{1500a(40-9k)}{k^2(12k^2a+18a+11k^3+20+12k)}>\frac{1250a}{k^2}>0$$ attained for $K=K_0:= (11k^3+20+12k)/(12k^2a+18a+11k^3+20+12k)<1$. On the other hand, $$-\Lambda_5\le -\min_{i=1,\cdots,5}\{K\lambda_{i,\delta}(\overline p )-\lambda_{i,\delta}( p )\}\le \max\{ l_1,l_2,l_3,l_4,l_5\}$$ for the following linear forms (in $K$) $$l_1:=-k^2\left(\frac{11a}{4}+3k\right)K-\frac{11a}{4}k^2+3k^3,$$ $$l_2:=\left(5a+4k-\frac{11}{2}\right)K+5a+\frac{11}{2}-4k,$$ $$l_3:= (5a+5+3k)K+5a-5-3k,$$ $$l_4:=\left(\frac{ak}{29}-\frac{11k^3}{2}\right)K+\frac{ak}{29}+\frac{11k^3}{2},$$ $$l_5:=((12-10k^2)a+12k)K+(12-10k^2)a-12k.$$ To refute our inequality it is sufficient to prove that $M_i(K_{j,k})> 0$ for any triple $(i,j,k)$ with $i,j\in \{1,2\},\; i\neq j, \; k\in \{1,2,3,4,5\}$ where $ l_k(K_{j,k})=M_j(K_{j,k}).$ Explicit calculations give (for the values $m_{ijk}:=\frac{M_i(K_{j,k})}{500ak^2}$) $$\frac{m_{121}}{3} = \frac{(9k^4+6k^6)a+10000+5k^7+20k^4+3k^5-2250k}{k^2((3k^4+3000)a+3k^5+2750k)}>0,$$ $$\frac{m_{211}}{3} = \frac{(9k^4+6k^6)a+10000+5k^7+20k^4+3k^5-2250k}{(4500-3k^6)a+5000+3000k-3k^7}>0,$$ $$m_{122} = \frac{60000-(60k^4+90k^2)a-192k^3+66k^4-13500k-101k^2-158k^5}{k^2((6000-10k^2)a-11k^2+8k^3+5500k)}>0,$$ $$m_{212} = \frac{60000-(60k^4+90k^2)a-192k^3+66k^4-13500k-101k^2-158k^5}{(10k^4+9000)a+11k^4-8k^5+10000+6000k}>0,$$ $$m_{123} = \frac{30000-(45k^2+30k^4)a-55k^2-6750k-33k^3+30k^4-37k^5} {k^2((3000-5k^2)a+2750k-5k^2-3k^3)} >0,$$ $$m_{213} = \frac{30000-(45k^2+30k^4)a-55k^2-6750k-33k^3+30k^4-37k^5} {(5k^4+4500)a+5k^4+3k^5+5000+3000k} >0,$$ $$m_{124} = \frac{3480000-(36k^3+24k^5)a-t(k)} {k^2((348000-4k^3)a+319000k+319k^5)} >0,$$ $$m_{214} = \frac{3480000-(36k^3+24k^5)a-t(k)} {(522000+4k^5)a+580000+348000k-319k^7} >0,$$ $$m_{215} = \frac{(9k^4+30k^6-54k^2)a+15000-u(k)} {5k^4a+1500a-6k^3+1375k-6k^2a} >0,$$ $$m_{125} = \frac{(9k^4+30k^6-54k^2)a+15000-u(k)} {(2250+6k^4-5k^6)a+2500+1500k+6k^5 } >0,$$ where $t(k):=783000k+ 80k^3+48k^4+44k^6+2871k^5+1914k^7<4\cdot 10^5$, $u(k):=120k^2-30k^5+3375k+18k^3- 100k^4-55k^7<2000$. Let, finally $K\ge 1$, then $$C\Lambda_1\ge C(K\mu_{1,\delta}(k+\alpha )-\mu_{1,\delta}( k-\beta))\ge \frac{5C }{2}(K-1)k^3a+3(K+1)Ck^2=$$ $$=L_1:= \frac{500(5kK+6a-5k)}{k^2}=\frac{2500(K-1)}{k}+\frac{3000a}{k^2},$$ and $$-\Lambda_5\le -\min_{i=2,\cdots,5}\{K\lambda_{i,\delta}(\overline p )-\lambda_{i,\delta}( p )\}\le\max\left\{L_2, L_3,L_4,L_5\right\}$$ for the following linears forms in $K$: $$L_2:=5(K+1)a+(1-K)(5-4k) = (5a-5+4k)(K-1)+10a,$$ $$L_3:=\frac{9}{2}(K+1)a+\frac{11+7k}{2}(K-1) =\frac{9a+11+7k}{2} (K-1)+ 9a,$$ $$L_4:=(K+1)a\frac{k}{29}-\frac{5k^3}{2}(K-1)=\left(\frac{ak}{29}-\frac{5k^3}{2}\right)(K-1)+\frac{2ak}{29},$$ $$L_5:=12k(K+1)a+\frac{9}{2}(K-1)=3\left(4ak+\frac{3}{2}\right)(K-1)+24ak.$$ One immediately sees that both the slope and the value at $K=1$ of $L_1$ are (much) bigger than those of $L_i, i=2,3,4,5$ which finishes the proof. [*Proof of Theorem 1.1.*]{} To prove the result it is sufficient to verify the condition (2.1) in Lemma 2.1, namely, that the five partial derivatives $\frac{\partial g}{\partial \lambda_i}, i=1,\ldots,5$ are strictly positive (and bounded which is automatic thank to compacity) on the symmetrized image $$M:=\bigcup_{\sigma \in \Sigma_{n} }\ T_{\sigma }\Lambda (B)\subset { \R}^n$$ of the unit ball under the map $\Lambda$, $$g(\lambda_1,\ldots,\lambda_5)=det(D^2w)- e_5(\Delta(w))^5 -e_3(\Delta(w))^3 S_2(w)-e_1\Delta(w) S_4(w)$$ being our equation. By homogeneity it is sufficient to show this on $M':=\Lambda (\S_1^4)$ which is an algebraic curve, the union of 120 curves $T_{\sigma }\Lambda (\S_1^4)$ and that it is sufficient, by symmetry, to verify the condition on the curve $\Lambda (\S_1^4)$ only. A brute force calculation shows then that $$g_1(p,\varepsilon ):=\frac{\partial g}{\partial \lambda_1}=\sum_{i=0}^{12} m_{i}p^{i} =$$ $$=m_{12}p^8(p^4-12p^2+54)+m_9 b^3+m_6p^4(p^2-\frac{3}{4})+m_2+m_0,$$ with $$m_{12} =3(\varepsilon^4+3\varepsilon^3-20\varepsilon^2 +12\varepsilon-56)(\varepsilon-2)^2(\varepsilon+2)^2, m_{10} =-12m_{12}, m_{8} =54m_{12},$$ $$m_{11}=m_{1}=0, m_9=3D(p,\varepsilon)(\varepsilon+7)(\varepsilon+2)(\varepsilon^2+2)(\varepsilon-2)^2,m_7=-9m_9,m_5=27m_9,$$ $$m_6= 108(2-\varepsilon)(3\varepsilon^7+17\varepsilon^6-54\varepsilon^5 -152\varepsilon^4+72\varepsilon^3-42\varepsilon^2+ 384\varepsilon+1344), m_3=27m_9,$$ $$m_4=-\frac{3m_6}{4},\;\;m_2= 1944 \varepsilon^2 (2-\varepsilon) (\varepsilon^2 - 7) (\varepsilon^2 +3),\;\; m_0 = -7776 \varepsilon^2 (\varepsilon^2 + 3)$$ for $D(p,\varepsilon):=\sqrt{(16-\varepsilon^2)(4-\varepsilon^2) b^2+144\varepsilon^2}, \; b=(p^2-3)p;$ $$g_2(p,\varepsilon ):=\frac{\partial g}{\partial \lambda_2}=\sum_{i=0}^{12} n_{i}p^{i}$$ with $$n_{12} =(\varepsilon+4)(\varepsilon+1)(4-\varepsilon^2)^2 , n_{10} =-(\varepsilon+2)(\varepsilon^4+19\varepsilon^3+86\varepsilon^2+182\varepsilon+96)(2-\varepsilon)^2,$$ $$n_{11}=n_{1}=0, \;\; n_{8} =9(\varepsilon+2)(\varepsilon^2+10\varepsilon+6)(\varepsilon^2+3\varepsilon+8)(2-\varepsilon)^2,$$ $$n_9=\varepsilon(\varepsilon+7)(\varepsilon+2)(\varepsilon^2+2)(2-\varepsilon)^2 \sqrt{3(4-p^2)},\; n_7=-9n_9,\; n_5=27n_9,$$ $$n_{6} =3(2-\varepsilon)(13\varepsilon^6+115\varepsilon^5+218\varepsilon^4+ 170\varepsilon^3-876\varepsilon^2-2856\varepsilon-1152),$$ $$n_4=9(\varepsilon-2)(11\varepsilon^6+62\varepsilon^5+33\varepsilon^4 -24\varepsilon^3-348\varepsilon^2-1176\varepsilon-288),$$ $$n_3=\varepsilon^3(2-\varepsilon)(\varepsilon+7)(3\varepsilon^2+2)\sqrt{3(4-p^2)},$$ $$n_2=108\varepsilon^2(2-\varepsilon) (\varepsilon^2+3)(\varepsilon^2+4\varepsilon-3), n_0=1296\varepsilon^2(\varepsilon^2+3);$$ $$g_3(p,\varepsilon ):= \frac{\partial g}{\partial \lambda_3}=\sum_{i=0}^{6} h_{2i}p^{2i}$$ with $$h_{12} = (\varepsilon + 4) (\varepsilon + 1) (4-\varepsilon^2) ^2, h_{10} = 2 (\varepsilon + 2)(\varepsilon^4 + \varepsilon^3 - 40 \varepsilon^2 - 70\varepsilon - 48)(2-\varepsilon )^2$$ $$h_{8} = -18 (\varepsilon + 2)(\varepsilon^4 + 4\varepsilon^3 - 19\varepsilon^2 - 28\varepsilon - 24)(2-\varepsilon )^2,$$ $$h_6 = 6 (\varepsilon - 2) (7\varepsilon^6 + 37\varepsilon^5 - 136 \varepsilon^4 - 274 \varepsilon^3 + 330\varepsilon^2 + 672\varepsilon + 576),$$ $$h_4 = 9 (\varepsilon - 2) (2 \varepsilon^6 - \varepsilon^5 + 27\varepsilon^4- 66\varepsilon^3 - 348\varepsilon^2 - 1176\varepsilon - 288),$$ $$h_2=108\varepsilon(2-\varepsilon)(\varepsilon^3+4\varepsilon^2- 15\varepsilon-84) (\varepsilon^2+3),$$ $$h_0 = -1296\varepsilon(\varepsilon^2 + 3) (\varepsilon^2 + 4\varepsilon - 14);$$ $$g_4(p,\varepsilon ):=\frac{\partial g}{\partial \lambda_4}=g_1(-p,\varepsilon ),$$ $$g_5(p),\varepsilon :=\frac{\partial g}{\partial \lambda_5}=g_2(-p,\varepsilon ),$$ and thus we need to consider only the functions $ g_1(p,\varepsilon ), g_2(p,\varepsilon ), g_3(p,\varepsilon )$ on the set $[-1,1]\times(0,1].$ We have to prove that for any fixed $\varepsilon\in (0,1]$ they are strictly positive. The technique of the proof is identical for all three derivatives, and we begin with $g_3$ wich is slightly simpler since it is a polynomial in two variables. One can rearrange it in the form $$g_3(p,\varepsilon )= g_{37}\varepsilon^7+g_{36}\varepsilon^6+g_{35}\varepsilon^5+ g_{34}\varepsilon^4+g_{33}\varepsilon^3+g_{32}\varepsilon^2+ g_{31}\varepsilon+g_{30}$$ with $$g_{37}=2 q^5+42 q^3-18 q^4-108 q+18 q^2,\; g_{36}=138 q^3-2 q^5- 216 q+q^6-36 q^4-45 q^2,$$ $$g_{35}=-92 q^5+558 q^4+2160 q+5 q^6-1296-1260 q^3+261 q^2,$$ $$g_{34}=-4 q^6+5184 q-12 q^3- 5184-1080 q^2+28 q^5-36 q^4,$$ $$g_{33}=-1944 q^2-2520 q^4-40 q^6 +14256-10692 q+5268 q^3+520 q^5,$$ $$g_{32}=-144 q^4+72 q^3+112 q^5+17496 q-4320 q^2-16 q^6-15552,$$ $$g_{31}=-736 q^5+80 q^6+2304 q^4-54432 q+54432-4608 q^3 +18576 q^2,\:g_{30}=64q^2(q-3)^4\ge 0$$ for $q=p^2\in [0,1]$. Therefore, $$g_3(p,\varepsilon )\ge \varepsilon(\bar g_{37}\varepsilon^6+\bar g_{36}\varepsilon^5+\bar g_{35}\varepsilon^4+ \bar g_{34}\varepsilon^3+ \bar g_{33}\varepsilon^2+\bar g_{32}\varepsilon+ \bar g_{31})$$ where $\bar g_{3i}:=\min_{q\in [0,1]}g_{3i}(q)$, and an elementary calculation gives $$\frac{g_3(p,\varepsilon )}{\varepsilon}\ge -64\varepsilon^5-160\varepsilon^4- 5184\varepsilon^3+ 4848\varepsilon^2-15552\varepsilon+15616 >1620$$ for $\varepsilon\in (0,\frac{9}{10}]$. For $\varepsilon\in [\frac{9}{10},1]$ we have ${g_3(p,\varepsilon )}\ge\sum_{i=0}^{6}\bar h_{2i}q^{i}$ where $\bar h_{2i}:=\min_{\varepsilon\in [\frac{9}{10},1]} h_{2i}$ and thus $${g_3(p,\varepsilon )}\ge -736q^5+80q^6+2304q^4-54432q+54432-4608q^3+18576q^2>4840.$$ Thus, finally ${g_3(p,\varepsilon )}\ge \min \{ 1620\varepsilon,4840\}.$ The function ${g_1(p,\varepsilon )}=s_1-t_1$ with $$s_1:=(q^3-6q^2+9q) \varepsilon^7+(5q^3-30q^2+45q-72)\varepsilon^6+ (108q^2-18q^3-162q)\varepsilon^5+$$ $$(-432q+288-48q^3+288q^2) \varepsilon^4+(24q^3-144q^2+216q)\varepsilon^3+$$ $$1512\varepsilon^2+ (1152q-768q^2+128q^3)\varepsilon+448q^2(q-3)^4\ge$$ $$-72\varepsilon^6-72\varepsilon^5+96\varepsilon^4+1512\varepsilon^2\ge 1440\varepsilon^2>0,$$ and $$t_1:=( \varepsilon^2-4)(\varepsilon+7) (\varepsilon^2+2)bD(p,\varepsilon);$$ simplifying $s_1^2-t_1^2=(s_1-t_1)(s_1+t_1)=g_1(p,\varepsilon )(s_1+t_1)$ one finds $$(540 q^2-1296 q-216 q^4-4 q^6+288 q^3+48 q^5) \varepsilon^{13}+$$ $$(-1944 q^4+3024 q^3+432 q^5-7776 q+5184+2268 q^2-36 q^6) \varepsilon^{12}+$$ $$(-2052 q^2+3240 q^4-5328 q^3-720 q^5+60 q^6+10368 q) \varepsilon^{11}+$$ $$(936 q^6+50544 q^4+55944 q^2-11232 q^5-41472-97776 q^3 +29808 q)\varepsilon^{10}$$ $$+(26136 q^2+120 q^6-15696 q^3+6480 q^4-24624 q -1440 q^5) \varepsilon^9+$$ $$(57456 q^5+156816 q-492372 q^2-4788 q^6-134784 +534528 q^3-258552 q^4)\varepsilon^8+$$ $$(180576 q^2-352 q^6-313632 q+ 3168 q^3+4224 q^5-19008 q^4)\varepsilon^7+$$ $$(54432 q^4-238464 q^3+ 859248 q^2-1166400 q+1008 q^6+870912-12096 q^5)\varepsilon^6+$$ $$(1118592 q^3-518400 q^4-9600 q^6-1268352 q^2+736128 q +115200 q^5) \varepsilon^5+$$ $$(1524096 q^4+207360 q+2286144+28224 q^6-338688 q^5+ 2147904 q^2-3025152 q^3) \varepsilon^4+$$ $$(10752 q^6+580608 q^4- 903168 q^3-677376 q^2+2322432 q-129024 q^5)\varepsilon^3+$$ $$(3640320 q^3-25344 q^6+304128 q^5-1368576 q^4+ 8128512 q- 7471872 q^2)\varepsilon^2+$$ $$(3096576 q^4+4644864 q^2+57344 q^6-688128 q^5-6193152 q^3) \varepsilon\ge$$ $$64\varepsilon^4(-10\varepsilon^9+ 18\varepsilon^8- 648\varepsilon^6- 141\varepsilon^5-2214\varepsilon^4- 2266\varepsilon^3+5760\varepsilon^2 +35721)\ge 2.2\cdot 10^6\varepsilon^4.$$ Since $s_1+t_1\le 10^6$ one gets ${g_1(p,\varepsilon )}>2\varepsilon^4.$ Similarly, ${g_2(p,\varepsilon )}=s_2-t_2$ with a polynomial $s_2\ge 3000\varepsilon^2$ and $$t_2 =\varepsilon (\varepsilon+7) (\varepsilon-2)t(\varepsilon,q) p^3\sqrt{3(4-q)}$$ where $$t(\varepsilon,q):= (\varepsilon^4-2 \varepsilon^2-8) q^3+9(- \varepsilon^4+2\varepsilon^2+8) q^2 +27 (\varepsilon^4-2\varepsilon^2-8) p^2-9(3\varepsilon^2+2) \varepsilon^2 .$$ Simplifying $s_2^2-t_2^2 =g_2(p,\varepsilon )(s_2+t_2)$ one gets a polynomial $\ge$ $$(-2560\varepsilon^9-18176\varepsilon^8-325632\varepsilon^6 -1254656\varepsilon^4 +2202112\varepsilon^2+15116544) \varepsilon^4\ge 1.5\cdot 10^7 \varepsilon^4$$ and $g_2(p,\varepsilon )\ge 15\varepsilon^4$ which finishes the proof. 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--- abstract: 'This paper presents the natural extension of Buckley-Feuring method proposed in [@BuckleyFeuring99] for solving fuzzy partial differential equations (FPDE) in a non-polynomial relation, such as the operator $\varphi(D_{x_1}, D_{x_2})$, which maps to the quotient between both partials. The new assumptions and conditions proceedings from this consideration are given in this document.' address: - 'Facultad de Matemáticas. Universidad de Sevilla, 41012 Sevilla, Spain' - '*Keywords: Fuzzy differential equations, Buckley-Feuring solution, non-polynomial*' - '*2000 Mathematics Subject Classification: 03E72, 46S40*' author: - | D. Gálvez and J. L. Pino\ [Departamento de Estadística e Investigación Operativa]{}\ [Universidad de Sevilla]{} title: 'The extension of Buckley-Feuring solutions for non-polynomial fuzzy partial differential equations' --- Introduction {#introduction .unnumbered} ============ Many approaches for obtaining non-numerical solutions of fuzzy differential equations (FDE) have been developed from the introduction of fuzzy set concept by Zadeh [@Zadeh65] . These ones give a diversity of definitions for FDE solution based on different notions of fuzzy derivative, such as Seikkala derivative, Buckley-Feuring derivative, Puri-Ralescu derivative, Kandel-Friedman-Ming derivative, Goetschel-Voxman derivative, or Dubois-Prade derivative . Some relations between these derivatives are presented by Buckley and Feuring in [@BuckleyFeuring00] . However, only a few of these fuzzy derivatives are valid in some contexts as FDE solution. For example, the Goetschel-Voxman derivative, or the Dubois-Prade derivative provide solutions that cannot be a fuzzy number, whereas Puri-Ralescu derivative, and Kandel-Friedman-Ming derivative, always exist and provide a fuzzy number as solution of the FDE, but making use of abstract subtractions of fuzzy concepts in their definitions, making difficult the interpretation of this solutions in some real applications. This paper uses the Buckley-Feuring derivative for solving FPDE. This derivative does not always exist, but if it does, provides a fuzzy number solution easily understandable in the context in which a specific FPDE has been developed. The authors which proposed this concept of derivative, developed a methodology for solving constant coefficients polynomial FPDE in [@BuckleyFeuring99] . This paper present the extension of this methodology to a non-polynomial expression in partial fuzzy derivatives.\ \ In the following lines, the components of a FPDE are enumerated: - $x_i,\quad i=1, 2,\quad x_1\in S_1\subset I_1=(0, M_1],\quad x_2\in S_2\subset I_2=(0, M_2]$. Other domain limits can be established in this subsets, such as $ x_1>x_2$. - $\tilde{\boldsymbol{\beta}}= (\tilde{\beta}_1, \tilde{\beta}_2,...,\tilde{\beta}_k)$, a triangular fuzzy number vector. - $\mu(\beta_j)$ is the membership function of $\beta_j\in \tilde{\beta_j}.$ - $\mu_{\beta_j}(\alpha)=\{\beta_j\mid \mu(\beta_j)\geq\alpha,\quad\alpha\in(0,1)\}$ set called $\alpha$-cut.\ \ These sets are closed and bounded, so that is possible to define, for a fuzzy number $\tilde{\beta_j}: \tilde{\beta_j}[\alpha]= [b_1(\alpha), b_2(\alpha)]$, where: - $b_1(\alpha)$ is the lower value $\beta_j$ in which $\mu(\beta_j)\geq \alpha,\quad \beta_j\in \tilde{\beta}_j$. - $b_2(\alpha)$ is the higher value $\beta_j$ in which $\mu(\beta_j)\geq \alpha,\quad \beta_j\in \tilde{\beta}_j$. and $\tilde{\boldsymbol{\beta}}[\alpha]= \prod_j\tilde{\beta_j}[\alpha]$ - $\tilde{V}(x_1,x_2,\tilde{\boldsymbol{\beta}})$ is a positive and continuous function in $(x_1, x_2)\in S_1 \times S_2$ with partials $D_{x_1}, D_{x_2}$. This function must be also strictly increasing or strictly decreasing in $x_2\in S_2$, that is $\tilde{V}(k,x_2,\tilde{\boldsymbol{\beta}})$ is strictly increasing or strictly decreasing for all constant $k\in\mathbb{R}$.\ The fuzzy character of $\tilde{V}(x_1,x_2,\tilde{\boldsymbol{\beta}})$ shown by the tilde placed over $V$ is fixed by $\tilde{\boldsymbol{\beta}}$, and support the use of Buckley-Feuring derivative for solving FPDE. - $\varphi(D_{x_1}, D_{x_2})$ is an expression with constant coefficients in $(D_{x_1}, D_{x_2})$ applied to $\tilde{V}(x_1,x_2,\tilde{\boldsymbol{\beta}})$. - $F(x_1, x_2,\tilde{\boldsymbol{\beta}})$ continuous function in $(x_1, x_2)\in S_1 \times S_2.$ The specific FPDE treated in this paper has the following form according with this notation: $$\varphi(D_{x_1}, D_{x_2})\tilde{V}(x_1,x_2,\tilde{\boldsymbol{\beta}})= \frac{\partial \tilde{V}/\partial x_1}{\partial \tilde{V}/\partial x_2}=F(x_1, x_2,\tilde{\boldsymbol{\beta}})$$ The Buckley-Feuring (B-F) solution {#sec:B-F} ================================== The Buckley-Feuring (B-F) solution uses a solution of the crisp partial differential equation: $$V(x_1,x_2)= G(x_1,x_2,\boldsymbol{\beta}),$$ with $G$ continuous $\forall (x_1,x_2)\in S_1\times S_2$.\ The next step is the fuzzification of $G$: $$\tilde{Y}(x_1,x_2)= \tilde{G}(x_1,x_2,\tilde{\boldsymbol{\beta}}),$$ with $\tilde{G}$ continuous $\forall (x_1,x_2)\in S_1 \times S_2$ and strictly monotone for $x_2\in S_2$. Note that $\tilde{Y}_i$ is only the fuzzy representation of $G$, but not necessary the solution of the fuzzy partial differential equation. If it finally happens and $\tilde{Y}(x_1,x_2)$ is a B-F solution, then $\tilde{V}(x_1,x_2,\tilde{\boldsymbol{\beta}})=\tilde{Y}(x_1,x_2)$. With this notation, it is possible to see that:\ \ $\tilde{Y}(x_1,x_2)[\alpha]= [y_1(x_1,x_2,\alpha), y_2(x_1,x_2,\alpha)]$, and\ \ $\tilde{F}(x_1,x_2,\tilde{\boldsymbol{\beta}})[\alpha]= [f_1(x_1,x_2,\alpha), f_2(x_1,x_2,\alpha)], \forall\alpha$.\ and, by definition: $y_1(x_1,x_2,\alpha)= \min\{G(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\tilde{\boldsymbol{\beta}}[\alpha]\}$,\ $y_2(x_1,x_2,\alpha)= \max\{G(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\tilde{\boldsymbol{\beta}}[\alpha]\}$ ,and\ \ $f_1(x_1,x_2,\alpha)= \min\{F(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\tilde{\boldsymbol{\beta}}[\alpha]\}$,\ $f_2(x_1,x_2,\alpha)= \max\{F(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\tilde{\boldsymbol{\beta}}[\alpha]\}$,\ $\forall x_1, x_2, \alpha$.\ If it is possible to apply the $\varphi(D_{x_1}, D_{x_2})$ operator to $y_i, i=1, 2 $, getting continuous expressions $\forall(x_1, x_2)\in S_1\times S_2,\forall\alpha$, then it will be feasible to define the following expression in this domain $\Gamma(x_1,x_2,\alpha)$: $$\Gamma(x_1,x_2,\alpha)=[\Gamma_1(x_1,x_2,\alpha),\Gamma_2(x_1,x_2,\alpha)]$$ with: $\Gamma_1(x_1,x_2,\alpha)= \varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,\alpha)$ $\Gamma_2(x_1,x_2,\alpha)=\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,\alpha)$\ \ For a B-F solution $Y$, it is necessary to be a fuzzy number for this one. If, for each pair $(x_1, x_2)\in S_1\times S_2$, $\Gamma(x_1,x_2,\alpha)$ defines an $\alpha$-cut of a fuzzy number, then, Buckley and Feuring [@BuckleyFeuring99] call to $\tilde{Y_i}(x_1,x_2)$ **differentiable**, and we can write: $$\varphi(D_{x_1}, D_{x_2})\tilde{Y}(x_1,x_2)[\alpha]=\Gamma(x_1,x_2,\alpha),$$ $\forall(x_1, x_2)\in S_1\times S_2,\quad\forall\alpha$.\ \ So that, it is necessary to test if $\Gamma(x_1,x_2,\alpha)$ really define an $\alpha$-cut of a fuzzy number and verify the differentiability of $\tilde{Y}(x_1,x_2)$. For a triangular fuzzy number, the conditions are [@GoetschelVoxman86] : 1. $\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,\alpha)$ is an increasing function of $\alpha$, for each $(x_1, x_2)\in S_1\times S_2$. 2. $\varphi(D_{x_1}, D_{x_2})y_2(x_1,x_2,\alpha)$ is a decreasing function of $\alpha$, for each $(x_1, x_2)\in S_1\times S_2$. 3. $\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,1)\leq\varphi(D_{x_1}, D_{x_2})y_2(x_1,x_2,1)$ for each $(x_1, x_2)\in S_1\times S_2$. Once delimited the differentiability concept of $\tilde{Y}(x_1,x_2)$, it is possible to define the Buckley-Feuring solution. $\tilde{Y_i}(x_1,x_2)$ is a Buckley-Feuring solution if the following conditions hold:\ 1. $\tilde{Y_i}(x_1,x_2)$ is differentiable. 2. $\varphi(D_{x_1}, D_{x_2})\tilde{Y}(x_1,x_2)= \tilde{F}(x_1, x_2,\boldsymbol{\tilde{\beta}})$.\ Obviously, if differentiability conditions hold by the candidate to B-F solution $\tilde{Y}(x_1,x_2)$, this one will be a fuzzy number. To complete the conditions for B-F solutions only is necesary to test that: $$\varphi(D_{x_1}, D_{x_2})\tilde{Y}(x_1,x_2)= \tilde{F}(x_1, x_2,\tilde{\boldsymbol{\beta}}),$$ or the equivalent condition: 1. $\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,\alpha)= F_1(x_1, x_2,\alpha).$ 2. $\varphi(D_{x_1}, D_{x_2})y_2(x_1,x_2,\alpha)= F_2(x_1, x_2,\alpha).$ $\forall(x_1, x_2)\in S_1\times S_2,\quad\forall\alpha$.\ \ In this case we can identify $\tilde{Y}(x_1,x_2)$ with $\tilde{V}(x_1,x_2,\tilde{\boldsymbol{\beta}})$. Boundary conditions {#sec:Bound. cond.} =================== The FPDE can be subject to certain boundary conditions in a big variety of forms depending on a constant vector $\boldsymbol{c}=(c_1,...c_n)\in C_1\times ...\times C_n$. The inclusion of this ones has not great consequences in the methodology exposed. The crisp solution $G$ acquires under boundary conditions the form $G(x_1,x_2,\boldsymbol{\beta},\boldsymbol{c})$. The fuzzification of $G$ can take $\boldsymbol{c}$ in a triangular fuzzy vector $\tilde{\boldsymbol{c}}=(\tilde{c_1},...\tilde{c_n})\in C_1\times ...\times C_n$ with $\tilde{\boldsymbol{c}}[\alpha]= \prod_i\tilde{C_i}[\alpha],\quad i=1,...n$ and: $$\tilde{Y}(x_1,x_2)= \tilde{G}(x_1,x_2,\tilde{\boldsymbol{\beta}},\tilde{\boldsymbol{c}})$$ In this environment with boundary conditions, it is necessary to add a new condition for a B-F solution: $\tilde{Y}(x_1,x_2)$ must satisfier these conditions. In this form $\tilde{Y_i}(x_1,x_2)$ is a Buckley-Feuring solution if the following conditionshold: 1. $\tilde{Y_i}(x_1,x_2)$ is differentiable. 2. $\varphi(D_{x_1}, D_{x_2})\tilde{Y}(x_1,x_2)= \tilde{F}(x_1, x_2,\tilde{\boldsymbol{\beta}})$. 3. $\tilde{Y}(x_1,x_2)$ satisfies the boundary conditions. Example {#sec:example} ======= The following FPDE is proposed: $$\dfrac{\partial {\tilde{V}}/\partial{x_1}}{\partial \tilde{V}/\partial x_2}= \tilde{\beta} x_1^{ -1}x_2,\quad \tilde{\beta}\in (0, 1),x_1\geq 1, x_2>0$$ In this example: $\tilde{F}(x_1, x_2,\boldsymbol{\tilde{\beta}})=\tilde{\beta} x_1^{ -1}x_2,\quad \tilde{\beta}\in (0, 1),x_1\geq 1, x_2>0$\ and, by definition, the operator\ $\varphi(D_{x_1}, D_{x_2})V(x_1,x_2) \longrightarrow\dfrac{\partial \tilde{V}/\partial x_1}{\partial \tilde{V}/\partial x_2},$\ \ A possible solution to this FPDE in a crisp environment is, without special boundary conditions: $$G(x_1,x_2,\boldsymbol{\beta})= x_1^{\beta}x_2+\gamma,\beta\in (0, 1),x_1\geq 1, x_2>0$$ with $\boldsymbol{\beta}=(\beta,\gamma)$ Applying the fuzzification in $\beta$ and $\gamma$, these ones acquire a triangular fuzzy number form and $\boldsymbol{\tilde{\beta}} =(\tilde{\beta},\tilde{\gamma}),\quad\tilde{\beta}\in (0, 1)$. While $G$ holds all the conditions required, the Buckley-Feuring solution candidate is: $$\tilde{Y}(x_1,x_2)=G(x_1,x_2,\tilde{\beta},\tilde{\gamma})= x_1^{\tilde{\beta}}x_2+\tilde{\gamma},\tilde{\beta}\in (0, 1),x_1\geq 1, x_2>0$$ The fuzzy parameters have a membership function associated $\mu(\beta)$ and $\mu(\gamma)$ respectiveness. From the $\alpha$-cuts, it is possible to define: $\tilde{\beta}[\alpha]= [b_1(\alpha), b_2(\alpha)]$,\ $\tilde{\gamma}[\alpha]= [g_1(\alpha), g_2(\alpha)]$, and\ $\tilde{\boldsymbol{\beta}}[\alpha]= \tilde{\beta}[\alpha] \times \tilde{\gamma}[\alpha]$\ And from this ones: $\tilde{Y}(x_1,x_2)[\alpha]= [y_1(x_1,x_2,\alpha), y_2(x_1,x_2,\alpha)]$, and\ $\tilde{F}(x_1,x_2,\tilde{\boldsymbol{\beta}})[\alpha]= [f_1(x_1,x_2,\alpha), f_2(x_1,x_2,\alpha)], \forall\alpha $.\ Where: $y_1(x_1,x_2,\alpha)= \min\{G(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\boldsymbol{\tilde{\beta}}[\alpha]\}= G(x_1,x_2,g_1(\alpha), b_1(\alpha)) $,\ $y_2(x_1,x_2,\alpha)= \max\{G(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\boldsymbol{\tilde{\beta}}[\alpha]\}=G(x_1,x_2, g_2(\alpha), b_2(\alpha))$\ \ and,\ \ $f_1(x_1,x_2,\alpha)= \min\{F(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\boldsymbol{\tilde{\beta}}[\alpha]\}=F(x_1,x_2, g_1(\alpha), b_1(\alpha))$,\ $f_2(x_1,x_2,\alpha)= \max\{F(x_1,x_2,\boldsymbol{\beta}),\quad \boldsymbol{\beta}\in\boldsymbol{\tilde{\beta}}[\alpha]\}= F(x_1,x_2, g_2(\alpha), b_2(\alpha))$,\ \ $\forall\alpha\quad x_1\geq 1, x_2>0 $\ \ In the $\tilde{G}$ function proposed, $\forall(x_1, x_2)x_1\geq 1, x_2>0$:\ $y_1(x_1,x_2,\alpha)=x_1^{b_1(\alpha)}x_2+{g_1(\alpha)},\quad b_1(\alpha)\in (0, 1)$\ $y_2(x_1,x_2,\alpha)=x_1^{b_2(\alpha)}x_2+{g_2(\alpha)},\quad b_2(\alpha)\in (0, 1)$\ \ $f_1(x_1,x_2,\alpha)=b_1(\alpha)x_1^{-1}x_2,\quad b_1(\alpha)\in (0, 1)$\ $f_2(x_1,x_2,\alpha)=b_2(\alpha)x_1^{-1}x_2,\quad b_2(\alpha)\in (0, 1)$\ \ Testing the differentiability of $\tilde{G}(x_1,x_2,\tilde{\boldsymbol{\beta}})$, from $\Gamma(x_1,x_2,\alpha)=[\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,\alpha), \varphi(D_{x_1}, D_{x_2})y_2(x_1,x_2,\alpha)]$ verifying if $\Gamma(x_1,x_2,\alpha)$ defines an $\alpha$-cut of a triangular fuzzy number for each pair $(x_1, x_2), x_1\geq 1, x_2>0$, the following conditions must be hold: 1. $\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,\alpha)= b_1(\alpha)x_1^{-1}x_2,\quad b_1(\alpha)\in (0, 1)$ is an increasing function of $\alpha$, for each pair $(x_1, x_2), x_1\geq 1, x_2>0$. It will happen if $\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,\alpha)$ has positive derivative on $\alpha$: $$d_{\alpha}\left(b_1(\alpha)x_1^{-1}x_2\right)= b_1(\alpha)'x_1^{-1}x_2$$ While $\beta$ is a triangular fuzzy number and $b_1(\alpha)$is defined from its $\alpha-cuts$, $b_1(\alpha)$ satisfies the condition and is an increasing function with $b_1(\alpha)'>0$, and $x_1^{-1}x_2>0$ for $ x_1\geq 1, x_2>0$, $d_{\alpha}\left(b_1(\alpha)x_1^{-1}x_2\right)>0$ and the condition holds.\ \ 2. $\varphi(D_{x_1}, D_{x_2})y_2(x_1,x_2,\alpha)= b_2(\alpha)x_1^{-1}x_2,\quad b_2(\alpha)\in (0, 1)$ is a decreasing function of $\alpha$, for each pair $(x_1, x_2), x_1\geq 1, x_2>0$. Again, it will happen if $\varphi(D_{x_1}, D_{x_2})y_2(x_1,x_2,\alpha)$ has negative derivative on $\alpha$: $$d_{\alpha}\left(b_2(\alpha)x_1^{-1}x_2\right)= b_2(\alpha)'x_1^{-1}x_2$$ We can now use that $\beta$ is a triangular fuzzy number and $b_2(\alpha)$ is defined from its $\alpha-cuts$, so that, in analogy, $b_2(\alpha)$ satisfies the condition and is a decreasing function with $b_2(\alpha)'<0$, and $x_1^{-1}x_2>0$ for $ x_1\geq 1, x_2>0$, $d_{\alpha}\left(b_1(\alpha)x_1^{-1}x_2\right)<0$ and the condition holds.\ \ 3. $\varphi(D_{x_1}, D_{x_2})y_1(x_1,x_2,1)\leq \varphi(D_{x_1}, D_{x_2})y_2(x_1,x_2,1) \forall x_1, x_2)\in I_1\times I_2$.\ In the example: $$b_1(1)x_1^{-1}x_2 \leq b_2(2)x_1^{-1}x_2$$\ This condition is holds automatically if we realize that $b_1(\alpha)$ and $b_2(\alpha)$ are defined from a triangular fuzzy number and $b_2(1)>b_1(1)$ At this point, it is possible to say that $\tilde{Y}(x_1,x_2)=G(x_1,x_2,\tilde{\beta},\tilde{\gamma})= x_1^{\tilde{\beta}}x_2+\tilde{\gamma},\tilde{\beta}\in (0, 1),x\geq 1$ is differentiable and good candidate for B-F solution. But it is necessary the last step, and it must satisfier that:\ \ $\varphi(D_{x_1}, D_{x_2})\tilde{Y}(x_1,x_2)= \tilde{F}(x_1, x_2,\tilde{\boldsymbol{\beta}}) \quad \beta, \gamma\in (0, 1), \forall(x_1, x_2), x_1\geq 1, x_2>0$.\ For $G(x_1,x_2,\tilde{\beta},\tilde{\gamma})= x_1^{\tilde{\beta}}x_2+\tilde{\gamma},\tilde{\beta}\in (0, 1),x\geq 1, x_2\geq 0$:\ \ $\varphi(D_{x_1}, D_{x_2})\tilde{Y}(x_1,x_2)= \tilde{\beta} x_1^{ -1}x_2,\quad \tilde{\beta}\in (0, 1),x_1\geq 1, x_2>0 $\ and,\ $\tilde{F}(x_1, x_2,\tilde{\boldsymbol{\beta}})=\tilde{\beta} x_1^{ -1}x_2,\quad \tilde{\beta}\in (0, 1),x_1\geq 1, x_2>0. $\ Thus $G(x_1,x_2,\tilde{\beta},\tilde{\gamma})= x_1^{\tilde{\beta}}x_2+\tilde{\gamma},\tilde{\beta}\in (0, 1),x\geq 1, x_2\geq 0$ is a Buckley-Feuring solution for this non-polynomial fuzzy partial differential equation. [99]{} Allahviranloo, T. Diference methods for fuzzy partial differential equations, Computational Methods in Applied Mathematics, Vol.2 (nº3) (2002), pp. 233-242. Buckley, J.J. and Feuring,T. Introduction to fuzzy partial differential equations, Fuzzy Sets and Systems, 105 (1999), pp.241-248. Buckley, J.J. and Feuring,T. Fuzzy differential equations, Fuzzy Sets and Systems, 110 (2000), pp.43-54. Goetschel, R. and Voxman, W. Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986), pp.319-330. Quan, H., Hua, Y. and Jones, J.D. A general method for calculating functions of fuzzy numbers, Applied Mathematics Letters, Vol.5 (nº6) (1992), pp.51-55. Seikkala, s. On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987), pp.31-43. Zadeh, L.A. Fuzzy Sets. Inf. Control, 8 (1965), pp.338-353.

.0in 8.5in 6.2in 0.12in 3.0ex PS. \#1[[**\#1**]{}]{} \#1[Nucl. Phys., ]{} \#1[Comm. Math. Phys., ]{} \#1[Phys. Lett., ]{} \#1[Phys. Rev., ]{} \#1[Phys. Rev. Lett., ]{} \#1[Proc. Roc. Soc., ]{} \#1[Prog. Theo. Phys., ]{} \#1[Sov. J. Nucl. Phys., ]{} \#1[Theor. Math. Phys., ]{} \#1[Annals of Phys., ]{} \#1[Proc. Natl. Acad. Sci. USA, ]{} **$Sp(N_c)$ Gauge Theories and M Theory Fivebrane** Changhyun Ahn$^{a,}$[^1], Kyungho Oh$^{b,}$[^2] and Radu Tatar$^{c,}$[^3] [*$^a$ Dept. of Physics, Seoul National University, Seoul 151-742, Korea*]{} [ *$^b$ Dept. of Mathematics, University of Missouri-St. Louis, St. Louis, Missouri 63121, USA*]{} [*$^c$ Dept. of Physics, University of Miami, Coral Gables, Florida 33146, USA*]{} Abstract 0.2in We analyze M theory fivebrane in order to study the moduli space of vacua of $N=1$ supersymmetric $Sp(N_c)$ gauge theories with $N_f$ flavors in four dimensions. We show how the $N=2$ Higgs branch can be encoded in M theory by studying the orientifold which plays a crucial role in our work. When all the quark masses are the same, the surface of the M theory spacetime representing a nontrivial ${\bf S^1}$ bundle over ${\bf R^3}$ develops $A_{N_f-1}$ type singularities at two points where D6 branes are located. Furthermore, by turning off the masses, two singular points on the surface collide and produce $A_{2N_f-1}$ type singularity. The sum of the multiplicities of rational curves on the resolved surface gives the dimension of $N=2$ Higgs branch which agrees with the counting from the brane configuration picture of type IIA string theory. By rotating M theory fivebranes we get the strongly coupled dynamics of $N=1$ theory and describe the vacuum expectation values of the meson field parameterizing Higgs branch which are in complete agreement with the field theory results. Finally, we take the limit where the mass of adjoint chiral multiplet goes to infinity and compare with field theory results. For massive case, we comment on some relations with recent work which deals with $N=1$ duality in the context of M theory. Introduction ============ One of the most interesting tools used to study nonperturbative dynamics of low energy supersymmetric gauge theories is to understand the D(irichlet) brane dynamics where the gauge theory is realized on the worldvolume of D brane. This work was pioneered by Hanany and Witten [@hw] where the mirror symmetry of $N=4$ gauge theory in 3 dimensions was interpreted by changing the position of the Neveu-Schwarz(NS)5 brane in spacetime. (see also [@bo1][@bo2]). They took a configuration of type IIB string theory which preserves 1/4 of the supersymmetry and consists of parallel NS5 branes with D3 branes suspended between them and D5 branes located between them. A new aspect of brane dynamics was the creation of D3 brane whenever a D5 brane and NS5 brane are crossing through each other. This was due to the conservation of the linking number(defined as a total magnetic charge for the gauge field coupled with the worldvolume of the both types of NS and D branes). By T-dualizing the above configuration on one space coordinate, the passage to $N=2$ gauge theory in 4 dimensions can be described as two parallel NS5 branes and D4 branes suspended between them in a flat space in type IIA string theory. When one change the relative orientation of the two NS5 branes [@bar] while keeping their common 4 spacetime dimensions intact, the $N=2$ supersymmetry is broken to $N=1$. The brane configuration[@egk; @egkrs] preserves 1/8 of the supersymmetry and this corresponds to turning on a mass of adjoint field because the distances between D4 branes suspended between the NS5 branes relate to the vacuum expectation values(vevs). The configuration of D4 branes gives the gauge group while the D6 branes give the global flavor group. Using this configuration they described and checked a stringy derivation of Seiberg’s duality for $N=1$ supersymmetric gauge theory with $SU(N_c)$ gauge group with $N_f$ flavors in the fundamental representation which was previously conjectured in [@se1]. This result was generalized to brane configurations with orientifolds which then give $N=1$ supersymmetric theories with gauge group $SO(N_c)$ or $Sp(N_c)$ [@eva; @egkrs]. In this case the NS5 branes have to pass over each other and some strong coupling phenomena have to be considered. Similar results were obtained in [@bh; @bsty; @t] where the moduli space of the supersymmetric gauge theories is geometrically encoded in the brane setup. Another approach was initiated by Ooguri and Vafa [@ov] where they considered the compactification of IIA string theory on a double elliptically fibered Calabi-Yau threefold. The wrapped D6 branes around three cycles of Calabi-Yau threefold filling also a 4 dimensional spacetime. The transition between electric theory and its magnetic dual appears when a change in the moduli space of Calabi-Yau threefold occurs. Their results were generalized in the papers of [@ao; @a; @ar; @aot] to various other models which reproduce field theory results studied previously. So far the branes in string theory were considered to be rigid without any bendings. When the branes are intersecting each other, a singularity occurs. In order to avoid that kind of singularities, a very nice simplification was obtained by reinterpreting brane configuration in string theory from the point of view of M theory as was showed by Witten in [@w1]. Then both the D4 branes and NS5 branes come from the fivebranes of M theory (the former is an M theory fivebrane wrapped over $\bf{S^1}$ and the latter is an M theory fivebrane on $\bf{R^{10} \times S^1}$). That is, D4 brane’s worldvolume projects to a five manifold in $\bf{R^{10}}$ and NS5 brane’s worldvolume is placed at a point in $\bf{S^1}$ and fills a six manifold in $\bf{R^{10}}$. To obtain D6 branes one has to use a multiple Taub-NUT space whose metric is complete and smooth. The $N=2$ supersymmetry in four dimensions requires that the worldvolume of M theory fivebrane is $\bf{R^{1,3}}\times \Sigma$ where $\Sigma$ is uniquely identified with the curves [@sw] that appear in the solutions to Coulomb branch of the field theory. The configurations involving orientifolds were considered in[@lll; @bsty1]. The method of brane dynamics was used to study supersymmetric field theories in several dimensions by many authors [@ah; @ba; @k; @cvj1; @hov; @hz; @mmm; @fs; @w2; @biksy; @gomez; @cvj2; @hk; @hsz; @nos; @hy; @noyy; @ss; @bo; @mi]. The original work [@w1] was suited to study the moduli space for $N=2$ supersymmetric theories. By rotating one of the NS5 branes the $N=2$ supersymmetry is broken to $N=1$ [@bar]. In [@w2; @hoo] (see also [@biksy; @ss; @bo]) this was seen from the point of the M theory interpretation, by considering the possible deformation of the curve $\Sigma$. In field theory, the supersymmetry is broken by giving a mass to the adjoint field and if this mass is finite, the $N=1$ field theory can be compared with the previous results obtained in [@ads]. These papers considered the case of unitary groups. Recently, the exact low energy description of $N=2$ supersymmetric $SU(N_c)$ gauge theories with $N_f$ flavors in 4 dimensions in the framework of M theory fivebrane have been found in [@hoo]. They constructed M fivebrane configuration which encodes the information of Affleck-Dine-Seiberg superpotential [@ads] for $N_f < N_c$. Later, this approach has been used to study the moduli space of vacua of confining phase of $N=1$ supersymmetric gauge theories in four dimensions [@bo]. In terms of brane configuration of IIA string theory, this corresponds to the picture of [@egk] by taking multiples of NS’5 branes rather than a single NS’5 brane. In the present paper we generalize to the case of symplectic group $Sp(N_c)$ with $N_f$ flavors. The new ingredient that is introduced is the orientifold. We find an interesting picture which differs from the one obtained for unitary group $SU(N_c)$ [@hoo]. This is expected because for $SU(N_c)$ groups we have both baryonic and non-baryonic branches, but in the case of $Sp(N_c)$ we cannot construct any baryon, so we have only non-baryonic branch. This paper is organized as follows. In section 2 we review the papers of [@ip; @aps; @hms] and study the moduli space of vacua of the $N=1$ theory which is obtained from the $N=2$ theory by adding a mass term to the adjoint chiral multiplet. We discuss for different values of the number of flavors with respect to the number of colors. We also introduce massive matter. In section 3, we start with the setup of M theory fivebrane and discuss the Higgs branches with the resolution of singularities. In section 4, we rotate brane configuration and obtain information about the strong coupling dynamics of $N=1$ theory. In section 5, we take the mass of adjoint field infinite and compare it with field theory results for massless or massive matter. We discuss $N=1$ duality and compare with similar work without D6 branes obtained in [@cs] recently. Finally in section 6, we conclude our results and comment on the outlook in the future directions. Field Theory Analysis ===================== Let us review and summarize field theory results already known in the papers of [@ip; @aps; @hms] for future developments. We claim no originality for most of results presented in this section except that we have found the property of meson field $M^{ij}$ having only one kind matrix element which will be discussed in detail later. $N=2$ Theory ------------ Let us consider $N=2$ supersymmetric $Sp(N_{c})$ gauge theory with matter in the $2N_c$ dimensional representation of $Sp(N_c)$. In terms of $N=1$ superfields, $N=2$ vector multiplet consists of a field strength chiral multiplet $W_{\alpha}^{ab}$ and a scalar chiral multiplet $\Phi_{ab}$, both in the adjoint representation of the gauge group $Sp(N_c)$. The quark hypermultiplets are made of a chiral multiplet $Q^{i}_{a}$ which couples to the Yang-Mills fields where $i = 1,\cdots ,2N_{f}$ are flavor indices( the number of flavors has to be even ) and $a = 1, \cdots , 2N_{c}$ are color indices. The $N=2$ superpotential takes the form: $$\label{super} W = \sqrt{2} Q^{i}_{a} \Phi^{a}_{b} J^{bc} Q^{i}_{c} + \sqrt{2} m_{ij} Q^{i}_{a} J^{ab} Q^{j}_{b},$$ where $J_{ab}$ is the symplectic metric ( [cc]{} 0 & 1\ -1 & 0 ) used to raise and lower $Sp(N_{c})$ color indices and $m_{ij}$ is the antisymmetric mass matrix \[mass\] ( [cc]{} 0 & -1\ 1 & 0 ) ( m\^f\_[1]{}, , m\^f\_[N\_f]{} ). Classically, the global symmetries are the flavor symmetry $O(2N_{f}) = SO(2N_{f})\times \bf{Z_{2}}$ in addition to $U(1)_{R}\times SU(2)_{R}$ chiral R-symmetry. The theory is asymptotically free for $N_{f}$ smaller than $2N_{c}+2$ and generates dynamically a strong coupling scale $\Lambda_{N=2}$ where we denote the $N=2$ theory by writing it in the subscript of $\Lambda$. The instanton factor is proportional to $\Lambda_{N=2}^{2N_{c}+2-N_{f}}$. Then the $U(1)_{R}$ symmetry is anomalous and is broken down to a discrete $\bf{Z_{2N_{c}+2-N_{f}}}$ by instantons. The moduli space contains the Coulomb and the Higgs branches. The Coulomb branch is parameterized by the gauge invariant order parameters u\_[2k]{}=&lt;(\^[2k]{})&gt;, k=1, , N\_c where $\phi$ is the scalar field in $N=2$ vector multiplet. Up to a gauge transformation $\phi$ can be diagonalized to a complex matrix, $<\phi>=\mbox{diag} ( A_1, \cdots, A_{N_c} )$ where $A_i= ( { a_i \atop 0 }{ 0 \atop -a_i} )$. At a generic point the vevs of $\phi$ breaks the $Sp(N_c)$ gauge symmetry to $U(1)^{N_c}$ and the dynamics of the theory is that of an Abelian Coulomb phase. The Wilsonian effective Lagrangian in the low energy can be made of the multiplets of $A_i$ and $W_i$ where $i=1, 2, \cdots, N_c$. If $k$ $a_i$’s are equal and nonzero then there exists an enhanced $SU(k)$ gauge symmetry. When they are also zero, an enhanced $Sp(k)$ gauge symmetry appears. On the other hand, the Higgs branches are described by gauge invariant quantities which are made from the squarks vevs and which can be written as the meson field $M^{ij} = Q^{i}_{a} J^{ab} Q^{j}_{b}$ because we do not have any baryons. Breaking $N=2$ to $N=1$ ----------------------- We want now to break $N=2$ supersymmetry down to $N=1$ supersymmetry by turning on a bare mass $\mu$ for the adjoint chiral multiplet $\Phi$. For the moment we consider that all the squarks are massless, so the terms of $m_{ij}$ in (\[super\]) will not enter into our superpotential. The superpotential is expressed as follows: $$\label{e1} W = \sqrt{2} Q^{i}_{a} \Phi^{a}_{b} J^{bc} Q^{j}_{c} + \mu \; \mbox{Tr}(\Phi^{2}).$$ When the mass of the adjoint chiral multiplet is much smaller than $\Lambda_{N=2}$, by turning a mass for the adjoint chiral multiplet, the structure of moduli space of vacua for $N=2$ theory is changed. Most of the Coulomb branch is lifted except $2N_{c} + 2 - N_{f}$ points which are related to each other by the action of ${\bf Z_{2N_{c} + 2 - N_{f}}}$ . When the mass $\mu$ is increased beyond $\Lambda_{N=2}$ we can integrate out the adjoint chiral multiplet in the low-energy theory. Below the scale $\mu$, by a one loop matching between the $N=1$ and $N=2$ theories we obtain the $N=1$ dynamical scale, $\Lambda_{N=1}$ to be: $$\label{scale} \Lambda_{N=1}^{2(3N_{c}+3-N_{f})} = \mu^{2N_{c} + 2}\Lambda_{N=2}^{2(2N_{c} + 2 -N_{f})}.$$ If $\mu$ is much larger than $\Lambda_{N=1}$ but finite, we can integrate out the heavy field $\Phi$ and to obtain a superpotential which is quartic in the squarks and proportional to $1/\mu$. The F-term equation for $\Phi$ from (\[e1\]) gives us to $$Q^{i}_{a}Q^{i}_{c} + \sqrt{2}\mu J_{ab}\Phi^{b}_{c} = 0$$ where we can read off $\Phi^b_c$ $$\Phi^{b}_{c} = \frac{1}{\sqrt{2}\mu} J^{ab} Q^{i}_{a} Q^{i}_{c}$$ or $$\Phi^{2} = - \frac{1}{2\mu^{2}} M^{2}. \label{fi}$$ We plug this into the superpotential equation and obtain $$\Delta~W = - \frac{1}{2\mu} \mbox{Tr}(M^{2}) \label{pot}$$ which is similar to the equation $(2.5)$ of [@hoo] but without the term involving $(\mbox{Tr}M)^{2}$ because $M$ is traceless antisymmetric in our case and with a minus sign due to (\[fi\]). Therefore, the system below the energy scale $\mu$ can be regarded as the $N=1$ SQCD with the tree level superpotential (\[pot\]) and with the dynamical scale $\Lambda_{N=1}$ given by (\[scale\]). When we take the limit of $\mu\rightarrow \infty$ keeping $\Lambda_{N=1}$ fixed, the superpotential (\[pot\]) vanishes. Let us start with the discussion for the various values of $N_{f}$ as a function of $N_{c}$. $\bullet \;\;\; 0\le N_{f}\le N_{c}$ In this range of the number of flavors, as it is well known, a superpotential is dynamically generated [@ads] by strong coupling effects. For a general value of $N_{f}$, the ADS superpotential is given by [@ip]: $$\label{ads} W_{ADS} = (N_{c} + 1 - N_{f}) \; \omega_{N_{c} + 1 - N_{f}} (\frac{2^{N_{c} - 1}\Lambda_{N=1}^{3(N_{c}+1)-N_{f}}}{\mbox{Pf}M})^{\frac{1} {N_{c}+1-N_{f}}}$$ where $\omega_{N_{c} + 1 - N_{f}}$ is an $N_{c}+1-N_{f}$ th root of unity and Pf(Pfaffian) of antisymmetric matrix $M$ has the following relation: $(\mbox{Pf}M)^2=\mbox{det} M$. For $N_{f} = N_{c}$, the gauge group is completely broken for $\mbox{Pf}<M>$ not zero and the ADS superpotential is generated by an instanton in the broken $Sp(N_{c})$. For large but finite values for $\mu$, the potential obtained after $N=2$ breaking into $N=1$ (\[pot\]) can be described as a perturbation theory to the ordinary $N=1$ theory. Then the total effective superpotential is the sum of $W_{ADS}$ and $\Delta ~W$ W\_[eff]{} = W\_[ADS]{} +  W. This form for $W_{eff}$ is exact for any non-zero value of $\mu$. The argument is based on the holomorphic property. The two terms appearing in an analytic function expanded with respect to $1/\mu$ are given by $W_{ADS}$ and $\Delta$W so a term that can be generated are of the form: \[term\] \^[-]{}M\^\_[N=1]{}\^[(3(N\_[c]{}+1)-N\_[f]{})]{} where $\alpha,\gamma$ are non-negative integers. In order to obtain $\alpha,\beta,\gamma$ we use the fact that (\[term\]) is invariant under the axial flavor symmetry $U(1)_{A}$ and has a charge 2 under the R-symmetry $U(1)_{R}$ where $U(1)_R$ is the anomaly free combination of the $U(1)$ R-symmetry group. The charges for $\Lambda_{N=1}, \mu$ and $M$ are given by: $\Lambda^{3N_{c}+3-N_{f}}_{N=1}$ M $\mu$ ------------ ---------------------------------- ---------------------------------- ----------------------------------- $U(1)_{R}$ 0 $\frac{2(N_{f}-N_{c}-1)}{N_{f}}$ $\frac{2(N_{f}-2N_{c}-2)}{N_{f}}$ $U(1)_{A}$ $2N_{f}$ 2 4 Using these values for the charges and applying them in (\[term\]), the condition that the superpotential has $U(1)_{A}$ charge $0$ gives us that $2\alpha -\beta=N_{f}\gamma$ and other condition that the superpotential has charge $2$ under $U(1)_{R}$ becomes $N_{f}(-\alpha+\beta-1) = (N_{c}+1)(-2\alpha+\beta)$. The combination of these two relations leads to: 1 - = (N\_[c]{} + 1 - N\_[f]{}). Because $\gamma\ge 0$ and we are considering the case of $N_{f}<N_{c}$, we have only two solutions for this equation, that is, $\alpha = 0,\gamma = 1/(N_{c}+1-N_{f})$ and $\alpha=1, \gamma=0$, which exactly correspond to the two terms which appear in (\[term\]). Therefore, it turns out that the superpotential (\[term\]) is exact. The moduli space of vacua is obtained by extrematizing this superpotential, and we get: $$\label{moduli} M^{2} = -\mu \; \omega_{N_{c}+1-N_{f}}( \frac{2^{N_{c}-1}\Lambda_{N=1}^{3(N_{c}+1)- N_{f}}}{\mbox{Pf}M})^{\frac{1}{N_{c}+1-N_{f}}}.$$ In this moment, by a similarity transformation, $M$ can be brought to different forms. In [@aps; @hms], $M$ has been brought to a form such that $M^{2}=0$ which is the right equation for $M$ whenever we do not consider the ADS potential. But in our case, the equation (\[moduli\]) tells us that $M^{2}$ is not equal with $0$, so we take another form for $M$ after a similarity transformation. We bring $M$ to the simplest form, i.e., with two top-right and bottom-left diagonal blocks, one being minus the other because $M$ is to be antisymmetric. Denote $m_{1}, \cdots, m_{N_{f}}$ by the elements of top-right diagonal block in $M$ and of course $-m_{1}, \cdots, -m_{N_{f}}$ by the elements of the bottom-left diagonal block. In this case we will have an equation like (\[moduli\]) for each $m_{i}$. Since the right hand side is the same for all the $m_{i}$’s, [*they have to be equal*]{}. So all the diagonal entries in the top-right and bottom-left diagonal blocks of [M]{} are equal. This is our [*new*]{} observation which will appear naturally in section 4 due to the symmetry of orientifolding and can be compared with the result of [@hoo] for $SU(N_c)$ case where there were two cases, one with equal diagonal entries and th other with two different entries on the diagonal. Now since all the top-right diagonal entries are equal with $m \equiv m_1= \cdots = m_{N_f}$, we find the value for $m$ by solving (\[moduli\]): $$\label{value} m = 2^{\frac{N_{c}-1}{2(N_{c}+1)-N_{f}}} \mu\Lambda_{N=2}$$ where we have used the renormalization group( RG ) matching equation (\[scale\]). The values of $m$ in equation (\[value\]) describe the moduli space of the $N=1$ theory in the presence of a perturbation to the ADS superpotential. When $\mu\rightarrow\infty$ and $\Lambda_{N=1}$ are finite, the solution diverges. In this case $\Delta$W is $0$ and divergence of the solution coincides with the fact that there is no supersymmetric vacua in this region of the flavor. Let us turn on quark mass terms like $\frac{1}{2}m^{ij}M_{ij}$. In this case the effective superpotential is given by W\_[eff]{} = W\_[ADS]{} +  W + m M where $W_{ADS}$ and $\Delta ~W$ are given by (\[ads\]) and (\[pot\]) and m is an antisymmetric matrix as in (ref[mass]{}) but where we take $m_{f}=m_{1}^{f} = \cdots = m_{N_{f}}^{f}$ In this case the equation (\[moduli\]) is modified to contain a term $ \mu m M/2$. As we consider the limit $\mu\rightarrow\infty$ by keeping $\Lambda_{N=1}$ finite, the system will be $N=1$ SQCD with massive flavors. Only terms which are proportional to $\mu$ will resist (so the term form the LHS of (\[moduli\]) will be neglected because it does not depend on $\mu$) and thus we obtain the solution for the moduli space to be: \_[N\_[c]{}+1-N\_[f]{}]{} ()\^ = m\_[f]{} m/2 with the solution m\^[N\_[c]{}+1]{}= giving $N_{c}+1$ vacua in accordance with the interpretation of the low energy physics as the pure $N=1$ Yang-Mills theory. Next we are now increasing the number of flavors. $\bullet \;\;\; N_{f} = N_{c}+1$. Now it is obvious that the ADS superpotential vanishes and the classical moduli space of vacua is changed quantum mechanically. It is parameterized by the meson satisfying the constraint[@ip] $$\mbox{Pf}M = 2^{N_{c}-1} \Lambda_{N=1}^{2(N_{c}+1)}. \label{cons}$$ Again the quartic term $\Delta ~W$ is small for large finite $\mu$ and can be considered as a perturbation to the ordinary $N=1$ theory. By introducing a Lagrange multiplier $X$ in order to impose the constraint(\[cons\]), the effective superpotential will be: $$\label{spt1} W_{eff} = X (\mbox{Pf}M - 2^{N_{c}-1}\Lambda_{N=1}^{2(N_{c}+1)}) - \frac{1}{2\mu} \mbox{Tr}(M^{2}).$$ &gt;From the derivative with respect to $M$, we get: $$M^{2} = \mu \; X \mbox{Pf}M.$$ For the case of $X \neq 0$, again we can bring $M$ by a similarity transformation to the same form as before and this tells us that again all the top-right diagonal entries are the same and we obtain: $$m^{N_{c}+1} = 2^{N_{c} - 1}\Lambda_{N=1}^{2(N_{c}+1)}$$ which leads to, after using the RG equation: $$m = 2^{\frac{N_{c}-1}{N_{c}+1}}\mu\Lambda_{N=2}$$ which gives the moduli space. $\bullet \;\;\; N_{f}=N_{c}+2$ In this case, the effective potential by adding $\Delta ~W$ is given by: $$W_{eff} = -\frac{\mbox{Pf}M}{2^{N_{c}-1}\Lambda_{N=1}^{2N_{c}+1}}- \frac{1}{2\mu} \mbox{Tr}(M^{2})$$ which give us after extrematizing: $$M^{2} = -\mu \frac{\mbox{Pf}M}{2^{N_{c}-1}\Lambda_{N=1}^{2N_{c}+1}}.$$ Again $M$ can be brought to a simple form by a similarity transformation and all the diagonal entries are equal. After using the RG equation we get the moduli space given by: $$m = 2^{\frac{N_c-1}{N_c}} \mu\Lambda_{N=2}.$$ $\bullet \;\;\; N_{f} > N_{c}+2$ The theory that we have discussed until now is the electric theory which for this range of the number of flavors has a dual description in terms of a $Sp(N_{f}-N_{c}-2)$ gauge theory with $N_{f}$ flavors $q^{i}$ in the fundamental $(i = 1, \cdots, 2N_{f})$, gauge singlets $M_{ij}$ and a superpotential $$W = \frac{1}{4\lambda}M_{ij}q^{i}_{c}q^{j}_{d} J^{cd}.$$ where the scale $\lambda$ relates the scale $\Lambda_{N=1}$ of the electric theory and the scale $\tilde{\Lambda}_{N=1}$ of the magnetic theory by: $$\Lambda_{N=1}^{3(N_{c}+1)-N_{f}}\tilde{\Lambda}_{N=1}^{3(N_{f}-N_{c}-1)-N_{f}} = C (-1)^{N_{f}-N_{c}-1} \lambda^{N_{f}}$$ where the constant $C$ was found in [@ip] to be $C=16$. The effective superpotential is given as: W\_[eff]{} = W + W\_[ADS]{} +  W. If the vevs for the magnetic quarks are $0$, then the analysis is identical to those for the case $N_{f} < N_{c}$. If the vevs are not zero, then as in [@hoo] we can take a limit to approach $\mbox{Pf}M=0$ and to use the corresponding formula in order to compare with the M theory approach. In the $SU(N_c)$ case, where baryon exist, a specific choice has been taken such that the baryons have a specific interpretation in the M theory picture. $N=2$ Higgs Branch from M Theory ================================= In this section we study the moduli space of vacua of $N=2$ supersymmetric QCD by analyzing M theory fivebranes. We will consider the Higgs branch in terms of geometrical picture. Let us first describe the Higgs branch in the type IIA brane configuration. Following the paper of [@egk], the brane configuration contains three kind of branes: the two parallel NS5 branes extend in the direction $(x^0, x^1, x^2, x^3, x^4, x^5)$, the D4 branes are stretched between two NS5 branes and extend over $(x^0, x^1, x^2, x^3)$ and are finite in the direction of $x^6$, and the D6 branes extend in the direction of $(x^0, x^1, x^2, x^3, x^7, x^8, x^9)$. In order to study symplectic or orthogonal gauge groups, we will consider an O4 orientifold which is parallel to the D4 branes in order to keep the supersymmetry and is not of finite extent in $x^6$ direction. The D4 branes is the only brane which is not intersected by this O4 orientifold. The orientifold gives a spacetime reflection as $(x^4, x^5, x^7, x^8, x^9) \rightarrow (-x^4, -x^5, -x^7, -x^8, -x^9)$, in addition to the gauging of worldsheet parity $\Omega$. The fixed points of the spacetime symmetry define this O4 planes. Each object which does not lie at the fixed points ( i.e. over the orientifold plane), must have its mirror image. Thus NS5 branes have a mirror in $(x^4, x^5)$ directions and D6 branes have a mirror in $(x^7, x^8, x^9)$ directions. Another important aspect of the orientifold is its charge, given by the charge of $H^{(6)}=d A^{(5)}$ coming from Ramond Ramond(RR) sector, which is related to the sign of $\Omega^2$. In the natural normalization, where the D4 brane carries one unit of this charge, the charge of the O4 plane is $ \pm 1$, for $\Omega^2= \mp 1$ in the D4 brane sector. With the above preliminary setup, let us discuss about the two different branches of the theory. The Coulomb branch can be described when all the D4 branes lie between NS5 branes where no squark has vevs. To go to the Higgs branch, the D4 branes are broken on the D6 branes and are suspended between D6 branes being allowed to move on the $(x^{7}, x^{8}, x^{9})$ directions. Together with the gauge field component $A_{6}$ in the $x^{6}$ coordinate this gives two complex parameters to parameterize the location of the D4 branes. In [@hoo], for $SU(N_{c})$ case, the Coulomb branch and the Higgs branch share common directions and this comes from the fact that there are two different eigenvalues for $M$ which correspond to $r$ equal eigenvalues and $N_{c}-r$ equal eigenvalues. By turning on vevs for $r$ squarks, this gives rise to make the $r$ dimensional block of $M$ be nonzero. In brane language, this describes breaking $r$ D4 branes on the D6 branes and suspending the remaining $N_{c}-r$ D4 branes between the two NS5 branes. In the case of $Sp(N_c)$ gauge theory, there are only two possibilities: ${\bullet}$ All D4 branes are suspended between the two NS5 branes where no squark has vevs. ${\bullet}$ Some of D4 branes are broken on D6 branes [^4]. We, for simplicity, restrict ourselves to the case of all D4 branes being broken on D6 branes. See [@extra; @extra1] for more general cases. The motion of D4 branes along D6 branes describes the Higgs branch and for each D4 brane suspended between two D6 branes there exist two massless complex scalars parameterizing the fluctuations of the D4 brane. Because of the O4 orientifold we have to take into account D4 branes stretched between two D6 branes. The s-rule [@hw] allows only one D4 brane (and its mirror) between a NS5 brane and a single D6 brane (and its mirror). For $N=2$ theory because we have two NS5 branes, for both of them we have to impose the s-rule. Also, in contrast with $N=1$ theory, there are no complex scalars which correspond to D4 branes stretched between NS’5 branes and D6 branes. However remember that for $N=1$ theory[@egkrs] there are no complex scalar corresponding to a D4 brane stretched between NS5 brane and a D6 brane. The dimension of the Higgs moduli space is obtained by counting all possible breakings of D4 branes on D6 branes as follows: the first D4 brane is broken in $N_{f}-1$ sectors between the D6 branes (therefore the complex dimension is the twice of $N_f-1$), the second D4 branes is broken in $N_{f}-3$ sectors (the complex dimension is twice of $N_f-3$) and so on. But, besides that we have to consider the antisymmetric orientifold projection which eliminates some degrees of freedom, as explained in [@egkrs]. Then the dimension of the Higgs moduli space is given by: \[higgsdim\] 2\[(2N\_[f]{}-2-1) + (2N\_[f]{}-6-1) + + (2N\_[f]{}-4N\_[c]{}+2-1)\]= 4N\_[c]{}(N\_[f]{}-N\_[c]{})-2N\_[c]{} or $4N_{c}N_{f} - 2N_{c}(2N_{c}+1)$ where in the previous equations we have explicitly extracted 1 as a result of the antisymmetric orientifold projection.The overall factor $2$ in the left hand side is due to the mirror D6 branes and the result is very similar to the field theory result except an extra multiplicative factor 2 in the right hand side, because we consider here complex dimensions. In field theory, because of the $N_{f}$ vevs, the gauge symmetry is completely broken and there are $4N_{c}N_{f}-2N_{c}(2N_{c}+1)$ massless neutral hypermultiplets for a $N=2$ supersymmetric theory which thus exactly gives the dimension of the Higgs moduli space. Thus, the field theory results match the brane configuration results. Let us discuss how the above brane configuration appears in M theory context in terms of a generically smooth single M fivebrane whose worldvolume is ${\bf R^{1,3}} \times \Sigma$ where $\Sigma$ is identified with Seiberg-Witten curves[@as] that determine the solutions to Coulomb branch of the field theory. As usual, we write $v=x^4+i x^5, s=(x^6+i x^{10})/R, t=e^{-s}$ where $x^{10}$ is the eleventh coordinate of M theory which is compactified on a circle of radius $R$. Then the curve $\Sigma$, describing $N=2$ $Sp(N_c)$ gauge theory with $N_f$ flavors, is given[@lll] by an equation in $(v, t)$ space \[ah0\] t\^2-(v\^2 B(v\^2, u\_k)+ \_[N=2]{}\^[2N\_c+2-N\_f ]{} \_[i=1]{}\^[N\_f]{} m\_i )t+ \_[N=2]{}\^[4N\_c+4-2N\_f]{} \_[i=1]{}\^[N\_f]{} (v\^2-[m\_i]{}\^2)=0 where $B(v^2)$ is a polynomial of $v^2$ of degree $N_c$ with the coefficients depending on the moduli $u_k$, $ v^{2N_c} + u_2 v^{2N_c -2} + \cdots + u_{2N_c} $ and $m_i$ is the mass of quark[^5]. Including D6 Branes -------------------- In M theory, the type IIA D6 branes are the magnetic dual of the electrically charged D0 branes, which are the Kaluza-Klein monopoles described by a Taub-NUT space. This is derived from a hyper-Kähler solution of the four-dimensional Einstein equation. But we will ignore the hyper-Kähler structure of this Taub-NUT space. Instead, we use one of the complex structures, which can be described by[@lll] \[ah1\] y z=\_[N=2]{}\^[4N\_c+4-2N\_f]{} \_[i=1]{}\^[N\_f]{} (v\^2-[m\_i]{}\^2) in $\bf C^3$. The D6 branes are located at $y=z=0, v=\pm m_i$. This surface, which represents a nontrivial $\bf S^1$ bundle over $\bf R^3$ instead of the flat four dimensional space ${\bf R^3} \times {\bf{S^1}}$ with coordinates $(x^4, x^5, x^6, x^{10})$, is the unfolding of the $A_{2n-1}$ ($n=N_f$) singularity in general. The Riemann surface $\Sigma$ is embedded as a curve in this curved surface and given by \[ah2\] y+z=v\^2 B(v\^2)+ \_[N=2]{}\^[2N\_c+2-N\_f ]{} \_[i=1]{}\^[N\_f]{} m\_i. which reproduces to eq. (\[ah0\]) as we identify $y$ with $t$. &gt;From the symmetries existent in the type II A brane configuration, not all of them are preserved in the M-theory configuration. Our type IIA brane configuration has $U(1)_{4,5}$ and $SU(2)_{7,8,9}$ symmetries interpreted as classical $U(1)$ ans $SU(2)$ R-symmetry groups of the 4 dimensional theory on the brane worldvolume. The classical brane configuration is invariant both under the rotations. One of them, only $SU(2)_{7,8,9}$ is preserved in M theory quantum mechanical configuration but $U(1)_{4,5}$ is broken. This is a the same as saying that the $U(1)_{R}$ symmetry of the $N=2$ supersymmetric field theory is anomalous being broken by instantons. As discussed in section 2, the instanton factor is proportional with $\Lambda_{N=2}^{2N_{c}+2-N_{f}}$. So we have to see what is the charge of this factor under $U(1)_{4,5}$. We see this from equations (\[ah1\]) and (\[ah2\]) by considering $v$ of charge 2. We list below the charges of coordinates and parameter in the table: $z$ $ y$ $ v$ $\Lambda_{N=2}^{2N_{c}+2-N_{f}}$ ------------ ------------ ------ ---------------------------------- $4N_{c}+4$ $4N_{c}+4$ 2 $4N_{c}+4-2N_{f}$ In this case, the full $U(1)_{4,5}$ symmetry is restored, by assigning the instanton charge $(4N_{C}+4-2N_{f})$ to the $\Lambda$ factor. Note that whenever some $m_i$ are the same, the smooth complex surface (\[ah1\]) develops A-type singularity. But this is misleading since the hyper-Kähler structure becomes singular only if the D6 branes have the same position in $x^6$ and not only in $v$. When D6 branes with the coincident $m_i$’s are separated in the $x^6$ direction, the singular surface (\[ah1\]) must be replaced by a smooth one which is the resolution of $A$-type singularity. We will briefly describe the resolution of the $A$-type singularity. On the resolved surface, we also describe the parity due to orientifolding. Resolution of the $A$-type Singularity -------------------------------------- When all bare masses are the same but not zero (say $ m=m_i$), the surface (\[ah1\]) $S$ develops singularities of type $A_{n-1}$ at two points $y=z=0, v=\pm m$. By succession of blowing ups, we obtain a smooth complex surface $\tilde{S}$ which isomorphically maps onto the singular surface $S$ except at the inverse image of the singular points. Over each singular point, there exist $n-1$ rational curves $\bf CP^1$’s on the smooth surface $\tilde{S}$. These rational curves are called the exceptional curves. Let us denote the exceptional curves over the point $y=z=0, v=m$ by $C_1, C_2, \cdots , C_{n-1}$ and those over the point $y=z=0, v=-m$ by $C'_1, C'_2, \cdots , C'_{n-1}$. Here $C_i$’s (resp. $C'_i$) are arranged so that $C_i$ (resp. $C'_i$) intersects $C_j$ (resp. $C'_j$) only if $i= j\pm 1$. The symmetry due to orientifolding yields the correspondence between $C_i$ and $C'_i$. When we turn off the bare mass, that is, $m_i =0$ for all $i$, the singularity gets worse. Two singular points on the surface $S$ collides to create the $A_{2n-1}$ singularity. Now there are ${2n-1}$ exceptional curves on the resolved surface, which may be considered as a union of two previous exceptional curves $C_i$ and $C'_i$ and a new rational curve, say $E$ which connects these two exceptional curves. The orientifold provides a reflection between $C_i$ and $C'_i$ while inducing a self-automorphism on $E$. The more precise picture of the resolved surface is as follows: It is covered by $2N_f$ complex planes $U_1, U_2, \cdots , U_{2n}$ with coordinates $(y_1 =y, z_1), (y_2, z_2) ,\cdots , (y_n, x_n =y)$ which are mapped to the singular surface $S$ by U\_i (y\_i, z\_i) { [l]{} y= y\^i\_i z\^[i-1]{}\_i\ z = y\_i\^[2N\_f-i]{}z\_i\^[2N\_f+1-i]{}\ v= y\_iz\_i . The planes $U_i$ are glued together by $z_iy_{i+1} =1$ and $y_iz_i = y_{i+1}z_{i+1}$. The exceptional curve $C_i$ is defined by the locus of $y_i=0$ in $U_i$ and $z_{i+1}=0$ in $U_{i+1}$, the exceptional curve $E$ by $y_n=0$ in $U_n$ and $z_{n+1} =0$ in $U_{n+1}$ and the exceptional curve $C'_i$ by $y_{2n-i}=0$ in $U_{2n-i}$ and $z_{2n-i+1}=0$ in $U_{2n-i+1}$. The separation of the D6 branes in the $x^6$ direction corresponds to the infinitesimal direction on the singular surface $S$. Hence the position of the $D6$-brane may be interpreted as the $2N_f$ intersection points of the exceptional curves. The Higgs Branch ---------------- In this section, all the bare masses are turned off. In M theory, the transition to the Higgs branch occurs when the fivebrane intersects with the $D6$-branes, which means that the curve $\Sigma$ given by (\[ah2\]) passes through the singular point $y=z=v=0$. As a special case, we will consider when all D4 branes are broken on $D6$ branes in type IIA picture. Write the right hand side of (\[ah2\]) as: \[C\] v\^2B(v\^2) = v\^2(v\^[2N\_c]{} + u\_2 v\^[2N\_c -2]{} + + u\_[2N\_c]{}). Then our case corresponds to $u_k =0$ for all $k$. To describe the Higgs branch, we will study how the curve y+z = v\^[2N\_c]{} looks like in the resolved $A_{2N_f -1}$ surface. Here we ignored the factor $v^2$ in the right hand side of (\[C\]) because it is always contained in the orientifold plane $O4$ and thus does not contribute to the Higgs branch. Away from the singular point $y=z=v=0$, we may regard the curve as embedded in the original $y-z-v$ space because there is no change in the resolved surface in this region. Near the singular point $y=z=v=0$, we have to consider the resolved surface. On the $i$-th patch $U_i$ of the resolved surface, the equation of the curve $\Sigma$ becomes y\^i\_i z\^[i-1]{}\_i + y\_i\^[2N\_f-i]{}z\_i\^[2N\_f+1-i]{} = y\_i\^[2N\_c]{}z\_i\^[2N\_c]{} Now we may factorize this equation according to the range of $i$: For $i=1,\ldots , 2N_c$, we have y\^i\_i z\^[i-1]{}\_i (1 + y\_i\^[2N\_f -2i]{}z\_i\^[2N\_f+ 2-2i]{} - y\_i\^[2N\_c-i]{} z\_i\^[2N\_c+1-i]{} ) =0 , for $ i=2N_c +1, \ldots , 2N_f -2N_c$, y\_i\^[2N\_c]{} z\_i\^[2N\_c]{} (y\_i\^[i-2N\_c ]{} z\_i\^[i-2N\_c-1]{} + y\_i\^[2N\_f -i -2N\_c]{}z\_i\^[2N\_f -i-2N\_c +1]{} - 1) = 0 , and for $ i=2 N_f- 2N_c +1,\ldots ,2N_f$, y\_i\^[2N\_f-i]{}z\_i\^[2N\_f+1-i]{} ( y\_i\^[2i-2N\_f]{} z\_i\^[2i-2 -2N\_f]{} + 1 - y\_i\^[2N\_c-2N\_f +i]{}z\_i\^[2N\_c-2N\_f +i -1]{}) =0. Thus the curve consists of several components. One component, which we call $C$, is the zero of the last factor of the above equations. This extends to the one in the region away from $y=z=v=0$ which we have already considered. The other components are the rational curves $C_1, \ldots , C_{n-1}, E, C'_1,\ldots , C'_{n-1}$ with some multiplicities. For convenience, we rename the exceptional curves $E_1, \ldots , E_{2n-1}$ so that $E_i$ is defined by $y_i=0$ on $U_i$ and $z_{i+1} =0$ on $U_{i+1}$. Hence we can see from the above factorization that the component $E_i$ has multiplicity $l_i = i$ for $i=1,\ldots , 2N_c $; $l_i =2N_c $ for $i=2N_c +1, \ldots , 2N_f -2N_c $; and $l_i = {2N_f-i}$ for $i=2 N_f- 2N_c +1,\ldots 2N_f-1$. Note that the component $C$ intersects with $E_{2N_c}$ and $E_{2N_f -2N_c}$. To count the dimension of the Higgs branch, recall that once the curve degenerates and $\bf CP^1$ components are generated, they can move in the $x^7, x^8, x^9$ directions [@w1]. This motion together with the integration of the chiral two-forms on such $\bf CP^1$’s parameterizes the Higgs branch of the four-dimensional theory. However, we have to omit the component $E_{N_f} =E$ because this corresponds to the D4 brane connecting a D6 brane and its mirror. (Recall that $E$ was created after collision of two singular points which were mirror to each other.) Such a D4 brane is eliminated by the antisymmetric orientifold projection. Hence we have to put $l_{N_f} =0$. Now, after consideration of $\bf Z_2$ symmetry, the quaternionic dimension of the Higgs branch is \_[i=1]{}\^[2N\_f-1]{} l\_i= \_[i=1]{}\^[2N\_c]{} i + (2N\_f-4N\_c )N\_c= 2N\_c(N\_f-N\_c) -N\_c, which is the half of the complex dimension given in (\[higgsdim\]). Perhaps, a more appropriate geometric setting would have been a double covering of a $A_{N_f-1}$ singular surface with the embedded Seiberg-Witten curve. We will leave this for future investigation. The Rotated Configuration ========================= As we have seen that $N=2$ supersymmetry can be broken to $N=1$ by inserting a mass term of the adjoint chiral multiplet in field theory approach, we analyze the corresponding configuration in M theory fivebranes. While in the context of IIA picture, this turns out to be the rotation of one of NS5 branes, in order to describe this configuration, let us introduce a complex coordinate w=x\^8+i x\^9. Of course, the fivebranes are positioned at $w=0$ before the rotation. Notice that the D4 brane corresponds classically to an M fivebrane at $v=w=0$ and is extended in the direction of $s$, the NS5 brane is at $s=w=0$ and extended in $v$ and NS’5 brane is at $v=0, s=s_0$ and extended in $w$. Now we rotate only the left NS5 branes and from the behavior of two asymptotic regions which correspond to the two NS5 branes with $v \rightarrow \infty$ this rotation leads to the following boundary conditions. \[bdy-cond\] & & w v v , t \~v\^[ 2N\_c+2]{}\ & & w 0 v , t \~\_[N=2]{}\^[2(2N\_c+2-N\_f)]{}v\^[2N\_f-2N\_c-2]{} where the left(right) NS5 brane is related to the first(second) asymptotic boundary condition. Far from the origin of the $(v, w)$ plane which is the location of D4 branes, the location of the NS5 brane in the $s$ plane can be described by $s(NS5)=-2 R(N_f-N_c-1) \ln v$ while the NS’5 brane in the $s$ plane by $ s(NS'5)=-2 R(N_c+1) \ln w$. We discuss about the R-symmetries of the rotated configuration. After rotation, $SU(2)_{7,8,9}$ is broken to $U(1)_{8,9}$. In order this to be true, because of the connection between $v$ and $w$ in (\[bdy-cond\]), $\mu$ has to have charges under $U(1)_{4,5}\times U(1)_{8,9}$. $v$ has charge 2 under $U(1)_{4,5}$ while 0 under $U(1)_{8,9}$. $w$ has charge 0 under $U(1)_{4,5}$ and 2 under $U(1)_{8,9}$. So $\mu$ has $(-2,2)$ charges under $U(1)_{4,5} \times U(1)_{8,9}$. From the equations (\[ah1\]), (\[ah2\]) and (\[bdy-cond\]) we find the following values for the R-symmetry charges: $v$ $w$ $y=t$ $z$ $\mu $ $\Lambda_{N=2}^{2N_{c}+2-N_{f}} $ -------------- ----- ----- -------------- -------------- -------- ------------------------------------ $U(1)_{4,5}$ 2 0 $4(N_{c}+1)$ $4(N_{c}+1)$ -2 $4(N_{c}+1)-2N_{f}$ $U(1)_{8,9}$ 0 2 0 0 2 0 Since the rotation is only possible at points in moduli space at which all 1-cycles on the curve $\Sigma$ are degenerate [@sw], the curve $\Sigma$ is rational, which means that the functions $v$ and $t$ can be expressed as a rational functions of $w$ after we identify $\Sigma$ with a complex plane $w$ with some deleted points. Because of the symmetry of $v \rightarrow -v, w \rightarrow -w$ due to orientifolding, we can write: v\^2=P(w\^2), t=Q(w\^2). Since $v$ and $t$ become infinity only if $w=0, \infty$, these rational functions are polynomials of $w$ up to a factor of some power of $w$: $P(w^2) = w^{2a}p(w^2), Q(w^2) = w^{2b}q(w^2)$ where $a$ and $b$ are some integers and $p(w^2)$ and $q(w^2)$ are some polynomials of $w$ with only even degree terms which we may assume non-vanishing at $w=0$. Near one of the points at $w=\infty$, $v$ and $t$ behave as $v\sim \mu^{-1}w$ and $t \sim t^{2N_c +2}$ by (\[bdy-cond\]). Thus the rational functions are of the form P(w\^2) = w\^[2a]{}(w\^[2-2a]{} + )/\^[2]{} Q(w\^2) = \^[-2N\_c -2]{}w\^[2b]{}(w\^[2N\_c +2-2b]{} + ). Around a neighborhood $w=0$, the Riemann surface $\Sigma$ can be parameterized by $1/v$ which goes to zero as $w\to 0$. Since $w$ and $1/v$ are two coordinates around the neighborhood $w=0$ in the compactification of $\Sigma$ and vanish at the same point, they must be linearly related $w \sim 1/v$ in the limit $w \to 0$. The function $P(w^2)$ then takes the form P(w\^2) = . However the equation $v^2 = P(w^2)$ implies that $P(w^2)$ must be a square. Hence we have $w_+ = w_-$ and by letting $w_0^2 =w_{\pm}$ \[rot2\] P(w\^2)= which is a square of $w^2-w_0^2/\mu w$. Since $t\sim v^{2N_f -2N_c -2}$ and $w\sim 1/v$ as $w\to 0$, we get $b= N_c +1 -N_f$ and thus, Q(w\^2) = \^[-2N\_c -2]{}w\^[2(N\_c+1 -N\_f)]{}(w\^[2N\_f]{} + ). For $N_f >0$, by the equation $yz = v^{2N_f}$ defining the space-time, $t=0$ (i.e. $y=0$) implies $v=0$. Therefore the zeros of the polynomial $w^{2n_f} + \cdots$ are $\pm w_0$ of $P(w^2)$. Hence we have Q(w\^2)=\^[-2N\_c-2]{} w\^[2(N\_c+1-N\_f)]{} (w\^2-w\_0\^2)\^[N\_f]{} The value of $w_0$ can be determined by the fact that $v^2$ and $t$ satisfy the relation t+\_[N=2]{}\^[4N\_c+4-2N\_f]{} v\^[2N\_f]{}/ t=v\^2 B(v\^2)+ \_[N=2]{}\^[2N\_c+2-N\_f ]{} \_[i=1]{}\^[N\_f]{} m\_i Then by plugging $v^2$ and $t$ into the above equation we can read off $w_0$ from the lowest order term in power of $w$ && \^[-2N\_c-2]{} w\^[2(N\_c+1-N\_f)]{} (w\^2-w\_0\^2)\^[N\_f]{}+ \_[N=2]{}\^[4N\_c+4-2N\_f]{} v\^[2N\_f]{} \^[2N\_c2]{} w\^[-2(N\_c+1-N\_f)]{} (w\^2-w\_0\^2)\^[-N\_f]{}=\ && - B(v\^2)- \_[N=2]{}\^[2N\_c+2-N\_f ]{} \_[i=1]{}\^[N\_f]{} m\_i We want to calculate now $w_{0}$ from the above equation. For this, we will match the lowest order term in powers of $w$ in this equation. Actually we will look for terms with $w^{0}$ i.e., constant terms. In the left hand side the first term will always have a power of $w$, so does not contribute to the lowest order term. In the right hand side the last term will contain at least $w^{2(N_{c}-1)}$ so again does not contribute. After using the expression for $v$, the second term in left hand side will be $$\label{w1} \Lambda_{N=2}^{4N_c+4-2N_f}(w^{2}-w_{0}^{2})^{N_{f}} \mu^{2N_{c}+2-2N_{f}}w^{-2N_{c}-2}$$ In the right hand side, from the explicit form of $B(v^{2})$, the only term that can be independent of $w$ is obtained when we take only the highest power of $v$ which then will give $v^{2N_{c}}$. The contribution of this in the right hand side is as follows: $$\label{w2} \frac{(w^{2}-w_{0}^{2})^{2N_{c}+2}}{\mu^{2N_{c}+2}}w^{-2N_{c}-2}$$ We now extract the lowest order from (\[w1\]) and (\[w2\]) and make them equal to obtain the relation for $w_{0}$ finally: $$\label{w3} (-1)^{N_{f}} \Lambda_{N=2}^{4N_c+4-2N_f} w_{0}^{2N_{f}}\mu^{4N_{c}+4-2N_{f}} = w_{0}^{4N_{c}+4}$$ This gives us the value for $w$ to be $$\label{w4} w_{0} = (-1)^{\frac{N_{f}}{4N_{c}+4-2N_{f}}}(\mu\Lambda_{N=2}).$$ This gives us $w_{0}$ up to a $\bf{Z_{4N_{c}+4-2N_{f}}}$ rotation. We have here $4N_{c}+4-2N_{f}$ instead of $2N_{c}+2-N_{f}$ because of the symmetry $w\leftrightarrow -w$ implied by the orientifold. The rotated curve is now completely determined. $N=1$ SQCD ========== We study the $\mu\rightarrow\infty$ limit of our M fivebrane configuration and compare it with the known results in $N=1$ supersymmetric gauge theory. We have considered the rotation of the left NS5-brane, which corresponds to the asymptotic region $t\sim v^{2N_{c}+2}$ before the rotation and to $w\rightarrow\infty, v\sim\mu^{-1} w$ and $t\sim\mu^{-2N_{c}-2}w^{2N_{c}+2}$ after the rotation. We expect the relation $t\sim v^{2N_{c}+2}$ to hold also in the $\mu\rightarrow \infty$ because the D4 branes still end on the left NS5-brane in this limit. In order to preserve this relation, we should rescale $t$ by a factor $\mu^{2N_c+2}$ and introduce a new variable \[new\] = \^[2N\_[c]{}+2]{} t which will have the same dependence on $ v$ as $ t$ had before the rotation and this corresponds to the shift of the origin in the direction of $(x^6, x^{10})$. After putting $y =t$ in (\[ah1\]), the space-time is described by z=\^[2N\_[c]{}+2]{}\_[N=2]{}\^[4N\_[c]{}+4-2N\_[f]{}]{}\_[i=1]{}\^[N\_[f]{}]{} (v\^[2]{}-m\_[i]{}\^[2]{}), where $\tilde{y} =\mu^{2N_{c}+2} y$. This equation describes a smooth surface in the limit $\mu \rightarrow\infty$ provided the product \^[2N\_[c]{}+2]{}\^[4N\_[c]{}+4-2N\_[f]{}]{} remains to be finite. We define this product as follows: \_[N=1]{}\^[2(3N\_[c]{}+3-N\_[f]{})]{} = \^[2N\_[c]{}+2]{} \_[N=2]{}\^[4N\_[c]{}+4-2N\_[f]{}]{} which is nothing but the RG matching condition of the four-dimensional field theory. Note that this spacetime and M fivebrane is under the rotation groups $U(1)_{4,5}$ and $U(1)_{8,9}$ in appropriate way discussed before. We want to see what are the deformations of the $N=2$ Coulomb branch after the rotation and the limit $\mu\rightarrow\infty$. Pure $Sp(N_{c})$ Theory ----------------------- Without matter, the curve (\[ah0\]) describing the $N=2$ Coulomb branch is given by: t\^[2]{} - C\_[N\_[c]{}+1]{}(v\^[2]{},u\_[k]{})t + \^[4N\_[c]{}+4]{}\_[N=2]{} = 0 where $C_{N_{c}+1}(v^{2},u_{k}) = v^2B(v^2, u_{k})$. This curve is completely degenerate at $(N_{c} +1)$ points on the Coulomb branch. At one these points, the curve has the following form v\^[2]{} = \_[N=2]{}\^[4]{} t\^[-1/(N\_[c]{}+1)]{} + t\^[1/(N\_[c]{}+1)]{}. Thus its rotation is v\^[2]{} &=& \^[2]{} \_[N=2]{}\^[4]{} w\^[-2]{} + \^[-2]{} w\^[2]{}\ t &=& \^[-2N\_[c]{}-2]{}w\^[2N\_[c]{}+2]{}. Now we rescale $t$ as $\tilde{t} = \mu^{2N_{c}+2} t$ and send $\mu$ to $\infty$ by keeping $\Lambda_{N=1}$ finite. It is easy to see that the curve becomes in this limit: v\^[2]{} &=& \_[N=1]{}\^[6]{} \^[-1/(N\_[c]{}+1)]{}\ w\^[2]{} &=& \^[1/(N\_[c]{}+1)]{}. where the RG matching condition is used. Introducing Massless Matter --------------------------- For the rotated configuration we use now the expressions for $v^{2}=P(w^{2})$ and $t=Q(w^{2})$ given before in terms of new variables. We again introduce the rescaled $\tilde{t}$ which is then given by \[rot1\] = w\^[2(N\_[c]{} + 1 - N\_[f]{})]{} (w\^[2]{} - w\_[0]{}\^[2]{})\^[N\_[f]{}]{}. For $ v$ and $ w$ we have the relation by remembering that the order parameters $u_k$ are independent of $\mu$, are powers of $\Lambda_{N=2}$ and vanish in the $\mu \rightarrow \infty$ \[rot3\] = \_[N=1]{}\^[6N\_[c]{}+6-2N\_[f]{}]{}v\^[2(N\_[f]{}-N\_[c]{}-1)]{}. When $\mu\rightarrow\infty$, the limit for (\[rot1\]) and (\[rot2\]) is given by the behavior of $w_{0}\sim \mu\Lambda_{N=2}$. By using the relation: \[rel1\] \_[N=2]{} = (\_[N=1]{}\^[3N\_[c]{}+3-N\_[f]{}]{} \^[N\_[c]{}+1-N\_[f]{}]{})\^, we have three regions for $N_f$. Let us see how the curves look like: $\bullet \;\;\; N_{f} < N_{c} + 1$ (\[rel1\]) tells us that $\mu\Lambda_{N=2}$ diverges and $w_{0}$ also diverges. Therefore the curve becomes infinite in the $x^{6}$ direction. So there is no field theory in four dimensions. This is just the same as saying that there is no supersymmetric vacua in the $N=1$ theory. $\bullet \;\;\; N_c+1 < N_{f} < 2(N_{c} + 1)$ In this case, from (\[rel1\]) it is easy too see that $\mu\Lambda_{N=2} = 0$ in the limit $\mu\rightarrow\infty$ and (\[rot1\]), (\[rot2\]) and (\[rot3\]) transform into: &=& w\^[2(N\_[c]{}+1)]{}\ v w &=& 0\ v\^[2(N\_[c]{}+1)]{} &=& \^[6(N\_[c]{}+1) - 2N\_[f]{}]{} v\^[2N\_[f]{}]{}.As explained in [@hoo], only the limit $\tilde{t}\ne 0, w=0$ is allowed, so the interpretation of the previous equation is that the curve splits into two components in this limit: $C_{L} (\tilde{t} = w^{2(N_{c} + 1)}, v = 0)$ and $C_{R} (\tilde{t} = \Lambda_{N=1}^{6N_{c}+6-2N_{f}} v^{2N_{f}-2N_{c}-2}, w=0)$ where the component $C_{L}$ corresponds to the NS’5 brane which was rotated and on the other hand, $C_{R}$ refers to the NS5 brane and the attached D4-branes. ${\bullet} \;\;\; N_{f} = N_{c} + 1$ In this case, the RG matching condition tells that $\mu\Lambda_{N=2}$ is equal to $\Lambda_{N=1}^{2}$. The equations (\[rot3\]), (\[rot1\]) and (\[rot2\]) become: &=& \_[N=1]{}\^[4(N\_[c]{}+1)]{}\ &=& (w\^[2]{}-w\_[0]{}\^[2]{})\^[N\_[c]{}+1]{}\ vw &=& 0. The correct interpretation of these equations is that the curve also splits into two components: $C_{L} (\tilde{t} = (w^{2} - w_{0}^{2})^{N_{c} + 1}, v = 0)$ and $C_{R} (\tilde{t} = \Lambda_{N=1}^{4(N_{c}+1)}, w = 0)$. For $SU(N_c)$ group the cases $N_{f} = N_{c} + 1$ and $N_{f} > N_{c} + 1$ differed from each other because the first different non-baryonic branch roots went to different limits and the second all non-baryonic branch roots have the same limit. That was determined by the fact that $M$, the meson matrix, had 2 different values for the diagonal entries. However in our case, for the $Sp(N_c)$ group, there is only one kind of top-right diagonal entry, so we do not see those difference appeared in $SU(N_c)$ gauge group and also do not have any baryonic branch. Massive Matter -------------- Before starting our discussion of introducing matter for our case, let us briefly examine the difference between the results of [@hoo] and [@biksy; @ss] for the case of massive matter when we consider $SU(N_c)$ gauge group. Actually we will just compare the equation (5.28) of [@hoo], and (4.6) and (5.1) of [@ss]. Notice that the second equation in (4.6) of [@ss] and the first one in (5.28) of [@hoo] are the same: v w = (m\_[f]{}\^[N\_[f]{}]{}\_[N=1]{}\^[3N\_[c]{}-N\_[f]{}]{})\^[1/N\_[c]{}]{}. The first equation of (4.6) in [@ss] is not the same as the second one of (5.28) in [@hoo]. Rather the equation of (5.28) looks like the equation (5.1) of [@ss] because there we have the dependence $t-w$. Recall that the vev for $M$ are given for equal squark masses by: m = m\_[f]{}\^\_[N=1]{}\^[ ]{}. The relation between $\tilde{t}$ and $w$ can be rewritten as: \[equi1\] w\^[N\_[f]{} - N\_[c]{}]{} = (w - m)\^[N\_[f]{}]{}. When we write it in terms of $t$ and $v$, this turns out: \[equi2\] v\^[N\_[c]{}]{} = (-1)\^[N\_f]{} \_[N=1]{}\^[3N\_[c]{} - N\_[f]{}]{} (v - m\_[f]{})\^[N\_[f]{}]{} The relations (\[equi1\]) and (\[equi2\]) are just the equivalent of (4.6) and (5.1) in [@ss]. In (\[equi2\]) we have a supplementary power of $\Lambda$ as compared with [@ss]. This is due to the fact that in [@hoo], $t$ has a dimension of mass by its definition, but in [@ss] $t$ is dimensionless. The power of $\Lambda$ is just used to match the dimension of mass. We then find that the same curve can be written in terms of $ t-w$ and $t-v$. The two descriptions correspond to the $N=1$ duality [@se1], between theories with gauge groups $SU(N_{c})$ and $SU(N_{f}-N_{c})$, as we can see from the dependence of $t$ as a function of $v$ and $w$ in (\[equi1\]) and (\[equi2\]). So the electric-magnetic duality can be observed also from the set-up of [@hoo]. The connection between [@hoo] and [@ss] may become clear only after we should introduce D6 branes in the set-up of [@ss] which is a very interesting direction to pursue and investigate. What happens when we consider our case? If all the quarks have equal mass $m_{f}$, then the curve for $\mu\rightarrow\infty$ becomes: \[s0\] v\^[2]{}w\^[2]{} &=& (m\_[f]{}\^[N\_[f]{}]{}\_[N=1]{}\^[3(N\_[c]{}+1)-N\_[f]{}]{})\^[2/(N\_[c]{}+1)]{}\ &=& w\^[2(N\_[c]{}+1-N\_[f]{})]{}(w\^[2]{}-()\^[2/(N\_[c]{}+1)]{})\^[N\_[f]{}]{}. By the relation which connects the vev of $M$ and the masses of quarks, we obtain: w\^[2(N\_[f]{}-N\_[c]{}-1)]{} = (w\^[2]{}-m\^[2]{})\^[N\_[f]{}]{}. \[s1\] When we write it in terms of $ t$ and $v$, this gives rise to: v\^[2N\_[c]{}+2]{} = (-1)\^[N\_f]{} \_[N=1]{}\^[6(N\_[c]{}+1)-2N\_[f]{}]{} (v\^[2]{}- 2\^ [m\_f]{}\^[2]{})\^[N\_[f]{}]{}. \[s2\] &gt;From the relations (\[s1\]) and (\[s2\]) we again see the duality between the theories with gauge groups $Sp(N_{c})$ and $Sp(N_{f}-N_{c}-2)$. The above relations are similar to the equations (2) and (6) of [@cs]. The equation between $ v$ and $w$ is just the same as those in [@cs]. The electric-magnetic duality appears as an interchange $v-w$, the curve describing the M-theory configuration being unique. Again, in [@cs] $t$ is dimensionless while in our case $\tilde{t}$ has a specific dimension. Therefore we see a power of $\Lambda$ in (\[s2\]). So (\[s1\]) and (\[s2\]) give the electric-magnetic duality in our approach. It would be very interesting to introduce D6 branes in [@cs] and to see the coincidence of the corresponding solution with ours. Now take the limit $m_{f}\rightarrow\infty$. After we integrate out the massive flavors and use the matching of the running coupling constant \_[N=1]{}\^[3(N\_[c]{}+1)]{}=m\_[f]{}\^[N\_[f]{}]{} \_[N=1]{}\^[3(N\_[c]{}+1)-N\_[f]{}]{} to rewrite the equation (\[s0\]) as v\^[2]{}w\^[2]{} &=& \^[6]{}\_[N=1]{}\ &=& w\^[2(N\_[c]{}+1-N\_[f]{})]{} ( w\^[2]{} - 2\^ )\^[N\_[f]{}]{}. If we keep $\tilde{\Lambda}_{N=1}$ finite while sending $m_{f}$ to $\infty$, this again reduces to the pure Yang-Mills result. Conclusions =========== In the present work we considered the M theory description of the supersymmetry breaking from $N=2$ to $N=1$ for the case of symplectic gauge group $Sp(N_c)$ obtaining many aspects of the strong coupling phenomena. In this case no baryon can be constructed so the only Higgs branch is the non-baryonic branch. In field theory approach, starting with a $N=2$ supersymmetric gauge theory and giving mass to the adjoint chiral multiplet, the extremum of the superpotential gave us a unique solution for the expectation value for the meson matrix $M$ in which $\mbox{Tr}M=0$ as opossed to the $SU(N_{c})$ case where $\mbox{M}\ne 0$. In the M theory fivebrane approach, we discussed first the unrotated configuration which corresponds to an $N=2$ theory, with or without D6 branes. For the case without D6 branes, M theory fivebrane configuration is a single fivebrane with the world volume ${\bf R^{1,3}} \times \Sigma$ where $\Sigma$ is the Seiberg-Witten curve of the gauge group $Sp(N_c)$. By introducing D6 branes, we have considered the complex structure of the corresponding Taub-NUT space which is the same one as those of the ALE space of $A_{2n-1}$-type and resolved the $A_{2n-1}$ singularity. One of the most important aspects of the $Sp(N_{c})$ gauge theory was the O4 orientifold which is parallel to D4 branes . Its antisymmetric projection which eliminates some degrees of freedom was essential in matching the dimension of the Higgs moduli space in IIA brane approach with the one of the field theory. This observation was used not only in type IIA picture when counting the number of D4 branes suspended between the D6 branes but also in the M theory picture when counting the multiplicities of the rational curves. It is known [@hov; @vafa] that A type singularity by imposing the ${\bf Z_2}$ symmetry, due to the orientifolding, leads to D type singularity. We expect that our M theory fivebrane argument starting from D type singularity can go similarly and will see how the interrelation between two types singularities plays a role. For rotated branes we used the coordinates $v=x^{4}+ix^{5}$ and $w=x^{8}+ix^{9}$, the position of the D4 branes in the $w$ direction being identified with the eigenvalues of the meson matrix $M$. We found at most one eigenvalue $w_{0}$ for the asymptotic position of the D4 branes which is consistent with the field theory result where only one eigenvalue for $M$ was found. We connected $w_{0}$ with the unique eigenvalue of $M$. In section 5 we have obtained the forms for the rotated curves, for pure gauge group and for massive and massless matter. In all of our discussions as well as in many exciting works which appeared recently, many results obtained in field theory were rederived in M theory which makes M theory approach an extraordinary laboratory to derive results which were very difficult to obtain only by pure field theory methods. Now we have a clearer view over the strongly coupled phenomena of supersymmetric theories. 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[**B461**]{} (1996) 437, hep-th/9509175. C. Vafa, hep-th/9707131. [^1]: chahn@ctp.snu.ac.kr [^2]: oh@arch.umsl.edu [^3]: tatar@phyvax.ir.miami.edu [^4]: There are recent papers on this issue [@extra; @extra1]. [^5]: Note that this $m_i$ is nothing to do with the element of meson field $M$. Unfortunately we used same notation.

--- abstract: 'We show that the Right-Angled Coxeter group $C=C(G)$ associated to a random graph $G\sim \mathcal{G}(n,p)$ with $\frac{\log n + \log\log n + \omega(1)}{n} \leq p < 1- \omega(n^{-2})$ virtually algebraically fibers. This means that $C$ has a finite index subgroup $C''$ and a finitely generated normal subgroup $N\subset C''$ such that $C''/N \cong \mathbb{Z}$. We also obtain the corresponding hitting time statements, more precisely, we show that as soon as $G$ has minimum degree at least 2 and as long as it is not the complete graph, then $C(G)$ virtually algebraically fibers. The result builds upon the work of Jankiewicz, Norin, and Wise and it is essentially best possible.' author: - Gonzalo Fiz Pontiveros - Roman Glebov - Ilan Karpas bibliography: - 'legal.bib' nocite: '[@*]' title: 'Virtually Fibering Random Right-Angled Coxeter Groups' --- Introduction ============ A group $K$ *virtually algebraically fibers* if there is a finite index subgroup $K'$ admitting a surjective homomorphism $K'\to \ZZ$ with finitely generated kernel. This notion arises from topology: a $3$-manifold $M$ is virtually a surface bundle over a circle precisely when the fundamental group of $M$ virtually algebraically fibers (see the result of Stallings [@Sta61]). A *Right-Angled Coxeter group* (RACG) $K$ is a group given by a presentation of the form $$\left\langle x_1, x_2, \ldots x_n \;|\;x_i^2, [x_i, x_j]^{\sigma_{ij}}\;: 1\leq i< j\leq n\right\rangle$$ where $\sigma_{ij}\in \{0,1\}$ for each $1\leq i <j\leq n$. One can encode this information with a graph $\Gamma_{K}$ whose vertices are the generators $x_1,\ldots, x_n$ and $x_i\sim x_j$ if and only if $\sigma_{ij}=1$. Conversely given a graph $G$ on $n$ vertices, we will denote the corresponding RACG by $K(G)$. Random Coxeter groups have been of heightened recent interest, see for instance Charney and Farber [@charney2012random], Davis and Kahle [@davis2014random], and Behrstock, Falgas-Ravry, Hagen, and Susse [@behrstock2015global]. Recently, Jankiewicz, Norin, and Wise [@Jankiewicz_Virtually] developed a framework to show virtual fibering of a RACG using Betsvina-Brady Morse theory [@Bestvina_Morse_1997] and ultimately translated the virtual fibering problem for $K$ into a combinatorial game on the graph $\Gamma_K$. The method was successful on many special cases and also allowed them to construct examples where Betsvina-Brady cannot be applied to find a virtual algbraic fibering. A natural question to consider is whether this approach is successful for a ‘generic’ RACG, i.e., given a probability measure $\mu_n$ on the set of RACG’s of rank at most $n$, is it true that a.a.s. as $n\to \infty$, a group sampled from $\mu_n$ virtually algebraically fibers. This question is also considered in [@Jankiewicz_Virtually], specifically they consider sampling $\Gamma_K$ from the Erdős-Renyi random graph model $\mathcal{G}(n,p)$ and they prove the following result: \[JNW\] Assume that $$\frac{(2\log{n})^{\frac{1}{2}}+\omega(n)}{n^{\frac{1}{2}}}\leq p < 1 -\omega(n^{-2}),$$ and let $G$ be sampled from $\mathcal{G}(n,p)$. Then, asymptotically almost surely, the associated Right-Angled Coxeter group $K(G)$ virtually algebraically fibers. In this paper we extend this result to the smallest possible range of $p$, in fact we prove a hitting time type result. Namely we show that as soon as $\Gamma_K$ has minimum degree $2$ then a.a.s. $K$ virtually algebraically fibers. \[Main\] Let $G_0,G_1,\ldots, G_{\binom{n}{2}}$ denote the random graph graph process on $n$ vertices where $G_{i+1}= G_i\cup \{e_i\}$ and $e_i$ is picked uniformly at random from the non-edges of $G_i$. Let $T=\min_{t}\;\{t\;: \delta(G_t)=2\}$, then a.a.s. the random graph process is such that $K(G_m)$ virtually algebraically fibers if and only if $T\leq m <\binom{n}{2}$. In particular for any $p$ satisfying $$\frac{\log{n}+\log\log{n}+\omega(n)}{n}\leq p < 1-\omega(n^{-2})$$ and $G~\mathcal{G}(n,p)$, the random Right-Angled Coxeter group $K(G)$ virtually algebraically fibers a.a.s. The paper is structured as follows. In Section \[sec:legalsystems\], we establish the graph-theoretic framework used in the remainder of the paper, and show that the minimum degree condition is in fact necessary for $n\geq 3$ and hence Theorem \[Main\] is best possible. In Section \[sec:dense\], we look at the opposite extreme and prove Theorem \[Main\] for very large $p$. The proof presented in Section \[WeakBound\] mainly serves to provide the reader with the concepts and the intuition used later; it shows Theorem \[Main\] for most of the range of the edge probability. In Section \[sec:construction\], we present the construction used for the final part of the proof of Theorem \[Main\]. Then in Section \[sec:pseudorandom\] we prove Theorem \[Main\] in the remaining case in the pseudorandom setting, i.e., we prove the statement for every graph satisfying certain (deterministic) properties. Finally, in Section \[sec:proof\] we put the pieces together, and show that indeed in the remaining interval for $p$ in Theorem \[Main\], the random graph a.a.s. satisfies the conditions required in Section \[sec:pseudorandom\], thus completing the proof. ?? Notation -------- $V$ always denotes the vertex set; floor/ceiling; $G(n,p)$ and relation to the random graph process; $\log$ is base $e$ Legal Systems {#sec:legalsystems} ============= In this section we follow the definitions in [@Jankiewicz_Virtually] to present the combinatorial game introduced in [@Jankiewicz_Virtually] used to construct virtual algebraic fiberings of Right-Angled Coxeter groups. Let $G=(V,E)$ be a graph. We say that a subset $S\subset V$ is a *legal state* if both $S$ and $V\setminus S$ are non-empty [*connected subsets*]{} of $V$, i.e., the corresponding induced graphs are connected and non-empty. For each $v \in V$, a *move at $v$* is a set $M_v\subseteq V$ satisfying the following: - $v\in M_v$ - $N(v)\cap M_v=\emptyset$ Let $\mathcal{M}=\{M_v \; : v \in V\}$ denote a set of moves. We will identify subsets of $V$ as elements of ${\mathbb{Z}_{2}}^{V}$ in the obvious way. Thus each state and each move correspond to elements of ${\mathbb{Z}_{2}}^{V}$ and we will think of moves acting on states via group multiplication (or addition in this case). For a graph $G$, a state $S\subseteq V(G)$, and a set of moves $\mathcal{M}=\{M_v \; : v \in V\}$, the triple $(G, S, \mathcal{M})$ is a *legal system* if for any element $g \in \langle\mathcal{M}\rangle$, $g(S)$ is a legal state of $G$. Let $(G,S,\mathcal{M})$ be a legal system, then the RACG $K(G)$ must virtually algebraically fiber. To elucidate the notion of a legal system, let us look at some toy examples (see Figure \[fig:examples\]) and ask whether each of these graphs contains a legal system. \[fig:examples\] {width="50.00000%"} \[cherry\] Let $G=(V,E)$ be a graph with three vertices $V=\{v,u_1,u_2\}$ and two edges $E=\{\{v,u_1\},\{v,u_2\}\}$. We show that $G$ has a legal system. Our initial legal state will be $S=\{u_1\}$. For our set of moves we choose $M_v=\{v\}$ (note that this is the only possible choice for the move at $v$), $M_{u_1}=M_{u_2}=\{u_1,u_2\}$. Then the group generated by the moves of the graph, written as a collection of sets, is $\langle\mathcal{M}\rangle=\{\{v\},\{u_1,u_2\},\{v,u_1,u_2\},\emptyset\}$. Hence, for any element $g\in \langle\mathcal{M}\rangle$, $g(S)$ is either a set of the form $\{u_i\}$ or $\{v,u_i\}$, for $i=1,2$, and in any case a legal state. Thus, $(G,S,\mathcal{M})$ is a legal system. The graph in Example \[cherry\] is unique in the sense that it is the only graph with a vertex of degree $1$ on at least $3$ vertices which contains a legal system. We prove this later in Proposition \[degree 1\]. Next, we look at an example of a graph without a legal system. We proceed by exhaustion. \[bowtie\] Let $V=\{v,u_1,u_2,w_1,w_2\}$, $E=\{\{v,u_i\},\{v,w_i\}, \{w_1,w_2\},\{u_1,u_2\}\}$, $i=1,2$. Let $G=(V,E)$. Assume by contradiction that $(G,S,\mathcal{M})$ is a legal system. Since $v$ is connected to all other vertices in the graph, we must have $M_v=\{v\}$. For the same reason, $v$ can not belong to any other move apart from $M_v$. Hence, we can assume without loss of generality that $v \notin S$. Since $S$ is a connected subset of $V$, we can again assume without loss of generality that $S=\{u_1\}$ or $S=\{u_1,u_2\}$. In the latter case, $M_{w_i}=\{u_1,u_2,w_i\}$ for $i=1,2$, because by the definition of a move, it must be the case that $\{w_i\}\subseteq M_{w_i} \subseteq \{w_i,u_1,u_2\}$, and if $u_1$ or $u_2$ would not belong to $M_{w_i}$, then $M_{w_i}S$ would not be a legal state. But then the set $\{w_1,w_2\} \in \langle\mathcal{M}\rangle$, and $\{w_1,w_2\}S=\{w_1,w_2,u_1,u_2\}$ is not a legal state. In the former case, from similar consideration, it must be the case that $M_{w_i}=\{w_i,u_1\}$ for $i=1,2$, but then again $\{w_1,w_2\}\in \langle\mathcal{M}\rangle$, and $\{w_1,w_2\}S=\{w_1,w_2,u_1\}$ is not a legal state. Next we show that Theorem \[Main\] is is essentially best possible. In fact, any graph on more than $3$ vertices with minimum degree at most $1$ does not have a legal system. \[degree 1\] Let $G$ be a graph on $n$ vertices with $n\geq 4$ and suppose that $\delta(G)\leq 1$. Then $G$ does not have a legal system. For graphs with isolated vertices the statement is obvious, therefore we can assume that $\delta(G)=1$. We argue by contradiction. Suppose there exists an $S\subset V(G)$ and a set of moves $\mathcal{M}$ such that the triple $(G,S, \mathcal{M})$ is a legal system. Let $v$ be a vertex with $d(v)=1$ in $G$ and let $u$ be its unique neighbour. Since $u\notin M_v$ and $v\notin M_u$ we may assume without loss of generality that both $u,v \in S$ (if not then simply take a suitable translate). Observe that $v\in M_u(S)$ and $u\notin M_u(S)$. Furthermore, by our assumption the set $M_u(S)$ is connected and thus $M_u(S)=\{v\}$. Recall that $M_u$ is a set of non-neighbours of $u$ together with $u$ itself, and hence $S=M_u(M_u(S))=\{u,v\}$ which in turn implies that $M_u=\{u\}$. For every $g \in \langle \mathcal{M} \rangle$, we have that either $$\label{or} g(S)\in \{\{v\},\{u,v\}\} \text{\; or \; } M_v(g(S))\in \{\{v\},\{u,v\}\}.$$ Note that $u$ either belongs to both sets $g(S)$ and $M_v(g(S))$ or to neither of them, since $u \notin M_v$, whereas $v$ belongs to exactly one of these sets. Assume without loss of generality that $v \in g(S)$. If $u \in g(S)$, then $M_u(g(S))=g(S) \setminus \{u\}$ is a connected set containing $v$ but not $u$, and thus must be equal to $\{v\}$. This means that $g(S)=\{u,v\}$, providing . If, on the other hand, $u \notin g(S)$, then $g(S)$ is a connected set which contains $v$ but not $u$, which again means that $g(S)=\{v\}$, again providing . Thus, at least half the sets in $\left\{ g(S):~g\in \langle \mathcal{M} \rangle\right\}$ are either $\{ v \}$ or $\{ u,v \}$, which means that $\left|\left\{ g(S):~g\in \langle \mathcal{M} \rangle\right\}\right| \leq 4$, and therefore $M_w\in \left\{M_v, M_v\cup\{u\}\right\}$ for any $w\neq u$. Hence, $w \in M_v$ for any $w \neq u$, which means that $M_v=V\setminus\{u\}$. Furthermore, as $G$ has no isolated vertices we must have that $M_w=V\setminus\{u\}$ for any $w\neq u$ an hence $G$ must be in fact a star. The only way $M_v(M_u(S))=V\setminus \{u,v\}$ can be connected is if $n\leq 3$, a contradiction. Very dense regime {#sec:dense} ================= In this section we show Theorem \[Main\] in the simpler range of very dense graphs. \[densethm\] Let $G \in \mathcal{G}(n,m)$, i.e., a graph with $m$ edges picked uniformly at random. Suppose that $0.98\binom{n}{2}\leq m< \binom{n}{2}$. Then a.a.s $G$ has a legal system. Let $H$ denote the complement of $G$ and observe that $H\sim \mathcal{G}(n,t)$ where $t={n\choose 2}-m$. The strategy to find a legal system is a simple one: first we find a maximal matching $F=\{\{u_1,v_1\},\ldots ,\{u_k,v_k\}\} \subset H$. Then let $S=\{u_i\;: 1\leq i \leq k\}$ and for each $1\leq i \leq k$ set $M_{u_i}=M_{v_i}=\{u_i,v_i\}$ and $M_v=\{v\}$ for all $v\notin F$. We claim that with high probability this defines a legal system for $G$. Note that $V \in \langle \mathcal{M} \rangle$ and hence for any $g \in \langle \mathcal{M} \rangle$, the complement of $g(S)$ can be expressed as $V \setminus g(S)=(Vg)(S)$, in other words the orbit of $S$ is closed under taking complements. In particular, to prove the claim, it is enough to show that for any $g \in \langle \mathcal{M} \rangle$, the set $g(S)$ is connected. Furthermore, since $H$ contains at least one edge, $F$ must also be non-empty and so $g(S)\neq \emptyset$ for any $g\in \langle\mathcal{M}\rangle$. Thus it is sufficient to show that for every $g\in \langle\mathcal{M}\rangle$, the set $g(S)$ is connected. By maximality of $F$, we know that $G[V\setminus F]$ is a clique in $G$ (equivalently an independent set in $H$). Hence, by our choice of moves, the only way that $g(S)$ can fail to be connected is if there exists some $v\in V$ such that $$\tag{$\star$} \label{cond} |N_{H}(v)\cap \{u_i,v_i\}|\geq 1 \text{ for at least $\lceil k/2\rceil$ indices } i\in [k].$$ We now consider two cases.\ [***Case 1***:]{} $t=o(n^{\frac{1}{2}})$. Observe that the expected number of paths of length two in $G(n,t)$ is at most $n^3(\frac{2t}{n^2})^2 \to 0$. In particular, by Markov, with high probability no two edges are incident in $H$. In particular cannot happen with high probability.\ [***Case 2***:]{} $t=\Omega(n^{\frac{1}{2}})$. Observe that the expected number of independent sets of size $l$ in $\mathcal{G}(n,t)$ is $$O\left(n^{l}\left(1-\frac{2t}{n^2}\right)^{l^2/2}\right) =O\left( n^{l}e^{-\frac{1}{2}n^{-\frac{3}{2}}l^2}\right).$$ In particular, with high probability $H$ has no independent set of size $\Omega(n^{\frac{3}{4}})$. It follows that with high probability $|F|=(1-o(1))n=2k$. On the other hand, if occurs, we must have that there exists $v \in V$ such that $d_H(v)\geq k/2$, and by Chernoff ??add reference to chernoff from somewhere, perhaps?? the probability of such high degree vertex is vanishingly small. Let $G\in \mathcal{G}(n,p)$ where $0.99\leq p< 1- \omega(n^{-2})$. Then a.a.s. $G$ has a legal system. Sampling from $G$ from $\mathcal{G}(n,p)$ is equivalent to first choosing a random number $m\sim \text{Bin}({n\choose 2},p)$ of edges and then sampling $G$ from $\mathcal{G}(n,m)$. For $p$ in the above range we have that a.a.s. $0.98{n\choose 2}\leq m < {n\choose 2}$ and the corollary follows follows from Theorem \[densethm\]. Observe that this upper bound is also optimal since for $p=1-cn^{-2}$, the probability that $G$ is in fact the complete graph is bounded away from $0$ and it is easy to see that the complete graph cannot have a legal system. A weaker bound {#WeakBound} ============== Before we attempt to prove the main result of the paper we will give here a simple proof for a slightly smaller range of $p$. Namely we will show the following: \[big-p\] Let $\frac{3\log n}{n}\leq p \leq 0.99$. Then a.a.s. $G\sim \mathcal{G}(n,p)$ has a legal system. This achieves several purposes. We will be able to already introduce some of the ideas and statements required for the following section, motivate definitions in the construction and also present simplified computations by having a more restricted range of $p$. An *equitable* colouring of a graph $G$ is a proper colouring of the vertices of $G$, where the sizes of any two colour classes differ by at most $1$. The [*equitable chromatic number*]{} of $G$ is the smallest integer $k$ such that there exists an equitable colouring of $G$ with $k$ colours. We use the following theorem of Krivelevich and Patkós [@KrivPat09]. \[K-P\] Let $G\sim \mathcal{G}(n,p)$. There exists a constant $C$ such that asymptotically almost surely the following holds: - If $ \frac{C}{n}\leq p\leq \log{n}^{-8}$, then $$\chi_{=}(G)\leq \frac{np}{(1-o(1)\log{(np)}}.$$ - If $ \log{n}^{-8}<p<0.99$, then $$\frac{n}{2\log_b{n}-\log{\log_b{n}}}\leq \chi_{=}(G)\leq \frac{n}{2\log_b{n}-8\log{\log_b{(np)}}},$$ where $b=\frac{1}{1-p}$. Note that when $p \to 0$, then $\log_b{n}-\log{\log_b{(np)}} \sim \frac{\log{(np)}-\log\log{(np)}}{p}$. We are now ready to prove Theorem \[big-p\]. By Theorem \[K-P\] we know that a.a.s. we can find an equitable colouring of $G$ with $m=\Theta\left(\frac{np}{\log{(np)}}\right)$ colours. Call the colour classes $C_1,\ldots, C_m$ and set $M_v=C_i$, where $C_i$ is the colour class that $v$ belongs to. So $v \in M_v$ and $N(v) \cap M_v=\emptyset$, as required. Let $S$ be a random subset of $V$ where each $v \in V$ is included into $S$ independently with probability $\frac{1}{2}$. Note that, as in the proof of \[densethm\], $V \in \langle \mathcal{M} \rangle$ and hence it is enough to show that for any $g \in \langle \mathcal{M} \rangle$, the set $g(S)$ is connected and non-empty. The following well known lemma essentially reduces the task to proving that none of these sets contains an isolated vertex. \[Csubsets\] Let $G \in \mathcal{G}(n,p)$ and $S\subset V(G)$ with $|S|\geq c n$ for some $c>0$. Then $${\mathbb{P}\left(S\text{ is not connected}\right)}= O\left({\mathbb{P}\left(S \text{ contains an isolated vertex}\right)}\right)=O\left(ne^{-c np}\right).$$ Notice that for every colour class $C_i$ and every state $g(S)$, the intersection $g(S)\cap C_i$ is either equal to $S\cap C_i$ or its complement $C_i \setminus S$. By well-known estimates on large deviation in binomial distribution, we observe that a.a.s. it is true that for almost every colour class $C_i$, we have $|S\cap C_i|\sim |C_i|/2$. Therefore, a.a.s. it is true that $|g(S)|> 2n/5$ for every state $g\in \langle\mathcal{M}\rangle$. Furthermore, the orbit of $S$ is of size $2^m$, where all moves only depend on the chosen equitable colouring of $G$ and not on $S$. The crucial observation here is that for any $g\in \langle\mathcal{M}\rangle$, the distribution of $g(S)$ is the same as that of $S$. Thus, by the union bound and Lemma \[Csubsets\], the probability that the triple $(G,S,\mathcal{M})$ is not a legal system is at most $$\sum_{g \in\langle\mathcal{M}\rangle} {\mathbb{P}\left(g(S) \text{ is not connected}\right)}\leq o(1)+ \exp\left(\frac{np}{\log{(np)}}-\frac{2}{5}np+\log{n} \right)=o(1).$$ Construction {#sec:construction} ============ The aim of this section is to outline our recipe to construct a legal system for $G \sim \mathcal{G}(n,p)$. The core idea behind the construction is the same as in §\[WeakBound\]. Ideally, we could simply choose a random initial set $S$, where each vertex in $G$ is added to the set with probability $\frac{1}{2}$. Then, the move at each vertex $v$ would be the colour class of vertex $v$ for an equitable colouring $C_1,\dots,C_m$ with $O(\log{n}/\log{\log{n}})$ colours, which we know exists w.h.p. from Theorem \[K-P\]. This is the approach taken in the proof of Theorem \[big-p\], but it does not work for all $p$ in the range of Theorem \[Main\]. The main obstruction in this range are vertices with only few neighbours in either $S\cap C_i$ or $C_i\setminus S$ for many of the colour classes $C_i$. This could happen for the obvious reason that a vertex simply has very few neighbours in $G$, or it is an unlikely (and unlucky, for that particular vertex) choice of the random set $S$. The idea is to show that one may deterministically modify our initial random set $S$ to take care of the problematic vertices. It is in this sets of vertices and their neighbourhoods that the modifications take place. The construction is as follows. - Let $D_0$ denote vertices of degree at most $\frac{\log{n}}{100}$. Assign two unique neighbours to each vertex of $D_0$. Call the set of such neighbours $N_0$, and set $V'=V(G) \setminus D_0$. - Partition $V'$ into large almost equitable independent sets $C_1, C_2, \ldots, C_m$ with $m\sim \frac{np}{{\log{np}}}\leq(1+o(1))\frac{\log n}{\log{\log{n}}} $. We can do this by first partitioning $V$ into equitable colour classes and then taking away vertices in $D_0\cup N_0$ as the size of this set will be negligible compared to the size of the colour classes. - Assign $+$ and $-$ signs to vertices of $G$ independently at random with probability $\frac{1}{2}$ and let $C_i^{+}=\{ v \in C_i:~ {\ensuremath{\mathrm{sign}}}(v)= +\}$ and $C_i^{-}=\{ v \in C_i :~ {\ensuremath{\mathrm{sign}}}(v)= -\}$. - Define the function $\kappa: 2^{V}\to \NN$ as $$\kappa(U)= \min_{\sigma \in \{+,-\}^{m}}\sum_{i} \left|U\cap C_i^{\sigma(i)}\right|,$$ and set $D_1=\left\{ v \in V' : \kappa(N(v))< \log{\log{n}}^2\right\}$. As before we assign a pair of unique neighbours to each vertex in $D_1$ with the further property that they both lie on the same colour class and not in $D_0\cup N_0$. We call this set of neighbours $N_1$. - We reassign to these pairs of vertices in $N_0$ and $N_1$ signs $+$ and $-$, so that for each pair one vertex is assigned $+$, and the other with $-$. Set $V''= V'\setminus D_1$. - For every vertex $v \in V''$, set $M_v= C_i$ for the unique $i$ such that $v\in C_i$, for every $v \in D_0\cup D_1$ set $M_v=\{v\}$ and for for every $v \in N_0$ set $M_v={u,v}$ where $u$ is the unique vertex in $N_0$ such that $N(u)\cap N(v)\neq \emptyset$. Furthermore, we set our initial activated set to be $S=\{v \in V: {\ensuremath{\mathrm{sign}}}(v)=+\}$. {width="40.00000%"} Proof of Main Theorem ===================== We will tackle the proof of the main theorem as follows: first we will give a small list of deterministic properties of a graph (which we call pseudorandom properties) that are sufficient to guarantee that the construction in the previous section indeed yields a legal system a.a.s. Finally we will complete the proof by showing that a random graph (at the appropriate density) a.a.s. presents all of the required pseudorandom properties. A caveat: there are two independent probability spaces at play in our approach: one is given by the random graph, and the other by the random 2-colouring in the construction. The first a.a.s. statement above is with respect to the latter space and the second with respect to the former. Sparse pseudorandom graphs {#sec:pseudorandom} -------------------------- \[pseudorandom\] For sufficiently large integer $n$, define $t=2n \log \log n / \log n$ and let $G$ be an $n$-vertex graph with $D_0:={\left}\{v\in V:~d(v)\leq \log n /100{\right}\}$ satisfying the following: (i) \[mindeg\] $\delta(G)\geq 2$, (ii) \[maxdeg\] $\Delta(G) =O(\log n)$, (iii) \[sized0\] $|D_0|\leq n^{0.9}$, (iv) \[noshortpath\] there exists no non-trivial path of length at most $4$ with both endpoints in $D_0$, (v) \[chromatic\] $m:=\chi_{=}(G) =O(\log n / \log \log n)$, (vi) \[mindeglarge\] every set $A\subseteq V(G)$ satisfying $\delta(G[A])> \log \log n^2/2$ is of size at least $t$, (vii) \[twolargesets\] between any two disjoint sets $A,B\subseteq V(G)$ of sizes at least $t$, there exists an edge in $G$ between $A$ and $B$. (viii) \[k23\] $G$ is $K_{2,3}$-free. Then $G$ has a legal system. Again, we start by assigning either $+$ or $-$ to every vertex of $G$ uniformly at random. As mentioned earlier, the subtle point where the proof of Theorem \[big-p\] cannot be applied here, are the few vertices that behave irregularly. Following the description in the sketch above, let us choose two neighbours $v_+, v_-$ for every vertex $v\in D_0$ such that no vertex is chosen twice - this is possible because of Properties  and . Denote the set of all such chosen neighbours by $N_0$, and reassign the signs of vertices in $N_0$ according to their subscripts. Furthermore, let us fix an arbitrary equitable colouring of $G$ with $m$ colours, and denote the colour classes by $D_1, \ldots, D_m$, set $C_i=D_i\setminus D_0$ and observe that $|C_i|=(1-o(1))|D_i|$ by property (\[sized0\]). As described in the sketch, for every $i\in [m]$ we define $C_i^+$ and $C_i^-$ to be the set of all vertices in $C_i$ with the corresponding sign. We would like to have a function that counts the minimum number of neighbours of any vertex in a set that contains either $C_i^+$ or $C_i^-$ for every $i\in [m]$. Towards that aim, we define $V'=V\setminus D_0$, $\kappa: 2^{V}\to \NN$ as $$\kappa(U)= \min_{\sigma \in \{+,-\}^{m}}\sum_{i} \left|U\cap C_i^{\sigma(i)}\right|,$$ and set $D_1=\left\{ v \in V' : \kappa(N(v))< \log{\log{n}}^2\right\}$. In order to work with the exceptional vertices in $D_1$, we need the following lemma, analogous to Property  for $D_0$. We remark here that the set $D_1$ is a *random subset* of $V'$ as it depends on the intial choice of 2-colouring. \[noshortpath1\] A.a.s. for every vertex $v\in V$, there are at most $1000$ paths of length $2$ between $v$ and vertices in $D_1$. Before we prove Lemma \[noshortpath1\], we need to make the following technical statements. \[domin\] Let $Y\sim {\ensuremath{\mathrm{Bin}}}(m, \frac{1}{2})$ where $m\geq 1$ and $X=\min\{Y, m-Y\}$ then $X$ dominates $Z$ where $Z\sim{\ensuremath{\mathrm{Bin}}}\left(\left\lfloor\frac{m}{2}\right\rfloor, \frac{1}{2}\right)$, that is for all $t\geq 0$ we have that $${\mathbb{P}\left(X\leq t\right)}\leq {\mathbb{P}\left(Z\leq t\right)}.$$ We argue by induction. For $m=1,2$ the claim obviously holds. Assuming that it holds for $m=m_0$, we show that it also holds for $m=m_0+2$. Observe that $Y\sim Y'+W$ where $Y'\sim {\ensuremath{\mathrm{Bin}}}(m_0,\frac{1}{2})$ and $W\sim {\ensuremath{\mathrm{Bin}}}(2,\frac{1}{2})$ are independent. Furthermore, observe that $X$ dominates $X_1+X_2$ where $X_1=\min\{Y',m_0-Y'\}$ and $X_2=\min\{W,2-W\}$. By the induction hypothesis, letting $Z_1\sim {\ensuremath{\mathrm{Bin}}}\left(\left\lfloor\frac{m}{2}\right\rfloor,\frac{1}{2}\right)$ and $Z_2\sim {\ensuremath{\mathrm{Bin}}}(1,\frac{1}{2})$ be independent random variables, we know that $X_i$ dominates $Z_i$ for $i=1,2$. Using the independence of $X_1$ and $X_2$ and of $Z_1$ and $Z_2$ it follows that $X_1+X_2$ dominates $Z_1+Z_2$ and hence $X$ also dominates $Z_1+Z_2\sim {\ensuremath{\mathrm{Bin}}}\left(\left\lfloor\frac{m+2}{2}\right\rfloor\right)$ as claimed. \[coupling\] Let $Y_1, \ldots, Y_k$ be independent random variables with $Y_i \sim {\ensuremath{\mathrm{Bin}}}\left(m_i, \frac{1}{2}\right)$ and $m_i\geq 1$ for every $i\in [k]$. Denote $X=\sum_{i=1}^{k}\min\{Y_i, m_i-Y_i\}$. Then $X$ dominates $Z\sim {\ensuremath{\mathrm{Bin}}}\left(\sum_i\left\lfloor\frac{m_i}{2}\right\rfloor , \frac{1}{2}\right)$. Let $X_i=\min\{Y_i, m_i-Y_i\}$, then the $X_i$’s are independent random variables and $X=\sum_i X_i$. By Claim \[domin\], we know that there exist independent random variables $Z_i\sim {\ensuremath{\mathrm{Bin}}}{\left}({\left}\lfloor\frac{m_i}{2}{\right}\rfloor, \frac{1}{2}{\right})$ such that each $X_i$ dominates $Z_i$ respectively. By independence of the $Z_i$’s, we then have that $X$ dominates $\sum_i Z_i\sim Z$. We can finally turn back our attention to the random set $D_1$. \[sized1\] For any $U\subset V'$, let $X=X(U)=\kappa(N(u))$. Suppose that $|U|\geq \frac{\log n}{101}$, then, $${\mathbb{P}\left(X(U)\leq 2 \log \log n^2\right)}\leq n^{-1/300}.$$ By Claim \[coupling\], we see that $X$ dominates $Y\sim {\ensuremath{\mathrm{Bin}}}{\left}(|N(u)|/2, \frac{1}{2}{\right})$ and thus $${\mathbb{P}\left(X< 2\log\log n^2\right)} \leq {\mathbb{P}\left(Y< 2\log\log n^2\right)} \leq {\mathbb{P}\left({\ensuremath{\mathrm{Bin}}}{\left}( \log n/202, \frac{1}{2} {\right}) < 2\log\log n^2\right)} <n^{-1/300}.$$ We are now ready to prove Lemma \[noshortpath1\]. By Property , every vertex $v\in V$ has $O(\log^2 n)$ vertices that are at distance at most $2$ from $v$. Therefore, if the statement of the lemma was to be wrong, by Property  there would be such $v$ where at least $1000$ of the $O(\log^2 n)$ vertices at distance at most $2$ from $v$ would be all in $D_1$. Although the events $X'(u_i)=\{u_i\in D_1 \}$ are not mutually independent, they are [*almost*]{} independent. Namely, for an arbitrary collection of $1000$ vertices $u_1, \ldots, u_{1000}$, the events $$X''(u_i) :=\mbox{``}\kappa{\left}(U_i {\right}) < 2 \log \log n^2 \mbox{''},$$ where $U_i=N(u_i)\setminus \bigcup_{j\neq i}N(u_j)$ are mutually independent since $U_i\cap U_j=\emptyset$ for $i\neq j$. Furthermore $X'(u_i) \implies X''(u_i)$ for every $i$. Finally, by Property  we see that $${\left}|N(u_i)\setminus \bigcup_{j\neq i}N(u_j) {\right}|> |N(u_i)|-2000 \geq \frac{\log n}{101},$$ and obtain $${\mathbb{P}\left( \bigwedge_{i\leq 1000}X'(u_i)\right)} \leq {\mathbb{P}\left(\bigwedge_{i\leq 1000}X''(u_i)\right)} = \prod_{i\leq 1000}{\mathbb{P}\left( X''(u_i) \right)} < \prod_{i\leq 1000}{\mathbb{P}\left(X(U_i)< 2\log\log n^2\right)} <n^{-1.1}.$$ The lemma now follows by a union bound over all choices for $v\in V$ and all choices of $1000$ vertices $u_i$ at distance at most $2$ from $v$. As before we assign a pair of unique neighbours to each vertex in $D_1$ with the further property that they both lie on the same colour class. This is possible since by Lemma \[noshortpath1\], for every $v\in V$ at most $1000$ vertices from $D_1$ have joint neighbours with $v$, and by Property  every such vertex has at most $2$ joint neighbours with $v$, whereas $v$ has a total of at least $\log n /100$ neighbours in $V$, out of which at most one is in $D_0\cup N_0$ by Property . As with the vertices in $N_0$, we assign to these two vertices signs $+$ and $-$, and set $V''= V'\setminus D_1$. As described in the sketch, for every vertex $v\in V''$, we set $M_v= C_i$ for the unique $i$ such that $v\in C_i$ for every $v \in D_0\cup D_1$ set $M_v=\{v\}$ and for for every $v \in N_0$ set $M_v={u,v}$ where $u$ is the unique vertex in $N_0$ such that $N(u)\cap N(v)\neq \emptyset$. To finish the proof, all that is left is to prove the following claim: Let $S=\{v\in V: {\ensuremath{\mathrm{sign}}}(v)=+\}$, the triple $(G,S, \mathcal{M})$ is a legal system. As in the proof of Theorem  \[big-p\], $V \in \langle \mathcal{M} \rangle$, thus to prove the claim it is enough to prove that $g(S)$ is connected for every $g \in \langle \mathcal{M} \rangle$. Observe that, by construction, for any $g\in \mathcal{M}$ and any vertex $v\in D_0\cup D_1$, out of the two vertices $v_+, v_- \in N_0\cup N_1$ exactly one is in $g(S)$. Therefore, for every $g \in \langle\mathcal{M}\rangle$, no vertex from $D_0\cup D_1$ is isolated in $g(S)$. Notice $$g(S)\cap V''=\bigcup_{i=1}^{m}C_i^{\sigma(i)}\cap V''$$ for some $\sigma\in \{+,-\}^m$. Suppose for the sake of contradiction that there exists such a vector $\sigma\in \{+,-\}^m$ for which the set $X=\bigcup_{i=1}^{m}C_i^{\sigma(i)}\cap V''$ is not connected. Then there must exist a subset $A\subset X$ such that $e(A,X\setminus A)=0$. Consider an arbitrary vertex $v\in A$. Since $v\in V''$, we have $$|N(v)\cap (g(S)\cup N_0\cup N_1)|\geq \log\log n^2.$$ Furthermore, by Property , ${\left}|N(v)\cap {\left}(D_0\cup N_0{\right}){\right}|\leq 1$, and by Lemma \[noshortpath1\], ${\left}|N(v)\cap {\left}(D_1\cup N_1{\right}){\right}|\leq 4000$. Therefore, $|N(v)\cap A|>\log \log n^2 /2$, or in other words $\delta(G[A])>\log \log n^2 /2$. By Property  this implies that $|A|\geq t$. Analogously, $|X\setminus A|\geq t$, and Property  guarantees the existence of an edge between $A$ and $X\setminus A$, a contradiction. Putting the pieces together {#sec:proof} --------------------------- Theorems \[big-p\] and \[JNW\] show that $G\sim {\ensuremath{\mathcal{G}}}(n,p)$ a.a.s. has a legal system for $p\geq 3 \log n/n$. Furthermore, for $p\leq \log n/n$, a.a.s. $G$ has a vertex of degree at most $1$, and by Proposition \[degree 1\] it does not have a legal system for $n\geq 4$. Therefore, it suffices to show that in the range $\log n/n<p< 3 \log n/n$, the graph $G$ a.a.s. satisfies Properties – from Theorem \[pseudorandom\]. Properties  and  are well-known to hold a.a.s. in this range of $p$. Furthermore, Property  holds a.a.s. as an immediate consequence of Theorem \[K-P\]. Property   also holds a.a.s. Indeed, this is just a special case of Claim 4.4 in [@BSKrSu11] and Theorem  4.2.9 in [@thesis]. We show the remaining Properties   and   in two separate lemmas. \[lem:mindeglarge\] Let $G\sim {\ensuremath{\mathcal{G}}}(n,p)$ with $\frac{\log n}{n}<p< \frac{3\log n}{n}$. Then a.a.s. every $A\subseteq V(G)$ satisfying $\delta(G[A])>\log \log n^2$ is of size at least $2n\log\log n /\log n$. By Chernoff’s inequality, the probability that a set $A$ of size $a\leq 2n\log\log n /\log n$ induces more than $a\log\log n^2/5$ edges, is $\exp{\left}[-\Omega(a\log\log n^2){\right}]$. Applying the union bound over all such sets provides the statement of the lemma. \[lem:twolargesets\] Let $G \sim {\ensuremath{\mathcal{G}}}(n,p)$ with $\frac{\log n}{n}<p$ and let $t=2n \log \log n / \log n$. Then for any two disjoint sets $A, B \subseteq V(G)$ of sizes at least $t$, there exists an edge in $G$ between $A$ and $B$. Observe that it is enough to prove the theorem for any two sets of size exactly $t$ (assume for simplicity $t$ is an integer). Call a pair of disjoint sets $A, B \subseteq V$ of size $t$ *bad*, if there is no edge between $A$ and $B$. The probability that such a given pair $A$ and $B$ is bad, is at most $$\label{eq:badpair} (1-p)^{t^2}<e^{-pt^2}<e^{-2t\log \log n }.$$ . The number of disjoint pairs of sets of size $t$ $A, B \subseteq V$ is at most $$\label{eq:numofpairs} {n \choose t}^2 <(en/t)^{2t}<(2e^{\log \log n -\log \log \log n})^{2t},$$ so by  ,   and the union bound, the probability that a bad pair in $G$ exists, is at most $e^{-t \log \log \log n}=o(1)$.

--- abstract: 'We prove that each superinjective simplicial map of the complex of curves of a compact, connected, nonorientable surface is induced by a homeomorphism of the surface, if $g + n \leq 3$ or $g + n \geq 5$, where $g$ is the genus of the surface and $n$ is the number of the boundary components.' author: - Elmas Irmak title: Superinjective Simplicial Maps of the Complexes of Curves on Nonorientable Surfaces --- Key words: Mapping class groups, simplicial maps, nonorientable surfaces MSC: 32G15, 20F38, 30F10, 57M99 Introduction ============ Let $N$ be a compact, connected, nonorientable surface of genus $g$ (connected sum of $g$ copies of projective planes) with $n$ boundary components. Mapping class group, $Mod_N$, of $N$ is defined to be the group of isotopy classes of all self-homeomorphisms of $N$. The *complex of curves*, $\mathcal{C}(N)$, on $N$ is an abstract simplicial complex defined as follows: A simple closed curve on $N$ is called [*nontrivial*]{} if it does not bound a disk, a mobius band, and it is not isotopic to a boundary component of $N$. The vertex set, $A$, of $\mathcal{C}(N)$ is the set of isotopy classes of nontrivial simple closed curves on $N$. A set of vertices forms a simplex in $\mathcal{C}(N)$ if they can be represented by pairwise disjoint simple closed curves. The geometric intersection number $i([a], [b])$ of $[a]$, $[b] \in A$ is the minimum number of points of $x \cap y$ where $x \in [a]$ and $y \in [b]$. A simplicial map $\lambda : \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ is called [*superinjective*]{} if the following condition holds: if $[a], [b]$ are two vertices in $\mathcal{C}(N)$ such that $i([a], [b]) \neq 0$, then $i(\lambda([a]), \lambda([b])) \neq 0$. The main result of this paper: Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. Suppose that either $g + n \leq 3$ or $g + n \geq 5$. If $\lambda : \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ is a superinjective simplicial map, then $\lambda$ is induced by a homeomorphism $h : N \rightarrow N$ (i.e $\lambda([a]) = [h(a)]$ for every vertex $[a]$ in $\mathcal{C}(N)$). ------------------------------------------------------------------------ [The author was supported by Faculty Research Incentive Grant, BGSU.]{} The mapping class groups and complex of curves on orientable surfaces are defined similarly as follows: Let $R$ be a compact, connected, orientable surface. Mapping class group, $Mod_R$, of $R$ is defined to be the group of isotopy classes of orientation preserving homeomorphisms of $R$. Extended mapping class group, $Mod_R^*$, of $R$ is defined to be the group of isotopy classes of all self-homeomorphisms of $R$. The complex of curves, $\mathcal{C}(R)$, on $R$ is defined as an abstract simplicial complex. The vertex set is the set of isotopy classes of nontrivial simple closed curves, where here nontrivial means it does not bound a disk and it is not isotopic to a boundary component of $R$. A set of vertices forms a simplex in $\mathcal{C}(R)$ if they can be represented by pairwise disjoint simple closed curves. Ivanov proved that the automorphism group of the curve complex is isomorphic to the extended mapping class group on orientable surfaces. As an application he proved that isomorphisms between any two finite index subgroups are geometric. Ivanov’s results were proven by Korkmaz in [@K1] for lower genus cases. Luo gave a different proof of these results for all cases in [@L]. After Ivanov’s work, mapping class group was viewed as the automorphism group of various geometric objects on orientable surfaces. These objects include Schaller’s complex (see [@Sc] by Schaller), the complex of pants decompositions (see [@M] by Margalit), the complex of nonseparating curves (see [@Ir3] by Irmak), the complex of separating curves (see [@BM1] by Brendle-Margalit, and [@MV] by McCarthy-Vautaw), the complex of Torelli geometry (see [@FIv] by Farb-Ivanov), the Hatcher-Thurston complex (see [@IrK] by Irmak-Korkmaz), and complex of arcs (see [@IrM] by Irmak-McCarthy). As applications, Farb-Ivanov proved that the automorphism group of the Torelli subgroup is isomorphic to the mapping class group in [@FIv], and McCarthy-Vautaw extended this result to $g \geq 3$ in [@MV]. On orientable surfaces: Irmak proved that superinjective simplicial maps of the curve complex are induced by homeomorphisms of the surface to classify injective homomorphisms from finite index subgroups of the mapping class group to the whole group (they are geometric except for closed genus two surface) for genus at least two in [@Ir1], [@Ir2], [@Ir3]. Behrstock-Margalit and Bell-Margalit proved these results for lower genus cases in [@BhM] and in [@BeM]. Brendle-Margalit proved that superinjective simplicial maps of separating curve complex are induced by homeomorphisms, and using this they proved that an injection from a finite index subgroup of $K$ to the Torelli group, where $K$ is the subgroup of mapping class group generated by Dehn twists about separating curves, is induced by a homeomorphism in [@BM1]. Shackleton proved that injective simplicial maps of the curve complex are induced by homeomorphisms in [@Sc] (he also considers maps between different surfaces), and he obtained strong local co-Hopfian results for mapping class groups. On nonorientable surfaces: For odd genus cases, Atalan proved that the automorphism group of the curve complex is isomorphic to the mapping class group if $g + r \geq 6$ in [@A]. Irmak proved that each injective simplicial map from the arc complex of a compact, connected, nonorientable surface with nonempty boundary to itself is induced by a homeomorphism of the surface in [@Ir4]. She also proved that the automorphism group of the arc complex is isomorphic to the quotient of the mapping class group of the surface by its center. Atalan-Korkmaz proved that the automorphism group of the curve complex is isomorphic to the mapping class group in [@AK] if $g + r \geq 5$. They also proved that two curve complexes are isomorphic if and only if the underlying surfaces are homeomorphic. In this paper we use some results from [@AK]. Our techniques give simpler proofs of the some of the results in [@AK]. Since an automorphism of $\mathcal{C}(N)$ is a superinjective simplicial map, our result implies that automorphisms of $\mathcal{C}(N)$ are induced by homeomorphisms of $N$ which was proved in [@AK]. Some small genus cases ====================== =1.2in =2.37in In this section we will prove our main results for $(g, n) \in \{(1, 0), (1, 1), (1, 2),$ $ (2, 0), (2, 1)\}$. We note that since simplicial maps preserve geometric intersection zero, superinjective simplicial maps preserve geometric intersection zero and nonzero properties. \[A\] Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. Suppose that either $(g, n) \in \{(1, 0), (1, 1), (1, 2),$ $(2, 0), (2, 1)\}$. If $\lambda : \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ is a superinjective simplicial map, then $\lambda$ is induced by a homeomorphism $h : N \rightarrow N$ (i.e $\lambda([a]) = [h(a)]$ for every vertex $[a]$ in $\mathcal{C}(N)$). If $(g, n) = (1, 0)$, $N$ is the projective plane. There is only one element (isotopy class of a 1-sided curve) in the curve complex. Hence, any superinjective simplicial map is induced by the identity homeomorphism. If $(g, n) = (1, 1)$, $N$ is Mobius band. There is only one element (isotopy class of a 1-sided curve) in the curve complex. Hence, any superinjective simplicial map is induced by the identity homeomorphism. If $(g, n) = (1, 2)$, there are only two elements in the curve complex (see [@Sc]). They are the isotopy classes of $a$ and $b$ as shown in Figure \[figure1\]. We see that $i([a], [b]) = 1$. So, the superinjective simplicial map cannot send both of these elements to the same element. If this simplicial map fixes each of them, then it is induced by the identity homeomorphism, if it switches them then it is induced by a homeomorphism that switches the curves $a$ and $b$. If $(g, n) = (2, 0)$, there are only three elements in the curve complex (see [@Sc]). They are the isotopy classes of $a$, $b$ and $c$ as shown in Figure \[figure1\] (ii). We see that $i([a], [b]) = 1$, $i([a], [c]) = 1$ and $i([b], [c]) = 0$. Since superinjective simplicial maps preserve geometric intersection zero and nonzero properties, $\lambda$ fixes $[a]$. If it also fixes each of $[b]$ and $[c]$ then it is induced by the identity homeomorphism, if it switches $[b]$ and $[c]$ then it is induced by a homeomorphism that switches the one sided curves $b$ and $c$, while fixing $a$ up to isotopy. =1.2in =3.6in =1.4in If $(g, n) = (2, 1)$, then the curve complex is given by Scharlemann in [@Sc] as follows: Let $a$ and $b$ be as in Figure \[figure2\]. We see that $i([a], [b]) = 1$. The vertex set of the curve complex is $\{[a], [b], t_a^m ([b]) : m \in \mathbb{Z} \}$, where $t_a$ is the Dehn twist about the 2-sided curve $a$. The complex is shown in Figure \[figure2\]. Since $i([a], t_a^m ([b])) \neq 0$ for any $m$, we have $i(\lambda([a]), \lambda(t_a^m ([b])) \neq 0$ for any $m$, i.e. $\lambda([a])$ will not be connected by an edge to any other vertex. For any $m$, the elements $t_a^m ([b])$ and $t_a^{m+1} ([b])$ cannot be send to the same element by $\lambda$ since they have different intersection information with $t_a^{m-1} ([b])$, and $\lambda$ preserves geometric intersection zero and nonzero by definition. Similarly, $t_a^m ([b])$ and $t_a^{n} ([b])$ cannot be send to the same element by $\lambda$ for any $m, n$ with $m \neq n$. For any $m$, $t_a^m ([b])$ cannot be send to $[a]$, since $i(t_a^{m-1} ([b]), t_a^m ([b])) = 0$ and $i(t_a^{m-1} ([b]), [a]) \neq 0$, and so we will have $i(\lambda(t_a^{m-1} ([b])), \lambda(t_a^m ([b]))) = 0$ and $i(\lambda(t_a^{m-1} ([b])), \lambda([a]) \neq 0$. So, either $\lambda(t_a^m ([b])) = t_a^k ([b])$, $\lambda(t_a^{m-1} ([b]))= t_a^{k-1} ([b])$, $\lambda(t_a^{m+1} ([b]))= t_a^{k+1} ([b])$ or $\lambda(t_a^m ([b])) = t_a^k ([b])$, $\lambda(t_a^{m-1} ([b]))= t_a^{k+1} ([b])$, $\lambda(t_a^{m+1} ([b]))= t_a^{k-1} ([b])$ for some $k \in \mathbb{Z}$. In either case, we also have that $\lambda$ fixes $[a]$. By cutting $N$ along $a$ we get a cylinder with one puncture as shown in Figure \[figure2a\]. There is a reflection of the cylinder interchanging front face with the back face which fixes $b$ pointwise. This gives us a homeomorphism $r$ of $N$ such that $r(b) = b$ and $r(a) = a^{-1}$. So, $(r)_\#([b]) = [b]$ and $(r)_\#(t_a([b])) = t_{a^{-1}}([b]) = {t_a}^{-1}([b])$. The map $r_\#$ reflects the graph in Figure \[figure2\] at $[b]$ and fixes $[a]$. Now it is easy to see that $\lambda$ is induced by $t_a^k$ or $t_a^k \circ r$ for some $k \in \mathbb{Z}$. Properties of Superinjective Simplicial Maps ============================================ Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. We will list some properties of superinjective simplicial maps. First we give some definitions. Let $P$ be a set of pairwise disjoint, nonisotopic, nontrivial simple closed curves on $N$. $P$ is called a [*pair of pants decomposition*]{} of $N$ if each component $\Delta$ of the surface $N_P$, obtained from $N$ by cutting $N$ along $P$, is a pair of pants. A pair of pants of a pants decomposition is the image of one of these connected components under the quotient map $q: N_P \rightarrow N$. Let $a$ and $b$ be two distinct elements in a pair of pants decomposition $P$. Then $a$ is called [*adjacent*]{} to $b$ w.r.t. $P$ iff there exists a pair of pants in $P$ which has $a$ and $b$ on its boundary. Let $P$ be a pair of pants decomposition of $N$. Let $[P]$ be the set of isotopy classes of elements of $P$. Note that $[P]$ is a maximal simplex of $\mathcal{C}(N)$. Every maximal simplex $\sigma$ of $\mathcal{C}(N)$ is equal to $[P]$ for some pair of pants decomposition $P$ of $N$. On orientable surfaces all maximal simplices have the same dimension $3g + n -4$ where $g$ is the genus and $n$ is the number of boundary components of the orientable surface. On nonorientable surfaces this is not the case. There are different dimensional maximal simplices. In Figure \[Fig0\] we see some of them in $g=7$ case. In the figure we see cross sings. This means that the interiors of the disks with cross signs inside are removed and the antipodal points on the resulting boundary components are identified. The following lemma is given in [@A] and [@AK]: \[dim\] Let $N$ be a nonorientable surface of genus $g \geq 2$ with $n$ boundary components. Let $a_r= 3r+n-2$ and $b_r = 4r +n -2$ if $g = 2r+1$, and let $a_r= 3r+n-4$ and $b_r= 4r +n -4$ if $g =2r$. Then there is a maximal simplex of dimension $q$ in $\mathcal{C}(N)$ if and only if $a_r \leq q \leq b_r$. Now we prove some properties of superinjective simplicial maps. \[inj\] Suppose that $(g, n)= (3, 0)$ or $g + n \geq 4$. A superinjective simplicial map $\lambda : \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ is injective. Let $[a]$ and $[b]$ be two distinct vertices in $\mathcal{C}(N)$. If $i([a], [b]) \neq 0$, then $i(\lambda([a]), \lambda([b])) \neq 0$, since $\lambda$ preserves nondisjointness. So, $\lambda([a]) \neq \lambda([b])$. If $i([a], [b]) = 0$, we choose a vertex $[c]$ of $\mathcal{C}(N)$ such that $i([a], [c]) = 0$, and $i([b], [c]) \neq 0$. Then $i(\lambda([a]), \lambda([b])) = 0$, $i(\lambda([b]), \lambda([c])) \neq 0$. So, $\lambda([a]) \neq \lambda([b])$. Hence, $\lambda$ is injective. \ If a maximal simplex has dimension $b_r = 4r +n -2$ if $g = 2r+1$, or $b_r = 4r +n -4$ if $g=2r$, we will call it a [*top dimensional maximal simplex*]{}. Since a superinjective map $\lambda$ is injective, $\lambda$ sends top dimensional maximal simplices to top dimensional maximal simplices. In the following lemmas we will see that adjacency and nonadjacency are preserved w.r.t. top dimensional maximal simplices. \[adjacent\] Suppose that $(g, n)= (3, 0)$ or $g + n \geq 4$. Let $\lambda : \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ be a superinjective simplicial map. Let $P$ be a pair of pants decomposition on $N$ which corresponds to a top dimensional maximal simplex in $\mathcal{C}(N)$. Let $a, b \in P$ such that $a$ is adjacent to $b$ w.r.t. $P$. There exists $a' \in \lambda([a])$ and $b' \in \lambda([b])$ such that $a'$ is adjacent to $b'$ w.r.t. $P'$ where $P'$ is a set of pairwise disjoint elements of $\lambda([P])$ containing $a', b'$. Suppose that $(g, n)= (3, 0)$ or $g + n \geq 4$. The statement is easy to see in $(g, n)= (3, 0)$ and $(g, n) = (1,3)$ cases as there are only two curves in a pair of corresponding to a top dimensional maximal simplex. Assume that $(g, n) \neq (3, 0)$ and $(g, n) \neq (1,3)$. Let $P$ be a pair of pants decomposition on $N$ which corresponds to a top dimensional maximal simplex in $\mathcal{C}(N)$. Let $a, b \in P$ such that $a$ is adjacent to $b$ w.r.t. $P$. We can find a simple closed curve $c$ on $N$ such that $c$ intersects only $a$ and $b$ nontrivially (with nonzero geometric intersection) and $c$ is disjoint from all the other curves in $P$. Let $P'$ be a set of pairwise disjoint elements of $\lambda([P])$. Since $\lambda$ is injective by Lemma \[inj\], $\lambda$ sends top dimensional maximal simplices to top dimensional maximal simplices. So, $P'$ corresponds to a top dimensional maximal simplex. Assume that $\lambda([a])$ and $\lambda([b])$ do not have adjacent representatives w.r.t. $P'$. Since $i([c], [a]) \neq 0$ and $i([c], [b]) \neq 0$, we have $i(\lambda([c]), \lambda([a])) \neq 0$ and $i(\lambda([c]), \lambda([b])) \neq 0$ by superinjectivity. Since $i([c], [e]) = 0$ for all $e \in P \setminus \{a, b\}$, we have $i(\lambda([c]), \lambda([e])) = 0$ for all $e \in P \setminus \{a, b\}$. But this is not possible because $\lambda([c])$ has to intersect geometrically essentially with some isotopy class other than $\lambda([a])$ and $\lambda([b])$ in $\lambda([P])$ to be able to make essential intersections with $\lambda([a])$ and $\lambda([b])$ since $\lambda([P])$ is a top dimensional maximal simplex. This gives a contradiction to the assumption that $\lambda([a])$ and $\lambda([b])$ do not have adjacent representatives. \[nonadjacent\] Suppose that $g + n \geq 4$. Let $\lambda : \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ be a superinjective simplicial map. Let $P$ be a pair of pants decomposition on $N$ which corresponds to a top dimensional maximal simplex in $\mathcal{C}(N)$. Let $a, b \in P$ such that $a$ is not adjacent to $b$ w.r.t. $P$. There exists $a' \in \lambda([a])$ and $b' \in \lambda([b])$ such that $a'$ is not adjacent to $b'$ w.r.t. $P'$ where $P'$ is a set of pairwise disjoint elements of $\lambda([P])$ containing $a', b'$. Suppose that $g + n \geq 4$. Let $P$ be a pair of pants decomposition on $N$ which corresponds to a top dimensional maximal simplex in $\mathcal{C}(N)$. Let $a, b \in P$ such that $a$ is not adjacent to $b$ w.r.t. $P$. Let $P'$ be a set of pairwise disjoint elements of $\lambda([P])$. We can find simple closed curves $c$ and $d$ on $N$ such that $c$ intersects only $a$ nontrivially and is disjoint from all the other curves in $P$, $d$ intersects only $b$ nontrivially and is disjoint from all the other curves in $P$, and $c$ and $d$ are disjoint. Since $\lambda$ is injective by Lemma \[inj\], $\lambda$ sends top dimensional maximal simplices to top dimensional maximal simplices. We have $i(\lambda([a]), \lambda([c]) \neq 0$, $i(\lambda([a]), \lambda([x]) = 0$ for all $x \in P \setminus \{a\}$, $i(\lambda([b]), \lambda([d]) \neq 0$, $i(\lambda([d]), \lambda([x]) = 0$ for all $x \in P \setminus \{b\}$, and $i(\lambda([c]), \lambda([d]) = 0$ by superinjectivity. This is possible only when $\lambda([a])$ and $\lambda([b])$ have representatives which are not adjacent w.r.t. $P'$. \ A simple closed curve is called [*1-sided*]{} if its regular neighborhood is a Mobius band. It is called [*2-sided*]{} if its regular neighborhood is an annulus. There are 1-sided simple closed curves as well as 2-sided simple closed curves on nonorientable surfaces. Some 1-sided simple closed curves have orientable complements and some have nonorientable complements. We will first prove some properties of $\lambda$. =1.8in =1.4in \[1-sided-co\] Let $g=1$. Suppose that $n \geq 3$. Let $\lambda : \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ be a superinjective simplicial map. If $a$ is a 1-sided simple closed curve on $N$, then $\lambda(a)$ is the isotopy class of a 1-sided simple closed curve on $N$. Let $a$ be a 1-sided simple closed curve on $N$. Since $g=1$, the complement of $a$ is orientable. Let $a' \in \lambda(a)$. Since $g=1$, $a'$ cannot be a 2-sided nonseparating simple closed curve on $N$. Suppose $a'$ is a 2-sided separating simple closed curve on $N$. Case (i): Suppose that $n=3$. We complete $a$ to a pair of pants decomposition $P=\{a, x\}$ as shown in Figure \[Fig1a\] (i). $P$ corresponds to a top dimensional maximal simplex. Let $P'$ be a set of pairwise disjoint elements of $\lambda([P])$ containing $a'$. Let $x' \in \lambda([x]) \cap P'$. Since $P'$ is a pants decomposition $N$, $g=1$ and $a'$ is separating, we see that $x'$ has to be a sided curve as shown in Figure \[Fig1a\] (ii). Consider the curves $z, t$ as shown in \[Fig1a\] (i). We have $[z] \neq [t]$, each of $z$ and $t$ intersects $x$ essentially, and each of them is disjoint from $a$. Let $z' \in \lambda([z]), t' \in \lambda([t])$ such that each of $z'$ and $t'$ intersects each of $a', x'$ minimally. Since $\lambda$ is injective by Lemma \[inj\], $[z'] \neq [t']$. Since $z', t'$ are not in $P'$ and they are disjoint from $a'$, they both have to lie in the subsurface, $M$, which is a projective plane with two boundary components containing $x'$, and having $a'$ as a boundary component as shown in Figure \[Fig1a\] (ii). This gives a contradiction as there are no two nontrivial simple closed curves with distinct isotopy classes such that each of them intersects $x'$ essentially in $M$. By Scharlemann’s result about complex of curves of projective plane with two boundary components in [@Sc], there are only two curves up to isotopy in $M$, (see Figure \[figure1\]). They are $x'$ and a 1-sided curve intersecting $x'$ only once. =2.2in =2.1in Case (ii): Suppose that $n=4$. We complete $a$ to a pair of pants decomposition $P=\{a, x, y\}$ as shown in Figure \[Fig1\] (i). $P$ corresponds to a top dimensional maximal simplex. Let $P'$ be a set of pairwise disjoint elements of $\lambda([P])$ containing $a'$. Let $x' \in \lambda([x]) \cap P'$. Since $a$ is adjacent to only $x$ w.r.t. $P$, by Lemma \[adjacent\] and Lemma \[nonadjacent\], $a'$ should be adjacent to only $x'$ w.r.t $P'$. Since $g=1$, $a'$ is separating and $a'$ is adjacent to only $x'$ w.r.t $P'$, there is a subsurface $T \subseteq N$ such that it is homeomorphic to sphere with four boundary components where three of the boundary components of $T$ are the boundary components of $N$, $x'$ is a boundary component of $T$, and $a'$ divides $T$ into two pair of pants as shown in Figure \[Fig1\] (ii). Let $y' \in \lambda([y]) \cap P'$. Since $x$ is adjacent to $y$, $x'$ is adjacent to $y'$. Since $n=4$ and $g=1$, we have $a', x', y', N$ as shown in Figure \[Fig1\] (ii). But this will give us a contradiction as follows: Consider the curves $z, t$ as shown in \[Fig1\] (i). We see that $[z] \neq [t]$, each of $z$ and $t$ intersects $y$ essentially, and each of them is disjoint from $a$ and $x$. Let $z' \in \lambda([z]), t' \in \lambda([t])$ such that each of $z'$ and $t'$ intersects each of $a', x', y'$ minimally. Since $\lambda$ is injective by Lemma \[inj\], $[z'] \neq [t']$. Since $z', t'$ are not in $P'$ and they are disjoint from $a'$ and $x'$, they both have to lie in the subsurface, $M$, which is a projective plane with two boundary components containing $y'$, and having $x'$ as a boundary component as shown in the figure. This gives a contradiction as there are no two nontrivial simple closed curves with distinct isotopy classes such that each of them intersects $y'$ essentially in $M$, (see [@Sc]). =3.7in =3.7in Case (iii): Suppose that $n > 4$. The argument is similar. We complete $a$ to a pair of pants decomposition $P= \{a, x_1, \cdots, x_{n-2}\}$ as shown in Figure \[Fig2\] (i). $P$ corresponds to a top dimensional maximal simplex. Let $P'$ be a set of pairwise disjoint elements of $\lambda([P])$ containing $a'$. Let $x_1' \in \lambda([x_1]) \cap P'$. Since $a$ is adjacent to only $x_1$ w.r.t. $P$, by Lemma \[adjacent\] and Lemma \[nonadjacent\], $a'$ should be adjacent to only $x_1'$ w.r.t $P'$. Since $g=1$, $n > 4$, $a'$ is separating and $a'$ is adjacent to only $x_1'$ w.r.t $P'$, we see that there is a subsurface $T \subseteq N$ such that it is homeomorphic to sphere with four boundary components where three of the boundary components of $T$ are the boundary components of $N$, $x_1'$ is a boundary component on $T$, and $a'$ divides $T$ into two pair of pants as shown in Figure \[Fig2\] (ii). Let $x_i' \in \lambda([x_i]) \cap P'$ for each $i= 2, \cdots, n-2$. Since $x_1$ is adjacent to only $x_2$, $x_1'$ is adjacent to only $x_2'$. Similar to the previous case using that adjacency and nonadjacency are preserved, $n > 4$ and $g=1$, we have elements of $P'$ and $N$ as shown in Figure \[Fig2\] (ii). But this will give us a contradiction as in the previous case (consider the curves $z, t$ as shown in Figure \[Fig2\] (i)). We showed that $a'$ cannot be a 2-sided simple closed curve on $N$. Hence, $a'$ is a 1-sided simple closed curve on $N$. =2.5in \[1-sided-cn\] Let $g \geq 2$. Suppose that $(g, n) = (3, 0)$ or $g+n \geq 4$. Let $\lambda : \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ be a superinjective simplicial map. If $a$ is a 1-sided simple closed curve on $N$ whose complement is nonorientable, then $\lambda(a)$ is the isotopy class of a 1-sided simple closed curve whose complement is nonorientable. Let $a$ be a 1-sided simple closed curve on $N$ whose complement is nonorientable. Let $a' \in \lambda([a])$. [**Case 1:**]{} Suppose $a'$ is a 1-sided simple closed curve whose complement is orientable. This case happens only if the genus of $N$ is odd. So, suppose $g =2r +1$, where $r \geq 1$. In this case $a$ can be put into a maximal simplex $\Delta$ of dimension $4r + n -2$. Since $\lambda$ is injective by Lemma \[inj\], $\lambda(\Delta)$ is a simplex of dimension $4r + n-2$. By using Euler characteristic arguments we see that the complement of $a'$ has genus $r$ and $n+1$ boundary components. So, in the complement of $a'$ there can be at most a $3r + n - 3$ dimensional simplex, hence there can be at most a $3r + n - 2$ dimensional simplex containing $a'$ on $N$. Since dim $\lambda(\Delta) = 4r + n - 2 > 3r + n - 2$ as $r \geq 1$, we get a contradiction. [**Case 2:**]{} Suppose $a'$ is a 2-sided nonseparating simple closed curve on $N$. [**(i)**]{} Suppose the genus of $N$ is even, $g=2r$ for some $r \geq 1$. We can put $a$ into a maximal simplex $\Delta$ of dim $4r+n-4$. Then $\lambda(\Delta)$ has dimension $4r+n-4$. \(a) Suppose now that the complement of $a'$ is nonorientable. This case happens when $g \geq 4$. The complement of $a'$ has genus $2r-2$ and it has $n+2$ boundary components. By using Lemma \[dim\] we get the following: In the complement of $a'$ there can be at most a $4(r-1) + (n+ 2) - 4 = 4r + n -6$ dimensional simplex. Hence, there can be at most a $4r + n - 5$ dimensional simplex containing $a'$ on $N$. Since dim $ \lambda(\Delta) = 4r + n -4 > 4r + n - 5$, we get a contradiction. \(b) Suppose that the complement of $a'$ is orientable. The genus of the complement is $r-1$ and the number of boundary component is $n + 2$. By using Lemma \[dim\] we get the following: In the complement of $a'$ there can be at most a $3(r-1) + (n+ 2) - 4 = 3r + n -5$ dimensional simplex. Hence, there can be at most a $3r + n - 4$ dimensional simplex containing $a'$ on $N$. Since dim $\lambda(\Delta) = 4r + n -4 > 3r + n - 4$, we get a contradiction (since $r \geq 1$). [**(ii)**]{} Suppose the genus of $N$ is odd, $g =2r + 1 $ for some $r \in \mathbb{Z}$. In this case complement of $a'$ is nonorientable. It has genus $2r-1$, and it has $n+2$ boundary components. By using Lemma \[dim\] we get the following: We can put $a$ into a maximal simplex $\Delta$ of dim $4r+n-2$. Then $\lambda(\Delta)$ has dimension $4r+n-2$. In the complement of $a'$ there can be at most a $4(r-1) + (n+ 2) - 2 = 4r + n - 4$ dimensional simplex by Lemma \[dim\]. Hence, there can be at most a $4r + n - 3$ dimensional simplex containing $a'$ on $N$. Since dim $ \lambda(\Delta) = 4r + n - 2 > 4r + n - 3$, we get a contradiction. =1.8in =1.55in [**Case 3:**]{} Suppose $a'$ is a separating simple closed curve on $N$. This case does not occur when $(g, n)=(3, 0)$ as there are no such nontrivial simple closed curves. So, we assume that $g \geq 2$ and $g+n\geq 4$. [**(i)**]{} Suppose $g \geq 4$. If $(g,n) = (4, 0)$, we complete $a$ to a pants decomposition $P$ as shown in Figure \[Fig4a\]. Let $P'$ be a set of pairwise disjoint elements in $\lambda([P])$ containing $a'$. Let $c' \in \lambda([c]) \cap P'$. If $a'$ is a separating simple closed curve on $N$, we get a contradiction by Lemma \[adjacent\], as $c'$ will be in one of the connected components in the complement of $a'$, and hence $c'$ cannot be adjacent to four curves w.r.t. $P'$ even though $c$ is adjacent to four curves w.r.t. $P$. Assume $(g, n) \neq (4, 0)$. We can choose a 1-sided curve $b$ on $N$ intersecting $a$ exactly once such that a regular neighborhood $R$ of $a \cup b$ will be a projective plane with two boundary components $x, y$ such that $N \setminus R$ is connected and there is a curve $z$ as shown in Figure \[Fig3\] such that $a, x, y, z$ can be completed to a pair of pants decomposition $P$, which corresponds to a top dimensional maximal simplex in $\mathcal{C}(N)$, on $N$, and $a$ is adjacent to $x$ and $y$ and $x$ and $y$ are both adjacent to $z$ as shown in the figure. Let $P'$ be a set of pairwise disjoint elements of $\lambda([P])$ containing $a'$. Since $a$ is only adjacent to two curves $x, y$ w.r.t. $P$, there exist $x' \in \lambda([x]) \cap P'$ and $y' \in \lambda([y]) \cap P'$ such that $a'$ is only adjacent to $x'$ and $y'$ w.r.t. $P'$ by Lemma \[adjacent\] and Lemma \[nonadjacent\]. Since $x$ and $y$ are also adjacent to $z$, there exists $z' \in \lambda([z]) \cap P'$ such that $x'$ and $y'$ are adjacent to $z'$. These imply that the only case that $a'$ could be a 2-sided separating curve is that $a'$ and two boundary components (if exists) would form a pair of pants on $N$ and $a', x', y'$ would be as shown in Figure \[Fig3\]. \(a) Suppose $g = 2r$ for some $r \in \mathbb{Z}$. Then the complement of $R$ (the projective plane with two boundary components $x, y$) in $N$ (see Figure \[Fig3\]) is a nonorientable surface with genus $2r -1$ and $n+2$ boundary components. By using Lemma \[dim\] we get the following: In this complement there is a $4(r-1) + (n+ 2) - 4 = 4r + n -6$ dimensional simplex. Hence, there is a $4r + n - 3$ dimensional simplex $\Delta$ containing $a, x, y$ on $N$. The complement of the four holed sphere shown in Figure \[Fig3\] is a connected nonorientable surface of genus $2r - 2$, and it has $n$ boundary components. So, in this complement there can be at most a $4(r-1) + n - 4 = 4r + n -8$ dimensional simplex. Hence, there can be at most a $4r + n - 5$ dimensional simplex containing $a', x', y'$ on $N$. Since dim $ \lambda(\Delta) = 4r + n - 3 > 4r + n -8$ we get a contradiction. \(b) Suppose $g = 2r + 1$ for some $r \in \mathbb{Z}$. Then the complement of $R$ (the projective plane with two boundary components $x, y$) in $N$ (see Figure \[Fig3\]) is a nonorientable surface with genus $2r $ and $n+2$ boundary components. By using Lemma \[dim\] we get the following: In this complement there is a $4r + n+ 2 - 4 = 4r + n -2$ dimensional simplex. Hence, there is a $4r + n +1$ dimensional simplex $\Delta$ containing $a, x, y$ on $N$. In this case the complement of the four holed sphere shown in Figure \[Fig3\] is a connected nonorientable surface of genus $2r - 1$, and it has $n$ boundary components. So, in this complement there can be at most a $4(r-1) + n - 2 = 4r + n -6$ dimensional simplex. Hence, there can be at most a $4r + n - 3$ dimensional simplex containing $a', x', y'$ on $N$. Since dim $\lambda(\Delta) = 4r + n + 1 > 4r + n - 3$, we get a contradiction. =1.6in =1.7in [**(ii)**]{} Now suppose $g < 4$. Since $g + n \geq 4$, we have $n \geq 1$. In this case, we complete $a$ to a curve configuration $a, x$ as shown in Figure \[Fig4\], using a boundary component $\partial_1$ of $N$. Let $T$ be the subsurface of $N$ having $x$ and $\partial_1$ as its boundary as shown in Figure \[Fig4\]. $T$ is a projective plane with two boundary components. The curves $a$ and $x$ can be completed to a pair of pants decomposition $P$ on $N$ which corresponds to a top dimensional maximal simplex in $\mathcal{C}(N)$. Let $P'$ be a set of pairwise disjoint elements of $\lambda([P])$ containing $a'$. Let $x' \in \lambda([x]) \cap P'$. Since $a$ is adjacent to only $x$ w.r.t. $P$, we must have that $a'$ is adjacent to only $x'$ w.r.t. $P'$. There is a curve $z$ in $P$ different from $a$ and $x$ such that $x$ is adjacent to $z$ w.r.t. $P$. The only case that $a'$ is a separating curve where $a'$ is only adjacent to $x'$, and $x'$ is adjacent to at least two curves is that $a'$ is as shown in Figure \[Fig4\], i.e. there must be a four holed sphere, $R$, having $x'$ and three boundary components of $N$ (if exists) as its boundary components such that $a'$ divides $R$ into two pair of pants as shown in the figure. Hence, $a'$ cannot be a separating curve if $(g, n) = (2, 2)$. =2.4in =2.2in \(a) Suppose $g = 2, n = 3$. We complete $a$ to a pair of pants decomposition $P=\{a, b, c, d\}$ which corresponds to a top dimensional maximal simplex as shown in Figure \[Fig5\] (i). We see that $a$ is adjacent to only $b$ w.r.t. $P$. Let $a' \in \lambda([a])$. Let $P'$ be a set of pairwise disjoint elements of $\lambda([P])$ containing $a'$. Let $b' \in \lambda([b]) \cap P'$, $c' \in \lambda([c]) \cap P'$, $d' \in \lambda([d]) \cap P'$. Since $a'$ is separating and $a'$ should be adjacent to only one curve w.r.t $P'$, there is a subsurface $T \subseteq N$ such that $T$ is a sphere with four boundary components where three of the boundary components of $T$ are the boundary components of $N$, $b'$ is a boundary component of $T$, and $a'$ divides $T$ into two pair of pants as shown in Figure \[Fig5\] (ii). Since $b$ is adjacent to $c$ and $d$, $b'$ is adjacent to $c'$ and $d'$. Since $c$ is adjacent to $d$, $c'$ is adjacent to $d'$. Let $z, t$ be as shown in Figure \[Fig5\] (i). We see that $[z] \neq [t]$, each of $z$ and $t$ intersects $d$ essentially and each of them is disjoint from $b$ and $c$. Let $z' \in \lambda([z]), t' \in \lambda([t])$ such that each of $z'$ and $t'$ intersects each of $a', c', d'$ minimally. Since $\lambda$ is injective by Lemma \[inj\], $[z'] \neq [t']$. Since $z'$ is not in $P'$ and $z'$ is disjoint from $a', b'$ and $c'$, it has to intersect $d'$. Similarly $t'$ is not in $P'$ and $t'$ is disjoint from $a', b'$ and $c'$, and it has to intersect $d'$. So, $z'$ and $t'$ are curves with distinct isotopy classes which are both disjoint from $b', c'$ and intersect $d'$ nontrivially. By cutting $N$ along $b'$ and $c'$ we get a projective plane with two boundary components, $R$, containing $d'$. The curves $z'$ and $t'$ have to lie $R$. We get a contradiction as there are only two elements in the complex of curves in a projective plane with two boundary components (by Scharlemann’s result in [@Sc]), and $d'$ is one of them. =3.65in =3.65in =3.65in =3.65in \(b) Suppose $g = 2, n > 3$. We complete $a$ to a pair of pants decomposition $P$, which corresponds to a top dimensional maximal simplex, as shown in Figure \[Fig7\] (i). We see that $a$ is adjacent to only $b$ w.r.t. $P$. Let $a' \in \lambda([a])$. Let $P'$ be a set of pairwise disjoint elements of $\lambda([P])$ containing $a'$. Let $b' \in \lambda([b]) \cap P'$, $c' \in \lambda([c]) \cap P'$, $x_i' \in \lambda([x_i]) \cap P'$ for $i = 1, 2, \cdots, n-2$. Since $a'$ is separating and $a'$ should be adjacent to only one curve w.r.t $P'$, we see that there is a subsurface $T \subseteq N$ such that it is homeomorphic to sphere with four boundary components where three of the boundary components of $T$ are the boundary components of $N$, $b'$ is a boundary component of $T$, and $a'$ divides $T$ into two pair of pants as shown in Figure \[Fig7\] (ii). Since $b$ is adjacent to $c$ and $x_1$, $b'$ is adjacent to $c'$ and $x_1'$. Since $c$ is not adjacent to any other curve, $c'$ will not be adjacent to any other curve. This implies that $c'$ is either a 1-sided curve or a separating curve as shown in Figures \[Fig7\] (ii), \[Fig8\] (ii). Suppose $c'$ is as shown in Figures \[Fig7\] (ii). By using the adjacency relations between $x_i$’s we get $P'$ as shown in Figure \[Fig7\] (ii). Let $z, t$ be as shown in Figure \[Fig7\] (i). We see that $[z] \neq [t]$, each of $z$ and $t$ intersects $x_{n-2}$ essentially and each of them is disjoint from all the other curves in $P$. Let $z' \in \lambda([z]), t' \in \lambda([t])$ such that each of $z'$ and $t'$ intersects each curve in $P'$ minimally. Since $\lambda$ is injective by Lemma \[inj\], $[z'] \neq [t']$. We see that $z', t'$ are not in $P'$, and they both are disjoint from all the curves in $P'$ except for $x_{n-2}'$. We get a contradiction again similar to case (a), by using Scharlemann’s result about complex of curves of projective plane with two boundary components. Similarly, if $c'$ is as shown in Figures \[Fig8\] (ii), by using the adjacency relations between $x_i$’s we get $P'$ as shown in \[Fig8\] (ii). By choosing curves $w, y$ as shown in Figures \[Fig8\] (i), we get a contradiction as in (a) and (b). =2.5in =2.3in =2.3in =2.3in \(c) Now suppose $(g,n) = (3, 1)$. We complete $a$ to a pants decomposition $P$ as shown in Figure \[Fig4b\]. Let $P'$ be a set of pairwise disjoint elements in $\lambda([P]$ containing $a'$. Let $c' \in \lambda([c]) \cap P'$. If $a'$ is a separating simple closed curve on $N$, we get a contradiction by Lemma \[adjacent\], as $c'$ will be in one of the connected components in the complement of $a'$, and hence $c'$ cannot be adjacent to three curves w.r.t. $P'$ even though $c$ is adjacent to three curves w.r.t. $P$. \(d) Suppose $g = 3, n = 2$. We complete $a$ to a pair of pants decomposition $P$, which corresponds to a top dimensional maximal simplex, as in Figure \[Fig9\] (i). Let $a, b, c, d, e$ be as shown in the figure. We see that $a$ is adjacent to $b$ and $e$ w.r.t. $P$. Let $a' \in \lambda([a])$. Let $P'$ be a set of pairwise disjoint elements of $\lambda([P])$ containing $a'$. Let $b' \in \lambda([b]) \cap P'$, $c' \in \lambda([c]) \cap P'$, $d' \in \lambda([d]) \cap P'$, $e' \in \lambda([e]) \cap P'$. Since $a'$ is separating and $a'$ should be adjacent to $b'$ and $e'$ w.r.t $P'$, and $b'$ is adjacent to four curves $a', c', d', e'$, there is a subsurface $T \subseteq N$ such that $T$ is a sphere with four boundary components where two of the boundary components of $T$ are the boundary components of $N$, $b'$ and $c'$ are the other two boundary components of $T$, and $a'$ divides $T$ into two pair of pants as shown in Figure \[Fig9\] (iii). Since $e'$ is not adjacent to any other curve, $e'$ is a 1-sided curve as shown in the figure. Since $b$ is adjacent to $c$ and $d$, $b'$ is adjacent to $c'$ and $d'$, and we see that $c'$ and $d'$ are both 1-sided curves. Let $z, t$ be as shown in Figure \[Fig9\] (ii). We have $[z] \neq [t]$, each of $z$ and $t$ intersects $e$ essentially and each of them is disjoint from $a, b, c$ and $d$. Let $z' \in \lambda([z]), t' \in \lambda([t])$ such that each of $z'$ and $t'$ intersects each of $a',b', c', d', e'$ minimally. Since $\lambda$ is injective by Lemma \[inj\], $[z'] \neq [t']$. Since $z'$ is not in $P'$ and $z'$ is disjoint from $a', b', c'$ and $d'$, it has to intersect $e'$. Similarly $t'$ is not in $P'$ and $t'$ is disjoint from $a', b', c'$ and $d'$, and it has to intersect $e'$. So, $z', t'$ are two curves with distinct isotopy classes which are both disjoint from $a'$ and $b'$, and intersect $e'$ nontrivially. This gives a contradiction as in (a), (b). =3.5in =3.5in \(e) Suppose $g = 3, n > 2$. We complete $a$ to a pair of pants decomposition $P= \{a, b, c, d, x_1, \cdots, x_{n-1}\}$ which corresponds to a top dimensional maximal simplex as shown in Figure \[Fig10\] (i). We see that $a$ is adjacent to $b$ and $x_1$ w.r.t. $P$. Let $a' \in \lambda([a])$. Let $P'$ be a set of pairwise disjoint elements of $\lambda([P])$ containing $a'$. Let $b' \in \lambda([b]) \cap P'$, $c' \in \lambda([c]) \cap P'$, $d' \in \lambda([d]) \cap P'$, $x_i' \in \lambda([x_i]) \cap P'$, for $i= 1, \cdots, n-1$. Since $a'$ is separating and $a'$ should be adjacent to $b'$ and $x_1'$ w.r.t. $P'$, there is a subsurface $T \subseteq N$ such that it is a sphere with four boundary components where two of the boundary components of $T$ are the boundary components of $N$, and $b'$ and $x_1'$ are the other two boundary components of $T$, and $a'$ divides $T$ into two pair of pants as shown in Figure \[Fig10\] (ii). Since $b$ is adjacent to $a, c, d$ and $x_1$, $b'$ is adjacent to $a', c', d'$ and $x_1'$. So, there is a pair of pants in $P'$ having $b', c', d'$ on its boundary. Since $x_1$ is adjacent to $x_2$ w.r.t. $P$, there is a pair of pants in $P'$ having $x_1'$ and $x_2'$ on its boundary. Since $x_1'$ is not adjacent to any other curve the third boundary component of this pair of pants must be a boundary component of $N$ as shown in the figure. Similarly since $x_2$ is adjacent to $x_3$ w.r.t. $P$, there is a pair of pants in $P'$ having $x_2'$ and $x_3'$ on its boundary. Since $x_2'$ is not adjacent to any other curve the third boundary component of this pair of pants must be a boundary component of $N$ as shown in the figure. Continuing like this we get a pair of pants in $P'$ such that it has $x_{n-2}'$ and $x_{n-1}'$ on its boundary. Since $g=3$, $x_{n-1}'$ is only adjacent to $x_{n-2}'$, the curves $c', d'$ are only adjacent to $b'$, and we already have $n-1$ boundary components, we see that $b', c', x_{n-1}'$ should be all be 1 sided curves as shown in the figure. Let $z, t$ be as shown in Figure \[Fig10\] (i). We have $[z] \neq [t]$, each of $z$ and $t$ intersects $x_{n-1}$ essentially, and each of them is disjoint from all the other curves in $P$. Let $z' \in \lambda([z]), t' \in \lambda([t])$ such that each of $z'$ and $t'$ intersects each curve in $P'$ minimally. Since $\lambda$ is injective by Lemma \[inj\], $[z'] \neq [t']$. We see that $z', t'$ are not in $P'$, and they both are disjoint from all the curves in $P'$ except for $x_{n-1}'$. This means that $z', t'$ both are in the projective plane with two boundary components, where one of the boundary components is $x_{n-1'}$ and the other is a boundary component of $N$ as shown in the figure. The curve $x_{n-2}'$ is a one sided curve in this subsurface. The existence of such $z', t'$ gives a contradiction as in cases (a), (b), (c). We have seen that $a'$ cannot be a 1-sided simple closed curve with orientable complement, it cannot be a 2-sided nonseparating simple closed curve and it cannot be a separating simple closed curve. Hence, $a'$ is a 1-sided simple closed curve whose complement is nonorientable. Proof of the Main Result ======================== In this section we will prove our main result when $(g, n) = (3, 0)$ or $g + n \geq 5$. Together with Theorem \[A\] this will complete our proof of the main theorem for $g + n \leq 3$ or $g + n \geq 5$.\ First we consider two graphs on $N$ as given in [@AK]: If $g=1$, let $\mathcal{A}$ be the set of isotopy classes of all 1-sided simple closed curves on $N$. If $g \geq 2$ let $\mathcal{A}$ be the set of isotopy classes of all 1-sided simple closed curves which have nonorientable complements on $N$. Let $X(N)$ be the graph with vertex set $\mathcal{A}$ such that two distinct vertices in $X(N)$ are connected by an edge if and only if they have representatives intersecting transversely at one point. Let $\widetilde{X}(N)$ be a subgraph of $X(N)$ with the vertex set $\mathcal{A}$. Two distinct vertices $\alpha$ and $\beta$ are connected by an edge in $\widetilde{X}(N)$ if $\alpha$ and $\beta$ have representatives a and b intersecting transversely at one point such that \(i) either $g \geq 4$ and the surface $N_{a \cup b}$ obtained by cutting $N$ along $a$ and $b$ is connected (Since $N_a$ and $N_b$ are nonorientable, it is easy to see that $N_{a \cup b}$ is also nonorientable in this case.), \(ii) or $1 \leq g \leq 3$ and the Euler characteristic of one of the connected components of $N_{a \cup b}$ is at most $-2$. We will use some connectivity results about these graphs given by Atalan-Korkmaz in [@AK].\ If $a$ is a nontrivial simple closed curve on $N$, [*the link*]{}, $L_a$ of $a$, is defined as the full subcomplex spanned by all the vertices of $\mathcal{C}(N)$ which have representatives disjoint from $a$. [*The star*]{} $St_a$ of $a$ is defined as the subcomplex of $\mathcal{C}(N)$ consisting of all simplices in $\mathcal{C}(N)$ containing $[a]$ and the faces of all such simplices. So, the link $L_a$ of $a$ is the subcomplex of $\mathcal{C}(N)$ whose simplices are those simplices of $St_a$ which do not contain $[a]$. A nontrivial simple closed curve $b$ is called [*dual*]{} to $a$ if they intersect transversely only once. Let $D_a$ be the set of isotopy classes of nontrivial 1-sided simple closed curves that are dual to $a$ on $N$. The following theorem is given by Atalan-Korkmaz in [@AK]. \[id\] Let $N$ be a connected, nonorientable surface of genus $g \geq 1$ with $n$ holes, and $g + n \geq 5$. Let $a$ and $b$ be two 1-sided simple closed curves that are dual to each other such that $[a]$ and $[b]$ are two vertices in $\widetilde{X}(N)$ which are connected by an edge in $\widetilde{X}(N)$. Let $h$ be a mapping class. If $h(\gamma) = \gamma$ for every vertex $\gamma$ in the set $(St_a \cup D_a) \cap (St_b \cup D_b)$, then $h$ is the identity. =2.5in =2.5in We prove a similar result below. Suppose $(g, n) = (3, 0)$. When we remove the identifications on $N$ that are shown in the first part of Figure \[Fig(3,0)\], we get a sphere with three boundary components, $z, t, w$, as shown in the second part. The reflection of this sphere about the $xy$-plane which changes the orientation on each curve in $\{z, t, w, x, y\}$ shown in the figure induces a homeomorphism $R$ on $N$. We call $R$ a reflection homeomorphism on $N$. Let $r = [R]$. \[id2\] Suppose $(g, n)= (3, 0)$. Let $a$, $b$ be two 1-sided simple closed curves with nonorientable complements such that they are dual to each other. Let $h$ be a mapping class. If $h(\gamma) = \gamma$ for every vertex $\gamma$ in the set $(St_a \cup D_a) \cap (St_b \cup D_b)$, then $h$ is the identity or $r$. Let $a$ and $b$ be two 1-sided simple closed curves with nonorientable complements such that they are dual to each other. Let $T$ be a regular neighborhood of $a \cup b$. $T$ is a genus one surface with two boundary components, say $x$ and $y$. Since $(g, n) = (3, 0)$, $a$ and $b$ are dual to each other, and $N_a$ and $N_b$ are nonorientable, we can see that each of $x$ and $y$ bounds a genus one surface with one boundary component, mobius band, as shown in the Figure \[Fig(3,0)\]. Let $h$ be a mapping class such that $h(\gamma) = \gamma$ for every vertex $\gamma$ in the set $(St_a \cup D_a) \cap (St_b \cup D_b)$. Let $H$ be a representative of $h$ such that $H$ fixes the curves $x, y, a$ and $b$. If $H$ fixes the orientation of $x$ then it also fixes the orientation of $y$ by Corollary 4.6 in [@K2] as it also fixes $a$ and $b$ up to isotopy. Then $H$ is isotopic to identity on $T$, and also $H$ is isotopic to identity on the two mobius bands on each side of $T$. Since the Dehn twists about $x$ and $y$ are trivial, we see that $H$ is isotopic to identity on $N$. If $H$ changes the orientation of $x$ then it also changes the orientation of $y$ by Corollary 4.6 in [@K2] as it also fixes $a$ and $b$ up to isotopy. In this case the composition $H \circ R$ fixes the orientation of both of $x$ and $y$ where $R$ is the reflection homeomorphism defined above. By the above argument $H \circ R$ is isotopic to identity. Hence, $H$ is isotopic to $R$. \ We prove our main result with the following theorem: Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. Suppose that $(g, n) =(3, 0)$ or $g + n \geq 5$. If $\lambda : \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ is a superinjective simplicial map, then $\lambda$ is induced by a homeomorphism $h : N \rightarrow N$ (i.e $\lambda([a]) = [h(a)]$ for every vertex $[a]$ in $\mathcal{C}(N)$). Assume that $g + n \geq 5$. We will first prove the result using induction on $g$. [**Case i:**]{} Suppose that $g=1$. Let $a$ be a 1-sided simple closed curve. Let $a' \in \lambda([a])$. By Lemma \[1-sided-co\] we know that $a'$ is a 1-sided simple closed curve. Since $g=1$, both of $N_a$ and $N_{a'}$ are orientable. So, there is a homeomorphism $f: N \rightarrow N$ such that $f(a) = a'$. Let $f_{\#}$ be the simplicial automorphism induced by $f$ on $\mathcal{C}(N)$. Then $f_{\#}^{-1} \circ \lambda $ fixes $[a]$. By replacing $f_{\#}^{-1} \circ \lambda $ by $\lambda$ we can assume that $\lambda([a]) = [a]$. The simplicial map $\lambda$ restricts to $\lambda_a : L_a \rightarrow L_a$, where $L_a$ is the link of $[a]$. It is easy to see that $\lambda_a$ is a superinjective simplicial map. Since $L_a \cong \mathcal{C}(N_a)$, we get a superinjective simplicial map $\lambda_a : \mathcal{C}(N_a)\rightarrow \mathcal{C}(N_a)$. By following the proof of Lemma \[inj\], we see that $\lambda_a$ is injective. Since $N_a$ is a sphere with at least five boundary components, by Shackleton’s result in [@Sh], there is a homeomorphism $G_a: N_a \rightarrow N_a$ such that $\lambda_a$ is induced by $(G_a)_{\#}$. Let $\partial_a$ be the boundary component of $N_a$ which came by cutting $N$ along $a$. We can see that $G_a (\partial_a) = \partial_a$ as follows: Since $g=1$ there are at least four boundary components of $N$. Let $b, c$ be the curves as shown in the Figure \[bdcor\] (i) ($b$ separates two of the boundary components of $N$). Complete $\{a, b, c\}$ to a top dimensional pair of pants decomposition $P$ on $N$. We assumed that $\lambda([a])=[a]$. Since $a, b, c$ are pairwise disjoint, there exist $b', c'$ some representatives of $\lambda(b)$ and $\lambda(c)$ respectively, such that $a, b', c'$ are pairwise disjoint. Let $P'$ be the set of pairwise disjoint representatives of $\lambda([P])$ containing $a, b', c'$. Since $a$ is adjacent to $b$ and $c$ w.r.t. $P$, $a$ is adjacent to $b'$ and $c'$ w.r.t. $P'$ by Lemma \[adjacent\]. Since $a$ is 1-sided there is a genus one subsurface with two boundary components, say $T$, such that $a$ is in $T$ and the boundary components of $T$ are $b', c'$. Since $b$ is adjacent to only $a$ and $c$ w.r.t. $P$, and nonadjacency is preserved by $\lambda$ by Lemma \[nonadjacent\], we see that $b'$ must be adjacent to only $a$ and $c'$. We know that $b'$ cannot be a 1-sided curve as $g=1$. This shows that $a, b', c'$ are as shown in the Figure \[bdcor\] (ii) (i.e. $b'$ separates two of the boundary components of $N$). Since $\lambda_a$ is induced by $(G_a)_{\#}$, they agree on $[b]$ and $[c]$. This shows that $G_a (\partial_a) = \partial_a$. By composing $G_a$ with a homeomorphism isotopic to identity, we can assume that $G_a$ maps antipodal points on the boundary $\partial_a$ to antipodal points. So, $G_a$ induces a homeomorphism $g_a: N \rightarrow N$ such that $g_a (a)= a$ and $(g_a)_{\#}$ agrees with $\lambda$ on every vertex of $[a] \cup L_a$. So, $(g_a)_{\#}^{-1} \circ \lambda$ fixes every vertex in $[a] \cup L_a$. Let $D_a$ be the set of isotopy classes of 1-sided simple closed curves that are dual to $a$. Claim 1: $(g_a)_{\#}$ agrees with $\lambda$ on every vertex of $\{[a]\} \cup L_a \cup D_a$. Proof of Claim 1: We already know that $(g_a)_{\#}$ agrees with $\lambda$ on every vertex of $\{[a]\} \cup L_a$. Let $d$ be a 1-sided simple closed curve that is dual to $a$. Let $T$ be a regular neighborhood of $a \cup d$. We see that $T$ is a real projective plane with two boundary components, say $x, y$. Since $(g_a)_{\#}^{-1} \circ \lambda$ is identity on ${[a]} \cup L_a$, $(g_a)_{\#}^{-1} \circ \lambda$ is identity on the complement of $T$. Since $(g_a)_{\#}^{-1} \circ \lambda$ is a superinjective simplicial map, there exists $d' \in \lambda([d])$ such that $d'$ is disjoint from the complement of $T$, so $d'$ is in $T$. Since $d$ is the only nontrivial simple closed curve in $T$ which is not isotopic to $a$ by Scharlemann’s Theorem in [@Sc], we see that $d'$ is isotopic to $d$. So, $((g_a)_{\#}^{-1} \circ \lambda) ([d]) = [d]$. Hence, $(g_a)_{\#}^{-1} \circ \lambda$ is identity on ${[a]} \cup L_a \cup D_a$. =2.2in =2.2in =2.2in =2.2in =2.2in Claim 2: Let $v$ be any 1-sided simple closed curve on $N$. Then, $(g_v)_{\#} = (g_a)_{\#}$ on $\mathcal{C}(N)$. Proof of Claim 2: Since $g=1$, $\widetilde{X}(N)$ is connected by Theorem 3.10 in [@AK]. So, we can find a sequence $a \rightarrow a_1 \rightarrow a_2 \rightarrow \cdots \rightarrow a_n=v$ of 1-sided simple closed curves connecting $a$ to $v$ such that each consecutive pair is connected by an edge in $\widetilde{X}(N)$. By Claim 1, $(g_a)_{\#}$ agrees with $\lambda$ on $\{[a]\} \cup L_a \cup D_a$, and $(g_{a_1})_{\#}$ agrees with $\lambda$ on $\{[a_1]\} \cup L_{a_1} \cup D_{a_1}$. So, we see that $(g_a)_{\#} ^{-1} (g_{a_1})_{\#}$ fixes every vertex in $([a] \cup L_a \cup D_a) \cap ([a_1] \cup L_{a_1} \cup D_{a_1})$. We note that this is the same thing as fixing every vertex in $(St_a \cup D_a) \cap (St_{a_1} \cup D_{a_1})$ since the vertex sets of them are equal. By Theorem \[id\], we get $(g_a)_{\#} = (g_{a_1})_{\#}$ on $\mathcal{C}(N)$. By using the sequence, we get $(g_a)_{\#} = (g_{v})_{\#}$ on $\mathcal{C}(N)$. Since genus is 1 there is no nonseparating 2-sided curve on $N$. Since every separating curve is in the link, $L_r$, of some 1-sided curve $r$, we see that $(g_a)_{\#}$ agrees with $\lambda$ on $\mathcal{C}(N)$. This proves the theorem for $g=1$. [**Case ii:**]{} Suppose that $g=2$. Since $g +n \geq 5$, $N$ has at least three boundary components. Let $a, b$ be as in Figure \[Fig11\]. Let $P(a)$, $P(b)$ be the connected components of $\widetilde{X}(N)$ that contains $a$, $b$ respectively. By Theorem 3.10 in [@AK], $\widetilde{X}(N)$ has two connected components $P(a)$ and $P(b)$. Since $\lambda$ is superinjective and $a, b$ are pairwise disjoint, there exist $a' \in \lambda([a])$, $b' \in \lambda([b])$ such that $a', b'$ are pairwise disjoint. By Lemma \[1-sided-cn\] we know that $a'$ and $b'$ are 1-sided simple closed curves with nonorientable complements since $a$ and $b$ are. So, there is a homeomorphism $f: N \rightarrow N$ such that $f(a) = a'$. Let $f_{\#}$ be the simplicial automorphism induced by $f$ on $\mathcal{C}(N)$. We see that $f_{\#}^{-1} \circ \lambda $ fixes $[a]$ as in case (i). By replacing $f_{\#}^{-1} \circ \lambda $ by $\lambda$ we can assume that $\lambda([a]) = [a]$. The simplicial map $\lambda$ restricts to a superinjective map $\lambda_{a} : L_{a} \rightarrow L_{a}$. Since $L_{a} \cong \mathcal{C}(N_{a})$, we get a superinjective simplicial map $\lambda_{a} : \mathcal{C}(N_{a})\rightarrow \mathcal{C}(N_{a})$. Since $N_a$ is a genus one nonorientable surface with at least four boundary components, by case (i) there is a homeomorphism $G_{a}: N_{a} \rightarrow N_{a}$ such that $\lambda_{a}$ is induced by $(G_{a})_{\#}$. Let $\partial_a$ be the boundary component of $N_a$ which came by cutting $N$ along $a$. We can see that $G_a (\partial_a) = \partial_a$ as follows: Since $g=2$ there at least three boundary components of $N$. Let $b, c$ be the curves as shown in the Figure \[bdcor\] (iii). We see that $b$ is a 1-sided curve and $c$ separates a genus two subsurface containing $a, b$. Complete $\{a, b, c\}$ to a top dimensional pair of pants decomposition $P$ on $N$. We assumed that $\lambda([a])=[a]$. Since $a, b, c$ are pairwise disjoint, there exist $b', c'$ some representatives of $\lambda(b)$ and $(\lambda(c)$ respectively, such that $a, b', c'$ are pairwise disjoint. Let $P'$ be the set of pairwise disjoint representatives of $\lambda([P])$ containing $a, b', c'$. Since $a$ is adjacent to $b$ and $c$ w.r.t. $P$, $a$ is adjacent to $b'$ and $c'$ w.r.t. $P'$ by Lemmma \[adjacent\]. Since $a$ is 1-sided there is a genus one subsurface with two boundary components, say $T$, such that $a$ is in $T$ and the boundary components of $T$ are $b', c'$. Since $b$ is adjacent to only $a, c$ w.r.t. $P$, and nonadjacency is preserved by $\lambda$ by Lemma \[nonadjacent\], we see that $b'$ must be only adjacent to and $a, c'$. We also know that $b'$ must be a 1-sided curve whose complement is nonorientable, as $b$ is such a curve (see Lemma \[1-sided-cn\]). This shows that $a, b', c'$ are as shown in the Figure \[bdcor\] (iv). Since $\lambda_a$ is induced by $(G_a)_{\#}$, they agree on $[b]$ and $[c]$. This shows that $G_a (\partial_a) = \partial_a$. We note that when $g \geq 3$, we can use the curves shown in Figure \[bdcor\] (iv) to get similar results. Now we continue as follows: As in case (i), by composing $G_a$ with a homeomorphism isotopic to identity, we can assume that $G_a$ maps antipodal points on the boundary $\partial_a$ to antipodal points. So, $G_a$ induces a homeomorphism $g_a: N \rightarrow N$ such that $g_a (a)= a$ and $(g_a)_{\#}$ agrees with $\lambda$ on every vertex of $[a] \cup L_a$. So, $(g_a)_{\#}^{-1} \circ \lambda$ fixes every vertex in $[a] \cup L_a$. =2.5in =2.5in As in the proof of case i, we get $(g_{a})_{\#}$ agrees with $\lambda$ on $\{[a]\} \cup L_{a} \cup D_{a}$. Also as in that proof, we can see that if $v$ is any 1-sided simple closed curve $v$ such that $[v] \in P(a)$ (where $P(a)$ is the connected component of $\widetilde{X}(N)$ that contains $a$), then $(g_a)_{\#} = (g_{v})_{\#}$ on $\mathcal{C}(N)$. Hence, we can see that there exists a homeomorphism $h_1$ such that our original superinjective map $\lambda$ agrees with $(h_1)_{\#}$, on every vertex of $\{[a]\} \cup L_{a} \cup D_{a} \cup \{[v]\} \cup L_{v} \cup D_{v}$ for any 1-sided simple closed curve $v$ such that $[v] \in P(a)$. Similarly, there exists a homeomorphism, $h_2$, such that $\lambda$ agrees with $(h_2)_{\#}$ on every vertex of $\{[b]\} \cup L_{b} \cup D_{b} \cup \{[w]\} \cup L_{w} \cup D_{w}$ for any 1-sided simple closed curve $w$ such that $[w] \in P(b)$. So, $(h_1)_{\#}^{-1} (h_2)_{\#}$ fixes everything in the intersection of these two sets, for every such $v, w$. Claim: $(h_1)_{\#} = (h_2)_{\#}$. Proof of Claim: Let $c, d, e$ be as in Figure \[Fig11\]. Let $T$ be the genus 2 subsurface containing $a$ and $b$ and bounded by $c$ as shown in the figure. Since $(h_1)_{\#}^{-1} (h_2)_{\#}$ fixes $[c]$, (by composing with a map isotopic to identity if necessary) we may assume that ${h_1}^{-1} h_2$ fixes $c$. By the above argument we know that ${h_1}^{-1} h_2$ fixes every nontrivial simple closed curve up to isotopy in the complement of $T$. Since the complement of $T$ together with $c$ is a sphere with at least four boundary components, ${h_1}^{-1} h_2$ fixes $c$ and every nontrivial simple closed curve up to isotopy in the complement of $T$, ${h_1}^{-1} h_2$ is isotopic to identity in the complement of $T$ by Lemma 7.1 in [@AK]. So, ${h_1}^{-1} h_2$ is isotopic to identity on $c$. Since ${h_1}^{-1} h_2$ fixes $a$, $b$ up to isotopy, it fixes the curve $e$ shown in the figure up to isotopy, see the proof of Theorem \[A\] and Figure \[figure2\], and recall that the vertex set of the curve complex in $T$ is $\{[e], [a], t_e^m ([a]) : m \in \mathbb{Z} \}$. By looking at the complex we see that the map ${h_1}^{-1} h_2$ actually fixes every nontrivial simple closed curve on $T$ up to isotopy. Since it is also isotopic to identity on $c$, ${h_1}^{-1} h_2$ is isotopic to identity on $T$. This implies that $(h_1)_{\#}^{-1} (h_2)_{\#} = t_c^m$ for some $m \in \mathbb{Z}$. Since $(h_1)_{\#}^{-1} (h_2)_{\#}$ also fixes $[d]$, we have $m=0$, so $(h_1)_{\#}^{-1} (h_2)_{\#} = [id]$. Let $h=h_1$. If $x$ is a 1-sided simple closed curve with nonorientable complement (i.e. $[x]$ is a vertex in $\widetilde{X}(N)$), then $[x]$ is in one of $P(a)$ or $P(b)$ by Theorem 3.10 in [@AK]. So, by the above arguments $\lambda$ agrees with $h_{\#}$ on $\{[x]\} \cup L_{x} \cup D_{x}$. Since any nontrivial simple closed curve is in the dual or link of a 1-sided simple closed curve whose complement is nonorientable, $\lambda$ agrees with $h_{\#}$ on $\mathcal{C}(N)$. =2.5in =2.5in [**Case iii:**]{} Suppose that $g=3$. $N$ has at least two boundary components. Let $a, b, c$ be as in Figure \[Fig12\]. Let $P(a)$, $P(b)$ and $P(c)$ be the connected components of $\widetilde{X}(N)$ that contains $a$, $b$ and $c$ respectively. By Theorem 3.10 in [@AK], $\widetilde{X}(N)$ has three connected components $P(a)$, $P(b)$ and $P(c)$. As in the proof of case (ii), there exist homeomorphisms $h_1, h_2, h_3$ such that $\lambda$ agrees with $(h_1)_\#$ on every vertex of $\{[a]\} \cup L_{a} \cup D_{a} \cup \{[v]\} \cup L_{v} \cup D_{v}$ for any 1-sided simple closed curve $v$ such that $[v] \in P(a)$, $\lambda$ agrees with $(h_2)_\#$ on every vertex of $\{[b]\} \cup L_{b} \cup D_{b} \cup \{[w]\} \cup L_{w} \cup D_{w}$ for any 1-sided simple closed curve $w$ such that $[w] \in P(b)$, and $\lambda$ agrees with $(h_3)_\#$ on every vertex of $\{[c]\} \cup L_{c} \cup D_{c} \cup \{[z]\} \cup L_{z} \cup D_{z}$ for any 1-sided simple closed curve $z$ such that $[z] \in P(c)$. Let $d, e, f, g$ be as in Figure \[Fig12\]. By following the proof of case (ii) and using the curves $d, e$ we can see that $(h_1)_\# = (h_2)_\#$. (We note that in this case the complement of $T$ will be a genus one surface with at least 3 boundary components, and the difference map will be isotopic to identity on the complement of $T$, see Lemma 7.1 in [@AK]. The rest of the proof will follow as in case (ii).) Similarly, by using the curves $f, g$ we can see that $(h_2)_\# = (h_3)_\#$. Hence, we have $(h_1)_\# = (h_2)_\# = (h_3)_\#$. Let $h=h_1$. If $x$ is a 1-sided simple closed curve with nonorientable complement (i.e. $[x]$ is a vertex in $\widetilde{X}(N)$), then $[x]$ is in one of $P(a)$ or $P(b)$ or $P(c)$ by Theorem 3.10 in [@AK]. By the above arguments and the proof of case (ii), we see that $\lambda$ agrees with $h_{\#}$ on $\{[x]\} \cup L_{x} \cup D_{x}$. Since any nontrivial simple closed curve is in the dual or link of a 1-sided simple closed curve whose complement is nonorientable, $\lambda$ agrees with $h_{\#}$ on $\mathcal{C}(N)$. [**Case iv:**]{} We will prove the remaining cases when $g + n \geq 5$ by using induction on $g$. Assume that the theorem is true for some genus $g-1$ where $g-1 \geq 3$. We will prove that it is true for genus $g$. Let $a$ be a 1-sided simple closed curve on $N$ such that $N_a$ is nonorientable. Let $a' \in \lambda([a])$. By Lemma \[1-sided-cn\], $a'$ is a 1-sided simple closed curve with nonorientable complement. There is a homeomorphism $f: N \rightarrow N$ such that $f(a) = a'$. Let $f_{\#}$ be the simplicial automorphism induced by $f$ on $\mathcal{C}(N)$. Then $f_{\#}^{-1} \circ \lambda$ fixes $[a]$. By replacing $f_{\#}^{-1} \circ \lambda $ by $\lambda$ we can assume that $\lambda([a]) = [a]$. The simplicial map $\lambda$ restricts to a superinjective simplicial map $\lambda_a : L_a \rightarrow L_a$ as before. Since $L_a \cong \mathcal{C}(N_a)$, we get a superinjective simplicial map $\lambda_a : \mathcal{C}(N_a)\rightarrow \mathcal{C}(N_a)$. By the induction assumption, there is a homeomorphism $G_a: N_a \rightarrow N_a$ such that $\lambda_a$ is induced by $(G_a)_{\#}$. By following the proof of case (i), we see the following: There is a homeomorphism $g_a$ such that $\lambda$ agrees with $(g_a)_\#$ on $\{[a]\} \cup L_a \cup D_a$, and if $v$ is a 1-sided simple closed curve with nonorientable complement such that $[v]$ is connected to $[a]$ by a path in $\widetilde{X}(N)$, then $(g_v)_{\#} = (g_a)_{\#}$ on $\mathcal{C}(N)$. We also see that $\lambda$ agrees with $(g_v)_\#$ on $\{[v]\} \cup L_v \cup D_v$. Let $w$ be any 1-sided simple closed curve with nonorientable complement. Between $[a]$ and $[w]$ there is a path in $\widetilde{X}(N)$ since $\widetilde{X}(N)$ is connected by Theorem 3.10 in [@AK]. So, we have $(g_a)_{\#} = (g_w)_{\#}$ on $\mathcal{C}(N)$. Since the isotopy class of every nontrivial simple closed curve is in the link or dual of some 1-sided simple closed curve with nonorientable complement, we see that $\lambda$ agrees with $(g_a)_\#$ on $\mathcal{C}(N)$. By induction, we get our result for $g + n \geq 5$.\ =2.5in Now we prove the remaining case for the theorem. Suppose $(g, n) = (3, 0)$. Consider the curves $a, b, c$ as shown in Figure \[Figlast\]. Let $Q(a), Q(b), Q(c)$ be the connected components of $a, b, c$ in $X(N)$ respectively. By using Lemma \[1-sided-cn\], Theorem \[A\], Theorem \[id2\], and following the proof of case iii, we see that there exist homeomorphisms $h_1, h_2, h_3$ such that $\lambda$ agrees with $(h_1)_\#$ on $\{[a]\} \cup L_{a} \cup D_{a} \cup \{[v]\} \cup L_{v} \cup D_{v}$ for any 1-sided simple closed curve $v$ such that $[v] \in Q(a)$, $\lambda$ agrees with $(h_2)_\#$ on $\{[b]\} \cup L_{b} \cup D_{b} \cup \{[w]\} \cup L_{w} \cup D_{w}$ for any 1-sided simple closed curve $w$ such that $[w] \in Q(b)$, and $\lambda$ agrees with $(h_3)_\#$ on $\{[c]\} \cup L_{c} \cup D_{c} \cup \{[z]\} \cup L_{z} \cup D_{z}$ for any 1-sided simple closed curve $z$ such that $[z] \in Q(c)$. Let $d, e$ be as in Figure \[Figlast\]. By following the proof of case iii, and using the curves $d, e$ we can see that $(h_1)_\# = (h_2)_\#$ or $(h_1)_\# = (h_2)_\# \circ R_\#$ where $R$ is the reflection homeomorphism that we described at the beginning of this section. Similarly, we can see that $(h_2)_\# = (h_3)_\#$ or $(h_2)_\# = (h_3)_\# \circ R_\#$. Since $R_\#$ fixes the isotopy class of every nontrivial simple closed curve, the action of all these maps $(h_1)_\#$, $(h_2)_\#$ and $(h_3)_\#$ are the same on $\mathcal{C}(N)$. Let $h=h_1$. If $x$ is a 1-sided simple closed curve with nonorientable complement, then $[x]$ is in one of $Q(a)$ or $Q(b)$ or $Q(c)$ by Theorem 3.9 in [@AK]. By the above arguments and the proof of case (i), we see that $\lambda$ agrees with $h_{\#}$ on $\{[x]\} \cup L_{x} \cup D_{x}$. Since any nontrivial simple closed curve is in the dual or link of a 1-sided simple closed curve whose complement is nonorientable, $\lambda$ agrees with $h_{\#}$ on $\mathcal{C}(N)$. This completes the proof. [**Acknowledgments**]{} We thank Peter Scott for some discussions and his comments about this paper. We thank Mustafa Korkmaz for some discussions. We also thank Ferihe Atalan for some discussions. 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--- abstract: 'Monolayer Graphene contains two inequivalent local minimum, valleys, located at $K$ and $K''$ in the Brillouin zone. There has been considerable interest in the use of these two valleys as a doublet for information processing. Herein I propose a method to resolve valley currents spatially, using only a weak magnetic field. Due to the trigonal warping of the valleys, a spatial offset appears in the guiding centre co-ordinate, and is strongly enhanced due to collimation. This can be exploited to spatially separate valley states. Based on current experimental devices, spatial separation is possible for densities well within current experimental limits. Using numerical simulations, I demonstrate the spatial separation of the valley states.' author: - 'Samuel S. R. Bladwell' bibliography: - 'Biblio.bib' title: Valley separation via trigonal warping --- [*Introduction*]{}: Due to the particular symmetry of the honeycomb lattice of monolayer graphene, the valence and conduction bands meet at 6 points. In the immediate vicinity of these points, the dispersion is linear and the Fermi surface consists of two inequivalent cones at points $K$ and $K'$ in the Brillouin zone. These valleys are independent and degenerate, and several works have proposed using these valleys as a doublet for information processing; referred to as valleytronics, in analogy with spintronics[@Rycerz2007; @Schaibley2016]. Since the first proposal over a decade ago, considerable progress has been made with regards to the generation of both static valley polarisations, and valley polarised currents[@Garcia2008; @Gunlycke2011; @Jiang2013; @Settnes2016]. Detecting valley polarisation on the other hand has proved to be difficult. An early proposal suggested using superconducting superconducting contacts[@Akhmerov2007]. More recently it has been shown that static valley polarisations can be induced and detected via second harmonic generation[@Golub2011; @Wehling2015]. Nonetheless, the detection of valley polarised currents remains an ongoing challenge, with implications for a wide variety of phenomena beyond valleytronics. In this letter, motivated by recent developments in electron optics in graphene I propose an approach for the detection of valley polarised currents in graphene. Over the past decade, a variety of improvements in material processing have allowed for high mobilities, with mean free paths of tens of microns[@Banszerus2016]. Very recently, several groups have considered how to form highly collimated electron beams in graphene; Barnard [*et al*]{} using absorptive metal contacts to form a pinhole aperture, and Liu [*et al*]{} using a parabolic [*p-n*]{} junction as a refinement of the Veselago lens[@Barnard2017; @Liu2017]. Herein I show that collimation, combined with the trigonal warping of the Dirac cone, results in a significant enhancement in the spatial separation between ballistic valley polarised currents. Combined with an appropriate device layout, this spatial separation can be exploited to individually address distinct valley states. Due to the significant enhancement, I find that the required trigonal warping is small, and the required density is well within current experimental limits. It thereby provides a novel method of detecting ballistic valley polarised currents. [*Valley separation*]{}: The effective Hamiltonian for graphene near the charge neutrality point is Dirac-like, ${\cal H} \sim {\bm k} \cdot \boldsymbol{\sigma}$, where $\boldsymbol{\sigma}$ are the usual Pauli matrices and reflect the two constituent sub-lattices. At low densities, there is a four-fold degeneracy, due to spin and valley degrees of freedom. The two inequivalent valleys are located at $K$ and $K'$ respectively in the Brillouin zone, and close to the charge neutrality point are cylindrically symmetric. For higher densities, the Fermi surface in each valley $K$ and $K'$ exhibits trigonal warping, the emergence of which is shown in Fig. \[trigonal\]. With an applied transverse magnetic field, ${\bm p } \rightarrow \boldsymbol{\pi} = \bm p + e \bm A$, where $\bm A$ is the vector potential. If the applied field is weak, and the electron or hole density is high, the charge carrier dynamics can be described semi-classically, starting from the Heisenberg equation of motion for the operators, $$\begin{aligned} \dot{\hat{\boldsymbol{\pi}}}= \left[{\cal H}, \hat{\boldsymbol{\pi}}\right] = e B \hat{\bm v }\times {\bf n} \label{eqmotion}\end{aligned}$$ where ${\bf n}$ is the unit vector normal to the graphene plane, and $B$ is the magnitude of the applied magnetic field. Note that $\hat{\boldsymbol{\pi}}$ and $\hat{\bm v}$ are operators. Eq. is general, and holds for a variety of dispersion relations[@Bladwell2015]. In the semiclassical limit, the operator equation, Eq. , is converted to a classical equation of the expectation values, which can then be trivially integrated to yield the real space motion of a electron under an applied transverse magnetic field, $$\begin{aligned} {\bm r}(t) = \frac{\boldsymbol{\pi} \times \bm n}{eB} \label{eqmotion1}\end{aligned}$$ where $\bm r = \left< {\hat{ \bm r}}\right>$ and $\boldsymbol{\pi} = \left<\hat{\boldsymbol{\pi}}\right> $. This is the equation of motion for cyclotron motion, with the electron following the equienergetiuc contours of the Fermi surface. Thus at high densities, the semi-classical cyclotron orbits of graphene are trigonally warped. [![The emergence of trigonal warping in the two valleys, with $K$ ($K'$) indicated in red (blue).[]{data-label="trigonal"}](Valley11.pdf "fig:"){width="11.00000%"}]{} [![The emergence of trigonal warping in the two valleys, with $K$ ($K'$) indicated in red (blue).[]{data-label="trigonal"}](Valley12.pdf "fig:"){width="11.00000%"}]{} [![The emergence of trigonal warping in the two valleys, with $K$ ($K'$) indicated in red (blue).[]{data-label="trigonal"}](Valley13.pdf "fig:"){width="11.00000%"}]{} [![The emergence of trigonal warping in the two valleys, with $K$ ($K'$) indicated in red (blue).[]{data-label="trigonal"}](Valley14.pdf "fig:"){width="11.00000%"}]{} [![The emergence of trigonal warping in the two valleys, with $K$ ($K'$) indicated in red (blue).[]{data-label="trigonal"}](Valley21.pdf "fig:"){width="11.00000%"}]{} [![The emergence of trigonal warping in the two valleys, with $K$ ($K'$) indicated in red (blue).[]{data-label="trigonal"}](Valley22.pdf "fig:"){width="11.00000%"}]{} [![The emergence of trigonal warping in the two valleys, with $K$ ($K'$) indicated in red (blue).[]{data-label="trigonal"}](Valley23.pdf "fig:"){width="11.00000%"}]{} [![The emergence of trigonal warping in the two valleys, with $K$ ($K'$) indicated in red (blue).[]{data-label="trigonal"}](Valley24.pdf "fig:"){width="11.00000%"}]{} In Eq. , guiding centres of the two valleys are identically located at $(x, y) = (0, 0)$. For an electron optics device, for example, the pin-hole collimator designed in Ref. [@Barnard2017], the initial position of the wave packet is at $(0,0)$, the location of the injector. In addition, for a perfectly collimated beam of electrons, the initial velocity is fully aligned along the $x$ axis, parallel to the channel of the injector. For a cylindrically symmetric Fermi surface, the location of the guiding centre co-ordinate is unchanged. When the Fermi contour becomes trigonally warped, the guiding centre co-ordinate becomes offset, proportional to the magnitude of the trigonal warping. This offset effect is presented in Fig. \[fig2\]. Since the two valleys exhibit triagonal warping with opposite signs, the guiding centre co-ordinates are offset above and below the $y$ axis. The magnitude of the offset is proportional to the magnitude of the trigonal warping; as trigonal warping increases, so does guiding centre co-ordinate offset. To illustrate the effect analytically, I consider the following approximation for the Fermi momentum, $p_F$, $$\begin{aligned} p_F \approx \hbar k (1 + s u \sin3\theta ) \label{momentum}\end{aligned}$$ where $\theta$ is the polar angle, $\theta = \tan^{-1} k_y/k_x$, and $k_0 = \sqrt{\pi n}$. Here $k_x$ and $k_y$ are chosen according to Fig. \[brill\]. The valley index, $s=\pm1$, with $u = k a/4$, where $a$ is the lattice constant of graphene. It is important to note that this analytic approach is only valid while $3u \ll 1$, that is, $u \sim 0.1$. The transverse velocity must vanish for collimated injection. From Eq. , the condition is ${\partial p_y}/{\partial \theta} = 0$, where $p_y = p_F \cos\theta$. The guiding centre co-ordinate is $$\begin{aligned} (x_{gc}, y_{gc}) \approx \frac{\hbar k_0}{eB} \left(-s 3 u , 1 - u \right) \label{offset} \end{aligned}$$ where $s$ is once again the valley index. This offset of the two valley states can be clearly seen in Fig. \[fig2\]b. [![Real space trajectories with collimated injection. Left panel: at low densities, the Fermi surface is nearly cylindrically symmetric, and the trajectory of states in each valley are offset only slightly. Right panel: at high densities trigonal warping results in a guiding centre co-ordinate offset for a collimated source.[]{data-label="fig2"}](Kspace.pdf "fig:"){width="22.00000%"}]{} [![Real space trajectories with collimated injection. Left panel: at low densities, the Fermi surface is nearly cylindrically symmetric, and the trajectory of states in each valley are offset only slightly. Right panel: at high densities trigonal warping results in a guiding centre co-ordinate offset for a collimated source.[]{data-label="fig2"}](Kspace1.pdf "fig:"){width="22.00000%"}]{} Focusing in the half plane, that is, with source and the detector located on the $y$ axis, does not yield any spatial separation between the valley states. However, if an orthogonal device setup is employed, as shown in Fig. \[fig3\]a, the states can be spatial separated. The real space separation at the collector is $$\begin{aligned} x_+ - x_- \approx 8 u \frac{\hbar k_0}{eB} \label{realspaceoffset}\end{aligned}$$ where $x_+$ and $x_-$ are the real space positions in the plane of the detector. Thus trigonal warping of the valleys in graphene induces a real space separation when combined with an appropriate device setup, with a significant enhancement. This real space separation from collimated injection is shown in Fig. \[fig3\]b. [*Resolution limits:*]{} On its own, Eq. tells us little about whether the spatial splitting can be [*observed*]{}. If the broadening of the beam exceeds the spatial separation, then the valley states will overlap and resolution of the distinct valleys currents will not be possible. There are three specific sources of broadening; (1) broadening due to the medium, (2) the spatial resolution of the detector, and (3), the imperfect collimation of the source. Beam broadening due to the medium limits the scale of the focusing device. The path length of the ballistic measurements used in Ref. [@Barnard2017] were $l_{path} \sim 3-4\mu$m. For an orthogonal device with a geometry equivalent to that of Fig. \[fig3\]a, this corresponds to a maximum focusing length, $l \sim 2\mu$m. The device should be made as large possible, as the relative resolution of the detector scales as $w/l$, where $w$ is the width of the aperture. The mean free path in h-BN encapsulated graphene can be upwards of $10\mu$m, implying that significantly larger device scales are possible. [![Top Left: Orthogonal focusing setup, with source and detector planes oriented orthogonally. Here $l$ is the focusing length, and $B_z$ the weak focusing field. Top Right: Pinhole collimator layout. The aperture width is $w$, and the length $L$, with the angular spread, $\Delta \theta$ given by the twice the projected angle. This is one possible collimating source in graphene, with current experimental devices producing a beam with a beam spread of $\Delta \theta \approx \ang{20}$. Lower Panels: Trajectory computation for a pinhole collimator source, and pinhole aperture detector, with $l=2\mu$m, $L= 1\mu$m, and $w = 250nm$. Trajectories are computed with a random initial condition (random guiding centre coordinate), with the transverse magnetic field, $B = \hbar k_0/eL$. Each valley has trajectories in red (blue). Left (right) panel has a density $n = 6\times 10^{12}$cm$^{-2}$ ($n = 3 \times 10 ^{13}$cm$^{-2}$). For the higher density, the spatial separation of the beam can be clearly seen. []{data-label="fig3"}](Setup.pdf "fig:"){width="22.00000%"}]{} [![Top Left: Orthogonal focusing setup, with source and detector planes oriented orthogonally. Here $l$ is the focusing length, and $B_z$ the weak focusing field. Top Right: Pinhole collimator layout. The aperture width is $w$, and the length $L$, with the angular spread, $\Delta \theta$ given by the twice the projected angle. This is one possible collimating source in graphene, with current experimental devices producing a beam with a beam spread of $\Delta \theta \approx \ang{20}$. Lower Panels: Trajectory computation for a pinhole collimator source, and pinhole aperture detector, with $l=2\mu$m, $L= 1\mu$m, and $w = 250nm$. Trajectories are computed with a random initial condition (random guiding centre coordinate), with the transverse magnetic field, $B = \hbar k_0/eL$. Each valley has trajectories in red (blue). Left (right) panel has a density $n = 6\times 10^{12}$cm$^{-2}$ ($n = 3 \times 10 ^{13}$cm$^{-2}$). For the higher density, the spatial separation of the beam can be clearly seen. []{data-label="fig3"}](Pinhole.pdf "fig:"){width="22.00000%"}]{} [![Top Left: Orthogonal focusing setup, with source and detector planes oriented orthogonally. Here $l$ is the focusing length, and $B_z$ the weak focusing field. Top Right: Pinhole collimator layout. The aperture width is $w$, and the length $L$, with the angular spread, $\Delta \theta$ given by the twice the projected angle. This is one possible collimating source in graphene, with current experimental devices producing a beam with a beam spread of $\Delta \theta \approx \ang{20}$. Lower Panels: Trajectory computation for a pinhole collimator source, and pinhole aperture detector, with $l=2\mu$m, $L= 1\mu$m, and $w = 250nm$. Trajectories are computed with a random initial condition (random guiding centre coordinate), with the transverse magnetic field, $B = \hbar k_0/eL$. Each valley has trajectories in red (blue). Left (right) panel has a density $n = 6\times 10^{12}$cm$^{-2}$ ($n = 3 \times 10 ^{13}$cm$^{-2}$). For the higher density, the spatial separation of the beam can be clearly seen. []{data-label="fig3"}](Trajectories2.pdf "fig:"){width="23.00000%"}]{} [![Top Left: Orthogonal focusing setup, with source and detector planes oriented orthogonally. Here $l$ is the focusing length, and $B_z$ the weak focusing field. Top Right: Pinhole collimator layout. The aperture width is $w$, and the length $L$, with the angular spread, $\Delta \theta$ given by the twice the projected angle. This is one possible collimating source in graphene, with current experimental devices producing a beam with a beam spread of $\Delta \theta \approx \ang{20}$. Lower Panels: Trajectory computation for a pinhole collimator source, and pinhole aperture detector, with $l=2\mu$m, $L= 1\mu$m, and $w = 250nm$. Trajectories are computed with a random initial condition (random guiding centre coordinate), with the transverse magnetic field, $B = \hbar k_0/eL$. Each valley has trajectories in red (blue). Left (right) panel has a density $n = 6\times 10^{12}$cm$^{-2}$ ($n = 3 \times 10 ^{13}$cm$^{-2}$). For the higher density, the spatial separation of the beam can be clearly seen. []{data-label="fig3"}](Trajectories4.pdf "fig:"){width="23.00000%"}]{} Next, let us determine the collimation limit for a pinhole style collimator setup in the orthogonal geometry of Fig. \[fig3\]a. This can be understood as the area containing all guiding centre co-ordinates, $(x_{gc}, y_{gc} \in {\cal S})$ satisfying the limits, $y = L/2, -L/2$, $l -w/2<x< l + w/2$. Here $L$ and $w$ represent the width and length of the pinhole collimator respectively, while $l$ is the focusing length. A schematic showing these is presented in Fig. \[fig3\]. Specifically, the value of interest is the range of $y_{gc}$, which must be less than the spatial separation of the valley states. Provided the trigonal warping is relatively small, $u < 0.1$, the radius of curvature in the local region about $\left<v_x\right> = 0$ is identical for both valleys, and the trajectories within the collimator can be approximated by cyclotron orbits. The resulting approximate limit is $|y_{gc}| < \sqrt{wr_c} - L/2$, while the “spread" of the beam is double this. In the absence of any addition collimation, the required density for spatial separation for $w = r_c/6$ and $L = r_c/2$ is $n \approx 3\times 10^{13}$cm$^{-2}$. Due to the comparatively high density, narrower apertures can be used without reaching the diffraction limit, which can improve the resolution. Alternatively, the spread can be determined from typical values in the literature. For a pinhole collimator like that of Barnard [*et al*]{}, the angular full width half maximum was $\Delta \theta \sim \pi/9$, corresponding to a spread of $\Delta x = \Delta \theta r_c$. For $x_+ - x_- > \Delta x$, $8u > \pi/9$, which yields a minimum density of $n \approx 3 \times 10^{13}$cm$^{-2}$. Parabolic [*p-n*]{} junctions, as proposed by Liu [*et al*]{} have a significantly more collimated beam shape, with $\Delta \theta \sim \ang{5}$, however the beam has larger spatial dimensions, and this is limited by distance of the source to the [*p-n*]{} parabola. Combinations of parabolic junctions and pinhole aperture can further improve the angular distribution and spatial extent of the beam[@Boggild2017], which would further lower the density. A halving of the beam spread would reduce the required density by a factor of four, placing individual valley resolution tantalisingly close to densities accessible via back-gated devices. [*Numerical simulations:*]{} This can be grounded more firmly via numerical simulation of the device, to determine the required $u$ and therefore $n$ for resolution of the individual valleys. Since Eqs. and are valid only for small values of trigonal warping, I consider the energy bands of the usual tight-binding hamiltonian (see, for example, [@Neto2009]) to determine the equienergetic contours and then use Eq. to determine the dynamics. As already noted, this approach is valid provided $k r_c > 100$, and device feature sizes much large than the Fermi wavelength. Plots are generated by considering randomised initial conditions, summing the number of states that pass through the pinhole collimator, and pinhole aperture of the detector plane. [![Results of numerical simulations of a device with $l = 2\mu$m, $L = 1000$nm, and $w = 250$nm, with increasing density. The injector consists of a pinhole collimator, with the detector a pinhole aperture. At $n = 3 \times 10^{13}$cm$^{-2}$, the individual valley peaks can be clearly resolved. Here the red and blue dashed curves correspond to the signal from each valley, with the green curve being the total signal. []{data-label="numerics"}](Valleysplitting1.pdf "fig:"){width="22.00000%"}]{} [![Results of numerical simulations of a device with $l = 2\mu$m, $L = 1000$nm, and $w = 250$nm, with increasing density. The injector consists of a pinhole collimator, with the detector a pinhole aperture. At $n = 3 \times 10^{13}$cm$^{-2}$, the individual valley peaks can be clearly resolved. Here the red and blue dashed curves correspond to the signal from each valley, with the green curve being the total signal. []{data-label="numerics"}](Valleysplitting2.pdf "fig:"){width="22.00000%"}]{} [![Results of numerical simulations of a device with $l = 2\mu$m, $L = 1000$nm, and $w = 250$nm, with increasing density. The injector consists of a pinhole collimator, with the detector a pinhole aperture. At $n = 3 \times 10^{13}$cm$^{-2}$, the individual valley peaks can be clearly resolved. Here the red and blue dashed curves correspond to the signal from each valley, with the green curve being the total signal. []{data-label="numerics"}](Valleysplitting3.pdf "fig:"){width="22.00000%"}]{} [![Results of numerical simulations of a device with $l = 2\mu$m, $L = 1000$nm, and $w = 250$nm, with increasing density. The injector consists of a pinhole collimator, with the detector a pinhole aperture. At $n = 3 \times 10^{13}$cm$^{-2}$, the individual valley peaks can be clearly resolved. Here the red and blue dashed curves correspond to the signal from each valley, with the green curve being the total signal. []{data-label="numerics"}](Valleysplitting4.pdf "fig:"){width="22.00000%"}]{} The results of the numerical procedure are presented in Fig. \[numerics\]. For a device with a pinhole collimator, and a detector with a simple pinhole aperture, the required density is $\sim 2 \times 10^{13}$cm$^{-2}$, with narrow apertures, and longer pinhole channels lowering the required density. This is well within current experimentally achievable densities for graphene devices[@Craciun2011]. Further improvements by reducing $w$ are possible, however, there is a fundamental limit due to diffraction at low densities. For $n = 1 \times 10^{13}$cm$^{-2}$, $\lambda_F \sim 10$nm $\ll w = 250$nm, and naïvely, the angular spread due to diffraction is $\theta \approx 1/20$, which is significantly less that the collimation limit of the pinhole collimator[@Barnard2017]. The resolution can be further improved via the use of a second pinhole collimator as a detector allowing for resolution of the distinct valleys at lower densities. The principle complication lies in the differing trajectory shapes of the two valleys. In conclusion, I have shown that the ballistic trajectories of electrons in different valleys of graphene can be spatial separated using a collimating source, and a weak magnetic field. The magnitude of the separation at $\sim 3 \times 10^{13}$cm$^{-2}$ is sufficient to fully resolve the individual peaks, and this carrier density is achievable with current gating methods. Additional improvements in resolution are possible with the addition of a second pinhole collimator, and further adjustments to both the width, $w$ and length, $L$ of the collimator. Alternative designs would offer further improvements in resolution. Finally, the basic principle of valley separation outlined here will work for any two dimensional systems where the valleys exhibit trigonal warping, including bilayer graphene, moire superlattices, and two dimensional transitional metal dicalcogenides. This research was partially supported by the Australian Research Council Centre of Excellence in Future Low-Energy Electronics Technologies (project number CE170100039) and funded by the Australian Government. SSRB would like to thank Oleg Sushkov for his critical reading and suggestions.

--- abstract: 'Let $k$ be a knot in $S^3$. In [@HS], H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover of $(S^3, k)$ from a thin position of $k$. In this article, we will prove that if a thin position of $k$ induces a thin decomposition of double branched cover of $(S^3,k)$ by Howards and Schultens’ method, then the thin position is the sum of prime summands by stacking a thin position of one of prime summands of $k$ on top of a thin position of another prime summand, and so on. Therefore, $k$ holds the nearly additivity of knot width (i.e. for $k=k_1\#k_2$, $w(k)=w(k_1)\# w(k_2)-2$) in this case. Moreover, we will generalize the hypothesis to the property a thin position induces a manifold decomposition whose thick surfaces consists of strongly irreducible or critical surfaces (so topologically minimal.)' author: - Jungsoo Kim date: '13, Jan, 2010' title: A note on the nearly additivity of knot width --- Introduction and result ======================= Let $k$ be a knot in $S^3$. In [@HS], H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover (abbreviate it as *DBC*, and call the method *the H-S method*) of $(S^3, k)$ (see section \[section-DBC\]) and they proved that for $2$-bridge knots and $3$-bridge knots in thin position DBC inherits thin manifold decomposition (note that a knot in a thin position may not induce a thin manifold decomposition by the H-S method in general, see [@HS] and [@HRS].) Indeed, if $k$ is a non-prime $3$-bridge knot, then $k=k_1\#k_2$ for $2$-bridge knots $k_1$ and $k_2$, and thin position of $k$ is the sum of $k_1$ and $k_2$ by stacking a thin position of one of the knots on top of a thin position of the other (see Corollary 2.5 of [@HS].) So $w(k)=w(k_1)+w(k_2)-2$, i.e. $k$ holds the *nearly additivity* of knot width (for $k=k_1\#k_2$, $w(k)=w(k_1)+w(k_2)-2$, see [@ST2] for more details on “the nearly additivity of knot width”.) So we get a question whether the property that a thin position of a knot induces a thin manifold decomposition of DBC by the H-N method implies the nearly additivity of knot width in general. If a thin position of $k$ is the sum of an ordered stack of prime summands of $k$ where each summand is in a thin position (“*a sum of an ordered stack of prime summands*” means like the left of Figure \[figure-thin\], where the bottom summand is a Montesinos knot $M(0; (2,1), (3,1), (3, 1), (5, 1))$ in a thin position (this figure is borrowed from Figure 5.2.(c) of [@HK]) and the top summands are trefoils in thin position,) then the sum of an ordered stack like that in a different order also determines a thin position of $k$. So $k$ must hold the nearly additivity of knot width by the uniqueness of prime factorization of $k$. In [@RS], Y. Rieck and E. Sedgwick proved that thin position of the sum of small knots is the sum of an ordered stack in that manner, i.e. it holds the nearly additivity of knot width. But M. Scharlemann and A. Thompson proposed a way to construct a example to contradict the nearly additivity of knot width (see [@ST2].) Although R. Blair and M. Tomova proved that most of Scharlemann and Thompson’s constructions do not produce counterexamples for the nearly additivity of knot width (see [@BT],) the question seems not obvious. In this article, we will prove that the question is true. \[theorem-1\] If a thin position of a knot $k$ induces a thin manifold decomposition of double branched cover of $(S^3,k)$ by the Howards and Schultens’ method, then the thin position is the sum of prime summands by stacking a thin position of one of prime summands of $k$ on top of a thin position of another prime summand, and so on. Therefore, $k$ holds the nearly additivity of knot width in this case. In section \[section-critical\], we will generalize Theorem 1.1 by using the concept *critical surface* originated from D. Bachman (see [@Bachman1] and [@Bachman3] for the original definition and the recently modified definition of “critical surface”.) So we will get the corollary.\ [**Corollary 5.3.**]{} *If a thin position of a knot $k$ induces a manifold decomposition of double branched cover $M$ of $(S^3,k)$ by the Howards and Schultens’ method where each thick surface $H_+$ of the manifold decomposition of $M$ is strongly irreducible or critical in $M(H_+)$, then the thin position of $k$ is the sum of prime summands by stacking a thin position of one of prime summands on top of a thin position of another prime summand, and so on. Therefore, $k$ holds the nearly additivity of knot width in this case.* Generalized Heegaard splittings =============================== In this section, we will introduce some definitions about generalized Heegaard splittings. We use the notations and definitions by D. Bachman in [@Bachman3] through this section for convenience. A *compression body* (a *punctured compression body* resp.) is a $3$-manifold which can be obtained by starting with some closed, orientable, connected surface, $H$, forming the product $H\times I$, attaching some number of $2$-handles to $H\times\{1\}$ and capping off all resulting $2$-sphere boundary components (some $2$-sphere boundaries resp.) that are not contained in $H\times\{0\}$ with $3$-balls. The boundary component $H\times\{0\}$ is referred to as $\partial_+$. The rest of the boundary is referred to as $\partial_-$. A *Heegaard splitting* of a $3$-manifold $M$ is an expression of $M$ as a union $V\cup_H W$, where $V$ and $W$ are compression bodies that intersect in a transversally oriented surface $H=\partial_+V=\partial_+ W$. If $V\cup_H W$ is a Heegaard splitting of $M$ then we say $H$ is a *Heegaard surface*. Let $V\cup_H W$ be a Heegaard splitting of a $3$-manifold $M$. Then we say the pair $(V,W)$ is a *weak reducing pair* for $H$ if $V$ and $W$ are disjoint compressing disks on opposite sides of $H$. A Heegaard surface is *strongly irreducible* if it is compressible to both sides but has no weak reducing pairs. \[definition-GHS\] A *generalized Heegaard splitting* (GHS)[^1] H of a $3$-manifold $M$ is a pair of sets of pairwise disjoint, transversally oriented, connected surfaces, $\operatorname{Thick}(H)$ and $\operatorname{Thin}(H)$ (in this article, we will call the elements of each of both *thick surfaces* and *thin surfaces*, resp.), which satisfies the following conditions. 1. Each component $M'$ of $M-\operatorname{Thin}(H)$ meets a unique element $H_+$ of $\operatorname{Thick}(H)$ and $H_+$ is a Heegaard surface in $M'$. Henceforth we will denote the closure of the component of $M-\operatorname{Thin}(H)$ that contains an element $H_+\in\operatorname{Thick}(H)$ as $M(H_+)$. 2. As each Heegaard surface $H_+\subset M(H_+)$ is transversally oriented, we can consistently talk about the points of $M(H_+)$ that are “above” $H_+$ or “below” $H_+$. Suppose $H_-\in \operatorname{Thin}(H)$. Let $M(H_+)$ and $M(H'_+)$ be the submanifolds on each side of $H_-$. Then $H_-$ is below $H_+$ if and only if it is above $H'_+$. 3. There is a partial ordering on the elements of $\operatorname{Thin}(H)$ which satisfies the following: Suppose $H_+$ is an element of $\operatorname{Thick}(H)$, $H_-$ is a component of $\partial M(H_+)$ above $H_+$ and $H'_-$ is a component of $\partial M(H_+)$ below $H_+$. Then $H_->H'_-$. Suppose $H$ is a GHS of a $3$-manifold $M$ with no $S^3$ components. Then $H$ is *strongly irreducible* if each element $H_+\in \operatorname{Thick}(H)$ is strongly irreducible in $M(H_+)$. The manifold decomposition of double branched cover of $(S^3,k)$ from a thin position of $k$\[section-DBC\] =========================================================================================================== In this section, we will describe the method to construct a manifold decomposition of DBC from a thin position of a knot originated from H.N. Howards and J. Schultens in [@HS] (see section 3 of [@HS] for more details.) We borrow the notions and definitions directly from [@HS]. Let $h:\{S^3 - (\text{two points})\} \to [0, 1]$ be a height function on $S^3$ that restricts to a Morse function on $k$. Choose a regular value $t_i$ between each pair of adjacent critical values of $h|k$. The *width* of $k$ with respect to $h$ is $\sum_i \#|k\cap h^{-1}(t_i)|$. Define the *width* of $k$ to be the minimum width of $k$ with respect to $h$ over all $h$. A *thin position* of $k$ is the presentation of $k$ with respect to a height function that realizes the width of $k$. A *thin level* for $k$ is a $2$-sphere $S$ such that the following hold: 1. $S = h^{-1}(t_0)$ for some regular value $t_0$; 2. $t_0$ lies between adjacent critical values $x$ and $y$ of $h$, where $x$ is a minimum of $k$ lying above $t_0$ and $y$ is a maximum of $k$ lying below $t_0$. A *thick level* is a $2$-sphere $S$ such that the following hold: 1. $S = h^{-1}(t_0)$ for some regular value $t_0$; 2. $t_0$ lies between adjacent critical values $x$ and $y$ of $h$, where $x$ is a maximum of $k$ lying above $t_0$ and $y$ is a minimum of $k$ lying below $t_0$. A *manifold decomposition* is a generalized version of GHS in Definition \[definition-GHS\][^2], where we permit thin surfaces to be 2-spheres (this “manifold decomposition” is originated from [@ST] by M. Scharlemann and A. Thompson.) So adjacent thick and thin surfaces in a manifold decomposition cobound a (possibly, punctured-) compression body. Let the *complexity* of a connected surface $S$ be $c(S)= 1-\chi(S) = 2 \operatorname{genus}(S)-1$ for $S$ of positive genus. Define $c(S^2) = 0$. For $S$ not necessarily connected define $c(S) = \sum\{c(S')| S'\text{ a connected component of }S\}$. Let the *width* of the manifold decomposition $H$ of $M$ be the set of integers $\{c(S_i)| 1\leq i \leq k, \text{ each }S_i\text{ is the thick surface of }H \}$. Order these integers in monotonically non-increasing order. Compare the ordered multi-sets lexicographically. Define the *width* $w(M)$ of $M$ to be the minimal width over all manifold decompositions using the above ordering of the sets of integers. A given manifold decomposition of $M$ is *thin* if the width of the manifold decomposition is the width of $M$. Let $k$ be a knot and denote DBC of $(S^3,k)$ by $M$. If $k$ is in a thin position, then $M$ inherits a manifold decomposition as follows: Denote the thick levels of $k$ by $S_1,\cdots,S_n$ and the thin levels by $L_1,\cdots,L_{n-1}$. Each $S_i$ and each $L_i$ is a sphere that meets the knot some (even) number of times. More specifically, each $S_i$ meets $k$ at least $4$ times and each $L_i$ meets $k$ at least $2$ times. Denote the surface in $M$ corresponding to $S_i$ by $\tilde{S}_i$ and the surface in $M$ corresponding to $L_i$ by $\tilde{L}_i$ . Each $\tilde{S}_i$ is a closed orientable surface of genus at least $1$ and each $\tilde{L}_i$ is a closed orientable surface. More specifically, if $S_i$ meets $k$ exactly $2l$ times, then $\tilde{S}_i$ is a closed orientable surface of genus $l-1$. And if $L_i$ meets $k$ exactly $2l$ times, then $\tilde{L}_i$ is a closed orientable surface of genus $l-1$. See Figure \[figure-thin\]. ![Thin and thick levels of $k$ and corresponding surfaces in $M$\[figure-thin\]](fig-thin.pdf){width="8.5cm"} The $3$-ball bounded by $S_1$ ($S_n$ resp.) in $S^3$ corresponds to a handlebody $H_1$ ($H_n$ resp.) in $M$, where $H_1$ ($H_n$ resp.) is bounded by $\tilde{S}_1$ ($\tilde{S}_n$ resp.) in $M$. Moreover, the submanifold between $S_i$ and $L_{i-1}$ in $S^3$ (between $L_i$ and $S_{i}$ resp.) corresponds to a (possibly punctured) compression body $C_i^1$ ($C_i^2$ resp.) in $M$, where $\partial_+ C_i^1 = \tilde{S}_i$ ($\partial_+C_i^2=\tilde{S}_{i}$ resp.) and $\partial_- C_i^1 = \tilde{L}_{i-1}$ ($\partial_-C_i^2=\tilde{L}_{i}$ resp.) See Figure \[figure-md\]. Moreover, $C_i^1$ ($C_i^2$ resp.) is not a trivial compression body, i.e. homeomorphic to $F\times I$ for a closed surface $F$. ![The manifold decomposition of $M$\[figure-md\]](fig-md.pdf){width="4cm"} Therefore, the manifold decomposition of $M$ by Howards and Schultens can be written as an ordered set of thin and thick surfaces, $$\{\tilde{L}_0=\emptyset, \tilde{S}_1, \tilde{L}_1,\tilde{S}_2,\cdots, \tilde{L}_{n-1}, \tilde{S}_n,\tilde{L}_n=\emptyset\},$$ where we denote the empty negative boundaries of $H_1$ and $H_n$ as $\tilde{L}_0$ and $\tilde{L}_n$ for convenience. Proof of Theorem \[theorem-1\]\[section-main-theorem\] ====================================================== Let us consider the assumption that the given thin position of $k$ induces a thin manifold decomposition of DBC $M$ of $(S^3,k)$. In Rule 6 of [@ST], Scharlemann and Thompson proved that each thick surface $S$ in a thin manifold decomposition is *weakly incompressible*, i.e. any two compressing disks for $S$ on opposite sides of $S$ intersect along their boundary, so every thick surface in a thin manifold decomposition is strongly irreducible in $M(S)$ (we can also check it from Proposition 4.2.3 of [@SSS].) Let $H=\{\tilde{L}_0=\emptyset, \tilde{S}_1, \tilde{L}_1,\tilde{S}_2,\cdots, \tilde{L}_{n-1}, \tilde{S}_n,\tilde{L}_n=\emptyset\}$ be the manifold decomposition of DBC of $(S^3, k)$ by the H-S method as in the end of section \[section-DBC\], where each $\tilde{S}_i$ for $i=1,\cdots,n$ ($\tilde{L}_i$’s for $i=1,\cdots, n-1$ resp.) comes from a thick level (thin level resp.) of the given thin position of $k$. If there exist $m$-thin levels which intersect $k$ in two points, then we get $m$-thin spheres in $H$. Since the construction of the manifold decomposition allows only one surface for each thick or thin level, these $m$-thin spheres must be $\tilde{L}_{j_1}$, $\cdots$, $\tilde{L}_{j_m}$ for some $1\leq j_1< \cdots< j_m\leq {n-1}$. Let $\tilde{L}_{j_0}$ be $\tilde{L}_0$ and $\tilde{L}_{j_{m+1}}$ be $\tilde{L}_{n}$ for convenience (so they are empty sets.) Now we cut DBC along the $m$-thin spheres, and cap off all $S^2$ boundaries with $3$-balls. Then we get $(m+1)$-manifolds, $M_0$, $M_1$, $\cdots$, $M_m$, where each $M_i$ comes from the submanifold of DBC bounded by $\tilde{L}_{j_{i}}$ and $\tilde{L}_{j_{i+1}}$ for $i=0,\cdots, m$. In addition, we can induce the canonical manifold decomposition $H_i$ of $M_i$ from $H$, where $H_i=\{\, \emptyset, \tilde{S}_{j_i+1}, \tilde{L}_{j_i+1},\cdots,\tilde{L}_{j_{i+1}-1},\tilde{S}_{j_{i+1}},\emptyset\}$. Now every thin surface of $H_i$ is not homeomorphic to $S^2$ for $i=0,\cdots,m$, so we can say that the manifold decomposition $H_i$ of $M_i$ is a GHS in Definition of \[definition-GHS\]. Moreover, it is obvious that the strongly irreducibility of each $\tilde{S}_j$ in $M(\tilde{S}_j)$ for $j=1,\cdots,n$ does not change after it becomes a Heegaard surface in $M_i(\tilde{S}_j)$ for some $i$. So either $H_i$ is a strongly irreducible GHS or $M_i$ is homeomorphic to $S^3$ for $i=0,\cdots,m$. By Lemma 4.7 of [@Bachman3], $M_i$ is irreducible or homeomorphic to $S^3$ (obviously, it is also irreducible) for $i=0,\cdots,m$ , i.e. no one is $S^2\times S^1$. Since the thin spheres $\tilde{L}_{j_1},\cdots,\tilde{L}_{j_m}$ are separating in $M$, no one of them is an essential sphere in a $S^2\times S^1$ piece in the prime decomposition of $M$. So we can assume that there is no $S^2\times S^1$ piece in the prime decomposition of $M$. But it is not obvious whether the sum $M_0\#_{\tilde{L}_{j_1}}M_1\#_{\tilde{L}_{j_2}}\#\cdots\#_{\tilde{L}_{j_m}}M_m$ does not have any trivial summand. So we need the following claim.\ [**Claim.** *No $M_i$ is homeomorphic to $S^3$ for $i=0,\cdots,m$.*\ ]{} Suppose that $M_l$ is homeomorphic to $S^3$ for some $l$. Let the thin levels in the thin position of $k$ corresponding to $\tilde{L}_{j_l}$ and $\tilde{L}_{j_{l+1}}$ be $L'$ and $L''$ (one of both may be empty level if $l=0$ or $m$), and the thick level in the thin position of $k$ corresponding to $\tilde{S}_{j_l+1}$ be $S$. The thin levels $L'$ and $L''$ intersect $k$ in two points, i.e. each of both $L'$ and $L''$ realizes a connected sum of $k$, i.e. $k=k_1 \#_{L'} k' \#_{L''} k_2$. Moreover, the thick level $S$ must intersect $k$ in $4$ or more points. Since $M_l\cong S^3$ is DBC of $(k',S^3)$ and $S^3$ has the unique representation as DBC of $S^3$ branched along a knot or a link (see [@Wa2], and this is also true for the other lens spaces, see [@HR] or Problem 3.26 of [@K],) $k'$ is an unknot. Since $k$ is in a thin position, the unknot summand $k'$ in $k=k_1 \#_{L'} k' \#_{L''} k_2$ must intersect all levels between $L'$ and $L''$ in two points, this contradicts the existence of the thick level $S$. So we get a contradiction. In the cases of $l=0$ and $m$, we get a contradiction by similar arguments for each case. This completes the proof of Claim. Now we can say that the thin spheres $\tilde{L}_{j_1}$, $\cdots$, $\tilde{L}_{j_m}$ determine the prime decomposition of $M$. Let us consider the thin levels $L_{j_1}$, $\cdots$, $L_{j_m}$ corresponding to $\tilde{L}_{j_1}$, $\cdots$,$\tilde{L}_{j_m}$. Then they cut $k$ into $m+1$ summands, i.e $k=k_0\#\cdots\#k_m$. In particular, each $M_i$ is DBC of $(S^3,k_i)$ for $i=0,\cdots,m$ and it is also a prime manifold as already proved. Moreover, each $k_i$ must be prime by Corollary 4 of [@KT]. Since $k$ is in a thin position, each summand $k_i$ must be in a thin position. This complete the proof of Theorem \[theorem-1\]. A generalization of Theorem \[theorem-1\]\[section-critical\] ============================================================= In this section, we will generalize Theorem \[theorem-1\] using the concept *critical surface*. D. Bachman introduced a concept “critical surface” in [@Bachman1] and prove several properties about critical surface and minimal common stabilization. Also he proved Gordon’s conjecture (see Problem 3.91 of [@K]) using critical surface theory in [@Bachman3]. In particular, he used the term *topological minimal surface* to denote the class of surfaces such that they are incompressible, strongly irreducible or critical in [@Bachman4]. \[definition-critical\] Let $H$ be a Heegaard surface in some $3$-manifold which is compressible to both sides. The surface $H$ is *critical* if the set of all compressing disks for $H$ can be partitioned into subsets $C_0$ and $C_1$ such that the following hold. 1. For each $i=0,1$ there is at least one weak reducing pair $(V_i,W_i)$, where $V_i$, $W_i\in C_i$. 2. If $V\in C_0$ and $W\in C_1$ then $(V,W)$ is not a weak reducing pair. Suppose $H$ is a GHS of a $3$-manifold $M$ with no $S^3$ components. Then $H$ is *psudo-critical* if each thick surface $H_+\in \operatorname{Thick}(H)$ is strongly irreducible or critical in $M(H_+)$.[^3] Now we introduce a generalization of Theorem \[theorem-1\]. \[corollary-1\] If a thin position of a knot $k$ induces a manifold decomposition of double branched cover $M$ of $(S^3,k)$ by the Howards and Schultens’ method where each thick surface $H_+$ of the manifold decomposition is strongly irreducible or critical in $M(H_+)$, then the thin position of $k$ is the sum of prime summands by stacking a thin position of one of prime summands on top of a thin position of another prime summand, and so on. Therefore, $k$ holds the nearly additivity of knot width in this case. Assume $H=\{\tilde{L}_0=\emptyset, \tilde{S}_1, \tilde{L}_1,\tilde{S}_2,\cdots, \tilde{L}_{n-1}, \tilde{S}_n,\tilde{L}_n=\emptyset\}$ be the manifold decomposition of DBC of $(S^3, k)$ by the H-S method, and $m$-thin spheres $\tilde{L}_{j_1}$, $\cdots$, $\tilde{L}_{j_m}$ cut $M$ into $M_0,\cdots,M_m$ with the canonical GHS $H_0,\cdots,H_m$ as in section \[section-main-theorem\]. In particular, every thick surface $H_+\in \operatorname{Thick}{H}$ is strongly irreducible or critical in $M(H_+)$ by the hypothesis. It is obvious that the strongly irreducibility or criticality of each $\tilde{S}_j$ in $M(\tilde{S}_j)$ for $j=1,\cdots,n$ does not change after it becomes a Heegaard surface in $M_i(\tilde{S}_j)$ for some $i$. So either $H_i$ is a psudo-critical GHS or $M_i$ is homeomorphic to $S^3$ for $i=0,\cdots,m$. Since the proofs of Lemma 4.6 and Lemma 4.7 of [@Bachman3] do not depend on the number of critical thick levels, and only depend on the property that each thick surface of the GHS is strongly irreducible or critical, we can extend both lemmas to psudo-critical GHS. So we get each $M_i$ is irreducible or homeomorphic to $S^3$. The remaining arguments of the proof of Corollary \[corollary-1\] are the same as those of Theorem \[theorem-1\]. This completes the proof of Corollary \[corollary-1\] [10]{} D. Bachman, *Critical heegaard surfaces*, Trans. Amer. Math. Soc. **354** (2002), no. 10, 4015–4042 (electronic). D. Bachman, *Connected sums of unstabilized Heegaard splittings are unstabilized*, Geom. Topol. **12** (2008), no. 4, 2327–2378. D. Bachman, *Barriers to topologically minimal surfaces*, arXiv:0903.1692v1 R. Blair and M. Tomova, *Companions of the unknot and width additivity*, arXiv:0908.4103v1 D.J. Heath and T. Kobayashi, *Essential tangle decomposition from thin position of a link*, Pacific J. Math. **179** (1997), no.1, 101–117. H. Howards, Y. Rieck, and J. Schultens, *Thin position for knots and 3-manifolds: a unified approach; Workshop on Heegaard Splittings*, Geom. Topol. Monogr. **12** (2007), 89–120. C. Hodgson and and J.H. Rubinstein, *Involutions and isotopies of lens spaces*, Knot theory and manifolds (Vancouver, B.C., 1983), 60–96, Lecture Notes in Math., 1144, Springer, Berlin, 1985. H. Howards and J. Schultens, *Thin position for knots and $3$-manifolds*, Topology Appl., **155** (2008), no. 13, 1371–1381. R. Kirby, *Problems in low-dimensional topology*, from: “Geometric topology (Athens, GA, 1993)”, AMS/IP Stud. Adv. Math. **2**, Amer. Math. Soc. (1997) 35–473. P. Kim and J. Tollefson *Splitting the PL involutions of nonprime 3-manifolds*, Michigan Math. J. **27** (1980) no. 3, 259–274. Y. Rieck and E. Sedgwick, *Thin position for a connected sum of small knots*, Algebr. Geom. Topol. **2** (2002), 297–309 (electronic). T. Saito, M. Scharlemann, and J. Schultens, *Lecture notes on generalized Heegaard splittings*, arXiv:math/0504167. M. Scharlemann and A. Thompson, *Thin position for $3$-manifolds*, A.M.S. Contemp. Math., 164:231-238, 1994. M. Scharlemann and A. Thompson, *On the additivity of knot width*, Proceedings of the Casson Fest, 135–144 (electronic), Geom. Topol. Monogr., **7**, Geom. Topol. Publ., Coventry, 2004. F. Waldhausen, *Über involutionen der 3-sphäre*, Topology **8** (1969) 81–91. [^1]: Note that D. Bachman did not allow thin surfaces in a GHS to be $2$-spheres in [@Bachman3]. In particular, he introduced more generalized concept “pseudo-GHS” in [@Bachman3] to deal with thin spheres and trivial compression bodies. [^2]: Many authors use the term “generalized Heegaard splitting” to denote “manifold decomposition” in [@ST] by M. Scharlemann and A. Thompson, but the author distinguished the terms “manifold decomposition” and “generalized Heegaard splitting” to use two different definitions at the same time. [^3]: This definition is weaker than the definition “*critical GHS*” in [@Bachman3].

--- abstract: 'We consider Gaussian multiple-input multiple-output (MIMO) channels with discrete input alphabets. We propose a non-diagonal precoder based on the X-Codes in [@Xcodes_paper] to increase the mutual information. The MIMO channel is transformed into a set of parallel subchannels using Singular Value Decomposition (SVD) and X-Codes are then used to pair the subchannels. X-Codes are fully characterized by the pairings and a $2\times 2$ real rotation matrix for each pair (parameterized with a single angle). This precoding structure enables us to express the total mutual information as a sum of the mutual information of all the pairs. The problem of finding the optimal precoder with the above structure, which maximizes the total mutual information, is solved by [*i*]{}) optimizing the rotation angle and the power allocation within each pair and [*ii*]{}) finding the optimal pairing and power allocation among the pairs. It is shown that the mutual information achieved with the proposed pairing scheme is very close to that achieved with the optimal precoder by Cruz [*et al.*]{}, and is significantly better than Mercury/waterfilling strategy by Lozano [*et al.*]{}. Our approach greatly simplifies both the precoder optimization and the detection complexity, making it suitable for practical applications.' author: - 'Saif Khan Mohammed,  Emanuele Viterbo,  Yi Hong,  and Ananthanarayanan Chockalingam,  [^1] [^2]' title: | Precoding by Pairing Subchannels to Increase\ MIMO Capacity with Discrete Input Alphabets --- Mutual information, MIMO, OFDM, precoding, singular value decomposition, condition number. Introduction ============ Many modern communication channels are modeled as a Gaussian multiple-input multiple-output (MIMO) channel. Examples include multi-tone digital subscriber line (DSL), orthogonal frequency division multiplexing (OFDM) and multiple transmit-receive antenna systems. It is known that the capacity of the Gaussian MIMO channel is achieved by beamforming a [*Gaussian input alphabet*]{} along the right singular vectors of the MIMO channel. The received vector is projected along the left singular vectors, resulting in a set of parallel Gaussian subchannels. Optimal power allocation between the subchannels is achieved by waterfilling [@Cover]. In practice, the input alphabet is [*not Gaussian*]{} and is generally chosen from a finite signal set. We distinguish between two kinds of MIMO channels: [*i*]{}) [*diagonal*]{} (or parallel) channels and [*ii*]{}) [*non-diagonal*]{} channels. For a diagonal MIMO channel with discrete input alphabets, assuming only power allocation on each subchannel (i.e., a diagonal precoder), Mercury/waterfilling was shown to be optimal by Lozano [*et al.*]{} in [@Lozano]. With discrete input alphabets, Cruz [*et al.*]{} later proved in [@cruz] that the optimal precoder is, however, non-diagonal, i.e., precoding needs to be performed across all the subchannels. For a general non-diagonal Gaussian MIMO channel, it was also shown in [@cruz] that the optimal precoder is non-diagonal. Such an optimal precoder is given by a fixed point equation, which requires a high complexity numeric evaluation. Since the precoder jointly codes all the $n$ inputs, joint decoding is also required at the receiver. Thus, the decoding complexity can be very high, specially for large $n$, as in the case of DSL and OFDM applications. This motivates our quest for a practical low complexity precoding scheme achieving near optimal capacity. In this paper, we consider a general MIMO channel and a non-diagonal precoder based on X-Codes [@Xcodes_paper]. The MIMO channel is transformed into a set of parallel subchannels using Singular Value Decomposition (SVD) and X-Codes are then used to pair the subchannels. X-Codes are fully characterized by the pairings and the 2-dimensional real rotation matrices for each pair. These rotation matrices are parameterized with a single angle. This precoding structure enables us to express the total mutual information as a sum of the mutual information of all the pairs. The problem of finding the optimal precoder with the above structure, which maximizes the total mutual information, can be split into two tractable problems: [*i*]{}) optimizing the rotation angle and the power allocation within each pair and [*ii*]{}) finding the optimal pairing and power allocation among the pairs. It is shown by simulation that the mutual information achieved with the proposed pairing scheme is very close to that achieved with the optimal precoder in [@cruz], and is significantly better than the Mercury/waterfilling strategy in [@Lozano]. Our approach greatly simplifies both the precoder optimization and the detection complexity, making it suitable for practical applications. The rest of the paper is organized as follows. Section \[SMPrecoding\] introduces the system model and SVD precoding. In Section \[optimalprecoding\], we provide a brief review of the optimal precoding with discrete inputs in [@cruz] and the relevant MIMO capacity. In Section \[PrecodingX\], we present the precoding using X-Codes with discrete inputs and the relevant capacity expressions. In Section \[two\_subch\], we consider the first problem, which is to find the optimal rotation angle and power allocation for a given pair. This problem is equivalent to optimizing the mutual information for a Gaussian MIMO channel with two subchannels. In Section \[multi\_subch\], using the results from Section \[two\_subch\], we attempt to optimize the mutual information for a Gaussian MIMO channel with $n$ subchannels, where $n>2$. Conclusions are drawn in Section \[conclusions\].Finally, in Section \[sec\_ofdm\] we discuss the application of our precoding to OFDM systems. [*Notations*]{}: The field of complex numbers is denoted by $\mathbb{C}$ and let ${\mathbb R}^+$ be the positive real numbers. Superscripts $^T$ and $^{\dag}$ denote transposition and Hermitian transposition, respectively. The $n\times n$ identity matrix is denoted by $\mathbf{I}_{n}$, and the zero matrix is denoted by $\mathbf{0}$. The ${{\mathbb E}}[\cdot]$ is the expectation operator, $\Vert \cdot \Vert$ denotes the Euclidean norm of a vector, and $\|\cdot\|_F$ the Frobenius norm of a matrix. Finally, we let $\mbox{tr}(\cdot)$ be the trace of a matrix. System model and Precoding with Gaussian inputs {#SMPrecoding} =============================================== We consider a $n_t\times n_r$ MIMO channel, where the channel state information (CSI) is known perfectly at both transmitter and receiver. Let ${\bf x} = (x_1,\cdots, x_{n_t})^T$ be the vector of input symbols to the channel, and let ${\bf H}=\{h_{ij}\}$, $i=1, \cdots, n_r$, $j=1, \cdots, n_t$, be a full rank $n_r\times n_t$ channel coefficient matrix, with $h_{ij}$ representing the complex channel gain between the $j$-th input symbol and the $i$-th output symbol. The vector of $n_r$ channel output symbols is given by $$\label{system_modeleq} {\bf y} = \sqrt {P_T}{\bf H}{\bf x} + {\bf w}$$ where ${\bf w}$ is an uncorrelated Gaussian noise vector, such that ${{\mathbb E}}[{\bf w}{\bf w}^\dag]= {\bf I}_{n_r}$, and $P_T$ is the total transmitted power. The power constraint is given by $$\label{tx_pow} {{\mathbb E}}[\Vert {\bf x} \Vert^2] = 1$$ The maximum multiplexing gain of this channel is $n = \min(n_r,n_t)$. Let ${\bf u}=(u_1,\cdots, u_{n})^T \in{\mathbb C}^{n}$ be the vector of $n$ information symbols to be sent through the MIMO channel, with ${{\mathbb E}}[\vert u_i \vert^2] = 1, i = 1, \cdots, n$. Then the vector ${\bf u}$ can be precoded using a $n_t \times n$ matrix ${\bf T}$, resulting in ${\bf x}={\bf T}{\bf u}$. The capacity of the deterministic Gaussian MIMO channel is then achieved by solving \[cap\_gaussian\] $$\begin{aligned} \label{cap_gaussian_mimo} C({\bf H},P_T) & = & \max_{ {\bf K}_{\bf x} | \mbox{tr}({\bf K}_{\bf x} ) = 1} I({\bf x} ; {\bf y} | {\bf H}) \\ \nonumber & \geq & \max_{ {\bf K}_{\bf u}, {\bf T} \,| \, \mbox{tr}({\bf T}{\bf K}_{\bf u}{\bf T}^\dag) = 1} I({\bf u} ; {\bf y} | {\bf H})\end{aligned}$$ where $I({\bf x} ; {\bf y} | {\bf H})$ is the mutual information between ${\bf x}$ and ${\bf y}$, and ${\bf K}_{\bf x} {\stackrel {\Delta} {=} }{{\mathbb E}}[{\bf x}{\bf x}^\dag]$, ${\bf K}_{\bf u} {\stackrel {\Delta} {=} }{{\mathbb E}}[{\bf u}{\bf u}^\dag]$ are the covariance matrices of ${\bf x}$ and ${\bf u}$ respectively. The inequality in (\[cap\_gaussian\_mimo\]) follows from the data processing inequality [@Cover]. Let us consider the singular value decomposition (SVD) of the channel ${\bf H}={\bf U}{\mathbf \Lambda}{\bf V}$, where ${\bf U} \in {\mathbb C}^{n_r \times n}$, ${\mathbf \Lambda} \in {\mathbb C}^{n \times n}$, ${\bf V} \in {\mathbb C}^{n \times n_t}$, ${\bf U}^\dag{\bf U}={\bf V}{\bf V}^\dag={\bf I}_{n}$, and ${\mathbf \Lambda} = \mbox{diag} (\lambda_{1}, \ldots, \lambda_{n})$ with $\lambda_1 \geq \lambda_2, \cdots, \geq \lambda_{n} \geq 0$. Telatar showed in [@tela99] that the Gaussian MIMO capacity $C({\bf H},P_T)$, is achieved when ${\bf x}$ is Gaussian distributed and ${\bf T}{\bf K}_{\bf x} {\bf T}^{\dag}$ is diagonal. Diagonal ${\bf T}{\bf K}_{\bf x} {\bf T}^{\dag}$ can be achieved by using the optimal precoder matrix ${\bf T} = {\bf V}^\dag {\bf P}$, where ${\bf P} \in ({\mathbb R^{+}})^{n}$ is the diagonal power allocation matrix such that $\mbox{tr}( {\bf P}{\bf P}^\dag) = 1$. Furthermore, $u_i, i = 1, \ldots, n$, are i.i.d. Gaussian (i.e., [*no coding is required across the input symbols $u_i$*]{}). With this, the second line of (\[cap\_gaussian\_mimo\]) is actually an equality. Also, projecting the received vector ${\bf y}$ along the columns of ${\bf U}$ is information lossless and transforms the non-diagonal MIMO channel into an equivalent diagonal channel with $n$ non-interfering subchannels. The equivalent diagonal system model is then given by $$\label{eq_diag_model_ch6} {\bf r} {\stackrel {\Delta} {=} }{\bf U}^{\dag}{\bf y} = \sqrt{P_T} {\mathbf \Lambda} {\bf P} {\bf u} + {\Tilde {\bf w}}$$ where ${\Tilde {\bf w}}$ is the equivalent noise vector, and has the same statistics as ${\bf w}$. The total mutual information is now given by $$\label{tot_mi_gauss_ch6} I({\bf x} ; {\bf y} | {\bf H}) = \sum_{i=1}^{n} \log_2(1 + {\lambda_i}^2 p_{i}^2 P_T)$$ Note that now the mutual information is a function of only the power allocation matrix ${\bf P}=$ diag$(p_1,\ldots,p_n)$, with the constraint tr$({\bf P}{\bf P}^{\dag}) = 1$. Optimal power allocation is achieved through waterfilling between the $n$ parallel channels of the equivalent system in (\[eq\_diag\_model\_ch6\]) [@Cover]. Optimal precoding with discrete inputs {#optimalprecoding} ====================================== In practice, discrete input alphabets are used. Subsequently, we assume that the $i$-th information symbol is given by $u_i \in {\mathcal U}_i$, where ${\mathcal U}_i \subset {\mathbb C}$ is a finite signal set. Let ${\mathcal S} {\stackrel {\Delta} {=} }{\mathcal U}_1 \times {\mathcal U}_2 \times \cdots \times {\mathcal U}_n$ be the overall input alphabet. The capacity of the Gaussian MIMO channel with discrete input alphabet ${\mathcal S}$ is defined by the following problem \[cap\_discrete\] $$\begin{aligned} \label{cap_gaussian_mimo2} C_{\mathcal S}({\bf H},P_T) = \max_{{\bf T} \,| \, {\bf u} \in {\mathcal S}, \Vert {\bf T} \Vert_F = 1} I({\bf u} ; {\bf y} | {\bf H})\end{aligned}$$ Note that there is no maximization over the pdf of ${\bf u}$, since we fix ${\bf K}_{\bf u} = {\bf I}_n$. The optimal precoder ${\bf T}^{*}$, which solves Problem \[cap\_discrete\], is given by the following fixed point equation given in [@cruz] $$\label{opt_precoder} {\bf T}^{*} = \frac {{\bf H}^\dag{\bf H}{\bf T}^{*} {\bf E} } {\Vert {\bf H}^\dag{\bf H}{\bf T}^{*} {\bf E} \Vert_F}$$ where ${\bf E}$ is the minimum mean-square error (MMSE) matrix of ${\bf u}$ given by $$\label{mmse_mat} {\bf E} = {{\mathbb E}}[ ( {\bf u} - {{\mathbb E}}[{\bf u}|{\bf y}]) ( {\bf u} - {{\mathbb E}}[{\bf u}|{\bf y}])^\dag ]$$ The optimal precoder is derived using the relation between MMSE and mutual information [@guo_mmse]. We observe that, with discrete input alphabets, it is no longer optimal to beamform along the column vectors of ${\bf V}^\dag$ and then use waterfilling on the parallel subchannels. Even when ${\bf H}$ is diagonal (parallel non-interfering subchannels), the optimal precoder ${\bf T}^{*}$ is [*non diagonal*]{}, and can be computed numerically (using a gradient based method) as discussed in [@cruz]. However, the complexity of computing ${\bf T}^{*}$ is prohibitively high for practical applications, especially when $n$ is large and/or the channel changes frequently. We propose a suboptimal precoding scheme based on X-Codes [@Xcodes_paper], which achieves close to the optimal capacity $C_{\mathcal S}({\bf H},P_T)$, at low encoding and decoding complexities. Precoding with X-Codes {#PrecodingX} ====================== X-Codes are based on a pairing of $n$ subchannels $\ell = \{ (i_k,j_k)\in [1,n]\times [1,n], i_k < j_k, k = 1, \ldots n/2 \}$. For a given $n$, there are $(n-1)(n-3) \cdots 3\,1$ possible pairings. Let ${\mathcal L}$ denote the set of all possible pairings. For example, with $n = 4$, we have $${\mathcal L} = \left\{ \{(1,4),(2,3)\} \,,\, \{(1,2),(3,4)\} \,,\, \{(1,3),(2,4)\} \right \}$$ X-Codes are generated by a $n \times n$ real orthogonal matrix, denoted by ${\bf G}$. When precoding with X-Codes, the precoder matrix is given by ${\bf T} = {\bf V}^\dag {\bf P}{\bf G}$, where ${\bf P} = \mbox{diag}(p_1, p_2, \cdots, p_n) \in {(\mathbb R^{+})}^{n}$ is the diagonal power allocation matrix such that $\mbox{tr}( {\bf P}{\bf P}^\dag) = 1$. The $k$-th pair consists of subchannels $i_k$ and $j_k$. For the $k$-th pair, the information symbols $u_{i_k}$ and $u_{j_k}$ are jointly coded using a $2 \times 2$ real orthogonal matrix ${\bf A}_k$ given by $$\label{akmat} {\bf A}_k = \left[\begin{array}{cc} \cos(\theta_k) & \sin(\theta_k) \\ -\sin(\theta_k) & \cos(\theta_k) \end{array} \right] \ \ \ k=1,\ldots n/2$$ The angle $\theta_k$ can be chosen to maximize the mutual information for the $k$-th pair. Each ${\bf A}_k$ is a submatrix of the code matrix ${\bf G}=(g_{i,j})$ as shown below $$\begin{aligned} \label{Xak} \begin{array}{ll} g_{{i_k},{i_k}} = \cos(\theta_k) & g_{{i_k},{j_k}} = \sin(\theta_k) \\ g_{{j_k},{i_k}} = -\sin(\theta_k) & g_{{j_k},{j_k}} = \cos(\theta_k) \end{array}\end{aligned}$$ It was shown in [@Xcodes_paper] that, for achieving the best diversity gain, an optimal pairing is one in which the $k$-th subchannel is paired with the $(n - k + 1)$-th subchannel. For example, with this pairing and $n$ = $6$, the X-Code generator matrix is given by $$\footnotesize {\mathbf G}\!\!=\!\!\left[ \!\!\begin{array}{cccccc} \cos(\theta_1) & ~~ & ~~ & ~~ & ~~ & \sin(\theta_1) \\ ~~ & \cos(\theta_2) & ~~ & ~~ & \sin(\theta_2) & ~~ \\ ~~ & ~~ & \cos(\theta_3) & \sin(\theta_3) & ~~ & ~~ \\ ~~ & ~~ & -\sin(\theta_3) & \cos(\theta_3) & ~~ & ~~ \\ ~~ & -\sin(\theta_2) & ~~ & ~~ & \cos(\theta_2) & ~~ \\ -\sin(\theta_1) & ~~ & ~~ & ~~ & ~~ & \cos(\theta_1) \end{array} \!\! \right]$$ The special case with $\theta_k = 0, k = 1,2, \cdots, n/2$, results in no coding across subchannels. Given the generator matrix ${\bf G}$, the subchannel gains ${\mathbf \Lambda}$, and the power allocation matrix ${\bf P}$, the mutual information between ${\bf u}$ and ${\bf y}$ is given by $$\begin{aligned} \label{cap4} &&\hspace{-5mm}I_{\mathcal S}({\bf u};{\bf y}|{\mathbf \Lambda},{\bf P} ,{\bf G}) = h({\bf y} | {\mathbf \Lambda},{\bf P},{\bf G}) - h({\bf w}) \\ \nonumber &&\hspace{-3mm}= -\!\!\int_{{\bf y} \in {\mathbb C}^{n_r}} \hspace{-4mm} p({\bf y}|{\mathbf \Lambda},{\bf P},{\bf G})\log_2(p({\bf y}|{\mathbf \Lambda},{\bf P},{\bf G})) d{\bf y} - n\log_2(\pi e )\end{aligned}$$ where the received vector pdf is given by $$\label{py2} p({\bf y}|{\mathbf \Lambda},{\bf P},{\bf G}) = \frac{1}{\vert {\mathcal S} \vert \pi^n} \sum_{{\bf u} \in {\mathcal S}} e^{-\Vert {\bf y} - \sqrt{P_T}{\bf U}{\mathbf \Lambda} {\bf P} {\bf G} {\bf u} \Vert ^2}$$ and when $n = n_r$ (i.e., $n_r \leq n_t$), it is equivalently given by $$\label{py} p({\bf y}|{\mathbf \Lambda},{\bf P},{\bf G}) = \frac{1}{\vert {\mathcal S} \vert \pi^n} \sum_{{\bf u} \in {\mathcal S}} e^{-\Vert {\bf r} - \sqrt{P_T}{\mathbf \Lambda} {\bf P} {\bf G} {\bf u} \Vert ^2}$$ where ${\bf r} = (r_1, r_2, \cdots, r_n)^T {\stackrel {\Delta} {=} }{\bf U}^\dag{\bf y}$. We next define the capacity of the MIMO Gaussian channel when precoding with ${\bf G}$. In the following, we assume that $n_r \leq n_t$, so that $I_{\mathcal S}({\bf u};{\bf y}|{\mathbf \Lambda},{\bf P} ,{\bf G}) = I_{\mathcal S}({\bf u};{\bf r}|{\mathbf \Lambda},{\bf P} ,{\bf G})$. Note that, when $n_r > n_t$, the receiver processing ${\bf r} = {\bf U}^\dag{\bf y}$ becomes information lossy, and $I_{\mathcal S}({\bf u};{\bf y}|{\mathbf \Lambda},{\bf P} ,{\bf G}) > I_{\mathcal S}({\bf u};{\bf r}|{\mathbf \Lambda},{\bf P} ,{\bf G})$. We introduce the following definitions. For a given pairing $\ell$, let ${\bf r}_k {\stackrel {\Delta} {=} }( r_{i_k}, r_{j_k} )^T$, ${\bf u}_k {\stackrel {\Delta} {=} }( u_{i_k} , u_{j_k} )^T$, ${\mathbf \Lambda}_k {\stackrel {\Delta} {=} }\mbox{diag}(\lambda_{i_k}, \lambda_{j_k})$, ${\bf P}_k {\stackrel {\Delta} {=} }\mbox{diag}( p_{i_k}, p_{j_k} )$ and ${\mathcal S}_k {\stackrel {\Delta} {=} }{\mathcal U}_{i_k} \times {\mathcal U}_{j_k}$. Due to the pairing structure of ${\bf G}$ the mutual information $I_{\mathcal S}({\bf u};{\bf r}|{\mathbf \Lambda},{\bf P} ,{\bf G})$ can be expressed as the sum of mutual information of all the $n/2$ pairs as follows: $$\begin{aligned} \label{cap4_prl} I_{\mathcal S}({\bf u};{\bf r}|{\mathbf \Lambda},{\bf P} ,{\bf G}) &=& \sum_{k = 1}^{n/2} I_{{\mathcal S}_k}({\bf u}_k;{\bf r}_k |{\mathbf \Lambda}_k,{\bf P}_k ,{\theta_k})\end{aligned}$$ Having fixed the precoder structure to ${\bf T} = {\bf V}^\dag {\bf P}{\bf G}$, we can formulate the following \[cap\_discrete\_Xcoded\] $$\begin{aligned} \label{cap_Xcoded} C_X({\bf H},P_T) = \max_{{\bf G}, {\bf P} \,| \, {\bf u} \in {\mathcal S}, \mbox{tr}({\bf P}{\bf P}^\dag) = 1} I_{\mathcal S}({\bf u};{\bf r}|{\mathbf \Lambda},{\bf P} ,{\bf G})\end{aligned}$$ It is clear that the solution of the above problem is still a formidable task, although it is simpler than Problem \[cap\_discrete\]. In fact, instead of the $n\times n$ variables of ${\bf T}$, we now deal with $n$ variables for power allocation in ${\bf P}$, $n/2$ variables for the angles defining ${\bf A}_k$, and the pairing $\ell\in {\cal L}$. In the following, we will show how to efficiently solve Problem \[cap\_discrete\_Xcoded\] by splitting it into two simpler problems. Power allocation can be divided into power allocation among the $n/2$ pairs, followed by power allocation between the two subchannels of each pair. Let ${\bar {\bf P}} = \mbox{diag}({\bar p_1} , {\bar p_2}, \cdots, {\bar p_{n/2}})$ be a diagonal matrix, where ${\bar p_k} {\stackrel {\Delta} {=} }\sqrt{p_{i_k}^2 + p_{j_k}^2}$ with ${\bar p_k}^2$ being the power allocated to the $k$-th pair. The power allocation within each pair can be simply expressed in terms of the fraction $f_k {\stackrel {\Delta} {=} }p_{i_k}^2 / {\bar p_k}^2$ of the power assigned to the first subchannel of the pair. The mutual information achieved by the $k$-th pair is then given by $$\begin{aligned} \label{cap4_prl1} &&\hspace{-15mm}I_{{\mathcal S}_k}({\bf u}_k;{\bf r}_k |{\mathbf \Lambda}_k,{\bf P}_k ,{\theta_k}) = I_{{\mathcal S}_k}({\bf u}_k;{\bf r}_k |{\mathbf \Lambda}_k,{\bar p_k},f_k ,{\theta_k}) \\ &&\hspace{-6mm} = -\int_{ {\bf r}_k \in {\mathbb C}^2 } p({\bf r}_k)\log_2 p({\bf r}_k) \, d{\bf r}_k - 2\log_2(\pi e ) \nonumber\end{aligned}$$ where $p({\bf r}_k)$ is given by $$\label{pdfrk} p({\bf r}_k) = \frac{1}{\vert {\mathcal S}_k \vert \pi^2} \sum_{{\bf u}_k \in {\mathcal S}_k} e^{-\Vert {\bf r}_k - \sqrt{P_T}{\bar p_k}{\mathbf \Lambda}_k {\bf F}_k {\bf A}_k {\bf u}_k \Vert ^2}$$ where ${\bf F}_k {\stackrel {\Delta} {=} }\mbox{diag}(\sqrt{f_k}, \sqrt{1 - f_k})$ and ${\bf A}_k$ is given by (\[akmat\]). The capacity of the discrete input MIMO Gaussian channel when precoding with X-Codes can be expressed as \[capxcodesapprox\] $$\begin{aligned} \label{cap_xcodes_approx} {C}_{X}({\bf H},P_T) = \max_{\ell \in {\mathcal L}, {\bar {\bf P}} | \mbox{tr}({\bar {\bf P}}{\bar {\bf P}}^\dag) = 1} \sum_{k = 1}^{n/2} C_{{\mathcal S}_k}(k,\ell,{\bar p_k})\end{aligned}$$ where $C_{{\mathcal S}_k}(k,\ell,{\bar p_k})$, the capacity of the $k$-th pair in the pairing $\ell$, is achieved by solving \[capxcodes\_pair\] $$\begin{aligned} \label{cap_xcodes_pair} C_{{\mathcal S}_k}(k,\ell,{\bar p_k}) = \max_{\theta_k,f_k} I_{{\mathcal S}_k}({\bf u}_k;{\bf r}_k |{\mathbf \Lambda}_k,{\bar p_k},f_k ,{\theta_k})\end{aligned}$$ In other words, we have split Problem \[cap\_discrete\_Xcoded\] into two different simpler problems. Firstly, given a pairing $\ell$ and power allocation between pairs ${\bar {\bf P}}$, we can solve Problem \[capxcodes\_pair\] for each $k = 1,2, \cdots, n/2$. Problem \[capxcodesapprox\] uses the solution to Problem \[capxcodes\_pair\] to find the optimal pairing $\ell^{*}$ and the optimal power allocation ${\bar {\bf P}}^{*}$ between the $n/2$ pairs. For small $n$, the optimal pairing and power allocation between pairs can always be computed numerically and by brute force enumeration of all possible pairings. This is, however, prohibitively complex for large $n$, and we shall discuss heuristic approaches in Section \[multi\_subch\]. We will show in the following that, although suboptimal, precoding with X-Codes will provide a close to optimal capacity with the additional benefit that the detection complexity at the receiver is highly reduced, since there is coupling only between pairs of channels, as compared to the case of full-coupling for the optimal precoder in [@cruz]. In the next section, we solve Problem \[capxcodes\_pair\], which is equivalent to finding the optimal rotation angle and power allocation for a Gaussian MIMO channel with only $n = 2$ subchannels. Gaussian MIMO channels with $n = 2$ {#two_subch} =================================== With $n = 2$, there is only one pair and only one possible pairing. Therefore, we drop the subscript $k$ in Problem \[capxcodes\_pair\] and we find $C_X({\bf H},P_T)$ in Problem \[cap\_discrete\_Xcoded\]. The processed received vector ${\bf r} \in {\mathbb C}^2$ is given by $$\label{rxn2} {\bf r} = \sqrt{P_T} {\mathbf \Lambda} {\bf F} {\bf A} {\bf u} + {\bf z}$$ where ${\bf z} = {\bf U}^\dag{\bf w}$ is the equivalent noise vector with the same statistics as ${\bf w}$. Let $\alpha {\stackrel {\Delta} {=} }\lambda_1^2 + \lambda_2^2$ be the overall channel power gain and $\beta {\stackrel {\Delta} {=} }\lambda_1/\lambda_2$ be the [*condition number*]{} of the channel. Then (\[rxn2\]) can be re-written as $$\label{rxn3} {\bf r} = \sqrt{{\Tilde P_T}} {\mathbf {\Tilde \Lambda}} {\bf F} {\bf A} {\bf u} + {\bf z}$$ where ${\Tilde P_T} {\stackrel {\Delta} {=} }P_T \alpha$ and ${\mathbf {\Tilde \Lambda}} {\stackrel {\Delta} {=} }{\mathbf \Lambda}/ \sqrt{\alpha} = \mbox{diag}( \beta/\sqrt{1 + \beta^2}, 1/\sqrt{1 + \beta^2})$. The equivalent channel ${\mathbf {\Tilde \Lambda}}$ now has a gain of $1$, and its channel gains are dependent only upon $\beta$. Our goal is, therefore, to find the optimal rotation angle $\theta^{*}$ and the fractional power allocation $f^{*}$, which maximize the mutual information of the equivalent channel with condition number $\beta$ and gain $\alpha=1$. The total available transmit power is now ${\Tilde P_T}$. It is difficult to get analytic expressions for the optimal $\theta^{*}$ and $f^{*}$, and therefore we can use numerical techniques to evaluate them and store them in lookup tables to be used at run time. For a given application scenario, given the distribution of $\beta$, we decide upon a few discrete values of $\beta$ which are representative of the actual values observed in real channels. For each such quantized value of $\beta$, we numerically compute a table of the optimal values $f^{*}$ and $\theta^{*}$ as a function of ${\Tilde P_T}$. These tables are constructed offline. During the process of communication, the transmitter knows the value of $\alpha$ and $\beta$ from channel measurements. It then finds the lookup table with the closest value of $\beta$ to the measured one. The optimal values $f^{*}$ and $\theta^{*}$ are then found by indexing the appropriate entry in the table with ${\Tilde P_T}$ equal to $P_T \alpha$. In Fig. \[palloc\_frac\], we graphically plot the optimal power fraction $f^*$ to be allocated to the stronger channel in the pair, as a function of $P_T$. The input alphabet is 16-QAM and $\beta = 1,1.5,2,4,8$. For $\beta = 1$, both channels have equal gains, and therefore, as expected, the optimal power allocation is to divide power equally between the two subchannels. However with increasing $\beta$, the power allocation becomes more asymmetrical. It is observed that at low $P_T$ it is optimal to allocate all power to the stronger channel. At high $P_T$ the opposite is true, and it is the weaker channel which gets most of the power. For a fixed $\beta$, as $P_T$ increases, the power allocated to the stronger channel is shifted to the weaker channel. For a fixed $P_T$, a higher fraction of the total power is allocated to the weaker channel with increasing $\beta$. In the high $P_T$ regime, these results are in contrast with the waterfilling scheme, where almost all subchannels are allocated equal power. In Fig. \[palloc\_theta\], the optimal rotation angle $\theta^*$ is plotted as a function of $P_T$. The input alphabet is 16-QAM and $\beta = 1.5,2,4,8$. For $\beta = 1$ the mutual information is independent of $\theta$ for all values of $P_T$. For $\beta = 1.5, 2$, the optimal rotation angle is almost invariant to $P_T$. For larger $\beta$, the optimal rotation angle varies with $P_T$ and approximately ranges between $30-40^\circ$ for all $P_T$ values of interest. Fig. \[MI\_vs\_pfrac\_17dB\_16qam\] shows the variation of the mutual information with the power fraction $f$ for $\alpha = 1$. The power $P_T$ is fixed at 17 dB and the input alphabet is 16-QAM. We observe that for all values of $\beta$, the mutual information is a concave function of $f$. We also observe that the sensitivity of the mutual information to variation in $f$ increases with increasing $\beta$. However, for all $\beta$, the mutual information is fairly stable (has a “plateau”) around the optimal power fraction. This is good for practical implementation, since this implies that an error in choosing the correct power allocation would result in a very small loss in the achieved mutual information. In Fig. \[MI\_vs\_theta\_17dB\_16qam\], we plot the variation of the mutual information w.r.t. the rotation angle $\theta$. The power $P_T$ is fixed at 17 dB and the input alphabet is 16-QAM. For $\beta = 1$, the mutual information is obviously constant with $\theta$. With increasing $\beta$, mutual information is observed to be increasingly sensitive to $\theta$. However, when compared with Fig. \[MI\_vs\_pfrac\_17dB\_16qam\], it can also be seen that the mutual information appears to be more sensitive to the power allocation fraction $f$, than to $\theta$. In Fig. \[thetavar\_4qam\], we plot the mutual information of X-Codes for different rotation angles with $\alpha = 1$ and $\beta = 2$. For each rotation angle, the power allocation is optimized numerically. We observe that, the mutual information is quite sensitive to the rotation angle except in the range 30-40$^\circ$. We next present some simulation results to show that indeed our simple precoding scheme can significantly increase the mutual information, compared to the case of no precoding across subchannels (i.e., Mercury/waterfilling). For the sake of comparison, we also present the mutual information achieved by the waterfilling scheme with discrete input alphabets. We restrict the discrete input alphabets ${\mathcal U}_i, i=1,2$, to be square $M$-QAM alphabets consisting of two $\sqrt{M}$-PAM alphabets in quadrature. Mutual information is evaluated by solving Problem \[capxcodes\_pair\] (i.e., numerically maximizing w.r.t. the rotation angle and power allocation). In Fig. \[beta2\_4\_16qam\], we plot the maximal mutual information versus $P_T$, for a system with two subchannels, $\beta=2$ and $\alpha = 1$. Mutual information is plotted for 4- and 16-QAM signal sets. It is observed that for a given achievable mutual information, coding across subchannels is more power efficient. For example, with 4-QAM and an achievable mutual information of $3$ bits, X-Codes require only $0.8$ dB more transmit power when compared to the ideal Gaussian signalling with waterfilling. This gap increases to $1.9$ dB for Mercury/waterfilling and $2.8$ dB for the waterfilling scheme with $4$-QAM as the input alphabet. A similar trend is observed with $16$-QAM as the input alphabet. The proposed precoder clearly performs better, since the mutual information is optimized w.r.t. the rotation angle $\theta$ and power allocation, while Mercury/waterfilling, as a special case of X-Code, only optimizes power allocation and fixes $\theta=0$. In Fig. \[beta124\_4qam\], we compare the mutual information achieved by X-Codes and the Mercury/waterfilling strategy for $\alpha = 1$ and $\beta=1,2,4$. The input alphabet is $4$-QAM. It is observed that both the schemes have the same mutual information when $\beta = 1$. However with increasing $\beta$, the mutual information of Mercury/waterfilling strategy is observed to degrade significantly at high $P_T$, whereas the performance of X-Codes does not vary as much. The degradation of mutual information for the Mercury/waterfilling strategy is explained as follows. For the Mercury/waterfilling strategy, with increasing $\beta$, all the available power is allocated to the stronger channel till a certain transmit power threshold. However, since finite signal sets are used, mutual information is bounded from above until the transmit power exceeds this threshold. This also explains the reason for the intermediate change of slope in the mutual information curve with $\beta=4$ (see the rightmost curve in Fig. \[beta124\_4qam\]). On the other hand, due to coding across subchannels, this problem does not arise when precoding with X-Codes. Therefore, in terms of achievable mutual information, rotation coding is observed to be more robust to ill-conditioned channels. For low values of $P_T$, mutual information of both the schemes are similar, and improves with increasing $\beta$. This is due to the fact that, at low $P_T$, mutual information increases linearly with $P_T$, and therefore all power is assigned to the stronger channel. With increasing $\beta$, the stronger channel has an increasing fraction of the total channel gain, which results in increased mutual information. In Fig. \[beta1248\_16qam\], the mutual information with X-Codes is plotted for $\beta = 1,2,4,8$ and with 16-QAM as the input alphabet. It is observed that at low values of $P_T$, a higher value of $\beta$ is favorable. However at high $P_T$, with 16-QAM input alphabets, the performance degrades with increasing $\beta$. This degradation is more significant compared to the degradation observed with 4-QAM input alphabets. Therefore it can be concluded that the mutual information is more sensitive to $\beta$ with 16-QAM input alphabets as compared to 4-QAM. Gaussian MIMO channels with $n > 2$ {#multi_subch} =================================== We now consider the problem of finding the optimal pairing and power allocation between pairs for different Gaussian MIMO channels with even $n$ and $n > 2$. We first observe that mutual information is indeed sensitive to the chosen pairing, and this therefore justifies the criticality of computing the optimal pairing. This is illustrated through Fig. \[n4pair\], for $n = 4$ with a diagonal channel ${\mathbf \Lambda} = \mbox{diag}(0.8, 0.4,0.4,0.2)$ and 16-QAM. Optimal power allocation between the two pairs is computed numerically. It is observed that the pairing $\{ (1,4), (2,3)\}$ performs significantly better than the pairing $\{ (1,3), (2,4)\}$. In Fig. \[cruz\_cmp\], we compare the mutual information achieved with optimal precoding [@cruz], to that achieved by the proposed precoder with 4-QAM input alphabet. The $4 \times 4$ full channel matrix (non-diagonal channel) is given by (42) in [@cruz]. For X-Codes, the optimal pairing is $\{ (1,4), (2,3)\}$ and the optimal power allocation between the pairs is computed numerically. It is observed that X-Codes perform very close to the optimal precoding scheme. Specifically, for an achievable mutual information of 6 bits, compared to the optimal precoder [@cruz], X-Codes need only 0.4dB extra power whereas 2.3dB extra power is required with Mercury/waterfilling. Another application is in wireless MIMO channels with perfect channel state information at both the transmitter and receiver. The channel coefficients are modeled as i.i.d complex normal random variables with unit variance. In Fig. \[mimo\_ergodic\_cap\], we plot the ergodic capacity (i.e., the mutual information averaged over channel realizations) for a $4 \times 4$ wireless MIMO channel. For X-Codes, the best pairing and power allocation between pairs are chosen numerically using the optimal $\theta$ and power fraction tables created offline. It is observed that at high $P_T$, simple rotation based coding using X-Codes improves the mutual information significantly, when compared to Mercury/waterfilling. For example, for a target mutual information of 12 bits, X-Codes perform 1.2dB away from the idealistic Gaussian signalling scheme. This gap from the Gaussian signalling scheme increases to 3.1dB for the Mercury/waterfilling scheme and to 4.4dB for the waterfilling scheme with 16-QAM alphabets. In this application scenario the low complexity of our precoding scheme becomes an essential feature, since the precoder can be computed on the fly using the look-up tables for each channel realization. Application to OFDM {#sec_ofdm} =================== In OFDM applications, $n$ is large and Problem \[capxcodesapprox\] becomes too complex to solve, since we can no more find the optimal pairing by enumeration. It was observed in Section \[two\_subch\], that for $n = 2$, a larger value of the condition number $\beta$ leads to a higher mutual information at low values of $P_T$ (low SNR). Therefore, we conjecture that pairing the $k$-th subchannel with the $(n/2 + k)$-th subchannel could have mutual information very close to optimal, since this pairing scheme attempts to maximize the minimum $\beta$ among all pairs. We shall call this scheme the “conjectured” pairing scheme, and the X-Code scheme, which pairs the $k$-th with the $(n-k+1)$-th subchannel, the “X-pairing" scheme. Note that the “X-pairing” scheme was proposed in [@Xcodes_paper] as a scheme which achieved the optimal diversity gain when precoding with X-Codes. Given a pairing of subchannels, it is also difficult to compute the optimal power allocation between pairs ${\bar {\bf P}}$. However, it was observed that for channels with large $n$, even waterfilling power allocation between the pairs (with ${\alpha}_k {\stackrel {\Delta} {=} }\sqrt{ {\lambda}_{i_k}^2 + {\lambda}_{j_k}^2}$ as the channel gain of the $k$-th pair) results in good performance. Apart from the “conjectured” and the “X-pairing” schemes, we propose the following scheme which is based on the “Hungarian" assignment algorithm [@Kuhn55] and which attempts to find a good approximation to the optimal pairing. We shall call this as the “Hungarian" pairing scheme. Before describing the “Hungarian” pairing scheme, we briefly review the Hungarian assignment problem as follows. Consider $m$ different workers and $m$ different jobs that have to be completed. Also let $C(i,j)$ be the cost involved when the $i$-th worker is assigned to the $j$-th job. We can therefore think of a cost matrix, whose $(i,j)$-th entry has the value $C(i,j)$. The Hungarian assignment problem, is to then find the optimal assignment of workers to jobs (each worker getting assigned to exactly one job) such that the total cost of getting all the jobs completed is minimized. It is easy to see, that a maximization job assignment problem could be posed into a minimization problem and vice versa. To find a good approximation to the optimal pairing, we split the $n$ subchannels into two groups [*i)*]{} Group-I : subchannels 1 to $n/2$, with the $j$-th subchannel in the role of the $j$-th job ($j =1,2, \cdots n/2$), [*ii)*]{} Group-II : subchannels $n/2+1$ to $n$, with the $(n/2+i)$-th subchannel in the role of the $i$-th worker ($i=1,2, \cdots n/2$). Therefore, there are $n/2$ workers and jobs. For a given SNR $P_T$, we initially assume uniform power allocation between all pairs and therefore assign a power of $2P_T/n$ to each pair. The value of $C(i,j)$ is evaluated by finding the optimal mutual information achieved by an equivalent $n=2$ channel with the $n/2+i$-th and the $j$-th subchannels as its two subchannels. This can be obtained by first choosing a table (see Section \[two\_subch\]) with the closest value of $\beta$ to the given $\lambda_j/\lambda_{n/2+i}$, and then indexing the appropriate entry into the table with SNR=$2P_T(\lambda_j^2 + \lambda_{n/2+i}^2)/n$. The Hungarian algorithm then finds the pairing with the highest mutual information. Power allocation between the pairs is then achieved through the waterfilling scheme. It was observed through monte-carlo simulations that, even uniform power allocation between the subchannels results in almost same mutual information as achieved through waterfilling between pairs. This can be explained from the fact that by separating into a group of stronger (Group-I) and a group of weaker channels (Group-II), any pairing would result in all pairs having almost the same channel gain ${\alpha}_k$. This therefore implies that the optimal power allocation scheme would allocate nearly equal power to all pairs, which both the uniform and the waterfilling schemes would also do. Henceforth, it can be conjectured that with the proposed separation of subchannels into 2 groups, both the uniform and the waterfilling power allocation schemes would have close to optimal performance, and any further improvement in mutual information by optimizing the power allocation would be minimal. This also supports the initial usage of uniform power $2P_T/n$ to compute the entries $C(i,j)$ before executing the Hungarian algorithm. Furthermore, the computational complexity of the Hungarian algorithm is $O(n^3)$ and is therefore practically feasible. To study the sensitivity of the mutual information to the pairing of subchannels, we also consider a “Random” pairing scheme. In the “Random" pairing scheme, we first choose a large number ($\approx$ 50) of random pairings. For each chosen random pairing we evaluate the mutual information (through monte-carlo simulations) with waterfilling power allocation between pairs. Finally the average mutual information is computed. This gives us insight into the mean value of the mutual information w.r.t. pairing. It would also help us in quantifying the effectiveness of the heuristic pairing schemes discussed above. We next illustrate the mutual information achieved by these heuristic schemes for an OFDM system with $n=32$ subchannels and 16-QAM. The channel impulse response is $[ -0.454 + {\mathfrak j}0.145, -0.258 + {\mathfrak j}0.198, 0.0783 + {\mathfrak j}0.069, -0.408 - {\mathfrak j}0.396, -0.532 - {\mathfrak j}0.224 ]$. For the “conjectured" and the “X-pairing" schemes also, power allocation is achieved through waterfilling between the pairs. In Fig. \[ofdm\_ex\_1\] the total mutual information is plotted as a function of the SNR per sub carrier. It is observed that the proposed precoding scheme performs much better than the Mercury/waterfilling scheme. The proposed precoder with the “Hungarian" pairing scheme performs within 1.1dB of the Gaussian signalling scheme for an achievable total mutual information of $96$ bits (i.e., a rate of 96/128 = 3/4). The proposed precoder with the “Hungarian" pairing scheme performs about 1.6dB better than the Mercury/waterfilling scheme. The “X-pairing" scheme performs better than the Mercury/waterfilling and worse than the “Hungarian" pairing scheme. Even at a low rate of 1/2 (i.e., a total mutual information of 64 bits), the proposed precoder with the “Hungarian" pairing scheme performs about 0.7dB better than the Mercury/waterfilling scheme. In Fig. \[ofdm\_ex\_2\], we compare the mutual information achieved by the various heuristic pairing schemes. It is observed that the “conjectured" pairing scheme performs very close to the “Hungarian" pairing scheme except at very high SNR. For example, even for a high mutual information of 96 bits, the “Hungarian" pairing scheme performs better than the “conjectured" pairing scheme by only about 0.2dB. However at very high rates (like 7/8 and above), the “Hungarian" pairing scheme is observed to perform better than the “conjectured" pairing scheme by about 0.7dB. Therefore for low to medium rates, it would be better to use the “conjectured" pairing since it has the same performance at a lower computational complexity. The mutual information achieved by the “Random" pairing scheme is observed to be strictly inferior than the “conjectured" pairing scheme at all values of SNR, and at low SNR it is even worse than the Mercury/waterfilling strategy. This, therefore implies that the total mutual information is indeed sensitive to the chosen pairing. Further, till a rate of 1/2 (i.e., a mutual information of 64 bits) it appears that any extra optimization effort would not result in significant performance improvement for the “conjectured" pairing scheme, since it is already very close to the idealistic Gaussian signalling schemes. However at higher rate and SNR it may still be possible to improve the mutual information by further optimizing the selection of pairing scheme and power allocation between pairs. This is however a difficult problem that requires further investigation. Conclusions =========== In this paper, we proposed a [*low complexity*]{} precoding scheme based on the pairing of subchannels, which achieves near optimal capacity for Gaussian MIMO channels with discrete inputs. The low complexity feature relates to both the evaluation of the optimal precoder matrix and the detection at the receiver. This makes the proposed scheme suitable for practical applications, even when the channels are time varying and the precoder needs to be computed for each channel realization. The simple precoder structure, inspired by the X-Codes, enabled us to split the precoder optimization problem into two simpler problems. Firstly, for a given pairing and power allocation between pairs, we need to find the optimal power fraction allocation and rotation angle for each pair. Given the solution to the first problem, the second problem is then to find the optimal pairing and the power allocation between pairs. For large $n$, typical of OFDM systems, we also discussed different heuristic approaches for optimizing the pairing of subchannels. The proposed precoder was shown to perform better than the Mercury/waterfilling strategy for both diagonal and non-diagonal MIMO channels. Future work will focus on finding close to optimal pairings, and close to optimal power allocation strategies between pairs. [1]{} S.K. Mohammed, E. Viterbo, Y. Hong, and A. Chockalingam, “MIMO Precoding with X- and Y-Codes,” [*submitted to IEEE Trans. on Information Theory*]{}, Nov 2009 (available at http://arxiv.org/abs/0912.1909v1). T.M. Cover and Joy A. Thomas, [*Elements of information theory*]{}, John Wiley and Sons, 2nd Ed., July 2006. A. Lozano, A.M. Tulino, and S. Verdu, “Optimum Power Allocation for Parallel Gaussian Channels With Arbitrary Input Distributions,” [*IEEE. Trans. on Information Theory*]{}, pp. 3033–3051, vol. 52, no. 7, July 2006. F.P. Cruz, M.R.D. Rodrigues and S. Verdu, “MIMO Gaussian Channels with Arbitrary Inputs : Optimal Precoding and Power Allocation,” accepted in [*IEEE Trans. on Information Theory*]{}. I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” [*European Trans. Telecommun.,*]{} vol. 10, no. 6, pp. 585–595, November 1999. D. Guo, S. Shamai, S. Verdu, “Mutual information and minimum mean-square error in Gaussian channels,” in [*IEEE Trans. on Information Theory*]{}, 51(4): 1261–1282 April 2005. H.W. Kuhn, “The Hungarian method for the assignment problem,” Naval Research Logistic Quarterly, 2:83-97, 1955. [^1]: S. K. Mohammed and A. Chockalingam are with Indian Institute of Science, Bangalore $560012$, India. E-mail: $\tt saifind2007$@$\tt yahoo.com$ and $\tt achockal$@$\tt ece.iisc.ernet.in$. Saif K. Mohammed is currently visiting DEIS, Università della Calabria, Italy. [^2]: Yi Hong and Emanuele Viterbo are with DEIS - Università della Calabria, via P. Bucci, 42/C, 87036 Rende (CS), Italy. E-mail: $\tt \{hong,viterbo\}$@$\tt deis.unical.it$.

--- abstract: 'In the $\phi $-mapping theory, the topological current constructed by the order parameters can possess different inner structure. The difference in topology must correspond to the difference in physical structure. The transition between different structures happens at the bifurcation point of the topological current. In a self-interaction two-level system, the change of topological particles corresponds to change of energy levels.' address: | $^1$ Institute of Applied Physics and Computational Mathematics,\ P.O. Box 8009(28), Beijing 100088, P.R. China\ $^2$ Institute of Theoretical Physics, Department of Physics,\ Lanzhou University, 730000, P. R. China author: - 'Li-Bin Fu$^1$, Jie Liu$^1$, Shi-Gang Chen$^1$, and Yi-Shi Duan$^2$' title: 'The configuration of a topological current and physical structure: an application and paradigmatic evidence' --- 0.5cm In recent years, topology has established itself as an important part of the physicist’s mathematical arsenal [@zh1]. The concepts of the topological particle and its current have been widely used in particle physics [@duan1; @hha] and topological defect theory [@zh4]. Here, the topological particles are regarded as abstract particles, such as monopoles and the points defects. In this paper, we give a new understanding in topology and physics. Many physics system can be described by employing the order parameters. By making use of the $\phi $-mapping theory, we find that the topological current constructed by the order parameters can possess different inner structure. The topological properties are basic properties for a physics system, so the difference in configuration of the topological current must correspond to the difference in physical structure. Considering a $(n+1)$-dimensional system with $n$-component vector order parameter field $\vec \phi ({\bf x}),$ where ${\bf x}=(x^0,x^1,x^2,\cdots x^n)$ correspond to local coordinates. The direction unit field of $\vec \phi $ is defined by $$n^a=\frac{\phi ^a}{||\phi ||},\quad a=1,2,\cdots n \label{c1unit}$$ where $$||\phi ||=(\phi ^a\phi ^a)^{1/2}.$$ The topological current of this system is defined by $$j^\mu (x)=\frac{\in ^{\mu \mu _1\cdots \mu _n}}{A(S^{n-1})(d-1)!}\in _{a_1\cdots a_n}\partial _{\mu _1}n^{a_1}\cdots \partial _{\mu _n}n^{a_n} \label{firstcurr}$$ where $A(S^{n-1})$ is the surface area of $(n-1)$-dimensional unit sphere $% S^{n-1}$. Obviously, the current is identically conserved, $$\partial _\mu j^\mu =0.$$ If we define a Jacobians by $$\in ^{a_1\cdots a_n}D^\mu (\frac \phi x)=\in ^{\mu \mu _1\cdots \mu _n}\partial _{\mu _1}\phi ^{a_1}\cdots \partial _{\mu _n}\phi ^{a_n}, \label{firstjac}$$ as has been proved before [@topc], this current takes the form as $$j^\mu =\delta (\vec \phi )D^\mu (\frac \phi x). \label{deltfirstcurr}$$ Then, we can obtain $$j^\mu =\sum_{i=1}^l\beta _i\eta _i\delta (\vec x-\vec z_i(x^0))\frac{% dz_i^\mu }{dx^0},$$ in which $z_i(x^0)$ are the zero lines where $\vec \phi ({\bf x)}=0,$ the positive integer $\beta _i$ and $\eta _i=sgnD(\frac \phi {\vec x})$ are the Hopf index and Brouwer degree of $\phi $-mapping [@zh17] respectively, and $l$ is the total number of the zero lines$.$ This current is similar to a current of point particles and the $i$-th one with the charge $\beta _i\eta _i,$ and the zero lines $z_i(x)$ are just the trajectories of the particles, for convenience we define these point particles as topological particles. Then the total topological charge of the system is $$Q=\int_Mj^0d^nx=\sum_{i=1}^l\beta _i\eta _i,$$ here $M$ is a $n$-dimensional spatial space for a given $x^0.$ This is a topological invariant and corresponds to some basic conditions of this physical system. However, it is important that the inner structure of the topological invariant can be constructed in different configurations, i.e., the number of the topological particles and their charge can be changed. This change in configuration of the topological current must correspond to some change in physical structure. All of the above discussion is based on the condition that $$D\left( \frac \phi x\right) =\left. D^0\left( \frac \phi x\right) \right| _{z_i}\neq 0.$$ When $\left. D\left( \frac \phi x\right) \right| _{z_i}=0$ at some points $% p_i^{*}=z^{*}(x_c^0)$ at $x^0=x_c^0$ along the zero line $z_i(x^0)$, it is shown that there exist several crucial cases of branch process, which correspond to the topological particle generating or annihilating at limit points and splitting, encountering or merging at the bifurcation points. A vast amount of literature has been devoted to discussing these features of the evolution of the topological particles [@bifur]. Here, we will not spend more attention on describing these evolution, but put our attention on the physical substance of these processes. As we have known before, all of these branch processes keep the total topological charge conserved, but it is very important that these branch processes change the number and the charge of the topological particles. i.e. change the inner structure of the topological current. In our point of view, the different configuration of topological current corresponds to the different physical structure. We consider $x^0$ as a parameter $\lambda $ of a physics system. Let us define $$\left. f_i(\lambda )=D^0\left( \frac \phi x\right) \right| _{z_i}$$ As $\lambda $ changing, the value of $f_i(\lambda )$ changes along the zero lines $z_i(\lambda ).$ At a critical point $\lambda =\lambda _c,$ when $% f_i(\lambda _c)=0,$ we know that the inner structure of the topological current will be changed in some way, at the same time the physical structure will also be changed, i.e., the physical structure when $\lambda <\lambda _c$ will be different from the one when $\lambda >\lambda _c.$ The transition between these structures occurs at the bifurcation points where $f_i(\lambda )=0.$ As an application and example, let us consider a self-interacting two-level model introduced in Ref. [@wuniu]. The nonlinear two-level system is described by the dimensionless Schrödinger equation $$i\frac \partial {\partial t}\left( \begin{array}{c} a \\ b \end{array} \right) =H(\gamma )\left( \begin{array}{c} a \\ b \end{array} \right) \label{a}$$ with the Hamiltonian given by $$H(\gamma )=\left( \begin{array}{cc} \frac \gamma 2+\frac C2(|b|^2-|a|^2) & \frac V2 \\ \frac V2 & -\frac \gamma 2-\frac C2(|b|^2-|a|^2) \end{array} \right) , \label{b}$$ in which $\gamma $ is the level separation, $V$ is the coupling constant between the two levels, and $C$ is the nonlinear parameter describing the interaction. The total probability $|a|^2+|b|^2$ is conserved and is set to be $1$. We assume $a=|a|e^{i\varphi _1(t)}$, $b=|b|e^{i\varphi _2(t)},$ the fractional population imbalance and relative phase can be defined by $$z(t)=|b|^2-|a|^2,\;\;\;\;\;\varphi (t)=\varphi _2(t)-\varphi _1(t). \label{bal}$$ From Eqs. (\[a\]) and (\[b\]), we obtain $$\frac d{dt}z(t)=-V\sqrt{1-z^2(t)}\sin [\varphi (t)] \label{ceq1}$$ $$\frac d{dt}\varphi (t)=\gamma +Cz(t)+\frac{Vz(t)}{\sqrt{1-z^2(t)}}\cos [\varphi (t)]. \label{ceq2}$$ If we chose $x=2|a||b|\cos (\varphi ),$ $y=2|a||b|\sin (\varphi )$, it is easy to see that $x^2+y^2+z^2=1$ by considering $|a|^2+|b|^2=1,$ which describes a unit sphere $S^2\ $with $z$ and $\varphi $ a pair of co-ordinates$.$ We define a vector field on this unit sphere: $$\phi ^1=-V\sqrt{1-z^2}\sin (\varphi ), \label{f1}$$ $$\phi ^2=\gamma +Cz+\frac{Vz}{\sqrt{1-z^2}}\cos (\varphi ) \label{f2}$$ Apparently, there are singularities at the two pole points $z=\pm 1,$ which make the vector $\vec \phi $ is discontinuous at these points. However, the direction unit vector $\vec n$ is continuous. In the $\phi $-mapping theory, we only need the unit vector $\vec n$ is continuous and differentiable on the whole sphere $S^2$ (at the zero points of $\vec \phi $, the differential of $\vec n$ is a general function)$,$ and the vector $\vec \phi $ is continuous and differentiable at the neighborhoods of its zero points. Then from $\phi $-mapping theory, we can obtain a topological current as $$\vec j=\sum_{i=1}^l\beta _i\eta _i\delta (\varphi -\varphi _i(\gamma ))\delta (z-z_i(\gamma ))\frac{dz_i}{d\gamma }\left| _{p_i}\right. ,$$ in which $p_i=p_i(z_i,\varphi _i)$ is the trajectory of the $i$-th topological particle $P_i$ and $$\eta _i=sgn\left( D(\gamma )\right) =sgn\left( \det \left. \left( \begin{array}{cc} \partial \phi ^1/\partial \varphi & \partial \phi ^2/\partial \varphi \\ \partial \phi ^1/\partial z & \partial \phi ^2/\partial z \end{array} \right) \right| _{p_i}\right) .$$ From Eqs. (\[f1\]) and (\[f2\]), it is easy to see that $\vec \phi $ is single-valued on $S^2,$ which states that Hopf index $\beta _i=1$ $% (i=1,2,\cdots ,l)$ here. It can be proved that the total charge of this system is just the Euler number of $S^2$ which is $2$ [@lishen]. This is a topological invariant of $S^2$ which corresponds to the basic condition of this system: $|a|^2+|b|^2=1.$ The following discussion can show that this topological invariant can possess different configuration when $\gamma $ changes. This difference in topology corresponds to the change in adiabatic levels of this nonlinear system. We can prove that every topological particle corresponds to an eigenstate of the nonlinear two-level system. By solving $\vec \phi =0$ form (\[f1\]) and (\[f2\]), we find there are two different cases for discussing. Case 1. For $|C/V|\leq 1,$ there only two topological particles $P_1$ and $% P_2$, which locate on the line $\varphi =0$ and $\varphi =\pi $ respectively as shown in the upper panel of Fig.1. All of them with topological charge $% +1,$ and $D(\gamma )|_{P_{1,2}}>0$ for any $\gamma .$ Correspondingly, in this case, there are only two adiabatic energy levels in this nonlinear two-level mode for various $\gamma $ [@wuniu], as shown in the lower panel of Fig.1, $P_1$ corresponds to the upper level and $P_2$ corresponds to the lower level. Case 2, For $C/V>1,$ two more topological particles can appear when $\gamma $ lies in a window $-\gamma _c<\gamma <\gamma _c$. The boundary of the window can be obtained by assuming $D(\gamma )|_{P_i}=0,$ yielding $$\gamma _c=(C^{2/3}-V^{2/3})^{3/2}. \label{gc}$$ The striking feature happens at $\gamma =-\gamma _c,$ there exists a critical point $T_1^{*}(z_c,\pi )$ with $D(\gamma _c)|_{T_1^{*}}=0,$ as we have shown in Ref. [@bifur], we can prove that this point is a limit point, and a pair of topological particles $P_3$ and $P_4$ generating with opposite charge $-1$ and $+1$ respectively$,$ both of the new topological particles lie on the line $\varphi =\pi $. One of the original topological particle, $P_2$ with charge $+1$ on the line $\varphi =\pi ,$ moves smoothly up to $\gamma =\gamma _c,$ where it collides with $P_3$ and annihilates with it at another limit point $T_2^{*}(-z_c,\pi ).$ The other, $P_1,$ which lies on the line $\varphi =0$, still moves safely with $\gamma .$ As pointed out by B. Wu and Q. Niu [@wuniu], when the interaction is strong enough $(C/V>1),$ a loop appears at the tip of the lower adiabatic level when $C/V>1$ while $-\gamma _c\leq \gamma \leq \gamma _c$. We show the interesting structure in Fig. 2 in which $C/V=2$. For $\gamma <-\gamma _c,$ there are two adiabatic levels, the upper level corresponds to the topological particle $P_1$, the lower one corresponds to the topological particle $P_2;$ for $\gamma >\gamma _c$, there are also only two adiabatical levels, but at this time the lower one corresponds to $P_4$ while the upper one still corresponds to $P_1.$ The arc part of the loop on the tip of lower level when $-\gamma _c<\gamma <\gamma _c$ just corresponds to $P_3,$ which merges with the level corresponding to $P_4$ at the point $M$ on the left and with the one corresponding to $P_1$ at the point $T$ on the right$.$ From the above discussion, one finds that when the structure of the topological current is changed by generating or annihilating a pair of topological particles ( the upper panel of Fig. 2), at the same time, the physical structure is changed by adding two energy levels or subtracting two energy levels respectively (the lower panel of Fig.2). The critical behaviors happen at the limit points where $D(\gamma _c)|_{T_{1,2}^{*}}=0$ . In fact, this nonlinear two-level model is of comprehensive interest for it associates with a wide range of concrete physical systems, e.g., BEC in an optical lattice[@becol1] or in a double-well potential[@dwpp1], and the motion of small polarons[@polar]. So, the relation between topological particles and physical inner structure can be observed in experimental methods. Here we propose a perspective system for observing the striking phenomenon: a Bose-Einstein condensate in a double-well potential[@dwpp1; @RS99]. The amplitudes of general occupations $N_{1,2}(t)$ and phases $\varphi _{1,2}$ obey the nonlinear two-mode Schrödinger equations, approximately[@RS99], $$\begin{aligned} i\hbar \frac{\partial \phi _1}{\partial t} &=&(E_1^0+U_1N_1)\phi _1-K\phi _2 \nonumber \\ i\hbar \frac{\partial \phi _2}{\partial t} &=&(E_2^0+U_2N_2)\phi _2-K\phi _1\end{aligned}$$ with $\phi _{1,2}=\sqrt{N_{1,2}}exp(i\varphi _{1,2})$, and total number of atoms $N_1+N_2=N_T$, is conserved. Here $E_{1,2}^0$ are zero-point energy in each well, $U_{1,2}N_{1,2}$ are proportional to the atomic self-interaction energy, and $K$ describes the amplitude of the tunnelling between the condensates. After introducing the new variables $z(t)=(N_2(t)-N_1(t))/N_T$ and $\varphi =\varphi _2-\varphi _1$, one also obtain an equations having the same form as Eqs. (\[a\]) and (\[b\]) except for the parameters replaced by $$\gamma =-[(E_1^0-E_2^0)-(U_1-U_2)N_T/2]/\hbar ,$$ $$V=2K/\hbar ,\;C=(U_1+U_2)N_T/2\hbar .$$ With these explicit expressions, our theory and results can be directly applied to this system without intrinsic difficulty. In this system the topological particles can be located by the stable occupation and relative phase $\varphi =0$ or $\varphi =\pi $ for a give parameter $\gamma .$ And, one can draw the zero line of each topological particle by giving different $% \gamma .$ We hope our discussions will stimulate the experimental works in the direction. We note that for a system the global property (topology) is given, the interesting feature is that under the same topology the topological configuration can be different, this difference must correspond the different physical structure. 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--- abstract: 'In Part I of this work we defined a generalization of the concept of effective resistance to directed graphs, and we explored some of the properties of this new definition. Here, we use the theory developed in Part I to compute effective resistances in some prototypical directed graphs. This exploration highlights cases where our notion of effective resistance for directed graphs behaves analogously to our experience from undirected graphs, as well as cases where it behaves in unexpected ways.' author: - 'George Forrest Young,  Luca Scardovi,  and Naomi Ehrich Leonard, [^1][^2][^3]' bibliography: - 'REFabrv.bib' - 'ReferenceList.bib' title: 'A New Notion of Effective Resistance for Directed Graphs—Part II: Computing Resistances' --- Graph theory, networks, networked control systems, directed graphs, effective resistance Introduction {#sec:intro} ============ In the companion paper to this work, [@Young13I], we presented a generalization of the concept of effective resistance to directed graphs. This extension was constructed algebraically to preserve the relationships for directed graphs, as they exist in undirected graphs, between effective resistances and control-theoretic properties, including robustness of linear consensus to noise [@Young10; @Young11], and node certainty in networks of stochastic decision-makers [@Poulakakis12]. Further applications of this concept to directed graphs should be possible in formation control [@Barooah06], distributed estimation [@Barooah07; @Barooah08] and optimal leader selection in networked control systems [@Patterson10; @Clark11; @Fardad11]. Effective resistances have proved to be important in the study of networked systems because they relate global network properties to the individual connections between nodes, and they relate local network changes (e.g. the addition or deletion of an edge, or the change of an edge weight) to global properties without the need to re-compute these properties for the entire network (since only resistances that depend on the edge in question will change). Accordingly, the concept of effective resistance for directed graphs will be most useful if the resistance of any given connection within a graph can be computed, and if it is understood how to combine resistances from multiple connections. Computation and combination of resistances are possible for undirected graphs using the familiar rules for combining resistors in series and parallel. In this paper, we address the problems of computing and combining effective resistances for directed graphs. In Section \[sec:back\] we review our definition of effective resistance for directed graphs from [@Young13I]. In Section \[sec:equalres\] we develop some theory to identify directed graphs that have the same resistances as an equivalent undirected graph. We use these results in Section \[sec:direct\] to recover the series-resistance formula for nodes connected by one directed path and the parallel-resistance formula for nodes connected by two directed paths in the form of a directed cycle. In Section \[sec:indirect\] we examine nodes connected by a directed tree and derive a resistance formula that has no analogue from undirected graphs. Background and notation {#sec:back} ======================= We present below some basic definitions of directed graph theory, as well as our definition of effective resistance. For more detail, the reader is referred to the companion paper [@Young13I]. A *graph* $\mathcal{G}$ consists of the triple $\left(\mathcal{V}, \mathcal{E}, A \right)$, where $\mathcal{V} = \left\{1, 2, \ldots, N \right\}$ is the set of nodes, $\mathcal{E} \subseteq \mathcal{V}\times\mathcal{V}$ is the set of edges and $A \in \mathbb{R}^{N\times N}$ is a weighted adjacency matrix with non-negative entries $a_{i,j}$. Each $a_{i,j}$ will be positive if and only if $\left( i,j \right) \in \mathcal{E}$, otherwise $a_{i,j} = 0$. The graph $\mathcal{G}$ is said to be *undirected* if $\left(i,j\right) \in \mathcal{E}$ implies $\left(j,i\right) \in \mathcal{E}$ and $a_{i,j} = a_{j,i}$. Thus, a graph will be undirected if and only if its adjacency matrix is symmetric. The *out-degree* of node $k$ is defined as $d_k^{\text{\emph{out}}} = \sum_{j=1}^N{a_{k,j}}$. $\mathcal{G}$ has an associated *Laplacian* matrix $L$, defined by $L = D - A$, where $D$ is the diagonal matrix of node out-degrees. A *connection* in $\mathcal{G}$ between nodes $k$ and $j$ consists of two paths, one starting at $k$ and the other at $j$ and which both terminate at the same node. A *direct connection* between nodes $k$ and $j$ is a connection in which one path is trivial (i.e. either only node $k$ or only node $j$) - thus a direct connection is equivalent to a path. Conversely, an *indirect connection* is one in which the terminal node of the two paths is neither node $k$ nor node $j$. The graph $\mathcal{G}$ is *connected* if it contains a globally reachable node. Equivalently, $\mathcal{G}$ is connected if and only if a connection exists between any pair of nodes. A *connection subgraph* between nodes $k$ and $j$ in the graph $\mathcal{G}$ is a maximal connected subgraph of $\mathcal{G}$ in which every node and edge form part of a connection between nodes $k$ and $j$ in $\mathcal{G}$. If only one connection subgraph exists in $\mathcal{G}$ between nodes $k$ and $j$, it is referred to as *the* connection subgraph and is denoted by $\mathcal{C}_\mathcal{G}(k,j)$. Let $Q \in \mathbb{R}^{(N-1)\times N}$ be a matrix that satisfies $$\label{eqn:propq} Q\mathbf{1}_N = \mathbf{0}, \; QQ^T = I_{N-1} \text{ and } Q^TQ = \Pi.$$ Using $Q$, we can compute the reduced Laplacian matrix for any graph as $$\label{eqn:lbar} \overline{L} = QLQ^T,$$ and then for connected graphs we can find the unique solution $\Sigma$ to the Lyapunov equation $$\label{eqn:lyap} \overline{L}\Sigma + \Sigma \overline{L}^T = I_{N-1}.$$ If we let $$\label{eqn:xdef} X \mathrel{\mathop :}= 2Q^T \Sigma Q,$$ the resistance between two nodes in a graph can be computed as $$\label{eqn:dirres} r_{k,j} \!=\! \left(\mathbf{e}_N^{(k)} \!-\! \mathbf{e}_N^{(j)}\right)^{\!T}\!\!\! X \!\left(\mathbf{e}_N^{(k)} \!-\! \mathbf{e}_N^{(j)}\right) \!=\! x_{k,k} + x_{j,j} - 2x_{k,j}.$$ Note that Definition \[P1:def:generalres\] in the companion paper [@Young13I] extends effective resistance computations to disconnected graphs as well. In some of the following results, we make use of *binomial coefficients*, defined as $$\label{eqn:bincoef} \binom{n}{k} = \frac{n!}{k! \left(n-k\right)!} \; n,k \in \mathbb{Z}, \; 0 \leq k \leq n.$$ Directed and undirected graphs with equal effective resistances {#sec:equalres} =============================================================== In this section we prove Proposition \[prop:DAP\], which provides sufficient conditions for the resistances in a directed graph to be the same as the resistances in an equivalent undirected graph. The proof relies on two lemmas, which we prove first. Recall that a *permutation matrix* is a square matrix containing precisely one entry of $1$ in each row and column with every other entry being $0$. \[lem:Pprop\] Let $P$ be a permutation matrix. Then $P$ has the following properties 1. \[eqn:Porthog\]P\^[-1]{} = P\^T, 2. \[eqn:PPi\]P= P 3. \[eqn:PIPi\](P - I)= (P - I) = P - I. <!-- --> 1. This follows from the fact that the rows (or columns) of $P$ form an orthonormal set. See, e.g. Theorem 2.1.4 in [@Horn85]. 2. Since $P$ contains precisely one $1$ in each row and column, $P\mathbf{1}_N = \mathbf{1}_N$ and $\mathbf{1}_N^T = \mathbf{1}_N^TP$. Thus $P\Pi = P - \frac{1}{N}P\mathbf{1}_N\mathbf{1}_N^T = P - \frac{1}{N}\mathbf{1}_N\mathbf{1}_N^TP = \Pi P$. 3. The first part follows from [(\[eqn:PPi\])]{}, and again using the fact that $P\mathbf{1}_N = \mathbf{1}_N$ and $\mathbf{1}_N^T = \mathbf{1}_N^TP$, we have $(P-I)\Pi = P - I - \frac{1}{N}P\mathbf{1}_N\mathbf{1}_N^T + \frac{1}{N}\mathbf{1}_N\mathbf{1}_N^T = P - I$. Since $P^T$ also satisfies the requirements of a permutation matrix, the results of Lemma \[lem:Pprop\] apply to $P^T$ as well (this can also be seen by simply transposing equations [(\[eqn:Porthog\])]{}, [(\[eqn:PPi\])]{} and [(\[eqn:PIPi\])]{}). The following lemma is required to prove Proposition \[prop:DAP\]. \[lem:APdiag\] Let $A$ be a square matrix and $P$ be a permutation matrix of the same dimension as $A$. Suppose that $AP$ is diagonal. Then 1. $PA$ is also diagonal, \[lempart:diagtoo\] 2. $A\left(P - I\right) + A^T\left(P^T - I\right)$ is symmetric, that is $$\begin{gathered} \label{eqn:APIsymm} A\left(P - I\right) + A^T\left(P^T - I\right) =\\ \left(P - I\right)A + \left(P^T - I\right)A^T \text{, and}\end{gathered}$$ 3. \[eqn:PAbars\]\^T \^T = \^T\^T. <!-- --> 1. Let $D \mathrel{\mathop :}= AP$, which is diagonal by assumption. Then, by Lemma \[lem:Pprop\], we can see that $A = D P^T$. Thus $PA = P D P^T$, which implies that $PA$ is formed by permuting the rows and columns of a diagonal matrix, and is therefore diagonal. 2. Since $AP$ and $PA$ are diagonal, they are both symmetric. Thus $A^T P^T$ (${} = \left(PA\right)^T = PA$) is symmetric too. Since $\left(-A - A^T\right)$ is also symmetric, the result follows. 3. First we note that as $A P$ is diagonal, it is symmetric and commutes with its transpose (i.e. itself). Thus $P^T A^T A P = A P P^T A^T = A A^T$ (by [(\[eqn:Porthog\])]{}). Similarly, by part (\[lempart:diagtoo\]), $P A$ is also diagonal and so it too is symmetric and commutes with its transpose. Hence $P A A^T P^T = A^T P^T P A = A^T A$ (by [(\[eqn:Porthog\])]{}). Using these facts, we can observe that $\left(P^T\! -\! I\right)\! A^T A\! \left(P\! -\! I\right) = \left(P\! -\! I\right)\!A A^T\! \left(P^T\! -\! I\right)$. Now, adding $\left(P\! -\! I\right)\!A^2\!\left(P\! -\! I\right)$ to both sides gives us $\left[\left(P - I\right)\!A + \left(P^T - I\right)\!A^T\right]\! A \left(P - I\right) = \left(P - I\right) A \left[A\!\left(P - I\right) + A^T\! \left(P^T - I\right)\right]$. But we can use [(\[eqn:APIsymm\])]{} to write this as $$\begin{gathered} A\left(P - I\right)A\left(P - I\right) + A^T\left(P^T - I\right)A\left(P - I\right) = \\ \left(P \!-\! I\right) A\left(P \!-\! I\right) A + \left(P \!-\! I\right) A\left(P^T \!-\! I\right) A^T. \label{eqn:APproof1}\end{gathered}$$ Now, by [(\[eqn:PIPi\])]{}, we can pre- or post-multiply any factor of $\left(P - I\right)$ or $\left(P^T - I\right)$ by $\Pi$ without changing the matrix. Therefore, we can subtract $\left(P - I\right)A\Pi A\left(P - I\right)$ from both sides of [(\[eqn:APproof1\])]{}, obtain a common factor of $\Pi A \left(P - I\right)$ on the left hand side and $\left(P - I\right)A\Pi$ on the right hand side, then use [(\[eqn:APIsymm\])]{} to obtain $$\left(P^T \!-\! I\right)A^T\Pi A \left(P \!-\! I\right) = \left(P \!-\! I\right) A \Pi A^T \left(P^T \!-\! I\right),$$ which is equivalent to (using [(\[eqn:PIPi\])]{} again) $$\begin{gathered} \left(P^T - I\right)\Pi A^T\Pi A \Pi \left(P - I\right) = \\ \left(P - I\right)\Pi A \Pi A^T \Pi \left(P^T - I\right).\end{gathered}$$ Finally, pre-multiplying by $Q$ and post-multiplying by $Q^T$ gives us our desired result. The following proposition provides sufficient conditions for a directed graph to have the same resistances between any pair of nodes as an equivalent undirected graph. Although the assumption for the proposition may seem relatively general, it is straightforward to show that this can only apply to directed path and cycle graphs. \[prop:DAP\] Suppose $\mathcal{G} = \left(\mathcal{V},\mathcal{E},A\right)$ is a connected (directed) graph with matrix of node out-degrees $D$. Furthermore, suppose there is a permutation matrix $P$ such that $D = AP$. Let $\mathcal{G}_u = \left(\mathcal{V}_u,\mathcal{E}_u,A_u\right)$ be the undirected graph with $\mathcal{V}_u = \mathcal{V}$, $\mathcal{E}_u$ such that $(i,j) \in \mathcal{E} \Rightarrow (i,j) \text{ and } (j,i) \in \mathcal{E}_u$, and $A_u = \frac{1}{2}\left(A + A^T\right)$. Then the effective resistance between two nodes in $\mathcal{G}$ is equal to the effective resistance between the same two nodes in $\mathcal{G}_u$. First we note that $\mathcal{G}_u$ must be an undirected graph since its adjacency matrix is symmetric. The Laplacian matrix $L$ of $\mathcal{G}$ is given by $L = D - A = A\left(P - I\right)$. Thus $\overline{L}$ is given by $\overline{L} = QA\left(P - I\right)Q^T$, which can be rewritten (using [(\[eqn:PIPi\])]{}) as $\overline{L} = \overline{A} \, \overline{\left(P - I\right)}$. Furthermore, since $\mathcal{G}$ is connected, $\overline{L}$ is invertible [@Young10]. Next, we claim that the Laplacian matrix of $\mathcal{G}_u$ is given by $L_u\! \mathrel{\mathop :}=\! \frac{1}{2}\!\left[A\left(P - I\right) + A^T\left(P^T - I\right)\right]$. To see this, we first note that we can rewrite $L_u$ as $L_u = \frac{1}{2}\left(D + A^TP^T\right) - A_u$. But by part (\[lempart:diagtoo\]) of Lemma \[lem:APdiag\], $PA$ is diagonal and therefore so is $A^TP^T$. Hence $D_u \mathrel{\mathop :}= \dfrac{1}{2}\left(D + A^TP^T\right)$ is a diagonal matrix. Furthermore, $L_u \mathbf{1}_N = \frac{1}{2}\left[A\left(P\mathbf{1}_N - \mathbf{1}_N\right) + A^T\left(P^T\mathbf{1}_N - \mathbf{1}_N\right)\right]$. But since $P$ is a permutation matrix, $P\mathbf{1}_N = \mathbf{1}_N$ and $P^T\mathbf{1}_N = \mathbf{1}_N$, and so $L_u \mathbf{1}_N = \mathbf{0}$. Therefore, $L_u$ is equal to a diagonal matrix minus $A_u$ and has zero row sums. Hence $D_u$ must be the diagonal matrix of the row sums of $A_u$, i.e. the matrix of out-degrees of $\mathcal{G}_u$. Since $\mathcal{G}_u$ contains every edge in $\mathcal{G}$ (in addition to the reversal of each edge) and $\mathcal{G}$ is connected, $\mathcal{G}_u$ must also be connected. Thus $\Sigma_{u} = \frac{1}{2}\overline{L}_u^{-1}$ is the solution to the Lyapunov equation [(\[eqn:lyap\])]{} for $\mathcal{G}_u$. Using our expression for $L_u$ and [(\[eqn:PIPi\])]{}, we can write $\Sigma_{u} = \left[\overline{A}\,\overline{\left(P - I\right)} + \overline{A}^T \, \overline{\left(P - I\right)}^T\right]^{-1}$. Since $\Sigma_u$ is symmetric, we can also write $\Sigma_{u} = \left[\overline{\left(P - I\right)}\,\overline{A} + \overline{\left(P - I\right)}^T\,\overline{A}^T\right]^{-1}$. Now, when we substitute $\Sigma_{u}$ into the left hand side of equation [(\[eqn:lyap\])]{} for $\mathcal{G}$, we obtain $$\begin{gathered} \overline{L}\Sigma_{u} + \Sigma_{u}\overline{L}^T = \left[I + \overline{A}^T\,\overline{\left(P - I\right)}^T\left(\overline{A}\,\overline{\left(P - I\right)}\right)^{-1}\right]^{-1} \\ {} + \left[I + \left(\overline{\left(P - I\right)}^T\,\overline{A}^T\right)^{-1}\overline{\left(P - I\right)}\,\overline{A}\right]^{-1}.\end{gathered}$$ Using the Matrix Inversion Lemma [@Woodbury50] applied to the first term, we can rewrite this as $$\begin{gathered} \label{eqn:SigmaPAbars} \overline{L}\Sigma_{u} + \Sigma_{u}\overline{L}^T = I - \left[I + \overline{A}\,\overline{\left(P - I\right)}\left(\overline{A}^T\overline{\left(P - I\right)}^T\right)^{-1}\right]^{-1} \\ {} + \left[I + \left(\overline{\left(P - I\right)}^T\overline{A}^T\right)^{-1}\overline{\left(P - I\right)}\,\overline{A}\right]^{-1}\!\!\!.\end{gathered}$$ But by [(\[eqn:PAbars\])]{}, $\overline{(P - I)}^T \overline{A}^T \overline{A}\,\overline{(P - I)} = \overline{(P - I)} \,\overline{A} \, \overline{A}^T\overline{(P - I)}^T$, so $\overline{A}\,\overline{(P - I)}\left(\overline{A}^T\overline{(P - I)}^T\right)^{-1} = \left(\overline{(P - I)}^T \overline{A}^T\right)^{-1}\overline{(P - I)} \,\overline{A}$ and the final two terms in [(\[eqn:SigmaPAbars\])]{} are equal (with opposite signs). Thus $\overline{L}\Sigma_{u} + \Sigma_{u}\overline{L}^T = I$, and so $\Sigma_{u}$ solves [(\[eqn:lyap\])]{} for $\mathcal{G}$. This implies $\Sigma = \Sigma_{u}$, $X = X_u$ and $r_{k,j} = r_{u \; k,j}$ for all nodes $k$ and $j$. Effective resistances from direct connections {#sec:direct} ============================================= In this section we compute the resistance in directed graphs between a pair of nodes that are only connected through a single direct connection, or two direct connections in opposite directions (i.e. the connection subgraph consists of either a directed path or a directed cycle). These two scenarios are analogous (in undirected graphs) to combining multiple resistances in series and combining two resistances in parallel. At present, we do not have general rules for combining resistances from multiple direct connections. The most basic connection is a single directed edge. Intuitively, since an undirected edge with a given weight is equivalent to two directed edges (in opposite directions) with the same weight, one would expect that the resistance of a directed edge should be twice that of an undirected edge with the same weight. The following lemma shows that this is indeed true. \[lem:edgeres\] If $\mathcal{C}_\mathcal{G}(k,j)$ consists of a single directed edge from node $k$ to node $j$ with weight $a_{k,j}$, then $$\label{eqn:edgeres} r_{k,j} = \frac{2}{a_{k,j}}.$$ If we take node $j$ to be the first node in $\mathcal{C}_\mathcal{G}(k,j)$ and node $k$ to be the second, then $\mathcal{C}_\mathcal{G}(k,j)$ has Laplacian matrix $L = \begin{bmatrix}0 & 0\\-a_{k,j} & a_{k,j}\end{bmatrix}$. In this case, there is only one matrix $Q$ (up to a choice of sign) which satisfies [(\[eqn:propq\])]{}, namely $Q = \begin{bmatrix}\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\end{bmatrix}$. Then we have $\overline{L} = Q L Q^T = a_{k,j}$, and hence $\Sigma = \frac{1}{2a_{k,j}}$. Thus $X = 2Q^T\Sigma Q = \begin{bmatrix}\frac{1}{2a_{k,j}} & -\frac{1}{2a_{k,j}}\\-\frac{1}{2a_{k,j}} & \frac{1}{2a_{k,j}}\end{bmatrix}$, and finally, $$r_{k,j} = \left(\mathbf{e}_2^{(1)} - \mathbf{e}_2^{(2)}\right)^T X \left(\mathbf{e}_2^{(1)} - \mathbf{e}_2^{(2)}\right) = \frac{2}{a_{k,j}}. \IEEEQEDhereeqn$$ As a result of Lemma \[lem:edgeres\], when we refer to the effective resistance of a single (directed) edge, we mean twice the inverse of the edge weight. Our next two results extend to some directed graphs the familiar rules from undirected graphs for combining resistances in series and parallel. These cover the cases when a pair of nodes is connected only by either a directed path or cycle. \[theo:pathres\] Suppose $\mathcal{C}_\mathcal{G}(k,j)$ consists of a single directed path. Then $r_{k,j}$ is given by the sum of the resistances of each edge in the path between the two nodes (where the resistance of each edge is computed as in Lemma \[lem:edgeres\]). Suppose we label the nodes in $\mathcal{C}_\mathcal{G}(k,j)$ from $1$ to $N$ in the order in which they appear along the path, starting with the root and moving in the direction opposite the edges. Then we can write the adjacency matrix of $\mathcal{C}_\mathcal{G}(k,j)$ as $A = \text{diag}^{(-1)}\!\left(\begin{bmatrix}a_1 & a_2 & \cdots & a_{N-1}\end{bmatrix}\right)$, and the matrix of node out-degrees as $D = \text{diag}\!\left(\begin{bmatrix}0 & a_1 & \cdots & a_{N-1}\end{bmatrix}\right)$. If we let $P$ be the permutation matrix containing ones above the main diagonal and in the lower left corner, we can observe that $D = AP$. Therefore, by Proposition \[prop:DAP\], the resistance between any two nodes in $\mathcal{C}_\mathcal{G}(k,j)$ is equal to the resistance between the same two nodes in an undirected graph with adjacency matrix $A_u = \frac{1}{2}\left(A + A^T\right)$. Now, $A_u$ is the adjacency matrix of an undirected path, with weights of $\dfrac{1}{2}a_i$ on each edge. But the resistance of an edge in an undirected graph is the inverse of the edge weight and so each edge has resistance $\dfrac{2}{a_i}$. Thus the edge resistances in this undirected graph match those in the original directed path graph (computed according to Lemma \[lem:edgeres\]). Furthermore, the resistance between two nodes connected by an undirected path is simply the sum of the resistances of the edges between them. Thus the same is true for two nodes connected by a directed path. \[theo:cycleres\] Suppose $\mathcal{C}_\mathcal{G}(k,j)$ consists of a single directed cycle. Then $r_{k,j}$ is given by the inverse of the sum of the inverses of the resistances of each path connecting nodes $k$ and $j$ (where the resistance of each path is computed as in Theorem \[theo:pathres\]). Suppose we label the nodes in $\mathcal{C}_\mathcal{G}(k,j)$ from $1$ to $N$ in the reverse of the order in which they appear around the cycle, starting with any node. Then we can write the adjacency matrix of $\mathcal{C}_\mathcal{G}(k,j)$ as $A = \text{diag}^{(N-1)}\!\left(\begin{bmatrix}a_1\end{bmatrix}\right) + \text{diag}^{(-1)}\!\left(\begin{bmatrix}a_2 & a_3 & \cdots & a_N\end{bmatrix}\right)$ and the matrix of node out-degrees as $D = \text{diag}\!\left(\begin{bmatrix}a_1 & a_2 & \cdots & a_N\end{bmatrix}\right)$. If we let $P$ be the permutation matrix containing ones above the main diagonal and in the lower left corner, we can observe that $D = AP$. Therefore, by Proposition \[prop:DAP\], the resistance between any two nodes in $\mathcal{C}_\mathcal{G}(k,j)$ is equal to the resistance between the same two nodes in an undirected graph with adjacency matrix $A_u = \frac{1}{2}\left(A + A^T\right)$. Now, $A_u$ is the adjacency matrix of an undirected cycle, with weights of $\dfrac{1}{2}a_i$ on each edge. But the resistance of an edge in an undirected graph is the inverse of the edge weight, so each edge has resistance $\dfrac{2}{a_i}$. Thus the edge resistances in this undirected graph match those in the original directed cycle graph (computed according to Lemma \[lem:edgeres\]). Furthermore, the resistance between nodes $k$ and $j$ connected by an undirected cycle is given by $$r_{u \, k,j} = \frac{1}{\frac{1}{r_1} + \frac{1}{r_2}},$$ where $r_1$ is the resistance of one path between nodes $k$ and $j$ and $r_2$ is the resistance of the other path. Thus the same is true for two nodes connected by a directed cycle, where (by Theorem \[theo:pathres\]) $r_1$ and $r_2$ are equal to the resistances of the two directed paths between nodes $k$ and $j$. Effective resistances from indirect connections {#sec:indirect} =============================================== Lemma \[lem:edgeres\] and Theorems \[theo:pathres\] and \[theo:cycleres\] suggest a very intuitive interpretation of effective resistance for directed graphs. A directed edge can be thought of as “half” of an undirected edge - either by noting that a directed edge allows half of the interaction to take place that occurs through an undirected edge, or by viewing an undirected edge as consisting of two directed edges with equal weights but in opposite directions. Thus, the resistance of a directed edge is twice the resistance of an undirected edge with the same weight. Then, in path and cycle graphs, resistances combine in exactly the ways (i.e. in series and in parallel) we are used to. However, connections in directed graphs can be more complicated than these. In particular, two nodes in a directed graph may be connected even if neither node is reachable from the other. This will occur when the only connections between the nodes consist of two non-zero length paths which meet at a distinct node. In Theorem \[theo:treeres\] we prove an explicit expression for resistances in the case when $\mathcal{C}_\mathcal{G}(k,j)$ is a directed tree with unit edge weights. Before doing so we prove two lemmas on the correspondence between resistances and the matrix $X$ from [(\[eqn:xdef\])]{}, and two lemmas on the resistance between two leaves in a directed tree. We also rely on the finite series expressions given and proved in Appendix \[apndx:finser\]. \[lem:Xres\] There is a one-to-one relationship between the effective resistances between nodes in a graph and the entries of the matrix $X$ from [(\[eqn:xdef\])]{}. In particular, $$\label{eqn:xtor} r_{k,j} = x_{k,k} + x_{j,j} - 2x_{k,j} \text{, and}$$ $$\label{eqn:rtox} x_{k,j} = \frac{1}{2N}\sum_{i = 1}^N{r_{k,i}} + \frac{1}{2N}\sum_{i = 1}^N{r_{j,i}} - \frac{1}{N^2}\sum_{i = 1}^{N-1}{\sum_{\ell = i+1}^N{r_{i,\ell}}} - \frac{1}{2}r_{k,j}.$$ [(\[eqn:xtor\])]{} is simply the definition of $r_{k,j}$. To derive [(\[eqn:rtox\])]{}, we first note that from [(\[eqn:propq\])]{} and [(\[eqn:xdef\])]{}, $X$ has the property that $X\mathbf{1}_N = \mathbf{0}$ and $\mathbf{1}_N^T X = \mathbf{0}^T$. That is, $X$ has zero row- and column-sums. Now, using [(\[eqn:xtor\])]{}, we can write $r_{k,i} = x_{k,k} + x_{i,i} - 2x_{k,i}$ for any $1 \leq i \leq N$. Then, by summing this equation over $i$, we obtain $\sum_{i = 1}^N{r_{k,i}} = Nx_{k,k} + \text{tr}\left(X\right)$ (since $X$ has zero row-sums). Next, by summing again over $k$, we find that $\text{tr}\left(X\right) = \frac{1}{2N} \sum_{k = 1}^N{\sum_{i = 1}^N{r_{k,i}}}$. But $r_{i,i} = 0 \; \forall i$ and $r_{i,k} = r_{k,i}$ (by Theorem \[P1:theo:metric\] in the companion paper [@Young13I]). Thus we can say that $$\label{eqn:traceX} \text{tr}\left(X\right) = \frac{1}{N} \sum_{i = 1}^{N-1}{\sum_{\ell = i+1}^N{r_{i,\ell}}}.$$ Combining [(\[eqn:traceX\])]{} with our expression for $\sum_{i = 1}^N{r_{k,i}}$ gives us $$\label{eqn:xkk} x_{k,k} = \frac{1}{N}\sum_{i = 1}^N{r_{k,i}} - \frac{1}{N^2}\sum_{i = 1}^{N-1}{\sum_{\ell = i+1}^N{r_{i,\ell}}}.$$ Substituting the expression from [(\[eqn:xkk\])]{} for $x_{k,k}$ and $x_{j,j}$ in [(\[eqn:xtor\])]{} produces [(\[eqn:rtox\])]{}. \[lem:Xpath\] Suppose $\mathcal{G}$ is a directed path with unit edge weights containing $N$ nodes, in which the nodes are labelled from $1$ to $N$ in the order in which they appear along the path, starting with the root. Let $X$ be the corresponding matrix from [(\[eqn:xdef\])]{}. Then the entries of $X$ are given by $$\begin{gathered} \label{eqn:xpath} x_{k,j} \!=\! \frac{2N^2 \!+ 3N \!+ 1 + 3k^2 \!+ 3j^2 \!- 3\left(N\!+\!1\right)k - 3\left(N\!+\!1\right)j}{3N} \\ {} - \left|k - j\right|.\end{gathered}$$ Suppose $k, j \, \in \, \left\{1,2,\ldots,N\right\}$. Then by Theorem \[theo:pathres\], we know that the resistance between nodes $k$ and $j$ in our directed path is equal to $2$ (the resistance of each edge) times the number of edges between them. Since the nodes are labelled in order along the path, this gives us $r_{k,j} = 2\left|k - j\right|$. Therefore, from Lemma \[lem:Xres\], we know that $$\begin{gathered} \label{eqn:xpathinit} x_{k,j} = \frac{1}{N}\sum_{i = 1}^N{\left|k - i\right|} + \frac{1}{N}\sum_{i = 1}^N{\left|j - i\right|} - \frac{2}{N^2}\sum_{i = 1}^{N-1}{\sum_{\ell = i+1}^N{\!\!\left|i - \ell\right|}} \\ {} - \left|k - j\right|.\end{gathered}$$ We now proceed by examining each summation in turn. The first sum can be broken into two parts and then simplified using [(\[eqn:sumint\])]{} to obtain $\sum_{i=1}^N{\left|k - i\right|} = \frac{2k^2 - 2\left(N+1\right)k + \left(N+1\right)N}{2}$. By replacing $k$ with $j$ in the previous expression, we observe that $\sum_{i=1}^N{\left|j - i\right|} = \frac{2j^2 - 2\left(N+1\right)j + \left(N+1\right)N}{2}$. In the third sum in [(\[eqn:xpathinit\])]{}, we observe that $\ell > i$ for every term. Thus $\left|i - \ell\right| = \ell - i$, and we can use [(\[eqn:sumint\])]{} and [(\[eqn:sumintsq\])]{} to obtain $\sum_{i = 1}^{N-1}{\sum_{\ell = i+1}^N{\left|i - \ell\right|}} = \frac{\left(N^2 - 1\right)N}{6}$. Finally, [(\[eqn:xpath\])]{} follows from substituting our expressions for each summation into [(\[eqn:xpathinit\])]{}. The following results are needed to prove Theorem \[theo:treeres\]. In them, we examine the resistance between the leaves of a tree containing two branches that meet at its root and with unit weights on every edge, $\mathcal{G}^\text{tree}_{n,m}$, as shown in Fig. \[fig:C\_G\_trees\]. The effective resistance between the two leaves of $\mathcal{G}^\text{tree}_{n,m}$ will be denoted by $r(n,m)$. \[lem:treeresn1\] The effective resistance between the two leaves of $\mathcal{G}^\text{tree}_{n,1}$ is given by $$\label{eqn:treeresn1} r(n,1) = 2(n-1) + 2^{2-n}.$$ The number of nodes in $\mathcal{G}^\text{tree}_{n,1}$ is $N = n+2$. Let us label the nodes in $\mathcal{G}^\text{tree}_{n,1}$ from $1$ to $n+1$, in the reverse order of the edges, along the branch of length $n$, starting with the root (thus the leaf of this branch is node $n+1$). Then the other leaf (with an edge connecting it to the root) will be node $N = n+2$. Thus the resistance we seek to find is $r(n,1) = r_{n+1,n+2}$. Let $A_{N_p}^\text{path}$, $D_{N_p}^\text{path}$ and $L_{N_p}^\text{path}$ denote the adjacency matrix, matrix of out-degrees and Laplacian matrix of a directed path containing $N_p$ nodes and unit weights on every edge. Let the nodes in this path be labelled from $1$ to $N_p$ in the reverse of the order in which they appear, starting with the root. Thus $A_{N_p}^\text{path} = \text{diag}^{(-1)}\!\left(\mathbf{1}_{N_p-1}\right)$, $D_{N_p}^\text{path} = \text{diag}\!\left(\begin{bmatrix}0 & \mathbf{1}_{N_p-1}^T\end{bmatrix}\right)$ and $L_{N_p}^\text{path} = \text{diag}\!\left(\begin{bmatrix}0 & \mathbf{1}_{N_p-1}^T\end{bmatrix}\right) - \text{diag}^{(-1)}\!\left(\mathbf{1}_{N_p-1}\right)$. From these, we can observe that $$\label{eqn:oneL} \mathbf{1}_{N_p}^T L_{N_p}^\text{path} = \mathbf{e}_{N_p}^{(N_p) T} - \mathbf{e}_{N_p}^{(1) T} \text{, and}$$ $$\label{eqn:eiL} \mathbf{e}_{N_p}^{(i)T} L_{N_p}^\text{path} = \begin{cases}\mathbf{e}_{N_p}^{(i) T} - \mathbf{e}_{N_p}^{(i-1) T} &\text{if } 1 < i \leq N_p,\\\mathbf{0}^T &\text{if } i = 1.\end{cases}$$ Next, we will let $Q_{N_p}$ be a $\left(N_p - 1\right)\times N_p$ matrix which satisfies [(\[eqn:propq\])]{}, and $\overline{L}_{N_p}^\text{path}$ and $\Sigma_{N_p}^\text{path}$ be derived from [(\[eqn:lbar\])]{} and [(\[eqn:lyap\])]{} using $L_{N_p}^\text{path}$ and $Q_{N_p}$. Let $X_{N_p}^\text{path} = 2Q_{N_p}^T \Sigma_{N_p}^\text{path} Q_{N_p}$, according to [(\[eqn:xdef\])]{}. Then, by Lemma \[lem:Xpath\], the entries of $X_{N_p}^\text{path}$ are given by [(\[eqn:xpath\])]{}. Now, we can write the adjacency matrix, matrix of out-degrees and Laplacian matrix of $\mathcal{G}^\text{tree}_{n,1}$ as $A = \begin{bmatrix}A_{n+1}^\text{path} & \mathbf{0}\\\mathbf{e}_{n+1}^{(1) T} & 0\end{bmatrix}$, $D = \begin{bmatrix}D_{n+1}^\text{path} & \mathbf{0}\\\mathbf{0}^T & 1\end{bmatrix}$, and $L = \begin{bmatrix}L_{n+1}^\text{path} & \mathbf{0} \\ -\mathbf{e}_{n+1}^{(1) T} & 1\end{bmatrix}$. Next, let $Q = \begin{bmatrix}Q_{n+1} & \mathbf{0}\\\alpha\mathbf{1}_{n+1}^T & -\beta\end{bmatrix}$, where $\alpha = \frac{1}{\sqrt{(n+1)(n+2)}}$ and $\beta = \sqrt{\frac{n+1}{n+2}}$. Then $Q$ satisfies [(\[eqn:propq\])]{}. We can use [(\[eqn:lbar\])]{}, [(\[eqn:oneL\])]{} and the facts that $L_{n+1}^\text{path} \mathbf{1}_{n+1} = \mathbf{0}_{n+1}$ and $\beta\left(\alpha + \beta\right) = 1$ to express $\overline{L}$ as $$\overline{L} = \begin{bmatrix}\overline{L}_{n+1}^\text{path} & \mathbf{0}\\ \left(\beta - \alpha\right)\mathbf{e}_{n+1}^{(1) T}Q_{n+1}^T + \alpha\mathbf{e}_{n+1}^{(n+1) T}Q_{n+1}^T & 1\end{bmatrix}.$$ In order to compute resistances in $\mathcal{G}^\text{tree}_{n,1}$, we must find the matrix $\Sigma$ which solves [(\[eqn:lyap\])]{}. Since we have partitioned $\overline{L}$ into a $2\times 2$ block matrix, we will do the same for $\Sigma$. Let $\Sigma = \begin{bmatrix} S & \mathbf{t}\\ \mathbf{t}^T & u\end{bmatrix}$, where $S \in \mathbb{R}^{n\times n}$ is a symmetric matrix, $\mathbf{t} \in \mathbb{R}^{n}$ and $u \in \mathbb{R}$. Then multiplying out the matrices in [(\[eqn:lyap\])]{} and equating blocks in this matrix equation gives us $$\begin{aligned} \overline{L}_{n+1}^\text{path} S + S\overline{L}_{n+1}^{\text{path}\, T} &= I_n, \label{eqn:Tn1lyapS} \\ \overline{L}_{n+1}^\text{path} \mathbf{t} + \mathbf{t} + \left(\beta - \alpha\right) S Q_{n+1} \mathbf{e}_{n+1}^{(1)} \hspace{1.3cm} &\nonumber\\ {} + \alpha S Q_{n+1}\mathbf{e}_{n+1}^{(n+1)} &= \mathbf{0} \text{, and} \label{eqn:Tn1lyapt} \\ 2u + 2\left(\beta - \alpha\right)\mathbf{e}_{n+1}^{(1) T}Q_{n+1}^T\mathbf{t} \hspace{2.3cm} & \nonumber\\ {} + 2\alpha \mathbf{e}_{n+1}^{(n+1) T}Q_{n+1}^T\mathbf{t} &= 1. \label{eqn:Tn1lyapu}\end{aligned}$$ From [(\[eqn:Tn1lyapS\])]{}, it is clear that $S = \Sigma_{n+1}^\text{path}$. In addition, we can rewrite [(\[eqn:Tn1lyapu\])]{} as $$\label{eqn:Tn1uform} u = \frac{1}{2} - \left(\beta - \alpha\right)\mathbf{e}_{n+1}^{(1) T}Q_{n+1}^T\mathbf{t} - \alpha \mathbf{e}_{n+1}^{(n+1) T}Q_{n+1}^T\mathbf{t}.$$ In order to find a complete solution for $\Sigma$, we must solve [(\[eqn:Tn1lyapt\])]{} for $\mathbf{t}$. However, resistances are computed from $X$, which, if we let $\mathbf{v} \mathrel{\mathop :}= Q_{n+1}^T \mathbf{t} = \left[v_i\right]$ and use [(\[eqn:xdef\])]{}, can be written as $$\begin{gathered} X = \left[\begin{matrix} X^\text{path}_{n+1} + 2\alpha \mathbf{v}\mathbf{1}_{n+1}^T + 2\alpha\mathbf{1}_{n+1} \mathbf{v}^T + 2\alpha^2 u \mathbf{1}_{n+1}\mathbf{1}_{n+1}^T \\ -2\beta \mathbf{v}^T - 2\alpha\beta u \mathbf{1}_{n+1}^T \end{matrix}\right.\\ \left.\begin{matrix} -2\beta \mathbf{v} - 2\alpha\beta u \mathbf{1}_{n+1}\\ 2\beta^2 u \end{matrix}\right].\end{gathered}$$ Hence, to compute resistances in $\mathcal{G}^\text{tree}_{n,1}$, we need only compute $\mathbf{v}$, not $\mathbf{t}$. We can also note that as $X$ does not depend on our choice of $Q$ (by Lemma \[P1:lem:indofq\] in the companion paper [@Young13I]), neither does $\mathbf{v}$. In fact, we can write [(\[eqn:Tn1uform\])]{} as $u = \frac{1}{2} + \left(\alpha - \beta\right)v_{1} - \alpha v_{n+1}$, and the resistance we seek as $$\begin{gathered} r(n,1) = x^\text{path}_{n+1 \, n+1,n+1} + \left(\alpha +\beta\right)^2 + 2\left(\alpha +\beta\right)^2\left(\alpha - \beta\right)v_1 \\ {} + 2\left(\alpha + \beta\right)\left[2 - \alpha(\alpha + \beta)\right] v_{n+1}. \label{eqn:rn1fromv}\end{gathered}$$ Thus we only need to find $v_1$ and $v_{n+1}$ in order to compute $r(n,1)$. Now, $v_i = \mathbf{e}_{n+1}^{(i) T}\mathbf{v} = \mathbf{e}_{n+1}^{(i) T} Q_{n+1}^T \mathbf{t}$. We will therefore proceed by left-multiplying [(\[eqn:Tn1lyapt\])]{} by $\mathbf{e}_{n+1}^{(i) T} Q_{n+1}^T$. Using the fact that $S = \Sigma_{n+1}^\text{path}$, we obtain $$\begin{gathered} \label{eqn:Tn1lyapv} \mathbf{e}_{n+1}^{(i) T} Q_{n+1}^T Q_{n+1} L_{n+1}^\text{path} \mathbf{v} + v_i + \frac{\beta - \alpha}{2} \mathbf{e}_{n+1}^{(i) T} X_{n+1}^\text{path} \mathbf{e}_{n+1}^{(1)} \\ {} + \frac{\alpha}{2} \mathbf{e}_{n+1}^{(i) T} X_{n+1}^\text{path} \mathbf{e}_{n+1}^{(n+1)} = 0.\end{gathered}$$ But $\mathbf{e}_{n+1}^{(i) T} Q_{n+1}^T Q_{n+1} \!=\! \mathbf{e}_{n+1}^{(i) T} \left(I_{n+1} \!-\! \frac{1}{n+1}\mathbf{1}_{n+1}\mathbf{1}_{n+1}^T\right) \!=\! \mathbf{e}_{n+1}^{(i) T} \!-\! \frac{1}{n+1}\mathbf{1}_{n+1}^T$ by [(\[eqn:propq\])]{}, so by [(\[eqn:oneL\])]{} and [(\[eqn:eiL\])]{}, $$\mathbf{e}_{n+1}^{(i) T} Q_{n+1}^T Q_{n+1}L_{n+1}^\text{path}\mathbf{v} = \begin{cases}\frac{1}{n+1}v_1 + v_i - v_{i-1} - \frac{1}{n+1}v_{n+1} \hspace{-1.4cm}& \\ &\hspace{-1.3cm} \text{if } 1 < i \leq n+1,\\ \frac{1}{n+1}v_1 - \frac{1}{n+1}v_{n+1} & \text{if } i = 1.\end{cases}$$ Furthermore, using [(\[eqn:xpath\])]{}, we observe that $$\begin{aligned} \mathbf{e}_{n+1}^{(i) T} X_{n+1}^\text{path} \mathbf{e}_{n+1}^{(1)} &= x^\text{path}_{n+1 \, i,1} \\ &= \frac{(2n+3)(n+2)}{3(n+1)} + \frac{i(i - 2n - 3)}{n+1} \text{, and}\end{aligned}$$ $$\mathbf{e}_{n+1}^{(i) T} X_{n+1}^\text{path} \mathbf{e}_{n+1}^{(n+1)} = x^\text{path}_{n+1 \, i,n+1} = -\frac{n(n+2)}{3(n+1)} + \frac{i(i-1)}{n+1}.$$ Substituting these expressions into [(\[eqn:Tn1lyapv\])]{} gives us $$\begin{aligned} v_i &= \frac{1}{2}v_{i-1} - \frac{1}{2(n+1)}v_1 + \frac{1}{2(n+1)}v_{n+1} + f + g(i) \nonumber\\ &\hspace{5cm} \text{ if } 1< i \leq n+1 \label{eqn:recurrvi} \\ v_1 &= \frac{1}{n+2}v_{n+1} + h,\label{eqn:recurrv1}\end{aligned}$$ where $f = \frac{\left[(3\alpha -2\beta)n + 3(\alpha-\beta)\right](n+2)}{12(n+1)}$, $g(i) = \frac{i\left[-\beta i + 2(\beta-\alpha) n - 2\alpha + 3\beta\right]}{4(n+1)}$, and $h = \frac{\alpha n}{6} + \frac{(\alpha - \beta)n(2n+1)}{6(n+2)}$. We can now recursively apply [(\[eqn:recurrvi\])]{} $n$ times, starting with $i = n+1$, to find $$\begin{gathered} \label{eqn:recurrvisoln} v_{n+1} = 2^{-n}v_{1} - \frac{v_1}{n+1}\sum_{k = 1}^{n}{2^{-k}} + \frac{v_{n+1}}{n+1}\sum_{k = 1}^{n}{2^{-k}} + 2f\sum_{k = 1}^{n}{2^{-k}} \\ {} + 2\sum_{k = 1}^{n}{g(n+2-k)2^{-k}}.\end{gathered}$$ But we can write $g(n+2-k) = g_1 + g_2 k + g_3 k^2$, where $g_1 = \frac{(\beta - 2\alpha)(n+2)}{4}$, $g_2 = \frac{2\alpha n + 2\alpha + \beta}{4(n+1)}$, and $g_3 = \frac{-\beta}{4(n+1)}$. Therefore, by [(\[eqn:sumtwos\])]{}, [(\[eqn:suminttwos\])]{} and [(\[eqn:sumintsqtwos\])]{}, the sum involving $g(n+2-k)$ is $$\begin{gathered} \sum_{k = 1}^{n}{g(n+2-k)2^{-k}} = g_1\left(1 \!-\! 2^{-n}\right) + g_2\left[2 \!-\! \left(n+2\right)2^{-n}\right] \\ {} + g_3\left[6 - \left(n^2 + 4n + 6\right)2^{-n}\right].\end{gathered}$$ Using this result and [(\[eqn:sumtwos\])]{}, [(\[eqn:recurrvisoln\])]{} becomes $$\begin{gathered} \frac{n + 2^{-n}}{n+1} v_{n+1} = \frac{\left(n+2\right)2^{-n} - 1}{n+1}v_1 + 2f\left(1 - 2^{-n}\right) \\ {} + 2g_1\left(1 - 2^{-n}\right) + 2g_2\left[2 - \left(n+2\right)2^{-n}\right] \\ {} + 2g_3\left[6 - \left(n^2 + 4n + 6\right)2^{-n}\right].\label{eqn:recurrvn1}\end{gathered}$$ But now [(\[eqn:recurrvn1\])]{} and [(\[eqn:recurrv1\])]{} form a pair of linear equations in $v_1$ and $v_{n+1}$. Using the expressions for $f$, $g_1$, $g_2$, $g_3$ and $h$, along with the definitions of $\alpha$ and $\beta$, their solution is given by $$\label{eqn:vsols} \begin{aligned}v_1 &= \frac{\alpha\left[-2n^2 + 5n - 6 + 6.2^{-n}\right]}{6} \text{ and}\\ v_{n+1} &= \frac{\alpha\left[n^2 + 2n - 12 + (6n+12) 2^{-n}\right]}{6}.\end{aligned}$$ Finally, using [(\[eqn:xpath\])]{} and [(\[eqn:vsols\])]{} in [(\[eqn:rn1fromv\])]{}, along with the expressions for $\alpha$ and $\beta$, gives us [(\[eqn:treeresn1\])]{}. \[lem:treeresnl1\] For positive integers $n$ and $\ell$, the resistance between the two leaves of $\mathcal{G}^\text{tree}_{n,\ell+1}$ satisfies the recurrence relation $$\begin{gathered} r(n,\ell+1) = \frac{-3n^2 + 3\ell^2 - 2n\ell -n + 5\ell + 2}{2(n+\ell+1)^2} \\ {} + \frac{\ell^2 + 2n\ell + 2n + 3\ell}{n+\ell+1}2^{-n} + \frac{n^2 + n + 2}{2(n+\ell+1)}2^{-\ell}\\ {} + \frac{1}{4(n+\ell+1)} \sum_{k=1}^\ell {\textstyle\left(4 - \frac{2}{n+\ell+1} - 2^{k-\ell}\right)r(n,k)} \\ {} - \frac{n+\ell+2}{2(n+\ell+1)} \sum_{k=1}^n {\textstyle\left(\frac{1}{n+\ell+1} - 2^{k-n}\right)r(k,\ell)}\\ {} - \frac{1}{4(n+\ell+1)} \sum_{k=1}^n {\sum_{j=1}^\ell {\left(2^{1+k-n} - 2^{j-\ell}\right)r(k,j)}}. \label{eqn:treeresnl1}\end{gathered}$$ The proof of Lemma \[lem:treeresnl1\] relies on similar ideas to the proof of Lemma \[lem:treeresn1\], and is given in Appendix \[apndx:treeres\]. We now proceed to solve the recurrence relation given by Lemmas \[lem:treeresn1\] and \[lem:treeresnl1\] using several finite series results given in Appendix \[apndx:finser\]. \[theo:treeres\] Suppose $\mathcal{C}_\mathcal{G}(k,j)$ consists of a directed tree with unit weights on every edge. Then $r_{k,j}$ is given by $$\label{eqn:treeres} r_{k,j} = 2\left(n - m\right) + 2^{3 - n - m}\sum_{i = 1}^{\left\lfloor\frac{m+1}{2}\right\rfloor}{i \binom{n+m+2}{n+2i+1}},$$ where $n$ is the length of the shortest path from node $k$ to a mutually reachable node and $m$ is the length of the shortest path from node $j$ to a mutually reachable node. Since every node in $\mathcal{C}_\mathcal{G}(k,j)$ is reachable from either node $k$ or node $j$, if $\mathcal{C}_\mathcal{G}(k,j)$ is a tree then only nodes $k$ and $j$ can be leaves. But every tree has at least one leaf, so suppose that node $k$ is a leaf. If node $j$ is not a leaf, then $\mathcal{C}_\mathcal{G}(k,j)$ must be a directed path and node $j$ is the closest mutually reachable node to both nodes $k$ and $j$. Then $m = 0$, $n$ is the path length from $k$ to $j$ and [(\[eqn:treeres\])]{} reduces to $r_{k,j} = 2n$, which follows from Theorem \[theo:pathres\]. Conversely, if node $j$ is a leaf but node $k$ is not, $\mathcal{C}_\mathcal{G}(k,j)$ must be a directed path and node $k$ is the closest mutually reachable node to both nodes $k$ and $j$. Then $n = 0$, $m$ is the path length from $j$ to $k$ and [(\[eqn:treeres\])]{} reduces to $r_{k,j} = -2m + 2^{3-m}\sum_{i = 1}^{\left\lfloor\frac{m+1}{2}\right\rfloor}{i \binom{m+2}{2i+1}}$. But by [(\[eqn:binoddsumi\])]{} and [(\[eqn:binoddsum\])]{} from Lemma \[lem:standardsums\], $\sum_{i = 1}^{\left\lfloor\frac{m+1}{2}\right\rfloor}{i \binom{m+2}{2i+1}} = m2^{m-1}$, and so [(\[eqn:treeres\])]{} becomes $r_{k,j} = 2m$, which follows from Theorem \[theo:pathres\]. Now, if both node $k$ and node $j$ are leaves, then $\mathcal{C}_\mathcal{G}(k,j)$ must be a directed tree with exactly two branches. Thus $\mathcal{C}_\mathcal{G}(k,j)$ must correspond to the tree shown in Fig. \[fig:C\_G\_trees\] and $n$ and $m$ are the path lengths from nodes $k$ and $j$, respectively, to the point where the two branches meet. Furthermore, both $n$ and $m$ are at least $1$. By Corollary \[P1:cor:removepath\] from the companion paper [@Young13I], we observe that the resistance between nodes $k$ and $j$ remains the same as we remove all the nodes of $\mathcal{C}_\mathcal{G}(k,j)$ from the root to the node where the two branches meet. Thus, $r_{k,j}$ can be computed as the resistance between the two leaves of the tree shown in Fig. \[fig:C\_G\_trees\]. Let this tree be called $\mathcal{G}^\text{tree}_{n,m}$, and since the only two parameters that define $\mathcal{G}^\text{tree}_{n,m}$ are $n$ and $m$, we can write $r_{k,j}$ as a function of $n$ and $m$ only. That is, $$r_{k,j} = \mathrel{\mathop :} r(n,m).$$ \[fig:C\_G\_trees\] In order to compute $r(n,m)$, we will begin by considering the case where $m = 1$. Substituting $m = 1$ into [(\[eqn:treeres\])]{} gives $r(n,1) = 2(n-1) + 2^{2-n}$, which follows from Lemma \[lem:treeresn1\]. Now, suppose that [(\[eqn:treeres\])]{} holds for all $n > 0$ and all $m \in \left\{1, 2, \ldots, \ell\right\}$ (for some $\ell > 0$). Then $r_{k,j}$ for $m = \ell + 1$ can be computed using Lemma \[lem:treeresnl1\]. In particular, all resistances in the right-hand side of [(\[eqn:treeresnl1\])]{} are given by [(\[eqn:treeres\])]{}. Therefore, we find that $r(n,\ell+1)$ matches the expression $s(n,\ell)$ given in Lemma \[lem:rsumsimp\]. Therefore, $r(n,\ell+1)$ can be expressed in the form given in [(\[eqn:rsumsimp\])]{}. Next, suppose that $\ell$ is odd. That is, $\ell = 2p + 1$ for some integer $p \geq 0$. Then [(\[eqn:rsumsimp\])]{} gives us $$\begin{gathered} r(n, 2p+2) = 2\left(n - 2p - 2\right) + 2^{1-n-2p}\sum_{i=1}^{p+1}{i\binom{n+2p+4}{n+2i+1}} \\ {} + \frac{g(n,p)}{n+\ell+1},\end{gathered}$$ where $g(n,p)$ is given by [(\[eqn:gdef\])]{} in Lemma \[lem:oddexpression\]. But by Lemma \[lem:oddexpression\], $g(n,p) = 0$ for any integers $n \geq 0$ and $p \geq 0$. Thus [(\[eqn:treeres\])]{} holds for $m = \ell + 1$. Finally, suppose that $\ell$ is even. That is, $\ell = 2p$ for some integer $p > 0$. Then [(\[eqn:rsumsimp\])]{} gives us $$\begin{gathered} r(n, 2p+1) = 2\left(n - 2p - 1\right) + 2^{2-n-2p}\sum_{i=1}^{p}{i\binom{n+2p+3}{n+2i+1}} \\ {} + \frac{4p^2 + 2np + 2n + 6p +2}{n+2p+1}2^{1-n-2p} + \frac{h(n,p)}{n+\ell+1},\end{gathered}$$ where $h(n,p)$ is given by [(\[eqn:hdef\])]{} in Lemma \[lem:evenexpression\]. But by Lemma \[lem:evenexpression\], $h(n,p) = 0$ for any integers $n \geq 0$ and $p \geq 0$. Thus, $$r(n, 2p+1) = 2\left(n - 2p - 1\right) + 2^{2-n-2p}\sum_{i=1}^{p+1}{i\binom{n+2p+3}{n+2i+1}},$$ and so [(\[eqn:treeres\])]{} holds for $m = \ell + 1$. Therefore, by induction we have that [(\[eqn:treeres\])]{} also holds for all $n > 0$ and $m > 0$. Equation [(\[eqn:treeres\])]{} is a highly non-intuitive result, not least because on initial inspection it does not appear to be symmetric in $n$ and $m$ (although we know that it must be, by Theorem \[P1:theo:metric\] in the companion paper). Therefore, it becomes easier to interpret [(\[eqn:treeres\])]{} if we reformulate it in terms of the shorter path length and the difference between the path lengths. Thus, if we let $n$ be the length of the longer path, that is, $n = m + d$ for some $d \geq 0$, [(\[eqn:treeres\])]{} becomes $$r_{k,j} = 2d + 2^{3-2m-d}\sum_{i=1}^{\left\lfloor\frac{m+1}{2}\right\rfloor}{i{\textstyle \binom{2m+d+2}{m+d+2i+1}}} =\mathrel{\mathop :} 2d + e(m,d).$$ Then, using [(\[eqn:bincoef\])]{}, we can write $$\begin{aligned} e(m,d+1) &= 2^{3-2m-d}\!\!\sum_{i=1}^{\left\lfloor\frac{m+1}{2}\right\rfloor}{i\frac{2m+d+3}{2m\!+\!2d\!+\!4i\!+\!4}{\textstyle\binom{2m+d+2}{m+d+2i+1}}} \\ &< \frac{2m+d+3}{2m+2d+4}e(m,d),\end{aligned}$$ and hence conclude that $\displaystyle\lim_{d\rightarrow\infty}e(m,d) = 0$. Thus, when the connection subgraph between two nodes is a directed tree, the resistance between them is twice the difference between the lengths of the paths connecting each node to their closest mutually reachable node, plus some “excess” that disappears as this difference becomes large. Conversely, the excess is significant when the path length difference is small, leading to a resistance that is greater than twice the difference. One common approach to the analysis of resistive circuits is to replace a section of the network that connects to the rest through a single pair of nodes by a single resistor with an equivalent resistance. The simplest example of this is the replacement of a path with a single edge with equivalent resistance. If this principle were to extend to the calculation of effective resistance in directed graphs, then $r_{2,3}$ in $\mathcal{G}^\text{star}_3$ (as shown in Fig. \[fig:indirectexample\]) would match the formula from Theorem \[theo:treeres\]. However, a simple calculation shows that in $\mathcal{G}^\text{star}_3$, $$r_{2,3} = 2\left(n + m\right) - \frac{2nm}{n+m},$$ which only matches [(\[eqn:treeres\])]{} for $n = m = 1$. Thus in more general cases of connection subgraphs like $\mathcal{G}^\text{tree}_{n,m}$ but with arbitrary weights on every edge, the resistance between the leaves does *not* depend only on the equivalent resistance of each path. {width="3cm"} \[fig:indirectexample\] Theorems \[theo:pathres\], \[theo:cycleres\] and \[theo:treeres\] by no means characterise all the possible connection subgraphs in a directed graph. Other connection subgraphs include multiple paths from $k$ to $j$ (some of which could coincide over part of their length), multiple paths from $k$ to $j$ and multiple paths from $j$ to $k$ (again, some of which could partially coincide), multiple indirect connections of the type analysed in Theorem \[theo:treeres\] (which could partially coincide) and a combination of indirect and direct (i.e. path) connections. Further analysis is needed to completely describe how to compute resistances in these situations. Conclusions {#sec:conc} =========== The results of Lemma \[lem:edgeres\] and Theorems \[theo:pathres\] and \[theo:cycleres\] demonstrate that in some situations our definition of effective resistance for directed graphs behaves as an intuitive extension of effective resistance in undirected graphs. In contrast, Theorem \[theo:treeres\] demonstrates a fundamental difference between effective resistance in directed and undirected graphs that arises from the fundamentally different connections that are possible only in directed graphs. Nevertheless, the results presented above show that our notion of effective resistance for directed graphs provides an approach that can relate the local structure of a directed graph to its global properties. The familiar properties of effective resistance allows for a firm analysis of directed graphs that behave similarly to undirected graphs, while the unfamiliar properties can provide insight for the design of directed networks which contain essential differences as compared to undirected networks. Proof of Lemma \[lem:treeresnl1\] {#apndx:treeres} ================================= As stated in the lemma, we will assume that $n$ and $\ell$ are positive integers throughout this proof. Let $N_{n,\ell}$ be the number of nodes in $\mathcal{G}^\text{tree}_{n,\ell}$. The branch of length $n$ contains $n$ nodes (excluding the root), while the other branch contains $\ell$ nodes (excluding the root). Therefore, we have $N_{n,\ell} = n+\ell+1$. Let us label the nodes in $\mathcal{G}^\text{tree}_{n,\ell}$ from $1$ to $n+1$ along the branch of length $n$, in reverse order of the edge directions and starting with the root (thus the leaf of this branch is node $n+1$). Then let us label the nodes in the branch of length $\ell$ from $n+2$ to $N_{n,\ell} = n+\ell+1$ in reverse order of the edge directions. Thus the second leaf is node $N_{n,\ell}$. In the following, we will denote the adjacency matrix of $\mathcal{G}^\text{tree}_{n,\ell}$ by $A_{n,\ell}$, its matrix of node out-degrees by $D_{n,\ell}$ and its Laplacian matrix by $L_{n,\ell}$. Furthermore, we will let $Q_{n,\ell}$ be a $\left(N_{n,\ell}-1\right)\times N_{n,\ell}$ matrix that satisfies [(\[eqn:propq\])]{} and $\overline{L}_{n,\ell}$ and $\Sigma_{n,\ell}$ be the corresponding matrices from [(\[eqn:lbar\])]{} and [(\[eqn:lyap\])]{} using $L_{n,\ell}$ and $Q_{n,\ell}$. Finally, $X_{n,\ell}$ will be the matrix from [(\[eqn:xdef\])]{}, computed using $\Sigma_{n,\ell}$ and $Q_{n,\ell}$. Then, by Lemma \[lem:Xres\], the entries of $X_{n,\ell}$ are related to the resistances in $\mathcal{G}^\text{tree}_{n,\ell}$ by [(\[eqn:rtox\])]{}. As in the proof of Lemma \[lem:treeresn1\], let $A_{N_p}^\text{path}$, $D_{N_p}^\text{path}$ and $L_{N_p}^\text{path}$ denote the adjacency matrix, matrix of out-degrees and Laplacian matrix of a directed path containing $N_p$ nodes and unit weights on every edge. Let the nodes in this path be labelled from $1$ to $N_p$ in the order in which they appear, starting with the root. Then we can write $A_{n,\ell}$, $D_{n,\ell}$ and $L_{n,\ell}$ in terms of $A_{N_p}^\text{path}$, $D_{N_p}^\text{path}$ and $L_{N_p}^\text{path}$ as follows: $A_{n,\ell} = \begin{bmatrix}A_{n+1}^\text{path} & {\ensuremath \left.0\mkern-6.5mu \raisebox{0.6pt}{\text{\small /}}\right.}\\ \mathbf{e}_{\ell}^{(1)}\mathbf{e}_{n+1}^{(1) T} & A_{\ell}^\text{path}\end{bmatrix}$, $D_{n,\ell} = \begin{bmatrix}D_{n+1}^\text{path} & {\ensuremath \left.0\mkern-6.5mu \raisebox{0.6pt}{\text{\small /}}\right.}\\ {\ensuremath \left.0\mkern-6.5mu \raisebox{0.6pt}{\text{\small /}}\right.}& D_{\ell}^\text{path} + \mathbf{e}_{\ell}^{(1)}\mathbf{e}_{\ell}^{(1) T}\end{bmatrix}$ and $L_{n,\ell} = \begin{bmatrix}L_{n+1}^\text{path} & {\ensuremath \left.0\mkern-6.5mu \raisebox{0.6pt}{\text{\small /}}\right.}\\ -\mathbf{e}_{\ell}^{(1)}\mathbf{e}_{n+1}^{(1) T} & L_{\ell}^\text{path} + \mathbf{e}_{\ell}^{(1)}\mathbf{e}_{\ell}^{(1) T}\end{bmatrix}$. Using these expressions as well as [(\[eqn:oneL\])]{} and [(\[eqn:eiL\])]{}, we can observe that $$\label{eqn:oneLnl} \mathbf{1}_{N_{n,\ell}}^T L_{n,\ell} = -2\mathbf{e}_{N_{n,\ell}}^{(1) T} + \mathbf{e}_{N_{n,\ell}}^{(n+1) T} + \mathbf{e}_{N_{n,\ell}}^{(N_{n,\ell}) T} \text{, and}$$ $$\label{eqn:eiLnl} \mathbf{e}_{N_{n,\ell}}^{(i)T} L_{n,\ell} = \begin{cases}\mathbf{e}_{N_{n,\ell}}^{(i) T} \!- \mathbf{e}_{N_{n,\ell}}^{(i-1) T} &\text{if } 1 < i \leq N_{n,\ell}, \, i \neq n+2,\\\mathbf{e}_{N_{n,\ell}}^{(n+2) T} \!- \mathbf{e}_{N_{n,\ell}}^{(1) T} &\text{if } i = n+2,\\\mathbf{0}^T &\text{if } i = 1.\end{cases}$$ Let us now consider $\mathcal{G}^\text{tree}_{n,\ell+1}$. By our labeling convention, the resistance between the two leaves of $\mathcal{G}^\text{tree}_{n,\ell+1}$ is given by $r(n,\ell+1) = r_{n+1,n+\ell+2}$. Now, we can write the adjacency matrix of $\mathcal{G}^\text{tree}_{n,\ell+1}$ in terms of $A_{n,\ell}$ as $A_{n,\ell+1} = \begin{bmatrix}A_{n,\ell} & \mathbf{0}\\\mathbf{e}_{N_{n,\ell}}^{(N_{n,\ell}) T} & 0\end{bmatrix}$. In a similar fashion, we can write the matrix of node out-degrees for $\mathcal{G}^\text{tree}_{n,\ell+1}$ as $D_{n,\ell+1} = \begin{bmatrix}D_{n,\ell} & \mathbf{0}\\\mathbf{0}^T & 1\end{bmatrix}$, and the Laplacian matrix as $L_{n,\ell+1} = \begin{bmatrix}L_{n,\ell} & \mathbf{0} \\ -\mathbf{e}_{N_{n,\ell}}^{(N_{n,\ell}) T} & 1\end{bmatrix}$. Now, let $Q_{n,\ell+1} = \begin{bmatrix}Q_{n,\ell} & \mathbf{0}\\\alpha\mathbf{1}_{N_{n,\ell}}^T & -\beta\end{bmatrix}$, where $\alpha = \frac{1}{\sqrt{N_{n,\ell}(N_{n,\ell}+1)}} = \frac{1}{\sqrt{(n+\ell+1)(n+\ell+2)}}$ and $\beta = \sqrt{\frac{N_{n,\ell}}{N_{n,\ell}+1}} = \sqrt{\frac{n+\ell+1}{n+\ell+2}}$. Then $Q_{n,\ell+1}$ satisfies [(\[eqn:propq\])]{}. We can therefore use [(\[eqn:lbar\])]{}, [(\[eqn:oneLnl\])]{} and the facts that $L_{n,\ell} \mathbf{1}_{N_{n,\ell}} = \mathbf{0}_{N_{n,\ell}}$ and $\beta\left(\alpha + \beta\right) = 1$ to compute $\overline{L}_{n,\ell}$ as $$\begin{gathered} \overline{L}_{n,\ell} \!\!=\!\!\! \left[\begin{matrix}\overline{L}_{n,\ell} \\ \!-2\alpha\mathbf{e}_{N_{n,\ell}}^{(1) T}Q_{n,\ell}^T \!+\! \alpha\mathbf{e}_{N_{n,\ell}}^{(n+1) T}Q_{n,\ell}^T \!+\! \left(\alpha \!+\! \beta\right)\mathbf{e}_{N_{n,\ell}}^{(N_{n,\ell}) T}Q_{n,\ell}^T \end{matrix}\right.\\ \left.\begin{matrix} \mathbf{0}\\ 1\end{matrix}\right].\end{gathered}$$ In order to compute resistances in $\mathcal{G}^\text{tree}_{n,\ell+1}$, we must find the matrix $\Sigma_{n,\ell+1}$ which solves [(\[eqn:lyap\])]{}. Since we have partitioned $\overline{L}_{n,\ell+1}$ into a $2\times 2$ block matrix, we will do the same for $\Sigma_{n,\ell+1}$. Let $\Sigma_{n,\ell+1} = \begin{bmatrix} S & \mathbf{t}\\ \mathbf{t}^T & u\end{bmatrix}$, where $S \in \mathbb{R}^{(N_{n,\ell}-1)\times (N_{n,\ell}-1)}$ is a symmetric matrix, $\mathbf{t} \in \mathbb{R}^{N_{n,\ell}-1}$ and $u \in \mathbb{R}$. Then multiplying out the matrices in [(\[eqn:lyap\])]{} and equating blocks in this matrix equation gives us $$\begin{aligned} \overline{L}_{n,\ell} S + S\overline{L}_{n,\ell}^T &= I_{N_{n,\ell}-1}, \label{eqn:Tnl1lyapS} \\ \overline{L}_{n,\ell} \mathbf{t} + \mathbf{t} - 2\alpha S Q_{n,\ell} \mathbf{e}_{N_{n,\ell}}^{(1)} + \alpha S Q_{n,\ell}\mathbf{e}_{N_{n,\ell}}^{(n+1)} \hspace{-0.2cm}& \nonumber\\ {} + \left(\alpha + \beta\right) S Q_{n,\ell}\mathbf{e}_{N_{n,\ell}}^{(N_{n,\ell})} &= \mathbf{0} \text{, and} \label{eqn:Tnl1lyapt} \\ 2u - 4\alpha\mathbf{e}_{N_{n,\ell}}^{(1) T}Q_{n,\ell}^T\mathbf{t} + 2\alpha \mathbf{e}_{N_{n,\ell}}^{(n+1) T}Q_{n,\ell}^T\mathbf{t} \hspace{0.6cm}& \nonumber\\ {} + 2\left(\alpha + \beta\right) \mathbf{e}_{N_{n,\ell}}^{(N_{n,\ell}) T}Q_{n,\ell}^T\mathbf{t} &= 1. \label{eqn:Tnl1lyapu}\end{aligned}$$ From [(\[eqn:Tnl1lyapS\])]{}, it is clear that $S = \Sigma_{n,\ell}$. In addition, we can rewrite [(\[eqn:Tnl1lyapu\])]{} as $$\begin{gathered} \label{eqn:Tnl1uform} u = \frac{1}{2} + 2\alpha\mathbf{e}_{N_{n,\ell}}^{(1) T}Q_{n,\ell}^T\mathbf{t} - \alpha \mathbf{e}_{N_{n,\ell}}^{(n+1) T}Q_{n,\ell}^T\mathbf{t} \\ {} - \left(\alpha + \beta\right) \mathbf{e}_{N_{n,\ell}}^{(N_{n,\ell}) T}Q_{n,\ell}^T\mathbf{t}.\end{gathered}$$ Thus in order to find a complete solution for $\Sigma_{n,\ell+1}$, we must solve [(\[eqn:Tnl1lyapt\])]{} for $\mathbf{t}$. However, resistances are computed from the entries of $X_{n,\ell+1}$, which, if we let $\mathbf{v} \mathrel{\mathop :}= Q_{n,\ell}^T \mathbf{t} = \left[v_i\right]$ and use [(\[eqn:xdef\])]{}, can be written as $$\begin{gathered} X_{n,\ell+1} \!=\! \left[\begin{matrix} X_{n,\ell} + 2\alpha \mathbf{v}\mathbf{1}_{N_{n,\ell}}^T \!\!+ 2\alpha\mathbf{1}_{N_{n,\ell}} \mathbf{v}^T \!\!+ 2\alpha^2 u \mathbf{1}_{N_{n,\ell}}\mathbf{1}_{N_{n,\ell}}^T \\ -2\beta \mathbf{v}^T - 2\alpha\beta u \mathbf{1}_{N_{n,\ell}}^T\end{matrix}\right.\\ \left.\begin{matrix} -2\beta \mathbf{v} - 2\alpha\beta u \mathbf{1}_{N_{n,\ell}}\\ 2\beta^2 u \end{matrix}\right].\end{gathered}$$ Hence, in order to compute resistances in $\mathcal{G}^\text{tree}_{n,\ell+1}$, we need only compute $\mathbf{v}$, not $\mathbf{t}$. We should also note that as $X_{n,\ell+1}$ does not depend on our choice of $Q_{n,\ell+1}$ (by Lemma \[P1:lem:indofq\] in the companion paper [@Young13I]), neither does $\mathbf{v}$. In fact, we can write [(\[eqn:Tnl1uform\])]{} as $u = \frac{1}{2} + 2\alpha v_{1} - \alpha v_{n+1} - \left(\alpha + \beta\right) v_{N_{n,\ell}}$, and the resistance we seek as $$\begin{gathered} r(n,\ell+1) = x_{n,\ell \, n+1,n+1} + \left(\alpha +\beta\right)^2 + 4\alpha\left(\alpha +\beta\right)^2 v_1 \\ {} + 2\left(\alpha + \beta\right)\left[2 - \alpha\left(\alpha + \beta\right)\right] v_{n+1} - 2(\alpha + \beta)^3 v_{N_{n,\ell}}. \label{eqn:rnl1fromv}\end{gathered}$$ Thus we only need to find $v_1$, $v_{n+1}$ and $v_{N_{n,\ell}}$ in order to compute $r(n,\ell+1)$. Now, $v_i = \mathbf{e}_{N_{n,\ell}}^{(i) T}\mathbf{v} = \mathbf{e}_{N_{n,\ell}}^{(i) T} Q_{n,\ell}^T \mathbf{t}$. We will therefore proceed by left-multiplying [(\[eqn:Tnl1lyapt\])]{} by $\mathbf{e}_{N_{n,\ell}}^{(i) T} Q_{n,\ell}^T$. Using the fact that $S = \Sigma_{n,\ell}$, we obtain $$\begin{gathered} \label{eqn:Tnl1lyapv} \mathbf{e}_{N_{n,\ell}}^{(i) T} Q_{n,\ell}^T Q_{n,\ell} L_{n,\ell} \mathbf{v} + v_i - \alpha \mathbf{e}_{N_{n,\ell}}^{(i) T} X_{n,\ell} \mathbf{e}_{N_{n,\ell}}^{(1)} \\ {}+ \frac{\alpha}{2} \mathbf{e}_{N_{n,\ell}}^{(i) T} X_{n,\ell} \mathbf{e}_{N_{n,\ell}}^{(n+1)} + \frac{\alpha + \beta}{2} \mathbf{e}_{N_{n,\ell}}^{(i) T} X_{n,\ell} \mathbf{e}_{N_{n,\ell}}^{(N_{n,\ell})} = 0.\end{gathered}$$ But $\mathbf{e}_{N_{n,\ell}}^{(i) T} Q_{n,\ell}^T Q_{n,\ell} = \mathbf{e}_{N_{n,\ell}}^{(i) T} \left(I_{N_{n,\ell}} - \frac{1}{N_{n,\ell}}\mathbf{1}_{N_{n,\ell}}\mathbf{1}_{N_{n,\ell}}^T\right) = \mathbf{e}_{N_{n,\ell}}^{(i) T} - \frac{1}{N_{n,\ell}}\mathbf{1}_{N_{n,\ell}}^T$ by [(\[eqn:propq\])]{}, and so by using [(\[eqn:oneLnl\])]{} and [(\[eqn:eiLnl\])]{}, we find $$\begin{gathered} \mathbf{e}_{N_{n,\ell}}^{(i) T} Q_{n,\ell}^T Q_{n,\ell}L_{n,\ell}\mathbf{v} = \frac{2}{N_{n,\ell}}v_1 - \frac{1}{N_{n,\ell}}v_{n+1} - \frac{1}{N_{n,\ell}}v_{N_{n,\ell}} \\ {} + \begin{cases}v_i - v_{i-1} & \text{if } 1 < i \leq N_{n,\ell}, \; i \neq n+2,\\ v_{n+2} - v_{1} & \text{if } i = n+2,\\ 0 & \text{if } i = 1.\end{cases}\end{gathered}$$ Furthermore, we observe that $\mathbf{e}_{N_{n,\ell}}^{(i) T} X_{n,\ell} \mathbf{e}_{N_{n,\ell}}^{(1)} = x_{n,\ell \, i,1}$, $\mathbf{e}_{N_{n,\ell}}^{(i) T} X_{n,\ell} \mathbf{e}_{N_{n,\ell}}^{(n+1)} = x_{n,\ell \, i,n+1}$, and $\mathbf{e}_{N_{n,\ell}}^{(i) T} X_{n,\ell} \mathbf{e}_{N_{n,\ell}}^{(N_{n,\ell})} = x_{n,\ell \, i,N_{n,\ell}}$. Substituting these expressions into [(\[eqn:Tnl1lyapv\])]{} gives us $$\begin{aligned} v_i &= \frac{1}{2}v_{i-1} - \frac{1}{N_{n,\ell}}v_1 + \frac{1}{2N_{n,\ell}}v_{n+1} + \frac{1}{2N_{n,\ell}}v_{N_{n,\ell}} \nonumber\\ & \hspace{0.0cm} {} + \frac{\alpha}{2}x_{n,\ell\, i,1} - \frac{\alpha}{4}x_{n,\ell\, i,n+1} - \frac{\alpha+\beta}{4}x_{n,\ell\, i,N_{n,\ell}} \label{eqn:recurrnlvi}\\ & \hspace{4cm }\text{ if } 1< i \leq N_{n,\ell}, \; i \neq n+2, \nonumber\\ v_{n+2} &= \frac{1}{2}v_{1} - \frac{1}{N_{n,\ell}}v_1 + \frac{1}{2N_{n,\ell}}v_{n+1} + \frac{1}{2N_{n,\ell}}v_{N_{n,\ell}} \nonumber\\ & \hspace{1.6cm} {} + \frac{\alpha}{2}x_{n,\ell\, n+2,1} - \frac{\alpha}{4}x_{n,\ell\, n+2,n+1} \nonumber\\ & \hspace{3.2cm} {} - \frac{\alpha+\beta}{4}x_{n,\ell\, n+2,N_{n,\ell}} \text{, and} \label{eqn:recurrnlvn2} \\ v_1 &= \frac{1}{N_{n,\ell}+2}v_{n+1} + \frac{1}{N_{n,\ell} + 2}v_{N_{n,\ell}} + \frac{\alpha N_{n,\ell}}{N_{n,\ell} + 2}x_{n,\ell\, 1,1} \nonumber\\ & \hspace{-0.5cm} {} - \frac{\alpha N_{n,\ell}}{2\left(N_{n,\ell} + 2\right)}x_{n,\ell\, 1,n+1} - \frac{\left(\alpha + \beta\right)N_{n,\ell}}{2\left(N_{n,\ell} + 2\right)}x_{n,\ell\, 1,N_{n,\ell}}.\label{eqn:recurrnlv1}\end{aligned}$$ We can now recursively apply [(\[eqn:recurrnlvi\])]{} $n$ times, starting with $i = n+1$, and simplify using [(\[eqn:sumtwos\])]{} to find $$\begin{gathered} \frac{N_{n,\ell} \!-\! 1 \!+\! 2^{-n}}{N_{n,\ell}}v_{n+1} \!=\! \frac{\left[-2 \!+\! \left(N_{n,\ell} \!+\! 2\right)2^{-n}\right]}{N_{n,\ell}}v_1 + \frac{1 \!-\! 2^{-n}}{N_{n,\ell}}v_{N_{n,\ell}} \\ {} + \alpha\sum_{k = 1}^{n}{x_{n,\ell\, n+2-k,1} 2^{-k}} - \frac{\alpha}{2}\sum_{k = 1}^{n}{x_{n,\ell\, n+2-k,n+1} 2^{-k}} \\ {} - \frac{\alpha + \beta}{2}\sum_{k = 1}^{n}{x_{n,\ell\, n+2-k,N_{n,\ell}} 2^{-k}}. \label{eqn:recurrnlvn1}\end{gathered}$$ Similarly, we can recursively apply [(\[eqn:recurrnlvi\])]{} $\ell-1$ times, starting with $i = N_{\ell,n} = n+\ell+1$, substitute in [(\[eqn:recurrnlvn2\])]{} and simplify using [(\[eqn:sumtwos\])]{} to find $$\begin{gathered} \frac{N_{n,\ell} \!-\! 1 \!+\! 2^{-\ell}}{N_{n,\ell}}v_{N_{n,\ell}} \!=\! \frac{\left[-2 \!+\! \left(N_{n,\ell} \!+\! 2\right)2^{-\ell}\right]}{N_{n,\ell}}v_1 + \frac{1 \!-\! 2^{-\ell}}{N_{n,\ell}}v_{n+1} \\ {} + \alpha\sum_{k = 1}^{\ell}{x_{n,\ell\, N_{n,\ell}+1-k,1} 2^{-k}} - \frac{\alpha}{2}\sum_{k = 1}^{\ell}{x_{n,\ell\, N_{n,\ell}+1-k,n+1} 2^{-k}} \\ {} - \frac{\alpha + \beta}{2}\sum_{k = 1}^{\ell}{x_{n,\ell\, N_{n,\ell}+1-k,N_{n,\ell}} 2^{-k}}. \label{eqn:recurrnlvN}\end{gathered}$$ Note that [(\[eqn:recurrnlvN\])]{} reduces to [(\[eqn:recurrnlvn2\])]{} when $\ell = 1$. But now [(\[eqn:recurrnlv1\])]{}, [(\[eqn:recurrnlvn1\])]{} and [(\[eqn:recurrnlvN\])]{} form a set of three of simultaneous linear equations in $v_1$, $v_{n+1}$ and $v_{N_{n,\ell}}$. Substituting their solution into [(\[eqn:rnl1fromv\])]{} and then multiplying by $N_{n,\ell}$ (and using the definitions of $\alpha$ and $\beta$) gives us $$\begin{gathered} \label{eqn:rnl1fromx} N_{n,\ell}\, r(n,\ell+1) = N_{n,\ell} + 1 + \left(2^{2-n} - 2^{1-\ell}\right)x_{n,\ell\, 1,1} \\ {} + \left(2^{-\ell} \!-\! 2^{1-n}\right)x_{n,\ell\, 1,n+1} + \left(N_{n,\ell} \!+\! 1\right)\left(2^{-\ell} \!-\! 2^{1-n}\right)x_{n,\ell\, 1,N_{n,\ell}}\\ {} + N_{n,\ell}x_{n,\ell\, n+1,n+1} + 4\sum_{k = 1}^{n}{x_{n,\ell\, n+2-k,1} 2^{-k}} \\ {} - 2\sum_{k = 1}^{\ell}{x_{n,\ell\, N_{n,\ell}+1-k,1} 2^{-k}} - 2\sum_{k = 1}^{n}{x_{n,\ell\, n+2-k,n+1} 2^{-k}} \\ {} \!+\! \sum_{k = 1}^{\ell}{\!x_{n,\ell\, N_{n,\ell}+1-k,n+1} 2^{-k}} \!- \left(2N_{n,\ell} \!+\! 2\right)\!\!\sum_{k = 1}^{n}{\!x_{n,\ell\, n+2-k,N_{n,\ell}} 2^{-k}} \\ {}+ \left(N_{n,\ell} + 1\right)\sum_{k = 1}^{\ell}{x_{n,\ell\, N_{n,\ell}+1-k,N_{n,\ell}} 2^{-k}}.\end{gathered}$$ Now, by [(\[eqn:dirres\])]{}, we can write $x_{n,\ell\, k,j} = \frac{1}{2}x_{n,\ell\, k,k} + \frac{1}{2}x_{n,\ell\, j,j} - \frac{1}{2}r_{k,j}$. Furthermore, by Theorem \[theo:pathres\] we know that $$\begin{gathered} \label{eqn:rpathintree} r_{k,j} = 2\left|k - j\right| \text{ if } 1\leq k,j \leq n+1, \\ \text{or } n+2 \leq k,j \leq N_{n,\ell} \text{, and}\end{gathered}$$ $$\label{eqn:rpath2intree} r_{1,j} = 2\left(j - n - 1\right) \text{ if } n+2 \leq j \leq N_{n,\ell}.$$ Finally, by the definition of $r(\cdot,\cdot)$, we can say that $$\begin{gathered} \label{eqn:rcrossintree} r_{k,j} = r(k-1,j-n-1) \text{ if } 1 < k \leq n+1 \\ \text{ and } n+2 \leq j \leq N_{n,\ell}.\end{gathered}$$ Therefore, we can substitute for each non-diagonal $x_{n,\ell\, k,j}$ term in [(\[eqn:rnl1fromx\])]{} and use [(\[eqn:sumtwos\])]{} and [(\[eqn:suminttwos\])]{}, along with the fact that $N_{n,\ell} = n + \ell + 1$ to find $$\begin{gathered} \left(n\!+\!\ell\!+\!1\right) r(n,\ell+1) \!=\! -4n + 2\ell + 4 + \left(\ell^2 + n\ell + 2\ell - 3\right)\!2^{1-n} \\ {} + \left(\ell + 4\right)2^{-\ell} + \left(n \!+\! \ell \!+\! 2\right)\sum_{k=1}^{n}{r(n+1-k,\ell)2^{-k}} \\ {} - \frac{1}{2}\sum_{k=1}^{\ell}{r(n,\ell+1-k)2^{-k}} + \left(n \!+\! \ell \!+\! \frac{1}{2}\right)x_{n,\ell\, n+1,n+1}\\ {} + \left[\left(n+\ell+1\right)\left(2^{-1-\ell} - 2^{-n}\right) + 1\right]x_{n,\ell\, 1,1} \\ {} - \frac{n+\ell+2}{2}x_{n,\ell\, n+\ell+1,n+\ell+1} \\ {} - \left(n + \ell + 1\right)\!\sum_{k=1}^{n}{x_{n,\ell \, n+2-k,n+2-k}2^{-k}} \\ {} + \frac{n + \ell + 1}{2}\!\sum_{k=1}^{\ell}{x_{n,\ell \, n+\ell+2-k,n+\ell+2-k}2^{-k}},\end{gathered}$$ or, by changing indices inside the sums, $$\begin{gathered} \label{eqn:rnl1fromdiagx} \left(n\!+\!\ell\!+\!1\right) r(n,\ell+1) \!=\! -4n + 2\ell + 4 + \left(\ell^2 + n\ell + 2\ell - 3\right)\!2^{1-n} \\ {} + \left(\ell + 4\right)2^{-\ell} + \frac{n + \ell + 2}{2}\sum_{k=1}^{n}{r(k,\ell)2^{k-n}} - \frac{1}{4}\sum_{k=1}^{\ell}{r(n,k)2^{k-\ell}} \\ {} + \left(n + \ell + \frac{1}{2}\right)x_{n,\ell\, n+1,n+1} \\ {} + \left[\left(n+\ell+1\right)\left(2^{-1-\ell} - 2^{-n}\right) + 1\right]x_{n,\ell\, 1,1} \\ {} \!-\! \frac{n\!+\!\ell\!+\!2}{2}x_{n,\ell\, n+\ell+1,n+\ell+1} \!-\! \frac{n \!+\! \ell \!+\! 1}{2}\sum_{k=1}^{n}{\!x_{n,\ell \, k+1,k+1}2^{k-n}}\\ {} + \frac{n + \ell + 1}{4}\sum_{k=1}^{\ell}{x_{n,\ell \, n+1+k,n+1+k}2^{k-\ell}}.\end{gathered}$$ Now, by [(\[eqn:rtox\])]{} from Lemma \[lem:Xres\], we know that $$\label{eqn:xnldiagform} x_{n,\ell\, i,i} = \frac{1}{N_{n,\ell}}\sum_{k = 1}^{N_{n,\ell}}{r_{i,k}} - \frac{1}{N_{n,\ell}^2}\sum_{k = 1}^{N_{n,\ell}-1}{\sum_{j = k+1}^{N_{n,\ell}}{r_{k,j}}}.$$ Using [(\[eqn:rpathintree\])]{}, [(\[eqn:rpath2intree\])]{} and [(\[eqn:rcrossintree\])]{} and then [(\[eqn:sumint\])]{}, we can write the first sum in [(\[eqn:xnldiagform\])]{} as $$\label{eqn:nlsumresrow} \sum_{k = 1}^{n+\ell+1}{\!\!\!r_{i,k}} = \begin{cases}n^2 + \ell^2 + n + \ell & \text{if } i = 1,\\ n^2 + (3-2i)n + 2(i-1)^2 + \displaystyle\sum_{k=1}^{\ell}{r(i-1,k)}\hspace{-4cm}&\\ & \text{if } 1 < i \leq n+1,\\ 2n^2 + \ell^2 + 2n\ell + (4-4i)n + (3-2i)\ell \hspace{-4cm}& \\ {} + 2(i-1)^2 + \displaystyle\sum_{k = 1}^{n}{r(k,i-n-1)} \hspace{-3cm}&\\& \text{if } n+2 \leq i \leq n+\ell+1.\end{cases}$$ We can note that the double sum in [(\[eqn:xnldiagform\])]{} is independent of $i$. Let $$f \mathrel{\mathop :}= \sum_{k = 1}^{N_{n,\ell}-1}{\sum_{j = k+1}^{N_{n,\ell}}{r_{k,j}}}.$$ Then, substituting [(\[eqn:xnldiagform\])]{} and [(\[eqn:nlsumresrow\])]{} into [(\[eqn:rnl1fromdiagx\])]{}, changing indices and using the results of Lemma \[lem:finser\] produces $$\begin{gathered} \label{eqn:Nrnl1} \left(n+\ell+1\right) r(n,\ell+1) = \frac{-3n^2 + 3\ell^2 - 2n\ell - n + 5\ell + 2}{2(n+\ell+1)} \\ {} + \left(\ell^2 + 2n\ell + 2n + 3\ell\right)2^{-n} + \left(n^2 + n + 2\right)2^{-1-\ell} \\ {} + \frac{1}{4}\sum_{k=1}^{\ell}{\left(4 - \frac{2}{n+\ell+1} - 2^{k-\ell}\right)r(n,k)}\\ {} - \frac{n+\ell+2}{2}\sum_{k=1}^{n}{\left(\frac{1}{n+\ell+1} - 2^{k-n}\right)r(k,\ell)} \\ {} - \frac{1}{4}\sum_{k=1}^{n}{\sum_{j=1}^{\ell}{\left(2^{1+k-n} - 2^{j-\ell}\right)r(k,j)}}.\end{gathered}$$ Finally, dividing [(\[eqn:Nrnl1\])]{} through by $n + \ell + 1$ produces our desired result. Finite series {#apndx:finser} ============= The following series are either well-known or special cases of well-known series. The first two and the general cases of the third and fourth usually appear in any introductory mathematical text that covers series (e.g. section 4.2 of [@Riley06]). The fifth is slightly more obscure. \[lem:finser\] For integer values of $n > 0$, 1. \[eqn:sumint\] \_[k = 1]{}\^[n]{}[k]{} = n(n+1), 2. \[eqn:sumintsq\] \_[k = 1]{}\^[n]{}[k\^2]{} = n(n+1)(2n+1), 3. \[eqn:sumtwos\] \_[k = 1]{}\^[n]{}[2\^[-k]{}]{} = 1 - 2\^[-n]{}, 4. \[eqn:suminttwos\] \_[k = 1]{}\^[n]{}[k2\^[-k]{}]{} = 2 - (n+2)2\^[-n]{} 5. \[eqn:sumintsqtwos\] \_[k = 1]{}\^[n]{}[k\^2 2\^[-k]{}]{} = 6 - (n\^2 + 4n + 6)2\^[-n]{}. Equations [(\[eqn:sumint\])]{} and [(\[eqn:sumintsq\])]{} are special cases of (6.2.1) in [@Hansen75], while [(\[eqn:sumtwos\])]{}, [(\[eqn:suminttwos\])]{} and [(\[eqn:sumintsqtwos\])]{} are special cases of (6.9.1) in [@Hansen75]. All are easily proved using induction. Finite series of binomial coefficients {#subsec:binsums} -------------------------------------- Although there are many interpretations and uses of binomial coefficients, we will simply assume two basic facts about them, namely *Pascal’s rule*; $$\label{eqn:pascal} \binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}, \; 1 \leq k \leq n-1,$$ and the *binomial formula*; $$\label{eqn:binform} \left(x + y\right)^n = \sum_{i = 0}^n{\binom{n}{i} x^i y^{n-i}}, \; n \geq 0.$$ Pascal’s rule follows easily from [(\[eqn:bincoef\])]{} while the binomial formula can be inductively proved using Pascal’s rule. Equations [(\[eqn:pascal\])]{} and [(\[eqn:binform\])]{} can also be found in standard introductory mathematics texts, such as sections 1.5–1.6 in [@Riley06]. We can use Pascal’s rule to derive some identities involving binomial coefficients. These identities include the two in the following lemma. \[lem:binsumnmk\] For integer values of $n, m$ and $k$, with $n > 0$, $m \geq 0$ and $0 \leq k \leq m$, 1. \[eqn:binsumnm\]\_[i=1]{}\^[n]{} = 2. \[eqn:binsumnmk\]\_[i=1]{}\^[n]{} = - . Both results can be easily proven using mathematical induction and Pascal’s rule. A special case of the binomial formula can be found by substituting $y = 1$ into [(\[eqn:binform\])]{}, which gives $$\label{eqn:binformx} \left(1 + x\right)^n = \sum_{i = 0}^n{\binom{n}{i} x^i}, \; n \geq 0.$$ Differentiating this expression with respect to $x$ gives us $$\label{eqn:binformder} n\left(1 + x\right)^{n-1} = \sum_{i = 0}^n{i\binom{n}{i} x^{i-1}}, \; n \geq 1.$$ In the following results, we will make use of a few “well-known” series of binomial coefficients (for example, the first two can be found in Chapter 3 of [@Spiegel09] and all can be solved by Mathematica). Since they are not as standard as the basic facts stated above, we will include a brief proof of them for the sake of completeness. \[lem:standardsums\] For integer values of $n$, 1. \[eqn:binevensum\]\_[i = 0]{}\^[[ ]{}]{} = 2\^[n-1]{}, n &gt; 0, 2. \[eqn:binoddsum\]\_[i = 0]{}\^[[ ]{}]{} = 2\^[n-1]{}, n &gt; 0, 3. \[eqn:binevensumi\]\_[i = 0]{}\^[[ ]{}]{}[2i]{} = n2\^[n-2]{}, n &gt; 1 4. \[eqn:binoddsumi\]\_[i = 0]{}\^[[ ]{}]{}[(2i+1)]{} = n2\^[n-2]{}, n &gt; 1. Substituting $x = \pm 1$ into [(\[eqn:binformx\])]{} gives us $\sum_{i = 0}^n{\binom{n}{i}} = 2^n$ and $\sum_{i = 0}^n{(-1)^i \binom{n}{i}} = 0$ for any $n > 0$. Equations [(\[eqn:binevensum\])]{} and [(\[eqn:binoddsum\])]{} can be found by taking the sum and difference of these two expressions and dividing by $2$. Similarly, substituting $x = \pm 1$ into [(\[eqn:binformder\])]{} gives us $\sum_{i = 0}^n{i\binom{n}{i}} = n2^{n-1}$ and $\sum_{i = 0}^n{(-1)^{i-1}i \binom{n}{i}} = 0$ for any $n > 1$. Equations [(\[eqn:binevensumi\])]{} and [(\[eqn:binoddsumi\])]{} can be found by taking the sum and difference of these two expressions and dividing by $2$. We can now use the results from Lemma \[lem:standardsums\] to derive some more specialised series. These are summarised in the following lemma. As a point of notation, we will assume that any sum not containing any terms (such as $\sum_{i=0}^{-1}a_i$) is equal to zero. \[lem:specialsums\] For integer values of $p \geq 0$, 1. \[eqn:ievenevensum4\]\_[i = 1]{}\^[p+1]{}[i]{} = p2\^[2p+2]{} + 1, 2. \[eqn:ievenoddsum\]\_[i = 1]{}\^[p]{}[i]{} = p2\^[2p]{}, 3. \[eqn:ievenoddsum4\]\_[i = 1]{}\^[p+1]{}[i]{} = (p+1)2\^[2p+2]{}, 4. \[eqn:ioddevensum\]\_[i = 1]{}\^[p]{}[i]{} = (2p-1)2\^[2p]{} + 1, 5. \[eqn:ioddoddsum\]\_[i = 1]{}\^[p]{}[i]{} = (2p+1)2\^[2p]{} - p - 1, 6. \[eqn:i2sum1\]\_[i = 1]{}\^[p]{}[\_[k = 2i - 1]{}\^[2p]{}[i2\^[-k]{}]{}]{} = p\^2 + p, 7. \[eqn:i2sum2\]\_[i = 1]{}\^[p]{}[\_[k = 2i - 1]{}\^[2p-1]{}[i2\^[-k]{}]{}]{} = p\^2 - p 8. \[eqn:i2sum3\]\_[i = 1]{}\^[p+1]{}[\_[k = 2i - 1]{}\^[2p+1]{}[i2\^[-k]{}]{}]{} = p\^2 + p + . <!-- --> 1. This follows by substituting $n = 2p + 4$ into [(\[eqn:binevensumi\])]{} and [(\[eqn:binevensum\])]{}, taking the difference between the first expression and twice the second, removing the $i = 0$ and $i = 1$ terms, shifting indices by $1$, and then dividing by $2$. Note that by the conditions on [(\[eqn:binevensumi\])]{} and [(\[eqn:binevensum\])]{}, this is true for $p \geq 0$. 2. This follows by substituting $n = 2p + 2$ into [(\[eqn:binoddsumi\])]{} and [(\[eqn:binoddsum\])]{}, taking the difference between these expressions and dividing by $2$. Note that by the conditions on [(\[eqn:binoddsumi\])]{} and [(\[eqn:binoddsum\])]{}, this is true for $p \geq 0$. 3. This follows by substituting $p+1$ for $p$ in [(\[eqn:ievenoddsum\])]{}. 4. This follows by substituting $p-1$ for $p$ in [(\[eqn:ievenevensum4\])]{}, adding this to [(\[eqn:ievenoddsum\])]{}, and using Pascal’s rule to say that $\binom{2p+2}{2i+1} + \binom{2p+2}{2i+2} = \binom{2p+3}{2i+2}$. 5. This follows by substituting $n = 2p + 3$ into [(\[eqn:binoddsumi\])]{} and [(\[eqn:binoddsum\])]{}, taking the difference between these expressions, dividing by $2$ and taking the final term out of the sum. Note that by the conditions on [(\[eqn:binoddsumi\])]{} and [(\[eqn:binoddsum\])]{}, this is true for $p \geq 0$. 6. Let $s(p)$ represent the value of this sum, as a function of $p$. That is,\ $s(p) = \sum_{i=1}^{p}{\sum_{k=2i-1}^{2p}{i2^{-k}\binom{k+2}{2i+1}}}$. Then we can see that $s(0) = 0$ and furthermore (using [(\[eqn:ioddoddsum\])]{} and [(\[eqn:ievenoddsum4\])]{}), $s(p+1) = s(p) +2(p+1) - \frac{1}{2}$. Thus, we can say that $s(p) = \sum_{k=1}^{p}{\left(2k-\frac{1}{2}\right)}$, which simplifies using [(\[eqn:sumint\])]{} to our desired result. 7. We can see that $$\begin{gathered} \sum_{i=1}^{p}{\sum_{k=2i-1}^{2p-1}{\!\!i2^{-k}\binom{k+2}{2i+1}}} = \sum_{i=1}^{p}{\sum_{k=2i-1}^{2p}{\!\!i2^{-k}\binom{k+2}{2i+1}}} \\ {} - 2^{-2p}\sum_{i=1}^{p}{i\binom{2p+2}{2i+1}},\end{gathered}$$ and then the result follows from [(\[eqn:i2sum1\])]{} and [(\[eqn:ievenoddsum\])]{}. 8. This follows by substituting $p+1$ for $p$ in [(\[eqn:i2sum2\])]{}. In addition to these series evaluations, the following series manipulations will prove to be useful. \[lem:manips\] For integer values of $p \geq 0$ and $n \geq 0$, 1. $\displaystyle\sum_{i = 1}^{p+1}{\!\sum_{k = 2i - 1}^{2p+1}{\!\!i2^{-k}\textstyle\binom{n+k+3}{n+2i+2}}} = \sum_{i = 1}^{p+1}{\!\sum_{k = 2i - 1}^{2p+1}{\!\!i2^{-k+1}\textstyle\binom{n+k+2}{n+2i+1}}} $ \[eqn:i2manipodd\] - 2\^[-2p-1]{}\_[i=1]{}\^[p+1]{}[i]{}, and 2. $\displaystyle\sum_{i = 1}^{p}{\!\sum_{k = 2i - 1}^{2p}{\!\!i2^{-k}\textstyle\binom{n+k+3}{n+2i+2}}} = \sum_{i = 1}^{p}{\!\sum_{k = 2i - 1}^{2p}{\!\!i2^{-k+1}\textstyle\binom{n+k+2}{n+2i+1}}}$ \[eqn:i2manipeven\] - 2\^[-2p]{}\_[i=1]{}\^[p]{}[i]{}. <!-- --> 1. First, let us suppose that $p \geq 0$, $n \geq 0$ and $i$ is an integer between $1$ and $p+1$ (inclusive). Then, we can use Pascal’s rule with $k > 2i - 1$ to write $\binom{n+k+3}{n+2i+2} = \binom{n+k+2}{n+2i+2} + \binom{n+k+2}{n+2i+1}$, while for $k = 2i - 1$ we can say $\binom{n+k+3}{n+2i+2} = \binom{n+2i+2}{n+2i+2} = 1 = \binom{n+2i+1}{n+2i+1} = \binom{n+k+2}{n+2i+1}$. With these two facts, we can write $$\begin{gathered} \label{eqn:proofmanipodd1} \sum_{k = 2i-1}^{2p+1}{2^{-k}\binom{n+k+3}{n+2i+2}} = \sum_{k = 2i}^{2p+1}{2^{-k}\binom{n+k+2}{n+2i+2}} \\ {} + \sum_{k = 2i-1}^{2p+1}{2^{-k}\binom{n+k+2}{n+2i+1}}.\end{gathered}$$ By shifting indices by $1$, substituting in [(\[eqn:proofmanipodd1\])]{}, and then rearranging, the first sum on the right becomes $$\begin{gathered} \sum_{k = 2i}^{2p+1}{2^{-k}\binom{n+k+2}{n+2i+2}} = \sum_{k = 2i-1}^{2p+1}{2^{-k}\binom{n+k+2}{n+2i+1}} \\ {} - 2^{-2p-1}\binom{n+2p+4}{n+2i+2},\end{gathered}$$ and so [(\[eqn:proofmanipodd1\])]{} becomes $$\begin{gathered} \sum_{k = 2i-1}^{2p+1}{2^{-k}\binom{n+k+3}{n+2i+2}} = 2\!\!\sum_{k = 2i-1}^{2p+1}{2^{-k}\binom{n+k+2}{n+2i+1}} \\ {} - 2^{-2p-1}\binom{n+2p+4}{n+2i+2}.\end{gathered}$$ Substituting this expression into the left hand side of [(\[eqn:i2manipodd\])]{} produces the desired result. 2. Again, let us suppose that $p \geq 0$, $n \geq 0$ and $i$ is an integer, now between $1$ and $p$ (inclusive). As above, we can use Pascal’s rule to write $$\begin{gathered} \label{eqn:proofmanipeven1} \sum_{k = 2i-1}^{2p}{2^{-k}\binom{n+k+3}{n+2i+2}} = \sum_{k = 2i}^{2p}{2^{-k}\binom{n+k+2}{n+2i+2}} \\ {} + \sum_{k = 2i-1}^{2p}{2^{-k}\binom{n+k+2}{n+2i+1}}.\end{gathered}$$ By shifting indices by $1$, substituting in [(\[eqn:proofmanipeven1\])]{}, and then rearranging, the first sum on the right becomes $$\begin{gathered} \sum_{k = 2i}^{2p}{2^{-k}\binom{n+k+2}{n+2i+2}} = \sum_{k = 2i-1}^{2p}{2^{-k}\binom{n+k+2}{n+2i+1}} \\ {} - 2^{-2p}\binom{n+2p+3}{n+2i+2},\end{gathered}$$ and so [(\[eqn:proofmanipeven1\])]{} becomes $$\begin{gathered} \sum_{k = 2i-1}^{2p}{2^{-k}\binom{n+k+3}{n+2i+2}} = 2\!\!\sum_{k = 2i-1}^{2p}{2^{-k}\binom{n+k+2}{n+2i+1}} \\ {} - 2^{-2p}\binom{n+2p+3}{n+2i+2}.\end{gathered}$$ Substituting this expression into the left hand side of [(\[eqn:i2manipeven\])]{} produces the desired result. Now, we can use Lemmas \[lem:specialsums\] and \[lem:manips\] to evaluate two more complicated expressions which will be necessary for the completion of our derivation. \[lem:oddexpression\] Let $p$ and $n$ be non-negative integers, and let $$\begin{gathered} \label{eqn:gdef} g(n,p) \mathrel{\mathop :}= \frac{4p^2 \!+\! 6p \!+\! 2}{n\!+\!2p\!+\!2} \!+\! 4p \!+\! \left(4p^2 \!+\! 4np \!+\! 4n \!+\! 10p \!+\! 6\right)2^{1-n} \\ {} + 2^{-2p}\\ {} + 2^{-n-2p}\sum_{i=1}^{p+1} {\textstyle \!i\!\left\{2\binom{n+2p+4}{n+2i+1} \!-\! \binom{n+2p+4}{n+2i+2} \!-\! \left(2n\!+\!4p\!+\!6\right)\binom{2p+4}{2i+1}\right\}} \\ {} + 2^{2-n}\sum_{i=1}^{p+1}{\sum_{k=2i-1}^{2p+1} {\textstyle\!\!\!\! i2^{-k}\!\left\{\frac{n+2p+1}{n+2p+2}\binom{n+k+2}{n+2i+1} \!-\! \binom{n+k+2}{n+2i} \!+\! \binom{k+3}{2i+1}\right\}}} \\ {} + 2^{-2p}\sum_{i=1}^{p+1}{\sum_{k=1}^n {\textstyle\! i2^{-k}\left\{\binom{k+2p+4}{k+2i+2} \!-\! \frac{2n+4p+6}{n+2p+2}\binom{k+2p+3}{k+2i+1}\right\}}}.\end{gathered}$$ Then $g(n,p) = 0 \;\; \forall \; n,p \geq 0$. First, we can use [(\[eqn:ievenoddsum4\])]{} to simplify the third term in the first sum. In addition, the third term in the second sum can be written as $\binom{k+2}{2i+1} + \binom{k+2}{2i}$ using Pascal’s rule for $k \geq 2i-1$. We can then apply [(\[eqn:i2sum3\])]{} to the $\binom{k+2}{2i+1}$ term. This gives us $$\begin{gathered} \label{eqn:gsimp} g(n,p) = \frac{4p^2\! +\! 6p\! +\! 2}{n\!+\!2p\!+\!2} + 4p - \left(2p^2\! +\! 7p\! +\! 5\right)2^{1-n} + 2^{-2p} \\ {} + 2^{-n-2p}\sum_{i=1}^{p+1} {i\textstyle\left\{2\binom{n+2p+4}{n+2i+1} - \binom{n+2p+4}{n+2i+2}\right\}} \\ {} + 2^{2-n}\sum_{i=1}^{p+1}{\sum_{k=2i-1}^{2p+1} {\!\!\!\! i2^{-k}\left\{\textstyle\frac{n+2p+1}{n+2p+2}\binom{n+k+2}{n+2i+1} \!-\! \binom{n+k+2}{n+2i} \!+\! \binom{k+2}{2i}\right\}}} \\ {} + 2^{-2p}\sum_{i=1}^{p+1}{\sum_{k=1}^n {\!i2^{-k}\left\{\textstyle\binom{k+2p+4}{k+2i+2} \!-\! \frac{2n+4p+6}{n+2p+2}\binom{k+2p+3}{k+2i+1}\right\}}}.\end{gathered}$$ Next, we will consider the case when $p = 0$. Using [(\[eqn:bincoef\])]{} and [(\[eqn:sumtwos\])]{}, we can simplify $g(n,0)$ to find that $g(n,0) = 0 \; \forall \; n \geq 0$. Thus, in the rest of the proof, we will assume that $p > 0$. Furthermore, when $n = 0$, we can use [(\[eqn:ievenevensum4\])]{}, [(\[eqn:ievenoddsum4\])]{} and [(\[eqn:i2sum3\])]{} to find that $g(0,p) = 0 \; \forall \; p > 0$. Next, let us consider $g(n+1,p)$. Substituting $n+1$ in for $n$ in [(\[eqn:gsimp\])]{}, taking the $k = n+1$ terms out of the final sum and applying [(\[eqn:pascal\])]{} and [(\[eqn:i2manipodd\])]{} gives us $$\begin{gathered} \label{eqn:gn1p} g(n\!+\!1,p)\! = \!\frac{4p^2\! +\! 6p\! +\! 2}{n\!+\!2p\!+\!3} \!+\! 4p \!-\! \left(2p^2\! +\! 7p\! +\! 5\right)2^{-n} \!+\! 2^{-2p} \\ {} + 2^{-n-2p-1}\sum_{i=1}^{p+1}{\! i\!\left\{\textstyle2\binom{n+2p+4}{n+2i+1} \!-\! 2\binom{n+2p+4}{n+2i+2}\right\}} \\ {} + 2^{1-n}\sum_{i=1}^{p+1}{\sum_{k=2i-1}^{2p+1} {\!\!\!\! i2^{-k}\left\{\textstyle\frac{n+2p+1}{n+2p+3}\binom{n+k+2}{n+2i+1} \!-\! \binom{n+k+2}{n+2i} \!+\! \binom{k+2}{2i}\right\}}} \\ {} + 2^{-2p}\sum_{i=1}^{p+1}{\sum_{k=1}^{n}{\! i2^{-k}\left\{\textstyle\binom{k+2p+4}{k+2i+2} \!-\! \frac{2n+4p+8}{n+2p+3}\binom{k+2p+3}{k+2i+1}\right\}}}.\end{gathered}$$ Now, let us define a new function, $a(n,p)$, as $$\label{eqn:adef} a(n,p) \mathrel{\mathop :}= \left(n\!+\!2p\!+\!3\right)g(n\!+\!1,p) \!-\! \left(n\!+\!2p\!+\!2\right)g(n,p).$$ Then, from [(\[eqn:gsimp\])]{} and [(\[eqn:gn1p\])]{}, we obtain $$\begin{gathered} \label{eqn:asimp} a(n,p) = 4p \!+\! \left(4p^3 \!+\! 2np^2 \!+\! 16p^2 \!+\! 7np \!+\! 5n \!+\! 17p \!+\! 5\right)2^{-n} \\ {} + 2^{-2p} + 2^{-n-2p}\sum_{i=1}^{p+1}{\textstyle\! i\left\{-\left(n\!+\!2p\!+\!1\right)\binom{n+2p+4}{n+2i+1} \!-\! \binom{n+2p+4}{n+2i+2}\!\right\}} \\ {} + \left(n\!+\!2p\!+\!1\right)\!2^{1-n}\!\sum_{i=1}^{p+1}{\!\sum_{k=2i-1}^{2p+1} {\textstyle\!\!\!\! i2^{-k}\!\left\{\!-\binom{n+k+2}{n+2i+1} \!+\! \binom{n+k+2}{n+2i}\right. }} \\ {\textstyle\left.{} \!-\! \binom{k+2}{2i}\!\right\}} \!+\! 2^{-2p}\sum_{i=1}^{p+1}{\sum_{k=1}^{n}{\textstyle\! i2^{-k}\!\left\{\!\binom{k+2p+4}{k+2i+2} \!-\! 2\binom{k+2p+3}{k+2i+1}\!\right\}}}.\end{gathered}$$ We can use [(\[eqn:ievenevensum4\])]{}, [(\[eqn:ievenoddsum4\])]{} and [(\[eqn:i2sum3\])]{} to show that $a(0,p) = 0$. In a similar manner as before, we will next consider $a(n+1,p)$. Substituting $n+1$ in for $n$ in [(\[eqn:asimp\])]{}, taking the $k = n+1$ terms out of the final sum and applying [(\[eqn:pascal\])]{} and [(\[eqn:i2manipodd\])]{} produces $$\begin{gathered} \label{eqn:an1p} a(n+1,p) \!=\! 4p +\! \left(4p^3 \!\!\!+\!\! 2np^2 \!\!\!+\! 18p^2 \!\!\!+\!\! 7np \!+\! 5n \!+\!\! 24p \!+\! 10\right)\!2^{-n-1} \\ {} \!+\! 2^{-2p} \!+\! 2^{-n-2p-1}\!\sum_{i=1}^{p+1} {\textstyle\! i\!\left\{\!-\!\left(n\!+\!2p\!+\!2\right)\!\binom{n+2p+4}{n+2i+1} \!-\! 2\binom{n+2p+4}{n+2i+2}\!\right\}} \\ {} + \left(n\!+\!2p\!+\!2\right)2^{-n}\!\sum_{i=1}^{p+1}{\!\sum_{k=2i-1}^{2p+1} {\textstyle\!\!\! i2^{-k}\left\{-\binom{n+k+2}{n+2i+1} + \binom{n+k+2}{n+2i}\right.}} \\ {\textstyle\left.{} \!-\! \binom{k+2}{2i}\!\right\}} \!+\! 2^{-2p}\sum_{i=1}^{p+1}{\sum_{k=1}^{n}{\textstyle i2^{-k}\!\left\{\!\binom{k+2p+4}{k+2i+2} \!-\! 2\binom{k+2p+3}{k+2i+1}\!\right\}}}.\end{gathered}$$ Once again, we will define a new function, $b(n,p)$, as $$\label{eqn:bdef} b(n,p) \mathrel{\mathop :}= \frac{a(n+1,p) - a(n,p)}{n+2p}.$$ Note that $b(n,p)$ is well-defined since its denominator is positive for all $p > 0$ and $n \geq 0$. Then, from [(\[eqn:asimp\])]{} and [(\[eqn:an1p\])]{}, we obtain $$\begin{gathered} \label{eqn:bsimp} b(n,p) = -\!\left(2p^2 \!+\! 7p \!+\! 5\right)2^{-n-1} + 2^{-n-2p-1}\!\sum_{i=1}^{p+1}{\textstyle i\binom{n+2p+4}{n+2i+1}}\\ {} \!-\! 2^{-n}\sum_{i=1}^{p+1}{\!\sum_{k=2i-1}^{2p+1} {\textstyle\!\!\!\! i2^{-k}\!\left\{\!-\binom{n+k+2}{n+2i+1} \!+\! \binom{n+k+2}{n+2i} \!-\! \binom{k+2}{2i}\!\right\}}}.\end{gathered}$$ Using [(\[eqn:ievenoddsum4\])]{} and [(\[eqn:i2sum3\])]{}, we find that $b(0,p) = 0$. Finally, we will follow our previous procedure once more and consider $b(n+1,p)$. Substituting $n+1$ in for $n$ in [(\[eqn:bsimp\])]{}, using [(\[eqn:pascal\])]{} and [(\[eqn:i2manipodd\])]{}, and comparing to [(\[eqn:bsimp\])]{} produces $$\label{eqn:bn1p} b(n+1,p) = \frac{1}{2}b(n,p).$$ Hence, from [(\[eqn:bn1p\])]{} and $b(0,p) = 0$, we conclude that $b(n,p) = 0 \;\; \forall \; n \geq0, \; p > 0$. Substituting this result into [(\[eqn:bdef\])]{} tells us that $a(n+1,p) = a(n,p) \;\; \forall \; n \geq0, \; p > 0$, which, along with the fact that $a(0,p) = 0$, allows us to conclude that $a(n,p) = 0 \;\; \forall \; n \geq0, \; p > 0$. Finally, we can substitute this result into [(\[eqn:adef\])]{} to find that $$g(n+1,p) = \frac{n+2p+2}{n+2p+3}g(n,p) \;\; \forall \; n \geq0, \; p > 0,$$ which, along with the facts that $g(0,p) = 0$ and $g(n,0) = 0$, gives us our desired result. \[lem:evenexpression\] Let $p$ and $n$ be non-negative integers, and let $$\begin{gathered} \label{eqn:hdef} h(n,p) \mathrel{\mathop :}= \frac{4p^2 + 2p}{n\!+\!2p\!+\!1} + 4p - 2 + \left(4p^2 \!+\! 4np \!+\! 2n \!+\! 6p \!+\! 2\right)2^{1-n} \\ {} + 2^{1-2p} - \left(4p^2 + 2np + 2n + 6p + 2\right)2^{1-n-2p}\\ {} \!+\! 2^{1-n-2p}\!\sum_{i=1}^{p} {\textstyle\! i\!\left\{2\binom{n+2p+3}{n+2i+1} \!-\! \binom{n+2p+3}{n+2i+2} \!-\! \left(2n\!+\!4p\!+\!4\right)\!\binom{2p+3}{2i+1}\!\right\}} \\ {} \!+\! 2^{2-n}\!\sum_{i=1}^{p}{\!\sum_{k=2i-1}^{2p} {\textstyle\!\!\!\! i2^{-k}\!\left\{\frac{n+2p}{n+2p+1}\binom{n+k+2}{n+2i+1} \!-\! \binom{n+k+2}{n+2i} \!+\! \binom{k+3}{2i+1}\!\right\}}} \\ {} \!+\! 2^{1-2p}\!\sum_{i=1}^{p}{\!\sum_{k=1}^n {\textstyle\!\! i2^{-k}\!\left\{\binom{k+2p+3}{k+2i+2} \!-\! \frac{2n+4p+4}{n+2p+1}\binom{k+2p+2}{k+2i+1}\!\right\}}}.\end{gathered}$$ Then $h(n,p) = 0 \;\; \forall \; n,p \geq 0$. This proof proceeds almost exactly as the proof of Lemma \[lem:oddexpression\]. The only differences are that we use [(\[eqn:pascal\])]{}, [(\[eqn:ioddevensum\])]{}, [(\[eqn:ioddoddsum\])]{}, [(\[eqn:i2sum1\])]{} and [(\[eqn:i2manipeven\])]{} to simplify expressions (rather than [(\[eqn:sumtwos\])]{}, [(\[eqn:pascal\])]{}, [(\[eqn:ievenevensum4\])]{}, [(\[eqn:ievenoddsum4\])]{}, [(\[eqn:i2sum3\])]{} and [(\[eqn:i2manipodd\])]{}) and our intermediate functions are defined as $$c(n,p) \mathrel{\mathop :}= \left(n\!+\!2p\!+\!2\right)h(n+1,p) - \left(n\!+\!2p\!+\!1\right)h(n,p) \text{, and}$$ $$d(n,p) \mathrel{\mathop :}= \frac{c(n+1,p) - c(n,p)}{n+2p-1},$$ where $d(n,p)$ is well-defined since its denominator is positive for all $p > 0$ and $n \geq 0$. Our final result covers some simplification required for the proof of Theorem \[theo:treeres\]. \[lem:rsumsimp\] Suppose $n$ and $\ell$ are positive integers, and let $$\begin{gathered} s(n,\ell) \mathrel{\mathop:}= \frac{-3n^2 + 3\ell^2 - 2n\ell - n + 5\ell + 2}{2(n+\ell+1)^2} \\ {} + \frac{\ell^2 + 2n\ell + 2n + 3\ell}{n+\ell+1}2^{-n} + \frac{n^2 + n + 2}{2(n+\ell+1)}2^{-\ell}\\ {} + \tfrac{1}{2(n+\ell+1)} s_1(n,\ell) + \tfrac{1}{n+\ell+1} s_2(n,\ell) - \tfrac{n+\ell+2}{n+\ell+1} s_3(n,\ell) \\ {} - \tfrac{n+\ell+2}{n+\ell+1} s_4(n,\ell) - \tfrac{1}{2(n+\ell+1)} s_5(n,\ell) - \tfrac{1}{n+\ell+1} s_6(n,\ell) \text{, where}\end{gathered}$$ $s_1(n,\ell) \!\mathrel{\mathop :}=\! \displaystyle\sum_{k=1}^\ell {\!\left(4\! -\! \tfrac{2}{n+\ell+1}\! -\! 2^{k-\ell}\right)\!\left(n\!-\!k\right)}$, $s_2(n,\ell) \!\mathrel{\mathop :}=\! \displaystyle\sum_{k=1}^\ell {\!\left(4\! -\! \tfrac{2}{n+\ell+1}\! -\! 2^{k-\ell}\right)\!2^{1-n-k}\!\!\sum_{i=1}^{\left\lfloor\frac{k+1}{2}\right\rfloor}{\textstyle\!\! i\binom{n+k+2}{n+2i+1}}}$, $s_3(n,\ell) \mathrel{\mathop :}= \displaystyle\sum_{k=1}^n {\left(\tfrac{1}{n+\ell+1} - 2^{k-n}\right)\left(k-\ell\right)}$, $s_4(n,\ell) \mathrel{\mathop :}= \displaystyle\sum_{k=1}^n {\left(\tfrac{1}{n+\ell+1} - 2^{k-n}\right)2^{2-k-\ell}\sum_{i=1}^{\left\lfloor\frac{\ell+1}{2}\right\rfloor}{\textstyle i\binom{k+\ell+2}{k+2i+1}}}$, $s_5(n,\ell) \mathrel{\mathop :}= \displaystyle\sum_{k=1}^n {\sum_{j=1}^\ell {\left(2^{1+k-n} - 2^{j-\ell}\right)\left(k-j\right)}}$, and $s_6(n,\ell) \mathrel{\mathop :}= \displaystyle\sum_{k=1}^n {\sum_{j=1}^\ell {\left(2^{1+k-n} - 2^{j-\ell}\right)2^{1-k-j}\sum_{i=1}^{\left\lfloor\frac{j+1}{2}\right\rfloor}{\textstyle i\binom{k+j+2}{k+2i+1}}}}$. Then $$\begin{gathered} \label{eqn:rsumsimp} s(n,\ell) = 2\left(n - \ell - 1\right) + 2^{2-n-\ell}\sum_{i=1}^{\left\lfloor\frac{\ell+1}{2}\right\rfloor}{ i\binom{n+\ell+3}{n+2i+1}} \\ {} + \frac{1}{n+\ell+1}\left[\frac{\ell^2 + \ell}{n+\ell+1} + 2\ell - 2 + 2^{1-\ell} \vphantom{\sum_{i=1}^{\left\lfloor\frac{\ell+1}{2}\right\rfloor}{i}}\right. \\ {} + \left(\ell^2 + 2n\ell + 2n + 3\ell + 2\right)2^{1-n} \\ {} + 2^{1-\ell}\sum_{i=1}^{\left\lfloor\frac{\ell+1}{2}\right\rfloor}{\sum_{k=1}^{n}{\textstyle i2^{-k}\left\{\binom{k+\ell+3}{k+2i+2} - \frac{2n+2\ell+4}{n+\ell+1}\binom{k+\ell+2}{k+2i+1}\right\}}} \\ {} \!+\! 2^{1-n-\ell}\!\!\sum_{i=1}^{\left\lfloor\frac{\ell+1}{2}\right\rfloor}{\textstyle\!\!\! i\!\left\{2\binom{n+\ell+3}{n+2i+1} \!-\! \binom{n+\ell+3}{n+2i+2} \!-\! \left(2n \!+\! 2\ell\!+\!4\right)\binom{\ell+3}{2i+1}\!\right\}} \\ \left.{} \!+\! 2^{2-n}\!\!\!\!\sum_{i=1}^{\left\lfloor\frac{\ell+1}{2}\right\rfloor}{\!\!\!\!\sum_{k=2i-1}^{\ell}{\textstyle\!\!\!\! i2^{-k}\!\left\{\!\frac{n+\ell}{n+\ell+1}\binom{n+k+2}{n+2i+1} \!-\! \binom{n+k+2}{n+2i} \!+\! \binom{k+3}{2i+1}\!\right\}}}\! \right].\end{gathered}$$ We can use [(\[eqn:sumint\])]{}, [(\[eqn:sumtwos\])]{} and [(\[eqn:suminttwos\])]{} to simplify each of $s_1(n,\ell)$, $s_3(n,\ell)$ and $s_5(n,\ell)$. This gives us $s_1(n,\ell) = \tfrac{-2\ell^3 + 4n^2\ell+2n\ell^2-2n^2-\ell^2-4n-\ell-2}{n+\ell+1} + \left(n+1\right)2^{1-\ell}$, $s_3(n,\ell) = \tfrac{-3n^2+4\ell^2-2n\ell+n+8\ell+4}{2\left(n+\ell+1\right)} - \left(\ell+1\right)2^{1-n}$ and $s_5(n,\ell) = -n^2 - 2\ell^2 + 6n\ell - 3n - 6\ell + \left(\ell^2 + 3\ell\right)2^{1-n} + \left(n^2 + 3n\right)2^{-\ell}$. To simplify each of $s_2(n,\ell)$, $s_4(n,\ell)$ and $s_6(n,\ell)$, we can first exchange the order of summation to make the sum over $i$ the outermost sum, and then apply [(\[eqn:pascal\])]{}, [(\[eqn:binsumnm\])]{} and/or [(\[eqn:binsumnmk\])]{} to obtain $$\begin{gathered} s_2(n,\ell) = \left(4 - \tfrac{2}{n+\ell+1}\right)2^{1-n}\sum_{i = 1}^{\left\lfloor\frac{\ell+1}{2}\right\rfloor}{\sum_{k=2i-1}^{\ell}{\textstyle i2^{-k}\binom{n+k+2}{n+2i+1}}}\\ {} - 2^{1-n-\ell}\sum_{i = 1}^{\left\lfloor\frac{\ell+1}{2}\right\rfloor}{\textstyle i\binom{n+\ell+3}{n+2i+2}},\end{gathered}$$ $$\begin{gathered} s_4(n,\ell) = \tfrac{1}{n+\ell+1}2^{2-\ell}\sum_{i = 1}^{\left\lfloor\frac{\ell+1}{2}\right\rfloor}{\sum_{k=1}^{n}{\textstyle i2^{-k}\binom{k+\ell+2}{k+2i+1}}}\\ {} - 2^{2-n-\ell}\sum_{i = 1}^{\left\lfloor\frac{\ell+1}{2}\right\rfloor}{\textstyle i\binom{n+\ell+3}{n+2i+1}} + 2^{2-n-\ell}\sum_{i = 1}^{\left\lfloor\frac{\ell+1}{2}\right\rfloor}{\textstyle i\binom{\ell+3}{2i+1}} \text{, and}\end{gathered}$$ $$\begin{gathered} s_6(n,\ell) = 2^{2-n}\sum_{i=1}^{\left\lfloor\frac{\ell+1}{2}\right\rfloor}{\!\sum_{k=2i-1}^{\ell}{\textstyle\! i2^{-k}\!\left\{\binom{n+k+2}{n+2i+1} + \binom{n+k+2}{n+2i}\right\}}} \\ {} - 2^{2-n}\!\sum_{i=1}^{\left\lfloor\frac{\ell+1}{2}\right\rfloor}{\!\!\sum_{k=2i-1}^{\ell}{\textstyle\!\! i2^{-k}\binom{k+3}{2i+1}}} - 2^{1-\ell}\!\sum_{i=1}^{\left\lfloor\frac{\ell+1}{2}\right\rfloor}{\!\sum_{k=1}^{n}{\textstyle i2^{-k}\binom{k+\ell+3}{k+2i+2}}}.\end{gathered}$$ Substituting our simplified expressions for $s_1, s_2, s_3, s_4, s_5$ and $s_6$ into the definition of $s(n,\ell)$ gives us our desired result. [^1]: This research was supported in part by AFOSR grant FA9550-07-1-0-0528, ONR grant N00014-09-1-1074, ARO grant W911NG-11-1-0385 and the Natural Sciences and Engineering Research Council (NSERC) of Canada. [^2]: G. F. Young and N. E. Leonard are with the Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA; e-mail: `gfyoung@princeton.edu`; `naomi@princeton.edu`. [^3]: L. Scardovi is with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, M5S 3G4, Canada; e-mail: `scardovi@scg.utoronto.ca`.

--- abstract: 'We performed a radio recombination line (RRL) survey to construct a high-mass star-forming region (HMSFR) sample in the Milky Way based on the all-sky Wide-Field Infrared Survey Explorer (*All-WISE*) point source catalog. The survey was observed with the Shanghai 65m Tianma radio telescope (TMRT) covering 10 hydrogen RRL transitions ranging from H98$\alpha$ to H113$\alpha$ (corresponding to the rest frequencies of 4.5$-$6.9 GHz) simultaneously. Out of 3348 selected targets, we identified an HMSFR sample consisting of 517 sources traced by RRLs, a large fraction of this sample (486) locate near the Galactic plane ($|$*b*$|$ $<$ 2$\degr$). In addition to the hydrogen RRLs, we also detected helium and carbon RRLs towards 49 and 23 sources respectively. We cross-match the RRL detections with the 6.7 methanol maser sources built up in previous works for the same target sample, as a result, 103 HMSFR sources were found to harbor both emissions. In this paper, we present the HMSFR catalog accompanied by the measured RRL line properties and a correlation with our methanol maser sample, which is believed to tracer massive stars at earlier stages. The construction of an HMSFR sample consisting of sources in various evolutionary stages indicated by different tracers is fundamental for future studies of high-mass star formation in such regions.' author: - 'Hong-Ying Chen' - Xi Chen - 'Jun-Zhi Wang' - 'Zhi-Qiang Shen' - Kai Yang bibliography: - 'ref.bib' title: 'A 4-6 GHz Radio Recombination Line Survey in the Milky Way[^1]' --- Introduction\[1\] ================= Formation of high-mass stars in the giant molecular clouds, though intensively studied, remains mysterious (see review papers, e.g., @ZY2007 [@tan2014]). To reveal the intrinsic of high-mass star formation (HMSF) at the very early stage, the fundamental and vital step is to construct a complete sample of high-mass star-forming regions (HMSFRs). Ultra-compact H regions (UCHRs) ($<$ 0.1 pc) are hot ionized gas surrounding an exciting central high-mass star. Such regions are excited by an early O$-$B star from which the ultra-violet photons are strong enough to ionize neutral hydrogen. H regions spread widely at a Galactic scale and have strong luminosity across multiple wavebands (ultraviolet, visible, infra-red and radio), therefore, they are ideal tracers of HMSFRs. H region surveys in the Milky Way were firstly studied in visible wavelengths [@sharpless1953; @sharpless1959; @gum1955; @rodgers1960]. However, the extinction in the optical largely limited the capability of such researches. The dust-free radio observations are therefore needed to construct a more complete sample of Galactic H regions. In 1965, radio recombination line (RRL) was firstly detected by @HM1965 from M 17 and Orion A. Its thin optical depth in centimeter wavelengths makes it an optimal tracer of H regions. RRL surveys were then performed in the next few decades, e.g. @MH1967 [@wilson1970; @reifenstein1970; @downes1980; @CH1987] and @lockman1989. The properties of the Galactic RRLs, such as their spatial distribution, line widths, LSR velocities and intensities are probes of the morphological, chemical and dynamical information of the Milky Way (see @anderson2011). Thus, RRL is important in a range of astrophysical topics, such as the Galactic structure (e.g. @HH2015 [@downes1980; @AB2009]) and metallicity gradient across the Galactic disk which helps understanding the Galactic chemical evolution (GCE) [@wink1983; @shaver1983; @quireza2006; @balser2011]. More recent RRL surveys were performed with high-sensitivity facilities (e.g. @liu2013 [@alves2015; @anderson2011; @anderson2014]). In particular, the recent Green Bank Telescope (GBT) H Region Discovery Survey (HRDS) detected 603 discrete RRL components from 448 targets which were considered to be H regions, thus doubled the number of known Galactic H regions [@anderson2011]. With the demonstration that H regions can be reliably identified by their mid-infrared (MIR) morphology, @anderson2014 extended the HRDS sample to $\sim~8000$ candidate sources based on the *all-sky Wide-Field Infrared Survey Explorer* (*WISE*) MIR images (hereafter the catalog). The catalog contains $\sim~1500$ confirmed H regions with observed RRL data in the literature, it is the most complete sample of H regions to date. The *WISE* data have four MIR bands: 3.4 $\mu$m, 4.6 $\mu$m, 12 $\mu$m and 22 $\mu$m, with angular resolutions of 6$\arcsec$.1, 6$\arcsec$.4, 6$\arcsec$.5 and 12$\arcsec$, respectively, which are sensitive to HMSFRs. Its complete sky coverage and up-to-date database provide an optimal target sample for identifying HMSFR candidates. To further extended the HMSFR sample traced by RRLs beyond the catalog, we conducted an RRL survey with the Shanghai 65m Tianma Radio Telescope (TMRT) based on the *WISE* point source catalog rather than the *WISE* MIR images. Since as H regions form and evolve they will expand, selecting targets from the point source catalog will make our sample to include more compact, and therefore younger sources. Comparing to other single-dish RRL surveys, we concentrate more on the correlation and association with methanol masers to signpost different periods of star-forming processes. Class methanol maser is a powerful tracer of the hot molecular cloud phase of HMSFR [@minier2003; @ellingsen2006; @xu2008], when there is significant mass accretion. H region generally appears in more evolved phases of star formation, well before the main sequence [@walsh1998; @BM1996]. As suggested by @churchwell2002, due to beam-blending and thick optical depth, the densest and earliest H regions are heavily obscured, more extended detectable UCH regions probably are only formed until the central star reaches main sequence, and no longer accreting significant mass. By cross-matching the RRL and class methanol maser samples, the evolutionary stages of their hosts may be specified more accurately. Therefore, simultaneous observation for both the RRLs and 6.7 GHz methanol masers were conducted to investigate their associations. Notably, due to beam dilution, RRL emissions from dense UCH regions at earlier stages will be undetectable, thus our detected RRL sources will trace more extended and evolved H regions. Previous studies have demonstrated that 6.7 GHz methanol masers can be excited in the UCH regions, including both extended and compact sources identified by radio continuum data (e.g @hu2016). Since RRL-detected UCH regions are generally more evolved than those without RRL emissions, RRL researches will be helpful for identifying which methanol maser sources are at more evolutionary stages. In this paper, we report the RRL detections with the measured line parameters, as well as the results of a correlation with the 6.7 GHz methanol maser sample towards the same target sample built by [@yang2017; @yang2019]. Section \[2\] describes the sample selection and observations. Section \[f3\] presents the results of the survey followed by a discussion in section \[4\]. We summarize our main conclusions in Section \[5\].\ Observations and Data Reduction\[2\] ==================================== Source Selection\[2.1\] ----------------------- RRLs and 6.7 GHz methanol masers were observed simultaneously in our survey. The targets were selected with the following methodology: firstly, a cross-matching was applied between the 6.7 GHz methanol maser catalog created by the Methanol Multi beam (MMB) Survey conducted with the Parkes telescope [@caswell2010; @caswell2011; @green2010; @green2012; @breen2015], and the *All-WISE* point source catalog. As a result, there are 502 MMB maser sources which have a *WISE* counterpart with a spatial offset within 7$\arcsec$. We only kept 473 sources with *WISE* data available from all four bands. A magnitude and color-color analysis was then applied to those 473 sources (see @yang2017 for details). 73$\%$ of those sources fell in the color region with well-constrained *WISE* color criteria: \[3.4\] $<$ 14 mag; \[4.6\] $<$ 12 mag; \[12\] $<$ 11 mag; \[22\] $<$ 5.5 mag; \[3.4\] - \[4.6\] $>$ 2, and \[12\] - \[22\] $>$ 2. To avoid repetition, we excluded sources locating in the MMB survey region ($20\degr < l < 186\degr$ and $|$*b*$|$ $>$ 2$\degr$). Due to the limitation of observing range, we also excluded those with a declination below $-30 \degr$. In total, 3348 *WISE* point sources were selected searching for RRLs and methanol maser emissions. In this sample, 1473 sources are located at a high Galactic latitude region with $|$*b*$|$ $>$ 2$\degr$ and 1875 sources fall within $\pm 2\degr$ of the Galactic Plane. Among the selected targets, @yang2017 [@yang2019] detected 6.7 GHz methanol masers from 241 sources, 209 of them are near the Galactic Plane where $|$*b*$|$ $<$ 1$\degr$.\ Observation & Data Reduction\[2.2\] ----------------------------------- The observations were performed between 2015 September and 2018 January with the 65m TMRT in Shanghai, China [@yang2017; @yang2019]. A cryogenically cooled C-band receiver (4-8 GHz) with two orthogonal polarizations was employed in this survey. We used an FPGA-based spectrometer Digital Backend System (DIBAS) (VEGAS; @bussa2012) to receive and record the signals. A total of 16 spectral windows were applied in the observations, each has 16384 channels and a bandwidth of 23.4 MHz supplying a velocity resolution of $\sim$ 0.1 km s$^{-1}$ at 4.5 GHz. Ten of the spectral windows were set up to cover hydrogen RRL transitions spanning from H98$\alpha$ to H113$\alpha$ as specified in Table \[t1\]. In addition to RRLs, the 6.7 GHz methanol maser (CH$_3$OH) emission line, 4.8 GHz H$_2$CO emission and absorption lines, as well as the 4.7 and 6.0 GHz excited-state OH maser transitions were also observed. The observed 6.7 GHz methanol maser results are reported in @yang2017 [@yang2019]. This paper mainly focuses on RRL detection and a cross-match between RRLs and methanol masers. In the observations, the system temperature is about 20 $\sim$  30 K and the main beam efficiency of the TMRT is $\sim~60\%$. The beam has a full width at half-maximum ([FWHM]{}) of $\sim~3\arcmin - 4\arcmin$ at the frequencies of RRLs. There is an uncertainty of $<~20\%$ in the detected flux densities for the sources estimated from the observed variation of the calibrators. [500pt]{}\[c\][&gt;p[2.5cm]{}|&gt;p[1.3cm]{}&gt;p[1.3cm]{}&gt;p[1.3cm]{}&gt;p[1.3cm]{}&gt;p[1.3cm]{}&gt;p[1.3cm]{}&gt;p[2.7cm]{}&gt;p[1.3cm]{}]{} Window Number& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\ $\nu_o$ (MHz) & 4497.78 & 4593.09 & 4618.79 & 4758.11 & 4829.66 & 4874.16 & 5008.92 & 5148.7\ Line Name & H113$\alpha$ & H$_2^{13}$CO & H112$\alpha$ & OH & H$_2$CO & H110$\alpha$ & H109$\alpha$ & H108$\alpha$\ Window Number& 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16\ $\nu_o$ (MHz) & 6016.75 & 6032.92 & 6049.08 & 6106.85 & 6289.14 & 6478.76 & 6672.30 (6676.08) & 6881.49\ Line Name & OH & OH & OH & H102$\alpha$ & H101$\alpha$ & H100$\alpha$ & CH$_3$OH $\&$ H99$\alpha$ & H98$\alpha$\ \ [cccccccc]{} 654 & 517 & 103 & 137 & 49 & 23 & 5 & 5\ A position-switching mode was applied in the observations. Each source was observed with two ON-OFF cycles, both the ON- and OFF-positions in each cycle take $\sim~2$ minutes. To ensure data reliability, sources with bad data quality (eg. radio frequency interferences (RFI), high noise level or abnormal signals) were re-observed with many more cycles. For each source, we primarily set the OFF-point to (0.0$\degr$, $-0.4\degr$) away from the ON-position in (R.A., Decl.). Sources with OFF-positions showing RRL emission were re-observed with a different OFF-position to exclude the background emissions. In total, the observations took $\sim$ 700 hours of observing time excluding calibrations. The data were processed with the [GILDAS/CLASS]{}[^2] software package [@pety2005; @gildas2013]. Adjacent Hn$\alpha$ transitions with large quantum numbers (n $> 50$) have similar line properties, such as line intensity and [FWHM]{} line width (see @balser2006). Therefore, the spectra of the observed 10 transitions through H98$\alpha$ to H113$\alpha$ can be averaged to achieve a higher S/N. After averaging over the 10 RRL transitions at two polarizations, a typical 3-$\sigma$ sensitivity of $\sim$ 7 mK per channel was achieved for the majority of our sources. Due to different rest frequencies, the 10 RRL transitions have different velocity resolutions in their individual spectrum. When averaging over the spectral windows, the [GILDAS/CLASS]{} software will resample the data with the coarsest resolution of them. Depending on the spectral quality, we typically smooth the averaged spectrum over 5 to 30 channels to have a spectral resolution of $\sim$ 0.4 to $\sim$ 2.4 km s$^{-1}$. After that, depending on the background fluctuations, the spectral baselines were subtracted by a first or multi-order ($<$ 4) polynomial fitting. Then the line profiles were fitted with a Gaussian model. For the multi-component sources, the number of Gaussian components was decided via a visual inspection. Notably, it is somehow hard to disentangle the blended line components which have very close central frequencies, so some of the blended sources may be identified as single emission with wide line width (see Section \[3.1\]). The Gaussian fitting results are reported in Section \[3\].\ Result\[3\] =========== RRL Detections\[3.1\] --------------------- A summary of RRL detections in this work and 6.7 GHz methanol maser detections from @yang2017 [@yang2019] are presented in Table \[t2\]. Out of the 3348 targets, we detected hydrogen (H) RRL emissions from 527 positions, corresponding to a detection rate of 15.7%. Excluding the potential Planetary Nebula (PNe) and Supernova Remnant (SNR) (see Sec. \[3.2\]), we built a sample of 517 HMSFRs based on the RRL detections. The derived line parameters and source information for the H RRLs from the HMSFR sample are listed in Table \[t3\]. The spectra for all the detected RRLs (including helium (He) and carbon (C) RRLs; see Section \[4.3\]) are given in Appendix A. Amongst the 517 HMSFR candidates, 488 of them reside within $|b| < 2 \degr$, only 28 are from higher Galactic latitude regions ($|b| > 2 \degr$). Combining with the 240 methanol maser sample in @yang2017 [@yang2019], excluding one associated with a potential SNR, there are 654 HMSFR sources traced by RRLs and/or methanol maser listed in Table \[t4\]. We cross-matched our detected sources with the catalog consisting of $\sim$ 8000 sources. Due to the lack of radio observations, only $\sim$ 1500 sources in their sample were confirmed to be “Known” H regions (denoted as “K" sources in Table 2 of @anderson2014). For targets which spatially associate with multiple sources in the catalog, “Known” sources would be designated preferentially. For targets associate with multiple “Known” sources or other types of sources, the closest counterpart would be designated. There are 467 HMSFR candidates in our sample which are associated with at least one of sources within a separation of 3$\arcmin$ (corresponding to the beam size of TMRT) plus the radius of the source. Amongst them, 425 were classified as “known” H regions, we thus confirmed the other 42 sources being H regions. For the sources included in the catalog, we label their type accordingly in Column (11) of Table \[t3\]. There are also 3 PNe and 4 SNR candidate sources associated with sources, due to the extended morphology of H regions, these targets may be overlapped with the sources along the line of sight (LOS). There are 133 sources showing multiple (typically two or three) H RRL emission components. Figure \[f1\]a shows an example spectrum for such sources. The multiple RRL components may arise from nearby H regions within the TMRT beam or overlapped H regions along the LOS. Diffused ionized gas leaked from nearby HMSFRs may also cause multi-components in the RRL spectra [@zavagno2007; @anderson2010; @OK1997]. Some components have line emissions with close peak velocities causing confusion with a wide line width or non-Gaussian profile. An example of this is presented in Figure \[f1\]b. Notably, RRL detections at adjacent observing positions with similar peak velocities are possibly from the same extended H region. Moreover, H regions with large angular size may have leaked RRL emission detected by the side lobe of the telescope when observing its nearby target points. Those RRL components have similar line velocities but much weaker line intensities comparing to the emission from the central position of the source. There are 5 RRL sources which were potentially detected by the side lobe, we label those emissions as possible duplicated sources by “SL" and the name of the real source in Column (11) in Table \[t3\]. Our RRL detections have peak intensities ranging from 0.01 to 2 K with an average value of 0.07 K, and an integrated intensity from 0.1 to 58 K$\cdot$km s$^{-1}$ with an average of 1.9 K$\cdot$km s$^{-1}$. Among the 517 HMSFR candidates, there are 12 weak sources, labeled with “?" in column (11) in Table \[t3\], which have a line intensity only slightly above our 3$\sigma$ detection threshold, no accurate Gaussian fitting results can be achieved for them. For those weak sources, we only give their peak intensity in Table \[t3\]. Further observations with longer integration time are required for them to get a higher S/N. Figure \[f1\]c illustrates an example of these weak sources. In addition to the 12 weak sources, for multi-component sources containing such weak line component(s), we only provide Gaussian fitting results for the stronger components. In addition to H RRL transitions, the 23.4 MHz spectral windows (see Section \[2.2\]) also simultaneously cover the rest frequencies of the nearby He and C RRLs, they have the same intrinsic quantum numbers with the H RRLs within each spectral window. In the velocity domain, He and C RRLs typically have a constant velocity offset with respect to H RRLs of $-$122 km s$^{-1}$ and $-$149 km s$^{-1}$, respectively. The derived line parameters of He and C RRLs are given in Table \[t5\]. Figure \[f2\] shows an example source with all the three atomic RRLs. In total, we found 49 He RRLs and 23 C RRLs in the observed sample (see Section \[4.3\] for further discussions).\ Other Sources\[3.2\] -------------------- In addition to HMSFRs, ionized gas associated with other astrophysical objects such as planetary Nebula (PNe) and Supernova Remnant (SNR) can also produce RRLs. We performed a matching analysis for our sample with the [SIMBAD]{}[^3] catalog to exclude previously known PNe and SNR sources with a positional criterion of $< 3 \arcmin$. We mark sources with an explicit PNe or SNR identifier as “PNe" or “SNR" in Tables \[t6\] and \[t7\], respectively. To maximize the reliability of our HMSFR sample, for those candidate PNe/SNR sources, which may be associated with both PNe/SNR and H regions or with PNe/SNR located near the edge of the detecting beam ($\sim3 \arcmin$ away), we remove them from the final HMSFR sample and denote them as “PNe?"/“SNR?" in Table \[t6\] / \[t7\]. Notably, though spatially associated with PNe or SNR, the RRLs detected from those target positions may originated from H regions along the LOS. PNe usually has an expanding shell of ionized gas ejected from red giant stars. It is the main contamination of H region samples traced by RRLs. RRLs from PNe usually have wider line width (typically 30 $\sim$  50 km s$^{-1}$) than those from H regions (typical line width of 20 $\sim$  30 km s$^{-1}$) due to their expansion (see @garay1989 [@balser1997]). A typical RRL spectrum from PNe with a line width of $\sim$ 60 km s$^{-1}$ is shown in Figure \[f3\]. The existence of unknown PNe may cause a bias in the statistical analysis of the line width distribution for H regions (see Section \[4.2.2\]). There are 2 sources (G84.913$-$3.505; G85.946$-$3.488) which are explicitly associated with the well-built PNe NGC 7027, as well as 3 possible PNe sources. The full list of PNe candidates is given in Table \[t6\]. SNRs are non-thermal radio sources, and generally have only weak RRLs with wide line width. @liu2019 suggested that the broad line width (mostly $>$ 50 km s$^{-1}$) of the RRLs toward SNRs implies high temperature or turbulent motions of the plasma. Former studies indicated that stimulated emission may be a possible origin for RRLs from SNRs (see @liu2019 and references therein). The potentially similar radio morphology of SNR and H regions is the major confusion when disentangling these two samples [@anderson2017]. There are 5 potential SNR candidates in our sample, all these sources are spatially associated with multiple sources thus no clear identifier can be designated. We list the potential SNR sources in Table \[t7\]. Figure \[f4\] shows an example of RRL spectra from SNR. There is one potential SNR source (G28.532+0.129) which also exhibit methanol maser emission, however, it has a small spatial offset (57.41$\arcsec$) to a possible SNR source G28.5167+0.1333, and has a very wide RRL line width of 70.2 km s$^{-1}$, thus there may have both HMSFR and SNR sources along the LOS of this target. To minimize contamination, this source was excluded from the HMSFR sample and classified as a potential SNR.\ Discussion\[4\] =============== Association and Correlation with Methanol Masers\[4.1\] ------------------------------------------------------- In addition to our RRL emissions, there are 241 6.7-GHz class methanol maser sources detected towards the same sample with TMRT [@yang2017; @yang2019] including one towards a potential SNR (G28.532+0.129; see Section \[3.2\]). The majority of the maser sample (224/241) are close to the Galactic Plane ($|\textit{b}| < 2\degr$). 6.7 GHz methanol masers are believed to appear at an earlier stage of star formation, while H regions typically exist at more evolved stages (see Section \[2.1\]). Thus the association of 6.7 GHz methanol masers and RRLs would be helpful on discriminating HMSFRs at different evolutionary stages [@walsh1998; @jordan2017]. Our survey observed RRLs and methanol masers simultaneously with the same pointing positions, providing the most accurate results of cross-matching the signals. In addition, combing data from different surveys would bring systematic error in the cross-matching caused by inconsistent sensitivities and resolutions. Since we mainly focus on building up a cross-matched HMSFR sample, to reach higher reliability, we do not combine our date with the literature for the analysis in this paper. The 6.7-GHz class methanol maser signal is observed and received by the 15$^{th}$ window in our survey (see Table \[t1\]). Out of the 517 HMSFR sample with RRL detections, there are 103 (20.1$\%$) sources associated with methanol masers, meanwhile, 43.2$\%$ (103/240) methanol maser sources (excluding one potential SNR) exhibit RRL emissions. The remaining 137 methanol maser sources without RRLs may correspond to a younger evolutionary stage compared to those with RRLs. We label the sources showing both emission features in Column (12) in Table \[t3\]. Notably, selecting targets from the point source catalog will bias the sample to include younger sources, and therefore may increase the number of sources associated with both RRL and maser. As a comparison, @anderson2011 found that only $\sim$ 10% (46/448) of their H region sample associated with methanol maser. The correlation of LSR velocities between RRLs and methanol masers is shown in Figure \[f5\]. The majority of the associated RRL and methanol emissions seem to have fairly similar velocities, which intrinsically represent the systemic motion of the sources. As shown in Figure \[f5\], there is one source (G25.395+0.033) which has a large offset between the $V_{\rm LSR}$ of RRL and maser. Due to the commonly extended morphology of H regions, RRL and maser emissions detected from the same target position but with a large velocity offset may from different sources along the LOS, or different regions in the same extended cluster in our $\sim 3~\arcmin$ beam. As the processes of star formation at early stages are poorly understood, comparing the physical properties of HMSFR in different evolutionary stages will help us to study the formation and early evolution of massive star in such sources. We briefly discuss the spatial and intensity distributions of the three sub-sampled sources in the following sections, a more detailed discussion on their physical properties will be given in our future works.\ ### Distance and Galactocentric Distance Distributions\[4.1.1\] Figure \[f6\] shows the normalized distance and galactocentric distance ($R_{{\rm Gal}}$) distributions of RRL-only sources, maser-only sources, and sources associated with both tracers. The average distances with standard deviations of the three samples are 5.62 $\pm$ 2.25, 5.72 $\pm$ 3.69 kpc, and 6.63 $\pm$ 3.03 respectively. As shown in Figure \[f6\]a, the majority of our sources are located at $\lesssim$ 8kpc, and there is no significant difference between the distance distribution of the three samples. This is consistent with previous studies that HMSFRs at different stages are distributed similarly with distance (e.g. @urquhart2014). The sources are more dispersed in the term of $R_{{\rm Gal}}$. As shown in Figure \[f6\]b, although the three sub-samples have similar average $R_{{\rm Gal}}$ values (6.25 $\pm$ 2.09, 6.95 $\pm$ 1.86 and 5.70 $\pm$ 1.77 kpc for RRL-only sources, maser-only sources, and sources with both RRL and maser, respectively), both RRL-only sources and sources associated with both tracers seem to be more adequate near the Galactic center. This can be explained by the fact that the thin gas at the outer Galaxy makes ionized hydrogen hard to be formed, and vice versa. As illustrated by Figure \[f6\]b, three peaks can be seen in the distribution of $R_{{\rm Gal}}$ at $\sim$ 3$-$4, 6$-$7 and 8 kpc. The peak at 3$-$4 kpc for RRL-only sources and sources with both tracers may associate with the 4-kpc molecular ring [@dame2001]. The peak at 6$-$7 kpc appears in the $R_{{\rm Gal}}$ distribution for all sources may coincident with the northern segment of Sagittarius arm. The sources at the 8-kpc peak may be mostly associated with the local Sagittarius–Carina arm.\ ### RRL and Maser Line Intensity Distributions\[4.1.2\] Figure \[f7\] presents the normalized distributions of peak line intensity corrected by distance of our samples, which is defined by the peak line intensity times the square of Bayesian distance (see Section \[4.2.1\] for more information). For the RRL sources, the peak line intensity used here are the main beam temperature (T$_{\rm mb}$) of the RRLs. The RRL-only sources have a mean RRL intensity of 2.88 $\pm$ 0.70 mK kpc$^2$, which is lower than the mean intensity of 3.29 $\pm$ 0.60 mK kpc$^2$ of the sources associated with both RRL and maser. Meanwhile, the maser-only sources and sources associated with both tracers have similar mean maser intensity of 1.95 $\pm$ 0.83 Jy kpc$^2$ and 2.11 $\pm$ 0.8 Jy kpc$^2$, respectively. @urquhart2014 performed similar analyses and found that clumps with methanol maser sources have lower bolometric luminosity than those with H regions. The authors suggested that this is because earlier stage massive stars with methanol masers are more heavily embedded than those at later stages without methanol maser. Although there is no significant difference between our maser line intensity distribution of maser-only and sources associated with both RRL and maser (Figure \[f7\]b), as shown in Figure \[f7\]a, RRL-only sources have lower RRL line intensities than that of sources exhibiting methanol maser on average, and there is a lack of maser detection for RRL sources below 2.0 mK kcp$^2$. As illustrated by @ouyang2019, H regions with methanol masers appear to have higher electron temperature and emission measure than those without, therefore methanol masers are more likely to be produced in regions with high gas densities and hence have a higher detection rate at more luminous H regions.\ Galactic Distribution --------------------- ### RRL Spatial Distribution\[4.2.1\] Figure \[f8\] represents the Galactic latitude and longitude distributions of the H-RRL-only sources, methanol-maser-only sources, sources associated with both tracers, and He RRL sources (see Section \[4.3\] for more details). As shown in this figure, a large fraction of the sources associated with both signals locate near the Galactic Plane, only 3 of them with a Galactic latitude larger than $\pm$ 2$\degr$, reflecting a sparse abundance in such regions. The LSR velocities of the RRLs can be used to calculate the distance of its host H regions using the Bayesian distance calculator built by @reid2016[^4]. @reid2016 combines various types of distance information (spiral arm mode, kinematic distance, Galactic latitude, and parallax source) in a Bayesian approach, and fits the combined probability density function with multiple Gaussian components. The distance and error are estimated by the peak probability density and width of the Gaussian fitted component with maximum integrated probability density. The kinematic distances augmented by H absorption spectra were used to resolve the near/far ambiguity for sources within the Solar circle. A user-adjustable prior probability (from 0 to 1) that the source is beyond the tangent point was set to the default value of 0.5. The result distances with error are presented in Column (9) and (10) in Table \[t3\]. In Figure \[f9\], we compare the velocity distribution along Galactic longitude of our HMSFRs to the longitude-velocity diagram retrieved from @vallee2008. For sources exhibiting multiple RRL components with different peak velocities, we only use the peak velocity of the strongest component to do the analysis. As shown in this figure, most of the sources located in the first quadrant and our data align with the model curves well in the first and second quadrants.\ ### Line Width Distribution\[4.2.2\] Figure \[f10\] shows the RRL line width distribution of our H region sample excluding those potential PNe, SNR and weak sources. These sources have an average line width with a standard deviation of 23.6 $\pm$ 2.0 km s$^{-1}$, which is very close to that of the catalog (22.3 $\pm$ 5.3 km s$^{-1}$). Thermal broadened RRL has a line profile with an [FWHM]{} width proportional to T$^{1/2}\nu_o$ [@BS1972], where T is the temperature of the host H region, $\nu_o$ is the rest central frequency. For a typical RRL-hosting source with T $\sim$ 5000 to 13000 K across the Galaxy [@balser2015], the corresponding thermal-broadened line width is $\sim$ 15.3 to 24.6 km s$^{-1}$. In addition to thermal broadening, pressure broadening is also significant for RRLs at centimeter wavelengths [@keto2008]. The ratio between pressure-broadened and thermal broadened line width is proportional to $n_e N^7$, where $n_e$ is the electron density and $N$ is the principal quantum number of the RRL transition [@BS1972; @grim1974; @keto1995; @keto2008]. As illustrated by @keto2008, since pressure broadening is proportional to the density of host H region and less significant at high frequencies, comparing the line widths of RRLs measured with high-resolution interferometer in various wavelengths can be used to measure the electron density of the natal H region. RRLs with a line width narrower than 15 km s$^{-1}$ are likely from comparably cold, sparse H regions and are broadened purely by thermal, thus they can be used to probe the temperature of the host H regions. There are 69 sources in our sample containing H RRLs with a line width of $<$ 15 km s$^{-1}$, for thermally broadened RRL source with such narrow line width, the host H regions have an upper limit to the temperature of $\lesssim 5000$ K. According to @shaver1970 and @shaver1979, RRLs with such narrow line width are from cold nebulae. Another possible explanation for the narrow line width is that the observed RRL line width is underestimated due to radiative transfer effect caused by strong free-free emission, which typically has a larger optical depth. We detected 51 sources with a very broad line width ($>$ 35 km s$^{-1}$). Out of these sources, 22 have a line width broader than 40 km s$^{-1}$. RRL with a line width $>$ 40 km s$^{-1}$ may be very dense H regions. For these RRLs, there may also exist large-scale motions around the central young stars (@sewilo2004 [@ridge2001; @RB2001] and references therein) such as nebular expansion, rotation of the inner parts of the accretion disk, infall of matter, shocks, bipolar jets or photo evaporating flows. Some of those broad line sources are possibly previously unknown PNe or SNR sources. As mentioned in Section \[3.2\], since PNe generally have larger line width due to expansion, their existence may affect the RRL line width distribution of our sample. For example, one of the sources, G28.393+0.085, with the widest line width in our sample was identified as a known H region (G028.394+00.076) in @anderson2011. However, its RRL emission with extremely large line width is potentially from an unknown PNe or SNR source along the LOS. In addition, blended multiple RRL components with very close peak velocities may also cause confusion with a single wide line component.\ Helium and Carbon RRLs\[4.3\] ----------------------------- ### Helium RRL Detections\[4.3.1\] In addition to H RRLs, there are 49 sources also exhibit He RRLs. Their spatial distribution is shown in Figure \[f8\]. He RRLs are believed to have the same origin with H RRLs. The atomic heliums are ionized by the UV photons emitted by a central O6 or hotter star [@mezger1978; @roshi2017]. Since higher ionizing energy is needed for He RRLs, they are generally from stars that are more massive than those emit H RRL only. He RRLs are usually weaker than H RRLs, they can only be detected from sources with strong H RRL intensities. In our sample, the mean peak temperature of H RRL from the sources with He RRL emissions is 374.7 mK, which is $\sim$ 8 times brighter than those without He RRL detections. The detected He RRLs have a mean peak temperature of 31.6 mK and an average value of $T_{p {\rm He}}$/$T_{p {\rm H}}$ $\sim$ 0.1, where $T_{p {\rm He}}$ and $T_{p {\rm H}}$ denote the peak temperatures of He and H RRLs. Due to sensitivity limitation, He RRLs with peak temperature lower than $\sim$ 7 mK are below our detection threshold, this results in a lower limit to the $T_{p {\rm H}}$ of 70 mK for He RRL to be detected. Figure \[f11\] shows the plot between $T_{p {\rm He}}$ and $T_{p {\rm H}}$. For sources without He RRL detections, the expected $T_{p {\rm He}}$ values are calculated by multiplying their $T_{p {\rm H}}$ by 0.1. There are 76 sources which have an H RRL peak temperature above 70 mK (black dots above the red dashed line in Figure \[f11\]) but had no He RRL detections, showing a low abundance of He$^+$. We averaged over the H-RRL-only sources with $T_{p {\rm H}}$ above and below 70 mK, as shown in Figure \[f12\]. After averaging, He RRLs can be seen with a $S_{i {\rm He}}$ to $S_{i {\rm H}}$ ratio of $\sim$ 0.01 and $\sim$ 0.02 for sources above and below 70 mK, respectively, where $S_{i {\rm He}}$ and $S_{i {\rm H}}$ denote the integrated intensities of He and H RRLs. These values are much lower than the mean $S_{i {\rm He}}$ to $S_{i {\rm H}}$ ratio of 0.06 for sources with He RRL detections. This fact may suggest that in addition to the selecting effect caused by the sensitivity limit, the non-detections of He RRLs for the H-RRL-only sources may be mostly originated from the low abundance of He$^+$. Those sources are possibly being ionized by a less massive, later OB-type star. That is, helium is under ionized with respect to hydrogen.\ ### Distance and Galactocentric Distance Distribution Figure \[f13\] shows the normalized distance distribution of the 76 H-RRL-only sources with $T_{p {\rm H}}$ $>$ 70 mK and 380 sources (excluding the weak sources, and for multi-component sources, only the strongest emission were taken into consideration.) with $T_{p {\rm H}}$ $<$ 70 mK comparing to the 49 sources with He RRL. No significant difference in the distance distributions were found for the two sub-samples in this figure. This fact may support the argument that the non-detections of He RRLs are mainly caused by the low abundance of He$^+$ rather than sensitivity limit caused by distance. Figure \[f14\] shows the normalized $R_{{\rm Gal}}$ distribution density of the 49 He RRL sources and the 76 H-RRL-only sources with $T_{p {\rm H}}$ above 70 mK. As shown in this figure, sources without He RRL emissions locate nearer to the Galactic center than those with He RRLs on average. Under the above assumption that He RRL sources are being ionized by a more massive star with respect to those without He RRL, we suggest that more massive SFRs locate at longer distances on average. This is contradicting to previous studies that due to higher gas density, there is a concentration of mass in the inner Galaxy (e.g., @green2011 [@casassus2000; @lepine2011]). A more plausible explanation for the lower abundance of ionized He is line-blanketing effect caused by higher metallicity in such regions. There is a known negative metallicity radial gradient along the Galactic disk [@HW1999], the higher metal content in the atmosphere of OB-type stars near the Galactic Center will cause line-blanketing and reduce the number of He-ionizing photons that escape the star. This may result in a lower abundance of ionized He in such regions. Nevertheless, both Figures \[f13\] and \[f14\] may suffer from statistical bias due to limited sample size.\ ### He$^+$ Abundance along the Galactic Plane\[4.3.3\] H and He RRLs with high principal quantum numbers in the radio act similarly, so their line intensity ratio y$^+$ can be used to diagnose the abundance ratio between $^4$He$^+$ and H$^+$ [@wenger2013; @balser2006]. y$^+$ is defined as the following: $${\rm y}^+ = \frac{T_{p {\rm He}^+}\Delta \nu_{{\rm He}^+}}{T_{p {\rm H}^+}\Delta \nu_{{\rm H}^+}}$$ where $T_{p {\rm He}^+}$ and $T_{p {\rm H}^+}$ are the peak line temperatures of He and H RRLs, $\Delta \nu_{{\rm He^+}}$ and $\Delta \nu_{{\rm H^+}}$ are the line widths of them, respectively. The distribution of y$^+$ values along the Galactic Plane for the sources exhibiting He RRLs is presented in Figure \[f15\]. Our sample has a mean y$^+$ value with a standard deviation of 0.062 $\pm$ 0.029, this value is similar to that of the catalog (0.068 $\pm$ 0.023) measured in [@wenger2013]. Our sample shows no significant trend on y$^+$ with $R_{{\rm Gal}}$, as presented by the black fitting line in Figure \[f15\], a small positive slope of 0.002 $\pm$ 0.015 is derived for the sources. There is a very low correlation coefficient of $\sim$ 0.09 between y$^+$ and $R_{{\rm Gal}}$, and a large standard error of $\pm$ 0.015 to the fitting line slope, both showing weak dependence of y$^+$ on $R_{{\rm Gal}}$. This is consistent with earlier studies that y$^+$ has a negative or no obvious gradient with $R_{{\rm Gal}}$ through our Galaxy (see @balser2001 for a review). A weak increasing trend was also found on y$^+$ (y$^+$ = 0.0035 $\pm$ 0.0016 kpc$^{-1}$) in [@wenger2013], however, they pointed out that this result only weakly constrains the actual y$^+$ gradient due to the large uncertainties in their data. Due to the limited sample size, our analysis may also suffer from statistical bias. There are two sources with prominently high y$^+$ values ($>$ 0.15) (G23.563+0.008 and G84.722$-$1.248; see Figure \[f15\]), as concluded by @balser2001, possible origins for the high y$^+$ value are mass loss of helium near the surface of the massive star or overestimated abundance due to the radiative transfer effect.\ ### MIR Color Distribution\[4.3.4\] Due to the emission from the dust heated by the central star, one would expect higher luminosities at longer MIR wavelength from more massive SFRs, thus there may exist a difference between the color-color distribution of H regions with different masses of their exciting stars. Figure \[f16\] shows the *WISE* \[12\] $-$ \[22\] vs. \[3.4\] $-$ \[4.6\] and \[4.6\] $-$ \[12\] vs. \[3.4\] $-$ \[4.6\] color distributions of sources with and without He RRL detections (above 70 mK). However, there is no significant separation found between the color distributions of these two sub-samples. This may suggest that *WISE* colors are insensitive to mass variation of UCH regions.\ ### Carbon RRLs\[4.3.5\] Carbon RRLs are from cooler gas in photo-dissociation regions (PDRs) or diffuse gas ionized by the interstellar UV radiation (see @alves2015 and references therein). For a target position with both H and C RRLs, if the C RRL is emitted from a cold gas with a different velocity to the H region, it can have a shifted velocity offset with respect to the H RRLs [@alves2012]. There are 23 sources showing C RRL in our sample, 3 of them are from the nearby Orion Molecular Cloud Complex (G208.894$-$19.313; G213.706$-$12.602; G213.885$-$11.832). The line properties of the 23 C RRLs are presented in Table \[t5\]. C RRLs are essential to the studies of morphology and physical properties of its host PDR (e.g., @RA2001 [@roshi2002; @wenger2013]). For example, the non-thermal component of the carbon RRL line width can be used to diagnose the magnetic field in the host PDR [@roshi2007; @balser2016]. For a thermally broadened C RRL, assuming a typical PDR temperature of $\sim$ 10$^3$ K, a line width of $\sim$ km s$^{-1}$ is expected. However, the C RRLs in our sample have an average line width of 8.9 km s$^{-1}$. The large average line width of C RRLs may indicate that there may exist non-thermal turbulence in the PDRs in our sample (@roshi2007 and @barrett1964).\ Summary\[5\] ============ \(1) Using the TMRT, we performed a Galactic RRL survey in C band toward 3348 targets, selected from the *WISE* point source catalog. Excluding 5 potential PNe and 5 potential SNR candidates, we built a sample of 517 HMSFRs traced by RRL. The peak flux densities of the detected hydrogen RRLs are in a range of $\sim$ 10 to 1900 mK. Though the majority of the sources have a line width within 15 $\sim$ 35 km s$^{-1}$, there are 82 of them show very narrow line width characteristic and 30 of them have line width larger than 35 km s$^{-1}$. Our sample further expanded the H region sample on the basis of previous surveys. \(2) Within the detected H region sample, 103 sources also harbor 6.7 GHz methanol masers emissions. Combining the sources traced by RRL and/or maser, we built up a sample of 654 HMSFRs, providing fundamental information to study the high-mass star formation evolutionary stages. According to the argument that methanol maser appears earlier than the formation of H region, our sources may be associated with HMSFR at various star-forming sequences. By comparing the physical properties of the RRL-only sources, maser-only sources and sources associated with both tracers, we found no significant difference in distance and $R_{{\rm Gal}}$ distribution of the three sub-samples. A slightly higher maser association rate was found for more luminous RRL sources. \(3) In addition to H RRL, we also detected He RRLs from 49 sources which may associate with more massive HMSFRs, no significant gradient on the He/H abundance along the galactocentric distance was found from the 49 He RRL sources. \(4) A sample of 23 C RRLs were also built in this survey, which provides a promising sample for future studies on the physical properties of PDRs surrounding H regions, the wide average line width of C RRLs may indicate non-thermal turbulence in their host PDRs.\ Acknowledgement {#acknowledgement .unnumbered} =============== We are thankful for the assistance from the operators of the TMRT during the observations, the funding and support from China Scholarship Council (CSC) (File No.201704910999), Science and Technology Facilities Council (STFC) and the University of Manchester. This work was supported by the National Natural Science Foundation of China (11590781, 11590783, 11590784 and 11873002) , and Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme (2019). [p[1.8cm]{}p[1.6cm]{}p[1.5cm]{}p[1cm]{}p[0.6cm]{}p[1.6cm]{}p[1.6cm]{}p[1.6cm]{}p[0.5cm]{}p[0.5cm]{}p[1.5cm]{}p[0.6cm]{}p[3cm]{}]{} G18.059+2.035 & 18:16:27.60 & $-$12:14:45.7 & 22/04/16 & 26 & 0.53(0.05) & 19.4(2.0) & 19.7(0.8) & 1.69 & 0.24 & & & (K) G018.426+01.922\ G18.559+2.029 & 18:17:26.97 & $-$11:48:32.0 & 15/12/17 & 49 & 1.26(0.05) & 24.4(1.1) & 32.7(0.5) & 1.77 & 0.26 & & & (K) G018.426+01.922 / IRAS 18146-1148\ G18.915+2.027 & 18:18:08.75 & $-$11:29:45.2 & 15/12/17 & 31 & 0.42(0.04) & 12.7(1.3) & 33.1(0.5) & 1.77 & 0.26 & & & (K) G018.426+01.922 / IRAS 18153-1131\ G20.495+0.157 & 18:27:54.06 & $-$10:58:37.3 & 04/07/16 & 52 & 1.10(0.04) & 19.7(0.9) & 26.1(0.4) & 14.01 & 0.31 & & & (K) G020.481+00.168\ G20.749$-$0.112 & 18:29:21.28 & $-$10:52:38.5 & 04/07/16 & 263 & 7.54(0.06) & 26.9(0.2) & 54.5(0.1) & 3.46 & 0.20 & & & (K) G020.728-00.105 / Kes 68\ G20.762$-$0.064 & 18:29:12.19 & $-$10:50:36.0 & 04/07/16 & 243 & 5.52(0.06) & 21.3(0.2) & 53.5(0.1) & 3.45 & 0.20 & & yes & (K) G020.728-00.105 / Kes 68\ G21.919$-$0.324 & 18:32:19.23 & $-$09:56:19.9 & 04/07/16 & 41 & 0.85(0.05) & 19.5(1.4) & 83.2(0.6) & 5.27 & 0.32 & & & (K) G021.884-00.318 / IRAS 18294-0959\ G22.355+0.066 & 18:31:44.05 & $-$09:22:17.3 & 15/07/16 & 39 & 0.80(0.05) & 19.1(1.3) & 84.7(0.6) & 5.41 & 0.37 & & yes & (K) G022.357+00.064\ G22.396+0.334 & 18:30:50.75 & $-$09:12:39.8 & 15/07/16 & 21 & 0.27(0.03) & 12.1(1.6) & 82.9(0.7) & 5.38 & 0.40 & & & (C) G022.424+00.337\ [lllllllll]{}\ G18.059+2.035 & G18.559+2.029 & G18.915+2.027 & G20.495+0.157 & G20.749$-$0.112 & G21.919$-$0.324 & G22.396+0.334 & G22.397+0.300 & G22.551$-$0.522\ G22.835$-$0.438 & G22.873$-$0.258 & G22.877$-$0.432 & G22.953$-$0.358 & G23.009$-$0.379 & G23.039$-$0.641 & G23.096$-$0.413 & G23.172$-$0.183 & G23.240$-$0.114\ G23.241$-$0.481 & G23.271$-$0.139 & G23.315$-$0.184 & G23.323$-$0.294 & G23.338$-$0.213 & G23.351$-$0.139 & G23.386$-$0.130 & G23.402+0.450 & G23.416$-$0.108\ G23.428$-$0.231 & G23.431$-$0.519 & G23.458+0.066 & G23.458$-$0.016 & G23.465+0.115 & G23.473$-$0.212 & G23.490$-$0.028 & G23.538$-$0.004 & G23.563+0.008\ G23.601$-$0.015 & G23.696+0.167 & G23.740+0.157 & G23.771+0.149 & G23.823+0.135 & G23.868$-$0.117 & G23.873+0.086 & G23.899$-$0.268 & G23.900+0.520\ G23.929+0.499 & G23.959+0.405 & G23.964+0.168 & G23.995$-$0.097 & G24.010+0.503 & G24.113$-$0.176 & G24.132+0.123 & G24.191$-$0.036 & G24.235$-$0.223\ G24.273$-$0.137 & G24.283$-$0.009 & G24.323+0.047 & G24.349+0.020 & G24.351$-$0.269 & G24.359+0.127 & G24.393+0.013 & G24.425+0.243 & G24.426+0.351\ G24.427+0.122 & G24.443$-$0.228 & G24.470+0.464 & G24.479$-$0.250 & G24.499+0.390 & G24.519$-$0.111 & G24.520$-$0.565 & G24.546$-$0.245 & G24.554+0.503\ G24.564$-$0.308 & G24.615+0.421 & G24.626$-$0.101 & G24.639$-$0.030 & G24.730+0.153 & G24.775+0.118 & G24.811+0.056 & G24.818$-$0.108 & G24.820+0.158\ G24.826$-$0.073 & G24.848$-$0.102 & G24.865+0.145 & G24.910+0.037 & G25.141$-$0.400 & G25.156$-$0.273 & G25.229+0.175 & G25.356+0.263 & G25.383$-$0.377\ G25.392$-$0.131 & G25.664$-$0.120 & G26.087$-$0.055 & G26.327+0.307 & G26.331+0.134 & G26.353+0.010 & G26.374+0.246 & G26.526+0.381 & G26.579$-$0.120\ G26.580+0.080 & G26.854$-$0.077 & G26.902$-$0.306 & G26.983$-$0.228 & G27.028+0.283 & G27.069+0.072 & G27.180$-$0.004 & G27.185$-$0.082 & G27.977+0.078\ G28.063$-$0.085 & G28.150+0.169 & G28.188$-$0.212 & G28.231+0.039 & G28.264$-$0.182 & G28.280$-$0.154 & G28.291+0.010 & G28.328$-$0.075 & G28.342+0.101\ G28.393+0.085 & G28.413+0.145 & G28.439+0.035 & G28.452+0.002 & G28.577$-$0.333 & G28.579+0.144 & G28.585+3.712 & G28.688$-$0.278 & G28.692+0.028\ G28.747+0.270 & G28.757+0.059 & G28.852+3.701 & G28.855$-$0.219 & G28.920$-$0.228 & G28.928+0.019 & G29.119+0.029 & G29.414+0.185 & G29.609+0.197\ G29.725+0.074 & G29.780$-$0.260 & G29.833$-$0.261 & G29.873+0.032 & G29.887$-$0.005 & G29.887$-$0.779 & G29.939$-$0.870 & G29.941$-$0.071 & G30.030$-$0.383\ G30.103$-$0.079 & G30.136$-$0.228 & G30.145$-$0.067 & G30.197+0.309 & G30.301$-$0.203 & G30.338$-$0.251 & G30.339$-$0.174 & G30.365+0.288 & G30.392+0.121\ G30.446$-$0.359 & G30.464+0.033 & G30.587$-$0.125 & G30.604+0.176 & G30.610+0.235 & G30.624$-$0.107 & G30.652$-$0.204 & G30.667$-$0.332 & G30.668+0.063\ G30.672+0.014 & G30.693+0.228 & G30.735$-$0.295 & G30.741$-$0.195 & G30.769+0.105 & G30.810+0.046 & G30.810+0.314 & G30.846$-$0.075 & G30.857+0.004\ G30.902$-$0.035 & G30.912+0.020 & G30.920+0.088 & G30.927+0.351 & G30.945+0.158 & G31.036+0.236 & G31.041$-$0.232 & G31.101+0.265 & G31.122+0.063\ G31.375+0.483 & G31.496+0.177 & G31.508$-$0.164 & G31.544$-$0.043 & G31.554$-$0.101 & G31.611+0.151 & G31.677+0.245 & G32.010$-$0.323 & G33.031+0.084\ G33.086+0.001 & G33.265+0.066 & G33.430$-$0.016 & G33.548+0.021 & G34.033$-$0.024 & G34.185+0.114 & G34.286+0.129 & G34.366$-$0.058 & G34.515+0.066\ G34.530$-$1.087 & G34.546+0.535 & G34.712$-$0.595 & G34.719$-$0.678 & G35.067$-$1.569 & G35.291+0.808 & G35.360$-$1.781 & G35.442$-$0.018 & G35.467+0.138\ G35.500$-$0.021 & G35.579$-$0.031 & G35.603$-$0.203 & G35.615$-$0.951 & G35.681$-$0.176 & G35.823$-$0.202 & G36.454$-$0.187 & G37.359$-$0.074 & G37.593$-$0.124\ G37.659+0.119 & G37.669$-$0.093 & G37.769$-$0.263 & G37.800$-$0.372 & G39.312$-$0.216 & G39.537$-$0.378 & G39.882$-$0.346 & G40.445+2.528 & G40.495+2.570\ “G40.545+2.596 & G40.592+ 2.509 & G41.355+0.406 & G43.177$-$0.008 & G43.181$-$0.056 & G43.262$-$0.045 & G44.241+0.152 & G45.124+0.136 & G45.525+0.012\ ” G45.541$-$0.016 & G48.628+0.214 & G48.632$-$0.587 & G48.652$-$0.315 & G48.655$-$0.728 & G48.742$-$0.512 & G48.888$-$0.410 & G48.923$-$0.445 & G48.946$-$0.331\ G48.961$-$0.396 & G49.025$-$0.526 & G49.028$-$0.217 & G49.072$-$0.327 & G49.224$-$0.334 & G49.268$-$0.337 & G49.341$-$0.337 & G49.368$-$0.303 & G49.391$-$0.235\ G49.406$-$0.372 & G49.461$-$0.551 & G49.958+0.126 & G50.042+0.260 & G50.094$-$0.677 & G50.817+0.242 & G51.341+0.065 & G51.371$-$0.045 & G51.383$-$0.007\ G52.234+0.759 & G52.260$-$0.521 & G52.355$-$0.588 & G52.399$-$0.936 & G52.540$-$0.927 & G52.921$-$0.621 & G53.575+0.069 & G54.110$-$0.081 & G59.474$-$0.185\ G59.582$-$0.147 & G63.153+0.442 & G75.840+0.367 & G75.841+0.425 & G76.659+1.922 & G77.973+2.236 & G78.161+1.871 & G78.231+0.905 & G78.259$-$0.017\ G78.377+1.020 & G78.405+0.609 & G78.633+0.979 & G78.641+0.672 & G78.662+0.266 & G78.670+0.184 & G78.697+1.234 & G78.728+0.946 & G78.840+0.695\ G78.873+0.754 & G78.881+1.427 & G78.901+0.661 & G79.024+2.449 & G79.027+0.436 & G79.075+3.462 & G79.128+2.278 & G79.170+0.396 & G79.207+2.146\ G79.246+0.451 & G79.312$-$0.654 & G79.330$-$0.800 & G79.362$-$0.131 & G79.385$-$1.564 & G79.393+1.782 & G79.545$-$1.057 & G79.561$-$0.766 & G79.699+1.019\ G79.843+0.890 & G79.854$-$1.495 & G79.877+2.476 & G79.886+2.552 & G79.887$-$1.481 & G79.900+1.111 & G79.998$-$1.454 & G80.820+0.405 & G80.859$-$0.083\ G80.865+0.342 & G80.865+0.420 & G80.939$-$0.127 & G81.111$-$0.145 & G81.250+1.123 & G81.252+0.982 & G81.264$-$0.136 & G81.266+0.931 & G81.337+0.824\ G81.341+0.759 & G81.435+0.704 & G81.469+0.023 & G81.512+0.030 & G81.525+0.218 & G81.548+0.095 & G81.582+0.103 & G81.601+0.291 & G81.663+0.465\ G81.683+0.541 & G81.685$-$0.040 & G81.722+0.021 & G81.840+0.917 & G81.876+0.734 & G81.898+0.809 & G81.918$-$0.010 & G82.069$-$0.309 & G82.186+0.100\ G82.278+2.209 & G82.434+1.785 & G84.586$-$1.111 & G84.638$-$1.140 & G84.649$-$1.089 & G84.707$-$0.270 & G84.708$-$1.285 & G84.716$-$0.848 & G84.722$-$1.248\ G84.724$-$1.138 & G84.753+0.253 & G84.773$-$1.046 & G84.826$-$1.137 & G84.835$-$1.187 & G84.841$-$1.085 & G84.852+3.697 & G84.854$-$0.744 & G84.856$-$0.500\ G84.870$-$1.073 & G84.929$-$1.095 & G84.941$-$1.126 & G84.941$-$1.162 & G85.019$-$1.131 & G85.021$-$0.157 & G85.081$-$0.215 & G85.082$-$1.159 & G85.112$-$1.207\ G85.171$-$1.169 & G85.481$-$1.176 & G92.670+3.072 & G94.442$-$5.478 & G107.222$-$0.893 & G108.763$-$0.948 & G110.081+0.081 & G111.526+0.803 & G111.567+0.752\ G113.603$-$0.616 & G118.038+5.108 & G123.035$-$6.355 & G123.050$-$6.310 & G126.645$-$0.786 & G133.690+1.113 & G133.716+1.207 & G133.718+1.137 & G133.750+1.198\ G133.948+1.065 & G134.004+1.144 & G134.219+0.721 & G134.239+0.639 & G134.279+0.856 & G134.469+0.431 & G136.918+1.067 & G137.585+1.351 & G138.297+1.556\ G138.327+1.570 & G149.383$-$0.361 & G150.525$-$0.930 & G169.174$-$0.921 & G173.615+2.732 & G182.339+0.249 & G206.573$-$16.362 & G208.675$-$19.191 & G208.724$-$19.192\ G208.726$-$19.232 & G208.760$-$19.216 & G208.792$-$19.243 & G208.824$-$19.256 & G208.894$-$19.313 & G209.184$-$19.494 & G213.752$-$12.616 & G213.885$-$11.832 &\ \ G20.762$-$0.064 & G22.355+0.066 & G23.010$-$0.410 & G23.185$-$0.380 & G23.271$-$0.256 & G23.389+0.185 & G23.436$-$0.184 & G23.653$-$0.143 & G23.680$-$0.189\ G23.899+0.065 & G23.965$-$0.110 & G24.313$-$0.154 & G24.328+0.144 & G24.362$-$0.146 & G24.485+0.180 & G24.528+0.337 & G24.633+0.153 & G24.790+0.084\ G24.943+0.074 & G25.177+0.211 & G25.346$-$0.189 & G25.395+0.033 & G25.709+0.044 & G26.545+0.423 & G28.147$-$0.004 & G28.287$-$0.348 & G28.320$-$0.012\ G28.609+0.017 & G28.804$-$0.023 & G28.832$-$0.250 & G28.862+0.066 & G29.320$-$0.162 & G29.835$-$0.012 & G29.927+0.054 & G30.004$-$0.265 & G30.250$-$0.232\ G30.403$-$0.297 & G30.419$-$0.232 & G30.536$-$0.004 & G30.589$-$0.043 & G30.662$-$0.139 & G30.789+0.232 & G30.807+0.080 & G30.810$-$0.050 & G30.823+0.134\ G30.866+0.114 & G30.897+0.163 & G30.959+0.086 & G30.973+0.562 & G30.980+0.216 & G31.076+0.458 & G31.159+0.058 & G31.221+0.020 & G31.237+0.067\ G31.413+0.308 & G31.579+0.076 & G32.118+0.090 & G32.798+0.190 & G32.992+0.034 & G33.092$-$0.073 & G33.143$-$0.088 & G33.393+0.010 & G33.638$-$0.035\ G34.411+0.235 & G35.141$-$0.750 & G35.194$-$1.725 & G35.398+0.025 & G35.578+0.048 & G37.479$-$0.105 & G37.602+0.428 & G38.076$-$0.266 & G38.119$-$0.229\ G38.202$-$0.068 & G38.255$-$0.200 & G38.258$-$0.074 & G41.121$-$0.107 & G42.692$-$0.129 & G43.076$-$0.078 & G43.089$-$0.011 & G43.148+0.013 & G43.178$-$0.519\ G43.890$-$0.790 & G45.454+0.060 & G48.905$-$0.261 & G48.991$-$0.299 & G49.466$-$0.408 & G50.779+0.152 & G52.199+0.723 & G53.618+0.036 & G59.498$-$0.236\ G75.770+0.344 & G78.882+0.723 & G78.969+0.541 & G79.736+0.991 & G80.862+0.383 & G81.752+0.591 & G81.871+0.779 & G84.951$-$0.691 & G84.984$-$0.529\ G97.527+3.184 & G111.532+0.759 & G173.596+2.823 & G213.706$-$12.602 & & & & &\ \ G16.872$-$2.154 & G16.883$-$2.186 & G17.021$-$2.402 & G173.482+2.446 & G173.617+2.883 & G174.205$-$0.069 & G183.349$-$0.575 & G20.234+0.085 & G20.363$-$0.014\ G20.926$-$0.050 & G21.023$-$0.063 & G21.370$-$0.226 & G213.752$-$12.615 & G22.050+0.211 & G24.148$-$0.009 & G24.634$-$0.323 & G25.256$-$0.446 & G25.410+0.105\ G25.498+0.069 & G25.613+0.226 & G25.649+1.050 & G25.837$-$0.378 & G26.421+1.686 & G26.598$-$0.024 & G26.623$-$0.259 & G26.645+0.021 & G27.220+0.261\ G27.222+0.136 & G27.287+0.154 & G27.725+0.037 & G27.784+0.057 & G27.795$-$0.277 & G28.180$-$0.093 & G28.393+0.085 & G28.843+0.494 & G29.281$-$0.330\ G29.941$-$0.070 & G30.370+0.483 & G30.770$-$0.804 & G30.788+0.203(R) & G30.819+0.273 & G30.972$-$0.141 & G31.253+0.003(L) & G31.253+0.003(R) & G32.045+0.059\ G32.773$-$0.059 & G32.828$-$0.315 & G33.229$-$0.018 & G33.322$-$0.364 & G33.425$-$0.315 & G33.641$-$0.228 & G33.726$-$0.119 & G34.096+0.018 & G34.229+0.133\ G34.757+0.025 & G34.789$-$1.392 & G34.974+0.365 & G35.149+0.809 & G35.197$-$0.729 & G35.225$-$0.360 & G35.247$-$0.237 & G35.792$-$0.174 & G36.137+0.564\ G36.634$-$0.203 & G36.705+0.096 & G36.833$-$0.031 & G36.919+0.483 & G37.043$-$0.035 & G37.430+1.517 & G37.554+0.201 & G37.763$-$0.215 & G38.598$-$0.213\ G38.933$-$0.361 & G39.100+0.491 & G39.387$-$0.141 & G40.282$-$0.220 & G40.425+0.700 & G40.597$-$0.719 & G40.622$-$0.138 & G40.964$-$0.025 & G41.307$-$0.169\ G42.035+0.191 & G43.037$-$0.453 & G43.808$-$0.080 & G45.070+0.124 & G45.360$-$0.598 & G45.493+0.126 & G45.804$-$0.356 & G49.043$-$1.079 & G49.265+0.311\ G49.537$-$0.904 & G49.599$-$0.249 & G50.034+0.581 & G51.678+0.719 & G52.663$-$1.092 & G52.922+0.414 & G53.022+0.100 & G53.141+0.071 & G53.485+0.521\ G54.371$-$0.613 & G56.963$-$0.234 & G58.775+0.647 & G59.436+0.820 & G59.634$-$0.192 & G59.785+0.068 & G59.833+0.672 & G62.310+0.114 & G69.543$-$0.973\ G71.522$-$0.385 & G73.063+1.796 & G74.098+0.110 & G75.010+0.274 & G76.093+0.158 & G78.122+3.633 & G81.794+0.911 & G82.308+0.729 & G84.193+1.439\ G85.394$-$0.023 & G89.930+1.669 & G90.921+1.487 & G94.609$-$1.790 & G98.036+1.446 & G99.070+1.200 & G108.184+5.518 & G108.758$-$0.986 & G109.839+2.134\ G109.868+2.119 & G110.196+2.476 & G111.256$-$0.770 & G121.329+0.639 & G123.035$-$6.355 & G123.050$-$6.310 & G124.015$-$0.027 & G134.029+1.072 & G136.859+1.165\ G137.068+3.002 & G149.076+0.397 & & & & & & &\ [p[1.9cm]{}p[1.7cm]{}p[1.6cm]{}p[1.2cm]{}p[0.7cm]{}p[1.7cm]{}p[1.7cm]{}p[1.7cm]{}p[0.6cm]{}p[0.6cm]{}p[0.8cm]{}p[0.6cm]{}p[3cm]{}]{} G20.749$-$0.112 & 18:29:21.28 & $-$10:52:38.5 & 04/07/16 & 20 & 0.70(0.06) & 33.4(3.5) & $-$68.4(1.3) & 3.46 & 0.20 & He & & (K) G020.728-00.105 / Kes 68\ G23.428$-$0.231 & 18:34:48.38 & $-$08:33:22.0 & 21/08/16 & 15 & 0.24(0.06) & 15.2(3.8) & $-$20.0(1.9) & 5.93 & 0.23 & He & & (K) G023.423-00.216\ G23.436$-$0.184 & 18:34:39.21 & $-$08:31:40.4 & 21/08/16 & 14 & 0.17(0.04) & 10.8(2.1) & $-$24.6(1.3) & 5.87 & 0.24 & He & yes & (K) G023.458-00.179\ G23.473$-$0.212 & 18:34:49.22 & $-$08:30:28.2 & 21/08/16 & 9 & 0.17(0.04) & 17.6(4.6) & $-$19.8(2.0) & 5.86 & 0.24 & He & & (K) G023.458-00.179\ G23.563+0.008 & 18:34:12.06 & $-$08:19:36.6 & 06/08/16 & 13 & 0.30(0.04) & 21.6(3.6) & $-$34.1(1.3) & 5.60 & 0.40 & He & & (K) G023.572-00.020\ G24.479$-$0.250 & 18:36:49.63 & $-$07:37:55.3 & 31/08/16 & 15 & 0.16(0.03) & 10.0(1.7) & $-$26.7(1.1) & 5.83 & 0.24 & He & & (K) G024.493-00.219\ G24.790+0.084 & 18:36:12.46 & $-$07:12:10.8 & 05/09/16 & 28 & 0.50(0.06) & 16.4(2.1) & $-$15.1(0.9) & 6.04 & 0.23 & He & yes & (K) G024.844+00.093\ G25.346$-$0.189 & 18:38:12.83 & $-$06:50:00.8 & 05/09/16 & 17 & 0.40(0.05) & 22.4(3.5) & $-$64.9(1.7) & 3.83 & 0.29 & He & yes & (K) G025.382-00.151 / IRAS 18354-0652\ G25.392$-$0.131 & 18:38:05.54 & $-$06:45:58.3 & 06/09/16 & 22 & 0.61(0.11) & 26.5(6.8) & $-$22.1(2.1) & 8.88 & 0.31 & He & & (K) G025.382-00.151\ & & & & 23 & 0.20(0.06) & 7.9(3.3) & $-$55.7(1.0) & & & C & &\ [p[1.9cm]{}p[1.7cm]{}p[1.6cm]{}p[1.2cm]{}p[0.7cm]{}p[1.7cm]{}p[1.7cm]{}p[1.7cm]{}p[0.6cm]{}p[0.6cm]{}p[0.8cm]{}p[3.5cm]{}]{} G30.035$-$0.002 & 18:46:09.37 & $-$02:34:44.4 & 17/09/16 & 47 & 1.32(0.07) & 26.3(1.6) & 97.1(0.6) & 6.82 & 1.31 & PNe? & PN G030.0+00.0 / (K) G030.022 / IRAS 18436-0239\ G78.911+0.792 & 20:29:08.19 & +40:15:26.5 & 16/09/16 & 18 & 0.40(0.04) & 20.8(2.9) & 10.7(1.3) & 2.72 & 0.97 & PNe? & PN Sd 1 / (K) G078.886+00.709\ G78.931+0.722 & 20:29:29.59 & +40:13:55.0 & 16/09/16 & 32 & 1.05(0.07) & 30.7(2.2) & 17.2(1.0) & 3.07 & 0.87 & PNe? & PN Sd 1 / (K) G078.886+00.709\ G84.913$-$3.505 & 21:07:00.13 & +42:13:01.8 & 14/12/17 & 35 & 2.24(0.05) & 58.7(1.5) & 25.5(0.7) & 1.28 & 0.07 & PNe & NGC 7027\ G84.946$-$3.488 & 21:07:03.27 & +42:15:12.1 & 15/12/17 & 38 & 2.23(0.09) & 55.6(2.4) & 22.4(1.0) & 1.28 & 0.07 & PNe & NGC 7027\ [p[1.9cm]{}p[1.7cm]{}p[1.6cm]{}p[1.2cm]{}p[0.7cm]{}p[1.7cm]{}p[1.7cm]{}p[1.7cm]{}p[0.6cm]{}p[0.6cm]{}p[0.8cm]{}p[3.5cm]{}]{} G30.726+0.103 & 18:47:02.74 & $-$01:54:56.2 & 18/08/16 & 75 & 1.86(0.01) & 23.5(0.4) & 117.8(0.4) & 7.19 & 0.82 & SNR? & SNR G030.3+00.7 / (K) G030.796+00.183 / IRAS 18445-0158\ & & & & 51 & 1.42(0.01) & 26.2(0.4) & 90.2(0.4) & & & &\ & & & & 36 & 0.91(0.01) & 23.7(0.4) & 42.0(0.4) & & & &\ G79.831+1.280 & 20:29:53.77 & +41:17:18.8 & 08/09/16 & 40 & 1.10(0.06) & 25.6(1.7) & -16.9(0.7) & 5.28 & 0.63 & SNR? & SNR G079.8+01.2 / (K) G080.362+01.212\ G28.532+0.129 & 18:42:56.49 & $-$03:51:21.7 & 19/09/16 & 26 & 0.83(0.06) & 30.1(2.1) & 101.0(1.0) & 8.13 & 0.31 & SNR? & MAGPIS SNR? G28.5167+0.1333 / (K) G028.581+00.145\ & & & & 19 & 1.45(0.08) & 70.2(4.6) & 26.0(2.0) & & & &\ G28.565+0.021 & 18:43:23.21 & $-$03:52:32.3 & 19/09/16 & 86 & 2.59(0.06) & 28.3(0.8) & 96.9(0.3) & 8.09 & 0.37 & SNR? & SNR G028.56+00.00 / (K) G028.607+00.019\ & & & & 19 & 0.34(0.05) & 16.4(2.5) & 39.5(1.1) & & & &\ G27.102+0.024 & 18:40:41.50 & $-$05:10:32.5 & 09/09/16 & 42 & 1.41(0.07) & 31.1(1.8) & 92.9(0.7) & 8.55 & 0.39 & SNR? & MAGPIS SNR? G27.1333+0.0333\ **Appendix A:** RRL spectra for the HMSFR sample.\ \ \ \ \ \ **Appendix B:** RRL spectra for the previously known and potential PNe sources.\ \ **Appendix C:** RRL spectra for the potential SNR sources.\ \ [^1]: a machine readable catalog accompanies this paper [^2]: https://www.iram.fr/IRAMFR/GILDAS/ [^3]: http://simbad.u-strasbg.fr/simbad/ [^4]: http://bessel.vlbi-astrometry.org/bayesian

--- author: - | Nicholas T. Jones and S.-H. Henry Tye\ Laboratory for Elementary-Particle Physics, Cornell University, Ithaca, NY 14853.\ E-mail: , title: 'An Improved Brane Anti-Brane Action from Boundary Superstring Field Theory and Multi-Vortex Solutions' --- Introduction ============ D-branes play a crucial role in string theory [@Polchinski:1998rq]. To understand the D-brane anti-D-brane ($\DD$) system involves off-shell physics. A powerful way to study it is to write down its effective space-time action from background-independent or boundary string field theory (BSFT) [@Witten:1992qy; @Shatashvili:1993kk; @Gerasimov:2000zp]. Following the work on the non-BPS D-brane effective action in open superstring theory [@Kutasov:2000aq], this program was carried out by two groups (KL [@Kraus:2000nj] and TTU [@Takayanagi:2000rz]). Here we seek to improve on their effective $\DD$ action and study its properties. The effective action in Refs[@Kraus:2000nj; @Takayanagi:2000rz] has a number of interesting properties. It includes all powers of the single derivative of the tachyon field $T$, a feature very important for time dependent, or rolling tachyon, solutions [@Sen:2002nu; @Sugimoto:2002fp]. This feature is also necessary to lead to the fact that the lower dimensional branes appear as soliton solutions in tachyon condensation. In particular, KL/TTU find a codimension-two BPS brane as a solitonic solution, with the correct brane tension and the correct RR charge [@Sen:1999mg; @Witten:1998cd]. However, that vortex solution does not have “magnetic” flux inside it, contrary to our intuition from the Abelian Higgs model. As written, the KL/TTU effective action that involves all powers of the first derivative of $T$ does not respect the $U(1)\times U(1)$ gauge symmetry of the $\DD$ system; the derivatives of $T$ do not generalize to covariant derivatives, as is necessary since the complex tachyon field $T$ is charged under the relative $U(1)$. Without the correct gauge covariant action, it is not clear whether the vortex solution, and more generally the multi-vortex solutions, should have “magnetic” flux inside them or not. We improve the $\DD$ effective action by restoring the covariance and the $U(1)\times U(1)$ gauge symmetry of the system so the tachyon field couples to one of the gauge fields as expected. This improved action is summarized in Eq.(\[action\]). Starting with this $\DD$ action we find analytic multi-vortex multi-anti-vortex solutions (all parallel with arbitrary positions and constant velocities), summarized in Eq.(\[general\_solution\]). The solution with $n$ vortices (*i.e.* $n$ parallel codimension-2 branes) and $m$ anti-vortices has total tension $\varepsilon_{p-2} = (n+m) \tau_{p-2}$ and Ramond-Ramond (RR) charge $\mu_{p-2} = (n-m) \tau_{p-2}g_s$ under the spacetime $(p-1)$-form potential. Here $\tau_{p-2}$ is the D$(p-2)$-brane tension and $g_s$ is the string coupling constant. The simplicity of the solution suggests that the $\DD$ effective action may be useful to study the brane dynamics. For $m=0$ and an appropriate choice of the magnetic flux, the solution is supersymmetric and corresponds to $n$ BPS D$(p-2)$-branes. These solutions have a curious degeneracy. Each unit of winding (*i.e.* a vortex corresponding to a D-brane) can have up to one unit of “magnetic” flux inside it. That is, both the tension and the RR charge are independent of the presence (or absence) of the “magnetic” flux. We expect this degeneracy to be lifted by the quantum corrections to the $\DD$ action and/or the corrections from the higher derivative and gauge field-strength terms. However, it is not clear exactly how the degeneracy will be lifted. One motivation to understand the $\DD$ system better is its role in cosmology. D-brane interaction in the brane world scenario provides a natural setting for an inflationary epoch in the early universe [@Dvali:1998pa; @Burgess:2001fx; @Garcia-Bellido:2001ky; @Jones:2002cv; @Herdeiro:2001zb] (see also [@Quevedo:2002xw] for a review and extensive list of references). There, the inflaton is simply the brane-brane separation while the inflaton potential comes from their interaction. The simplest such scenario involves a brane-anti-brane pair[@Dvali:2001fw; @Burgess:2001fx]. Toward the end of inflation, as the brane and the anti-brane approach each other and collide, a tachyon emerges and tachyon condensation (*i.e.* the tachyon field rolling down its potential) is expected to reheat the universe and produce solitons (even codimensional branes) that appear as cosmic strings in our universe [@Jones:2002cv]. The cosmic string density is estimated to be compatible with present day observations, but will be critically tested by cosmic microwave background radiation and gravitational wave detectors in the near future[@Sarangi:2002yt]. To study inflation and how it ends, we also construct the $(\DD)_p$ effective action when the brane is separated from the anti-brane. We find a separation dependent tachyon potential which predicts that the $\DD$ system is classically stable when the brane and anti-brane are further than $\frac{2\pi^2\ap}{2\ln2}$ apart, but can quantum mechanically decay with the tachyon tunneling through its potential. The critical separation agrees with the result known from other methods [@Banks:1995ch] aside from the factor of $2\ln2$. The paper is organized as follows. In §2, we briefly review the BSFT derivation of the $\DD$ action. Then we use Lorentz and gauge symmetry to complete the terms in the effective action. As a check, we expand it to next to leading order and show agreement with known results. In §3, we present the general multi-vortex multi-anti-vortex solutions, with zero and non-zero gauge field strengths inside the vortices. We calculate the RR charge and the total energy of these solutions and reveal the degeneracy. We discuss how this degeneracy may be lifted. In §4, we construct the effective action when the D$p$-brane and the ${ \overline{\text{D}} }p$-brane are separated. The barrier potential to tunelling is evaluated. §5 is the conclusion. Brane Anti-Brane Effective Actions ================================== Linear Tachyon Action from BSFT ------------------------------- We summarize the brane anti-brane effective action from BSFT calculated by KL and TTU [@Kraus:2000nj; @Takayanagi:2000rz]. We restrict attention to D9-branes in type IIB theory, and generalize using T-Duality later. BSFT essentially extends the sigma-model approach to string theory[@Tseytlin:1989rr], in that (under certain conditions [@Witten:1992qy; @Gerasimov:2000zp]) the disc world-sheet partition function with appropriate boundary insertions gives the classical spacetime action. This framework for the bosonic BSFT was extended to the open superstring in [@Kutasov:2000aq] and formally justified in [@Marino:2001qc]. In the NS sector the spacetime action is $$\begin{aligned} \label{definition_S} S_{\text{spacetime}} &= -\int \mathcal DX\mathcal D\psi \mathcal D\tilde\psi\;e^{-S_\Sigma-S_{\partial\Sigma}}.\end{aligned}$$ where $\Sigma$ is the worldsheet disc and $\partial\Sigma$ is its boundary. The worldsheet bulk disc action is the usual one $$\begin{aligned} S_\Sigma &= \frac1{2\pi\ap}\int d^2z\; \partial X^\mu{ \overline{\partial} } X_\mu + \frac1{4\pi}\int d^2z\left(\psi^\mu{ \overline{\partial} }\psi_\mu + \tilde\psi^\mu\partial\tilde\psi_\mu\right)\\ &= \hf\sum_{n=1}^\infty nX_{-n}^\mu X_{n\;\mu} + i\sum_{r=\hf}^\infty\psi_{-r}^\mu\psi_{r\;\mu},\end{aligned}$$ after expanding the fields in the standard modes. To reproduce the Dirac-Born-Infeld (DBI) action for a single brane, the appropriate boundary insertion is the boundary pullback of the $U(1)$ gauge superfield to which the open string ends couple; for the $N$ brane $M$ anti-brane system, the string ends couple to the superconnection [@Quillen; @Witten:1998cd], hence the boundary insertion should be $$\begin{aligned} \label{boundary_insertion} e^{-S_{\partial\Sigma}} &= \Tr\hat P\exp\left[\int d\tau d\theta \mathcal M({\mathbf{X}})\right],& \mathcal M({\mathbf{X}}) &= \left(\begin{array}{cc} iA^1_\mu({\mathbf{X}})D{\mathbf{X}}^\mu&\sqrt\ap T^\dagger({\mathbf{X}})\\ \sqrt\ap T({\mathbf{X}})&iA^2_\mu({\mathbf{X}})D{\mathbf{X}}^\mu \end{array}\right)\end{aligned}$$ where the insertion must be supersymmetrically path ordered to preserve supersymmetry and gauge invariance. $A^{1,2}$ are the $U(N)$ and $U(M)$ connections, and $T$ is the tachyon matrix transforming in the $(N,{ \overline{M} })$ of $U(N)\times U(M)$. The lowest component of $\mathcal M$ is proportional to the superconnection. To proceed, it is simplest to perform the path-ordered trace by introducing boundary fermion superfields [@Marcus:1987cm]; we refer the reader to [@Kraus:2000nj] for details. The insertion (\[boundary\_insertion\]) can then be simplified to be $$\begin{aligned} \label{NM_insertion} \Tr P\exp\left[i\ap\int d\tau \left(\begin{array}{cc} F^1_{\mu\nu}\psi^\mu\psi^\nu+iT^\dagger T + \frac1\ap A^1_\mu\dot X^\mu& -iD_\mu T^\dagger\psi^\mu\\ -iD_\mu T\psi^\mu& F^2_{\mu\nu}\psi^\mu\psi^\nu+iTT^\dagger + \frac1\ap A^2_\mu\dot X^\mu \end{array}\right)\right],\end{aligned}$$ where the tachyon covariant derivatives are $$\begin{aligned} D_{\mu}T = \partial_{\mu} T + iA^1_{\mu}T-iTA^2_{\mu}.\label{T_cderiv}\end{aligned}$$ This expression reproduces the expected results when the tachyon and its derivatives vanish: the only open string excitations will be the gauge fields on the branes and the anti-branes, for each of which the action is the standard DBI action. For instance, with $N=M=1$, $DT=T=0$, the partition function (\[definition\_S\]) with the insertion (\[NM\_insertion\]) leads to $$\begin{aligned} \label{F_only} S_{\DD} &= -\tau_9\int d^{10}x\left[\sqrt{-\det(g+2\pi\ap F^1)} +\sqrt{-\det(g+2\pi\ap F^2)}\right].\end{aligned}$$ The measure in (\[definition\_S\]) was defined to reproduce the correct tension for the D9-branes, $\tau_9 = 1/[(2\pi)^9 g_s \ap^5]$[^1]. Unfortunately (\[NM\_insertion\]) cannot in general be simplified, but for a single brane anti-brane pair, $N=M=1$, demanding that the gauge field to which the tachyon couples vanishes, $A^-\equiv A^1-A^2=0$, the path-ordered trace can be performed using worldsheet boundary fermions. Writing $A^+=A^1+A^2$, we have [@Kraus:2000nj] $$\begin{aligned} \label{1DD_insertion} S_{\partial\Sigma} &= -\int d\tau\left[\ap T{ \overline{T} } + \ap^2(\psi^\mu\partial_\mu T)\frac1{\partial_\tau} (\psi^\nu\partial_\nu{ \overline{T} }) + \frac i2\left(\dot X^\mu A^+_\mu +\hf\ap F^+_{\mu\nu}\psi^\mu\psi^\nu\right)\right].\end{aligned}$$ The operator $1/\partial_\tau$ acting on a function $f(\tau)$ is defined to be the convolution of $f$ with sgn$(\tau)$ over the worldsheet boundary. For linear tachyon profiles, gauge and spacetime rotations allow us to write $T=u_1X^1+iu_2X^2$, and (\[definition\_S\]) can be calculated, since the functional integrals are all Gaussian. The result when $A^+=0$ is derived in [@Kraus:2000nj; @Takayanagi:2000rz]: $$\begin{aligned} \label{noncov_action} S_{\DD} = -2\tau_9\int d^{10}X_0\;\exp&\left[ -2\pi\ap[(u_1X_0^1)^2+(u_2X_0^2)^2]\right]\operatorname{\EuScript{F}}(4\pi\ap^2 u_1^2) \operatorname{\EuScript{F}}(4\pi\ap^2 u_2^2).\end{aligned}$$ where the function $\operatorname{\EuScript{F}}(x)$ is given by[@Kutasov:2000aq] $$\begin{aligned} \label{F_definition} \operatorname{\EuScript{F}}(x) = \frac{4^xx\Gamma(x)^2}{2\Gamma(2x)} = \frac{\sqrt{\pi}\Gamma(1+x)}{\Gamma(\hf+x)}.\end{aligned}$$ Note that $\operatorname{\EuScript{F}}(x)=0$ at $x=-1/2$, and $$\begin{aligned} \operatorname{\EuScript{F}}(x) = \begin{cases} 1 + (2\ln2)x + \left[{2(\ln2)^2-\frac{\pi^2}6}\right]x^2 +\mathcal O(x^3),& 0<x\ll1,\\\label{F_series} \sqrt{\pi x}\left[1 + \frac1{8x} + \mathcal O(\frac1{x^2})\right],& x\gg1, \\ -1/(1+x), & x \rightarrow -1. \end{cases}\end{aligned}$$ This action exhibits all the intricate properties of the $\DD$ system expected from Sen’s conjectures: the tachyon potential at its minima $T\to\infty$ completely cancels the brane tensions; even codimension solitons can appear on the D9-brane worldvolume, with exactly the correct tension to be lower dimensional D-branes; odd codimension solitons on which tachyonic fields reside can appear, with exactly the tension of the unstable non-BPS branes of type II string theories [@Sen:1999mg]. BSFT can also give the analogue of the D-brane Chern-Simons action for the $\DD$ system, defined similarly to (\[definition\_S\]), but with all fermions in the Ramond sector. The bulk contribution to the partition sum can be written as the wave-functional [@Kraus:2000nj; @Takayanagi:2000rz] $$\begin{aligned} \Psi^{RR}_{\text{bulk}} &\propto\exp\left[ -\hf\sum_{n=1}^\infty nX_{-n}^\mu X_{n\;\mu} - i\sum_{n=1}^\infty\psi_{-n}^\mu\psi_{n\;\mu}\right]C,\\ C&=\sum_{\text{odd }p}\frac{(-i)^{\frac{9-p}2}}{(p+1)!} C_{\mu_0\cdots\mu_p}\psi_0^{\mu_0}\cdots\psi_0^{\mu_p}.\end{aligned}$$ The $\psi_0^\mu$ are the zero modes of the Ramond sector fermions, and $C_{\mu_0\cdots\mu_p}$ are the even RR forms of IIB string theory. The normalization of $\Psi$ can be set later by demanding that the correct brane charge is reproduced. The Chern-Simons action is then defined by $$\begin{aligned} S_{\text{CS}} &= \int\mathcal DX\mathcal D\psi\;\Psi^{RR}_{\text{bulk}} \Str P e^{-S_{\partial\Sigma}},\end{aligned}$$ in which the trace given by $$\begin{aligned} \Str O \equiv \text{Tr}\;\left[\left( \begin{array}{cc}\mathbbm{1}_{N\times N}&0\\ 0&-\mathbbm{1}_{M\times M}\end{array}\right)O\right]\end{aligned}$$ results from the periodicity of the worldsheet fermion superfield which was necessary to implement to the supersymmetric path ordering. Again $e^{-S_{\partial\Sigma}}$ can be written as (\[NM\_insertion\]), with Ramond sector fermions. This expression can be viewed as a one dimensional supersymmetric partition function on $\mathbbm S^1$, and because the Ramond sector fermions are periodic, this is equivalent to $\Tr (-1)^Fe^{-\beta H}$. By Witten’s argument [@Witten:1982df], only the zero modes contribute to the partition sum, giving [@Kennedy:1999nn; @Kraus:2000nj; @Takayanagi:2000rz] $$\begin{aligned} \label{general_CS} S_{\text{CS}} &= \tau_9g_s\int C\wedge\Str e^{2\pi\ap i\mathcal F},\\ \mathcal F &= \left(\begin{array}{cc} F^1 + iT^\dagger T & -i(DT)^\dagger\\ -iD T & F^2 + iTT^\dagger \end{array}\right)\nonumber\end{aligned}$$ $\mathcal F$ is the curvature of the superconnection, and as usual, the fermion zero modes form the basis for the dual vector space and all forms above are written with $\psi_0^\mu \to dx^\mu$. This expression is exact[^2] and although it was derived for $2^{m-1}$ brane anti-brane pairs in [@Kraus:2000nj; @Takayanagi:2000rz] it appears to have the correct properties for the general $N$ brane $M$ anti-brane case. As for the action (\[noncov\_action\]), this result affirms Sen’s conjectures in that it exhibits appropriate coupling to the RR 10-form potential, and the even codimension solitons have the correct couplings to the relevant RR forms to be identified as lower dimensional branes. An Improved $\DD$ Action ------------------------ As written, the action (\[noncov\_action\]) for a single brane anti-brane pair does not manifest the necessary gauge covariance, and this form of the action is valid only for linear tachyon profiles[^3]. We now generalize the pure tachyon action of KL and TTU. Note that there are precisely two independent Lorentz and $U(1)$ invariant expressions in terms of first derivatives of the complex tachyon $T$ [@Takayanagi:2000rz; @Hashimoto:2002xe], $$\begin{aligned} \X &\equiv 2\pi\ap^2g^{\mu\nu}\partial_\mu T\partial_\nu{ \overline{T} },& \Y &\equiv\left(2\pi\ap^2\right)^2 \Big(g^{\mu\nu}\partial_\mu T\partial_\nu T\Big) \Big(g^{\alpha\beta}\partial_\alpha{ \overline{T} }\partial_\beta{ \overline{T} }\Big),\end{aligned}$$ (with the normalizations chosen for convenience). For the linear profile $T=u_1x_1+iu_2x_2$, the only translation invariant way to reexpress $u_{1,2}$ is as $u_{1,2} = \partial_{1,2} T^{1,2}$; then with $g^{\mu\nu}=\eta^{\mu\nu}$ we can calculate $\X$ and $\Y$, $$\begin{aligned} \X&=2\pi\ap^2(u_1^2 +u_2^2),\\ \Y&=\left(2\pi\ap^2\right)^2(u_1^2-u_2^2)^2,\end{aligned}$$ so the arguments of $\operatorname{\EuScript{F}}$ in (\[noncov\_action\]) can be written as $$\begin{aligned} 4\pi\ap^2u_1^2 &= \X+\sqrt{\Y},\\ 4\pi\ap^2u_2^2 &= \X-\sqrt{\Y}.\end{aligned}$$ This provides a unique way to covariantize (\[noncov\_action\]) as $$\begin{aligned} \label{no_gauge_action} S_{\DD} = -2\tau_9\int d^{10}x\sqrt{-g}\;&e^{-2\pi\ap T{ \overline{T} }} \operatorname{\EuScript{F}}(\X+\sqrt \Y)\operatorname{\EuScript{F}}(\X-\sqrt \Y),\end{aligned}$$ which reduces to (\[noncov\_action\]) when $T$ is linear in two spacetime coordinates. We shall see in §\[sol\_no\_gauge\] that restoring the spherical and gauge symmetry in this expression allows us to construct multiple codimension-2 BPS solitons as expected from the K-theory arguments [@Witten:1998cd]. Further, we can restore the $A^+$ dependence of the action, since (\[1DD\_insertion\]) remains quadratic when $A^+\ne0$ if $F^+$ is constant and the partition function (\[definition\_S\]) will be Gaussian. A similar calculation was performed for the non-BPS brane action [@Andreev:2000yn], and borrowing that result gives the extended tachyon and gauge field action $$\begin{aligned} \label{A+T_action} S_{\DD} &= -2\tau_9\int d^{10}x\;e^{-2\pi\ap T{ \overline{T} }} \sqrt{-\G}\;\operatorname{\EuScript{F}}(\X+\sqrt \Y)\operatorname{\EuScript{F}}(\X-\sqrt \Y),\end{aligned}$$ where now $\G_{\mu\nu}=g_{\mu\nu}+\pi\ap F^+_{\mu\nu}$ forms the effective metric for the tachyon, as is usual for open string states in the presence of a gauge connection [@Seiberg:1999vs]: $$\begin{aligned} \X &\equiv 2\pi\ap^2\G^{\{\mu\nu\}}\partial_\mu T\partial_\nu{ \overline{T} }& \Y &\equiv\left|2\pi\ap^2\G^{\mu\nu}\partial_\mu T\partial_\nu T\right|^2.\end{aligned}$$ Indices are raised and lowered with respect to $\G$: $\G^{\mu\nu}\G_{\nu\alpha} = \delta^\mu_\alpha$, and $\G^{\{\mu\nu\}}$ indicates the symmetric part of $\G$; this symmetrization is necessary to obtain a real action. [^4] This coupling to $F^+$ can be confirmed considering that the $\DD$ system reduces to the non-BPS brane system under the spacetime IIA $\leftrightarrow$ IIB quotient $(-1)^{F_L}$ [@Sen:1999mg], which in this system is applied by setting ${ \overline{T} }=T$, $F^1=F^2$: $$\begin{aligned} S_{\DD} \xrightarrow[A^1=A^2]{{ \overline{T} }=T} -2\tau_9\int d^{10}x\; e^{-2\pi\ap T^2}\sqrt{-\G}\;\operatorname{\EuScript{F}}(2\X)\operatorname{\EuScript{F}}(0) = \sqrt2 S_{\text{nBPS}}.\end{aligned}$$ The overall normalization of the action must be divided by $\sqrt2$ to compensate for the extra boundary fermion in the $\DD$ system which was integrated over, which is superfluous in the non-BPS brane system. The action (\[A+T\_action\]) is still incomplete in that $A^-=A^1-A^2$, the $U(1)$ connection to which the tachyon couples, was set to zero in its derivation. We can conjecture the extension to $A^-\ne0$ based on the following information: - Gauge covariance demands that all tachyon derivatives must be replaced by covariant derivatives. $A^-$ cannot appear outside a covariant derivative, so (\[A+T\_action\]) with $\partial T \to DT$ can only suffer corrections for non-constant $A^-$ (and of course, the higher $T$ and $A^+$ derivative corrections). - (\[F\_only\]) should be reproduced for $T=DT=0$. - We expect the gauge connections to appear in the matrix form $$\begin{aligned} \left(\begin{array}{cc}\hf F^+&0\\0&\hf F^+\end{array}\right) \to \left(\begin{array}{cc}F^1&0\\0&F^2\end{array}\right), \end{aligned}$$ when we restore $F^- = F^1 - F^2 \ne 0$. We can insert this into the action (\[A+T\_action\]) and trace over the $U(2)$ indices. This leads us to the next improvement to (\[A+T\_action\]), $$\begin{aligned} S_{\DD} &= -\tau_9\int d^{10}x\;e^{-2\pi\ap T{ \overline{T} }} \Bigg[\begin{array}{l} \sqrt{-\det[\G_1]}\operatorname{\EuScript{F}}(\X_1+\sqrt \Y_1)\operatorname{\EuScript{F}}(\X_1-\sqrt \Y_1)\\ +\sqrt{-\det[\G_2]}\operatorname{\EuScript{F}}(\X_2+\sqrt \Y_2)\operatorname{\EuScript{F}}(\X_2-\sqrt \Y_2) \end{array}\Bigg],\label{action}\\\nonumber (\G_{\mu\nu})_{1,2} &\equiv \left(g_{\mu\nu} + 2\pi\ap F^{1,2}_{\mu\nu}\right), \quad\quad\quad \begin{array}{rl} \X_{1,2} &\equiv 2\pi\ap^2\G^{\{\mu\nu\}}_{1,2} D_\mu TD_\nu{ \overline{T} },\\ \Y_{1,2} &\equiv\left|2\pi\ap^2\G^{\mu\nu}_{1,2} D_\mu TD_\nu T\right|^2, \end{array}\end{aligned}$$ The tachyon is charged only under $A^-$: $$\begin{aligned} &D_{\mu}T =\partial_{\mu}T+iA^-_{\mu}T, &A^\pm_\mu = A^1_\mu \pm A^2_\mu,\end{aligned}$$ and the function $\operatorname{\EuScript{F}}(x)$ is defined in (\[F\_definition\]). This is the effective action which shall be studied in this work. Corrections to this action will include higher derivative terms in $T$ and $F^{\pm}$. Possible terms like $(F^{-})^{n}T{ \overline{T} }$ may be included in higher tachyon derivatives since $[D_\mu,D_\nu]=iF_{\mu\nu}$. Being non-supersymmetric, there will be quantum corrections to the action as well. In the $\ap$ expansion, using (\[F\_series\]), we have $$\begin{aligned} \operatorname{\EuScript{F}}(\X+\sqrt \Y)\operatorname{\EuScript{F}}(\X-\sqrt \Y) = 1 + 4 (\ln 2) \X + \left[8 (\ln 2)^2 - \frac{\pi^2}{3}\right] \X^2 - \frac{\pi^2}{3} \Y + \ldots\end{aligned}$$ which agrees with the terms that have four powers of the single derivative of $T$ calculated in TTU. This provides a non-trivial check on the above improved action. Note that the action is invariant under $\sqrt{\Y} \rightarrow -\sqrt{\Y}$, so in the above Taylor expansion, only integer powers of $\Y$ appears in the action. For time-dependent tachyon fields, $T \rightarrow t/(\sqrt{2\pi}\ap)$, we have $\X-\sqrt \Y \rightarrow 0$ and $\X+\sqrt \Y = -\dot T^2 \rightarrow -1$, so $\operatorname{\EuScript{F}}(\X+\sqrt \Y)\operatorname{\EuScript{F}}(\X-\sqrt \Y) \rightarrow -1/(1 - \dot T^2)$. This justifies the approximation used for the rolling tachyon in Ref. [@Shiu:2002xp]. Solitons on Brane Anti-Brane Systems ==================================== Here we shall study the solitonic solutions of the improved $\DD$ effective action (\[action\]). Since $T$ is complex, the solitonic solutions will be vortices, corresponding to D7-branes with brane tension $\tau_7$. Let $z = x^1 + i x^2$ be the coordinate in the complex plane transverse to the D7-branes. The KL/TTU solution for a single vortex is given by $T = \lim_{u\to\infty}uz$, $A^\pm = 0$. We shall discuss solutions for parallel vortices and anti-vortices. The $n$ vortices are located at $\{z_i\}_{i=1,\ldots n}$ while the $m$ anti-vortices are located at $\{z_j^{\prime}\}_{j=1,\ldots m}$. Since $T$ is uncharged under $A^+ = A^1+A^2$ only solutions with $A^+=0$ are studied. We shall consider an ansatz where the energy density $\varepsilon_7 = (n+m) \tau_7$ while the total RR charge is $\mu_7=(n-m)\tau_7g_s$. We find that there are such solutions with and without an $A^-$ “magnetic” flux associated with each winding number. In §3.1, we show that the RR charge is independent of the gauge field, or magnetic flux. It is a function of the winding (minus the anti-winding) number only. In §3.2, we calculate the energy density $\varepsilon_7$ for vortices with and without magnetic flux. The general solution (\[general\_solution\]) can be found at the end of this subsection. To understand better the properties of the solutions, we consider the multi-vortex case more closely in §3.3. For an appropriate choice of magnetic flux, the multi-vortex solution is supersymmetric, though the degeneracy still persists. We note that the solution for multiple D7-branes without gauge flux we find was first studied by worldsheet methods in [@Hori:2000ic], and the tensions for multi-kink solitons on non-BPS brane worldvolumes were calculated in [@Hashimoto:2001rk]. Ramond-Ramond Charge of Multi-Soliton Solutions {#sol_no_gauge} ----------------------------------------------- We can gain more insight into the form of the solution giving multi-soliton branes by looking at the Chern-Simons action (\[general\_CS\]), which is known exactly. Multi-soliton solutions can be constructed with trivial gauge fields, just as in the single soliton case. In fact, we show that for soliton solutions, the RR charge is completely independent of the gauge field to which the tachyon couples. Starting with Eq. (\[general\_CS\]), for D7-brane solitons on a single brane anti-brane pair, we consider only nonzero RR field $C_8$ , and set to zero the gauge field under which $T$ is inert, $A^+=0$, $F^+=0$: $$\begin{aligned} \label{C8_ne_0} S_{\text{CS}} &= \tau_9g_s\int e^{-2\pi\ap T{ \overline{T} }}(-iC_8)\wedge\left[ 2\pi\ap iF^--(2\pi\ap)^2DT\wedge D{ \overline{T} }\right].\end{aligned}$$ The coupling to the field strength, $F^-$, is the standard one, giving the unstable 9-brane system coupling to 7-branes. The second term gives the soliton coupling, and the system can decay to solitons with trivial gauge fields. For brevity, we can extract the RR charge $\mu_7$ of the soliton under a $C_8$ which is constant in the plane in which $T$ condenses $$\begin{aligned} \label{RR_charge} &\mu_7 = -i\frac{\tau_7g_s}{2\pi}\int\limits_{\mathbbm{R}^2} e^{-2\pi\ap T{ \overline{T} }}\left[ iF^--(2\pi\ap)DT\wedge D{ \overline{T} }\right], &\tau_7=4\pi^2\ap\tau_9.\end{aligned}$$ The single D7-brane solution [@Kraus:2000nj; @Takayanagi:2000rz], can be written in polar coordinates on $\mathbbm{R}^2$ as $A^\pm=0$, $F^{1,2}=0$, $T = uz = ure^{i\theta}$: $$\begin{aligned} \mu_7 &= i\frac{\tau_7g_s}{2\pi} \int e^{-2\pi\ap u^2r^2}(2\pi\ap)u^2(-2ir) dr\wedge d\theta\\ & = \tau_7g_s.\end{aligned}$$ Unlike in the kinetic term, here $u$ can take any real value without altering the RR charge of the soliton. We can construct multi-centered soliton solutions, and in general $\mu_7$ is independent of the gauge field winding about them. To prove this, we require only that: - $T=0$ at the center of each soliton, and the tachyon fields winds about each of these centers. - $T\to\infty$ far from the solitons, so that there the tachyon potential and hence the D9-brane anti-brane energy density vanishes. Away from the solitons, the D9-brane and anti-brane have annihilated and the ground state is indistinguishable from the closed string vacuum. - $A^-$ can wind only about the soliton centers, in analogy to vortices in the Abelian Higgs model. We note first that the terms in (\[RR\_charge\]) can be rewritten using $$\begin{aligned} d\left(e^{-2\pi\ap T{ \overline{T} }} A^- \right) &= e^{-2\pi\ap T{ \overline{T} }}\left[F^- + 2\pi\ap A^- \wedge \left({ \overline{T} }dT + Td{ \overline{T} }\right)\right],\\ d\left(\hf e^{-2\pi\ap T{ \overline{T} }}\left[\frac{dT}{T} - \frac{d{ \overline{T} }}{{ \overline{T} }}\right]\right) &= e^{-2\pi\ap T{ \overline{T} }}\left(2\pi\ap dT\wedge d{ \overline{T} } + \hf d\left[\frac{dT}{T} - \frac{d{ \overline{T} }}{{ \overline{T} }}\right]\right).\end{aligned}$$ The final term is naïvely zero, but receives contributions from the poles in $dT/T-d{ \overline{T} }/{ \overline{T} }$ which result from the zeros or singularities of $T$; if $T$ is just a polynomial in $z$ and ${ \overline{z} }$, then this term is just $2\pi\delta^{(2)}(T,{ \overline{T} })$, and each zero (soliton) contributes equally to this $\delta$-function. The integrand of (\[RR\_charge\]) is then $$\begin{aligned} \nonumber \mu_7 &= i\frac{\tau_7g_s}{2\pi} \int\limits_{\mathbbm{C}} \left\{d\left(\hf e^{-2\pi\ap T{ \overline{T} }} \left[\frac{DT}{T}-\frac{D{ \overline{T} }}{{ \overline{T} }}\right]\right) - \hf e^{-2\pi\ap T{ \overline{T} }} d\left[\frac{dT}{T} - \frac{d{ \overline{T} }}{{ \overline{T} }}\right]\right\},\\ &= -i\frac{\tau_7g_s}{2\pi}\int\limits_{\mathbbm{C}} \hf e^{-2\pi\ap T{ \overline{T} }}d\left[\frac{dT}{T} - \frac{d{ \overline{T} }}{{ \overline{T} }}\right].\label{CS_exterior}\end{aligned}$$ The first term in the first line is the total derivative of a one-form which vanishes at the boundary at infinity, hence its integral vanishes. The remaining term (\[CS\_exterior\]) is independent of the gauge field and essentially counts the zeros and poles of $T$. Although this expression might not appear to be gauge invariant, the number of zeros and singularities of $T$ is a manifestly gauge invariant quantity. Hence $\mu_7$ is not only gauge invariant, but completely independent of $A^-$ and its curvature. As an example, if we construct a solution with $n$ holomorphic and $m$ anti-holomorphic zeros $$\begin{aligned} T = u\prod_{i=1}^n(z-z_i) \prod_{j=1}^m ({ \overline{z} }-{ \overline{z} }_j^\prime),\end{aligned}$$ then the total RR D7-brane charge is $$\begin{aligned} \mu_7 &= \tau_7g_s\int\limits_{\mathbbm C} e^{-2\pi\ap T{ \overline{T} }}\delta^{(2)}(T,{ \overline{T} })dT\wedge d{ \overline{T} } = (n-m)\tau_7g_s.\label{CS_D_Dbar}\end{aligned}$$ Physically, every soliton contributes one topological unit to the total RR charge of the solution as we expect. More complicated solutions include tachyon fields which are multiply wound about their zeros, $$\begin{aligned} \nonumber T &= u\prod_{i=1}^N \left(\frac{z-z_i}{{ \overline{z} }-{ \overline{z} }_i}\right)^{w_i/2} (z-z_i)^{l_i/2}({ \overline{z} }-{ \overline{z} }_i)^{l_i/2}, \tag{$w_i\in\mathbbm Z$, $l_i\in\mathbbm R^+$}\\ \mu_7 &= -i\frac{\tau_7g_s}{2\pi}\int\limits_{\mathbbm C} e^{-2\pi\ap T{ \overline{T} }}\hf d\left[\sum_{i=1}^Nw_i\left( \frac{dz}{z-z_i}-\frac{d{ \overline{z} }}{{ \overline{z} }-{ \overline{z} }_i}\right)\right] = \tau_7g_s\sum_{i=1}^Nw_i.\label{mult_wind}\end{aligned}$$ As we shall see, for this solution to be BPS it must be accompanied by a gauge field which winds about each $\{z_i\}$; one can explicitly check that the solution for $A^-$ does not provide any contribution to $\mu_7$. These calculations reveal that (\[CS\_exterior\]) behaves in an intuitive manner; holomorphic or “positively wound” zeros of $T$ correspond to D7-branes, and contribute one topological unit to $\mu_7$, whereas anti-holomorphic or “negatively wound” zeros of $T$ represent anti-D7-branes, and contribute oppositely to the RR charge, the sign arising from the antisymmetry of the volume element. As for the single soliton case, it is not necessary to take $u\to\infty$ to get the exact answer; this is not so when we consider the $\DD$ action. Multi-Soliton Tensions ---------------------- We now turn to the tension or energy density of the solitons, beginning with (\[action\]) and setting $F^+$ to zero. To obtain the lowest energy solution it is necessary to take $u$, the overall multiplying constant in the ansatz for $T$, to $\infty$. On the worldsheet, this limit corresponds to the infrared conformal limit, or equivalently to on-shell physics. In the effective theory, the limit allows the tension to be calculated exactly, and since we are searching for solutions representing classical D-branes, which have zero width, the regions in the plane at which we require that $V(T{ \overline{T} }) = 1$ must be points with all other regions having $V(T{ \overline{T} })=0$. Since $V(T{ \overline{T} })=\exp[-2\pi\ap T{ \overline{T} }]$, the potential will be maximal at the zeros of $T$, and shall vanish elsewhere when $u\to\infty$. We seek to calculate the tension of various solitons; the energy per 7-volume of the action (\[action\]) is $$\begin{aligned} \nonumber \varepsilon_7 &\equiv \int d^2x \frac2{\sqrt{-g}} \frac{\delta S_{\DD}}{\delta g^{0\nu}}g^{\nu0}\\ &= \tau_9\int d^2x\;e^{-2\pi\ap T{ \overline{T} }} \sqrt{\det(\delta+\pi\ap F^-)}\left[\begin{array}{l} \operatorname{\EuScript{F}}(\X_++\sqrt \Y_+)\operatorname{\EuScript{F}}(\X_+-\sqrt \Y_+)\\ +\operatorname{\EuScript{F}}(\X_-+\sqrt \Y_-)\operatorname{\EuScript{F}}(\X_--\sqrt \Y_-) \end{array}\right],\label{e7}\end{aligned}$$ when we assume $F^+=0$, all fields are time independent, and work with flat spacetime. $\X_\pm,\Y_\pm$ are the tachyon derivative terms containing the open string metrics $\G = g\pm\pi\ap F^-$ respectively. When $F^-$ has only a $B$-field component in the plane of tachyon condensation, (\[F\_series\]) and some simple manipulations yield $$\begin{aligned} \lim_{DT\to\infty}&\operatorname{\EuScript{F}}(\X+\sqrt \Y)\operatorname{\EuScript{F}}(\X-\sqrt \Y) = \pi\sqrt{\X^2-\Y},\\ &= 2\pi^2\ap^2\sqrt{ \left(D_zTD_{{ \overline{z} }}{ \overline{T} }-D_{{ \overline{z} }}TD_z{ \overline{T} }\right) \left((\G^{\{z{ \overline{z} }\}})^2D_zTD_{{ \overline{z} }}{ \overline{T} } -(\G^{\{{ \overline{z} }z\}})^2D_{{ \overline{z} }}TD_z{ \overline{T} }\right)}.\end{aligned}$$ Until this point, few conditions needed to be placed on the form of the solutions. Now there are two simplifying constraints we can impose; $A^-=0$ motivated from the fact that it seems to be possible to construct sensible soliton solutions without gauge field winding, in direct contrast to the solitons of standard field theory. Secondly, the condition $D_{{ \overline{z} }}T=0$ was found in [@Hori:2000ic] by worldsheet methods to be the condition which must be satisfied if $\mathcal N=2$ worldsheet supersymmetry and hence spacetime supersymmetry is to be preserved; configurations of multiple parallel branes are mutually BPS and must preserve some spacetime supersymmetry. We begin by considering examples that satisfy these conditions and proceed to other cases. The first example is one satisfying both conditions; assume $T$ is a holomorphic function with $n$ zeros at the points $\{z_j\}$, $T=\lim_{u\to\infty} u\prod_{j=1}^n (z-z_j)$. Then $T$ represents $n$ separated D7-branes, although the result is identical when some D7-brane locations coincide. The gauge field is trivial, hence $\G^{z{ \overline{z} }} = 2$ and the tension (\[e7\]) becomes (after taking $u\to\infty$) $$\begin{aligned} \varepsilon_7 &= 2\tau_9\int e^{-2\pi\ap T{ \overline{T} }}4\pi^2\ap^2\; |\partial_zT\partial_{{ \overline{z} }}{ \overline{T} }|\;\frac i2dz\wedge d{ \overline{z} },\\ &= i\tau_9\int e^{-2\pi\ap T{ \overline{T} }}(2\pi\ap)^2dT\wedge d{ \overline{T} }.\end{aligned}$$ Since $\partial_zT\partial_{{ \overline{z} }}{ \overline{T} }$ is always positive, the absolute value could be ignored. This is identical (up to the factor of $g_s$) to the D7-brane charge under the RR 8-form field (\[C8\_ne\_0\]); it was shown above that this is always equal to $n\tau_7g_s$, hence the multi-solitons have the correct tension ($\varepsilon_7=n\tau_7$) and exhibit BPS properties. Further, we can formulate solutions with vortices moving at constant velocities, $z_j = z_{j,0}+v_j t$. Of course, the second line of (\[e7\]) is no longer valid when the solution has time dependence, so the first line must be used to give the general form when $T$ is time dependent. The velocity dependence leads to the special relativistic $\gamma$-factors in the energy density of the resultant solution, $$\begin{aligned} \label{gamma-factors} \varepsilon_7 = \tau_7\sum_{j=1}^n\frac1{\sqrt{1-|v_j|^2}}.\end{aligned}$$ This result relies on the form of the kinetic action; for instance, in the case where $n=1$, $\X\sim 2-|v_1|^2$ and $\Y\sim |v_1|^4$, so only the combination $\sqrt{\X^2 -\Y} \sim \sqrt{1-|v_1|^2}$ together with the variations of $\operatorname{\EuScript{F}}(\X\pm\sqrt\Y)$ with respect to $g^{00}$ give the correct dilation factors above. Another example is condensation to a D7-brane anti-brane pair which is obviously non-supersymmetric; in this case we still expect tension of $2\tau_7$, whereas (\[CS\_D\_Dbar\]) shows that the total RR charge is zero. Placing the soliton and anti-soliton on the real axis at $x_0$ and $-x_0$ respectively $T=u(z-x_0)({ \overline{z} }+x_0)$, after taking $u\to\infty$ we have $$\begin{aligned} \varepsilon_7 &= 2\tau_9\int e^{-2\pi\ap T{ \overline{T} }} (2\pi\ap)^2 |\partial_zT\partial_{{ \overline{z} }}{ \overline{T} } - \partial_{{ \overline{z} }}T\partial_z{ \overline{T} }|\;\frac i2dz\wedge d{ \overline{z} }.\end{aligned}$$ For $x_0=0$, the brane and anti-brane are coincident, there is no winding of the tachyon field, and the total tension is zero (the tachyon derivative terms cancel). For any $x_0>0$, the absolute value gives two regions of integration with opposite signs $$\begin{aligned} \varepsilon_7 &= i\tau_9\left(\;\int\limits_{\Re(z)>0} - \int\limits_{\Re(z)<0}\right) e^{-2\pi\ap T{ \overline{T} }} (2\pi\ap)^2dT\wedge d{ \overline{T} }.\end{aligned}$$ By the arguments of the previous section, the first integral receives only the positive contribution from the zero of $T$ at $z=x_0$, the second only the negative contribution from the zero at ${ \overline{z} } = -x_0$, giving $2\tau_7$ as the total tension. Care must be taken because the arguments of the previous section relied on Stoke’s theorem, and here we are introducing a new boundary along the imaginary axis, however since we work in the limit $u\to\infty$ and the boundary integrand is proportional to $\exp[-2\pi\ap T{ \overline{T} }]$, the boundary term vanishes, and the result remains valid. We can use the understanding gained from this experience to calculate the tension of the configuration $T = \lim_{u\to\infty}u \prod_{i=1}^n(z-z_i) \prod_{j=1}^m({ \overline{z} }-{ \overline{z} }_j^\prime)$, representing $n$ D7-branes and $m$ anti-branes (all parallel). We assume no brane and anti-brane position coincides $z_i\ne z_j^\prime,\forall \{i,j\}$. The tension is $$\begin{aligned} \varepsilon_7 &= 2\tau_9\int e^{-2\pi\ap T{ \overline{T} }}(2\pi\ap)^2 T{ \overline{T} }\left|\sum_{i,k=1}^n\frac1{z-z_i}\frac1{{ \overline{z} }-{ \overline{z} }_k} - \sum_{j,l=1}^m\frac1{z-z_j^\prime}\frac1{{ \overline{z} }-{ \overline{z} }_l^\prime} \right|\;\frac i2dz\wedge d{ \overline{z} }.\end{aligned}$$ In regions about each $z_i$ or $z_j^\prime$ (D7-brane or D7-anti-brane) the term in absolute values is positive or negative respectively. Denoting these regions by $\Gamma_i$ and $\Gamma_j^\prime$ the tension is $$\begin{aligned} \varepsilon_7 &= i\tau_9\left(\sum_{i=1}^n\;\int\limits_{\Gamma_i} - \sum_{j=1}^m\;\int\limits_{\Gamma_j^\prime}\right) e^{-2\pi\ap T{ \overline{T} }}(2\pi\ap)^2dT\wedge d{ \overline{T} }.\end{aligned}$$ Each integral precisely resembles (\[C8\_ne\_0\]) and so the value of the integral is proportional to $+1$ for each holomorphic zero and $-1$ for each anti-holomorphic zero of $T$ in the region. The boundary terms again vanish because we take $u\to\infty$, giving the expected result $$\begin{aligned} &\varepsilon_7 = (n+m)\tau_7, &\mu_7 = (n-m)\tau_7g_s,\end{aligned}$$ where the result of the RR charge calculated earlier has been included for comparison. Vortex Tension with Magnetic Flux --------------------------------- More complicated solutions include gauge field winding about zeros of the tachyon field. For such solutions the RR 8-form charge is given by (\[mult\_wind\]), irrespective of the behavior of the gauge field. The tachyon fields wind more than once around each zero when $T$ is not entirely holomorphic. This represents multiple D7-branes which preserve $\mathcal N = 1$ spacetime supersymmetries; the necessary conditions to obtain such BPS configurations are $D_{{ \overline{z} }}T = 0$ and $F_{zz} = F_{{ \overline{z} }{ \overline{z} }} = 0$[@Hori:2000ic]. The later condition is trivial in this system, but imposing the former determines the form of $A^-$ and its curvature $$\begin{aligned} T &= \lim_{u\to\infty} u\prod_{i=1}^N \left(\frac{z-z_i}{{ \overline{z} }-{ \overline{z} }_i}\right)^{w_i/2} (z-z_i)^{l_i/2}({ \overline{z} }-{ \overline{z} }_i)^{l_i/2}, \tag{$w_i\in\mathbbm Z^+$}\\ A^-_{{ \overline{z} }} &= -\frac i2\sum_{i=1}^N\frac{a_i} {{ \overline{z} }-{ \overline{z} }_i},\quad\quad\quad F^-_{z{ \overline{z} }} = -2\pi\sum_{i=1}^Na_i \delta^{(2)}(z-z_i,{ \overline{z} }-{ \overline{z} }_i),\label{mult_wind_gauge}\end{aligned}$$ For this configuration, since $$\begin{aligned} D_{{ \overline{z} }}T = \frac T2\sum_{i=1}^N \frac{l_i-(w_i-a_i)} {{ \overline{z} } - { \overline{z} }_i},\end{aligned}$$ we must have $l_i = w_i-a_i, \forall i$ to be a BPS configuration. In order to obtain a solution with just one unit of magnetic flux for each winding number, $w_i = a_i$, $l_i$ must be zero; this can be achieved by taking the limit $l_i\to0$ in such a way that $\lim_{l_i\to0,u\to\infty} ul_i \to \infty$. To calculate the tension of this ansatz, we must apply a regularization of the $\delta$-functions, and the tension is regularization dependent; we choose that which gives the tension to be independent of the gauge field winding, to show that such a solution is possible. Formally this requires that we write the $\delta$-function in $F^-$ as a Gaussian of width $\epsilon$; requiring that $\frac1\epsilon\sim u^{n>2}$ implies taking $\epsilon\to0$ before $u\to\infty$, and the tension of the solution will be equal to the RR charge. In this regularization, before taking $u\to\infty$ we split the integral into regions about each zero of $T$ as before, and split each region $\Gamma_i$ into one about the pole at $z_i$ $(\Gamma_{i,\le\epsilon})$ and one over the rest of the region $(\Gamma_{i,>\epsilon})$ $$\begin{aligned} \varepsilon_7 &= \lim_{u\to\infty}\lim_{\epsilon\to0} \sum_{i=1}^N\left(\;\int\limits_{\Gamma_i\le\epsilon} +\int\limits_{\Gamma_i>\epsilon}\right)\lag.\end{aligned}$$ The first integral can be evaluated using $$\begin{aligned} \sqrt{\det(\delta+\pi\ap F^-)} &= \sqrt{1+\hf(\pi\ap F^-)^2} &\longrightarrow&&&4\pi^2\ap\left|a_i\right| \delta^{(2)}(z-z_i,{ \overline{z} }-{ \overline{z} }_i),\\ ({\G}^{-1})^{\mu\nu} &= \frac{g^{\mu\nu}-\pi\ap F^{-\mu\nu}} {1+\hf(\pi\ap F^-)^2} &\longrightarrow&&&0,\end{aligned}$$ about $z=z_i$, which gives for the contribution to the tension from $\Gamma_{i,\le\epsilon}$ $$\begin{aligned} 2\tau_9\int\limits_{\Gamma_{i,\le\epsilon}} 4\pi^2\ap\left|a_i\right| \delta^{(2)}(z-z_i,{ \overline{z} }-{ \overline{z} }_i)\frac i2 dz\wedge d{ \overline{z} } = \left|a_i\right|\tau_7.\end{aligned}$$ The remaining integral is over that part of $\Gamma_i$ further than $\epsilon$ from $z_i$. Naïvely this should be zero because we have argued that only the zeros of $T$ contribute to the soliton tension and charge, but the specific regularization of the solution using $\frac1\epsilon\sim u^{n>2}$ gives us $\tau_7|w_i-a_i|$, it being necessary to take $u\to\infty$ after taking $\epsilon\to0$. The total energy density of the soliton is the sum over the integrals in both regions for all $\Gamma_i$ and is $$\begin{aligned} \label{tension_w_gf} \varepsilon_7 &= \tau_7 \sum_{i=1}^N\left(|w_i-a_i|+\left|a_i\right|\right).\end{aligned}$$ Since $w_i$ are positive (allowing them to take negative values would change some branes to anti-branes), when all $w_i \ge a_i \ge 0$ the solution has minimal energy and the tension is equivalent to the RR 8-form charge (\[mult\_wind\]), $\varepsilon_7 = \tau_7 \sum_{i=1}^Nw_i = \mu_7/g_s$. Therefore we appear to have multiple solutions representing certain brane systems, with different degrees of gauge field winding but with identical energy and RR charge. At this level in the effective theory, it is a curious degeneracy of the soliton solutions, which is likely lifted by higher derivative and/or quantum corrections to the effective action. To summarize, we give the general ansatz for a tachyon field representing a set of parallel $n$ D7-branes and $m$ anti-branes. The energy per 7-volume of this solution is $\varepsilon_7=(n+m)\tau_7$ and its RR charge under the spacetime 8-form potential is $\mu_7=(n-m)\tau_7g_s$. $$\begin{aligned} T &= \lim_{u\to\infty}u\prod_{i=1}^N \left(\frac{z-z_i}{{ \overline{z} }-{ \overline{z} }_i}\right)^{w_i/2} |z - z_i|^{l_i}\prod_{j=1}^M \left(\frac{{ \overline{z} }-{ \overline{z} }_j^\prime}{z-z_j^\prime}\right)^{w_j^\prime/2} |z - z_j^\prime|^{l_j^\prime},\nonumber\\ A^-_{{ \overline{z} }} &= -\frac i2\sum_{i=1}^N\frac{a_i} {{ \overline{z} }-{ \overline{z} }_i} - \frac i2\sum_{j=1}^M\frac{a_j^\prime} {{ \overline{z} }-{ \overline{z} }_j^\prime},\label{general_solution}\end{aligned}$$ where $z_i$ ($z_j^\prime$) are the constant positions of the (anti-)branes. Single valuedness of $T$ requires that $\{w_i,w_j^\prime\}$ is some set of positive integers, and we have defined $\sum_{i=1}^Nw_i=n, \sum_{j=1}^Mw_j^\prime = m$. $\{a_i, a_j^\prime\}$ must satisfy $0\le a_i \le w_i$, $0\le a_j^\prime \le w_j^\prime$ in order to obtain the minimal energy solution as in (\[tension\_w\_gf\]). If the brane or anti-brane were to move at constant velocities, the tensions would pick up the relativistic $\gamma$-factors as in (\[gamma-factors\]). Discussion ---------- The richness of vortex solutions examined in this section is vindication of the gauge covariant form of the tachyon kinetic terms used; the tensions of all solitons are as expected, the results can be calculated in any coordinate system, and by persuading the solitons to move at constant velocities the necessary special relativistic factors arise. All this evidence depends crucially on the $\X\pm\sqrt \Y$ structure of the action. The usual topological arguments suggest the vortices of the action (\[action\]) are stable and we verify this by perturbing a characteristic solution representing $n$ coincident D7-branes, $$\begin{aligned} &T = \lim_{u\to\infty}u\left(\frac z{{ \overline{z} }}\right)^{n/2} \left(z{ \overline{z} }\right)^{l/2} + t(z,{ \overline{z} }), &A^-_{{ \overline{z} }} = -\frac{ia}{2{ \overline{z} }}.\end{aligned}$$ The first order perturbations vanish for all values of $[l,n,a]$, so these are solutions of the equations of motion. When the condition for $\mathcal N = 2$ worldsheet supersymmetry is satisfied, $$\begin{aligned} \label{BPS_condition} &D_{{ \overline{z} }}T = 0, &\text{or}&& l = n-a,\end{aligned}$$ the second order perturbations are $$\begin{gathered} \frac{\delta^2 S}{\delta T^2}t^2 + \frac{\delta^2 S}{\delta { \overline{T} }^2}{ \overline{t} }^2 + 2\frac{\delta^2 S}{\delta T\delta{ \overline{T} }}t{ \overline{t} } \propto -le^{-2\pi\ap u^2|z|^{2l}}\left( \hf\eta^{ij}\partial_it\,\partial_j{ \overline{t} } + a\delta^{(2)}(z,{ \overline{z} })t{ \overline{t} }\right)\\ - \underbrace{le^{-2\pi\ap u^2|z|^{2l}}\Big[\overbrace{ \left(\partial_z t\partial_{{ \overline{z} }}{ \overline{t} } +\partial_{{ \overline{z} }}t\partial_z{ \overline{t} }\right) }^{\text{from }\X^2} - \overbrace{ 2\partial_{{ \overline{z} }}t\partial_z{ \overline{t} }}^{\text{from }\Y} +\ldots\Big]}_{\text{total derivative}}\end{gathered}$$ The terms other than the kinetic terms in the off-brane directions conspire to form a total derivative (again because of the form of the tachyon kinetic terms, $\X\pm\sqrt \Y$) leaving just the modes in the directions along the D7-branes. When $a=0$, they represent the two massless fluctuations of the set of branes in the two transverse dimensions. When $a\ne0$ the gauge field must be perturbed similarly otherwise the fluctuations are massive; checks reveal that the gauge field perturbations are likewise stable. Let us recall Derrick’s theorem (see [@Coleman:1985] for instance); in a field theory of a set of scalar fields, suppose there is a time-independent solitonic solution $T(x)$ with codimension $d_c$. Consider the one-parameter family of field configurations defined by $T(x;\lambda) \equiv T(\lambda x)$ where $\lambda$ is positive and real. In general the energy is $E(\lambda) = \lambda^{-d_c}P + \lambda^{-d_c+2}K_2 + ... $, where $P$ is the potential energy contribution (defined so $P \ge 0$) and $K_2$ is the two derivative kinetic energy term, and extra terms may be present if there are more than two derivative terms in the theory. By Hamilton’s principle, this must be stable at $\lambda=1$, that is, ignoring possible extra terms, $(d_c-2)K_2 + d_c P =0$. Since both $P$ and $K_2$ are positive in an ordinary field theory, only codimenion $d_c=1$ soliton is possible. Vortices (codimension two) in Abelian Higgs model are possible due to the presence of magnetic flux. Now we look at vortices in the $\DD$ system. The energy of a time-independent soliton $T(x)$ with codimension $d_c$, in the limit $DT\to\infty$ is $E = 2\pi\tau_9\int d^{(d_c)}x V(T{ \overline{T} }) \sqrt{\X^2-\Y}$. That is $P=0$ which implies $E$ scales like a two derivative term and $$\begin{aligned} (d_c-2) E = 0\end{aligned}$$ Therefore only codimension two solitons are possible. In contrast to the Abelian Higgs model, a magnetic flux is not necessary for the existence of the vortices in the $\DD$ system, as we have seen. Returning to the issue of the apparent degeneracy between vortices with and without $A^-$ flux, we repeat that this degeneracy is expected to be lifted by corrections to the effective action. The effective action has at least two sources of corrections: [[**.**]{}]{}[-5pt]{} Classically there are higher derivative terms. Since the $\DD$ system is non-supersymmetric, there will be quantum corrections. Such corrections should lift the degeneracy. For any solution that involves both vortices and anti-vortices, supersymmetry is clearly broken; a more careful calculation would reveal tachyonic modes in such systems. Considering only vortices which satisfy the BPS condition $l=n-a$, the solution with no gauge field ($a=0, l=n$) can only receive corrections from multiple (anti-)holomorphic derivatives of $T$ (${ \overline{T} }$). Since $T\sim z^n$, only the first $n$ holomorphic derivatives of $T$ are non-zero. Importantly the degeneracy is already present in the single vortex solution : the $n=1$ solution may have flux $a=0$ or up to $a=1$, with the same energy and RR charge. Note that the $n=1$ solution without flux [@Kraus:2000nj; @Takayanagi:2000rz] is classically exact: there are no gauge field derivative corrections and all second and higher derivatives of the tachyon field vanish. If the degeneracy is lifted, one may naïvely conclude that this zero-flux solution will be the stable one. If this is the exact BPS solution, then putting $n$ of them together should be an exact solution too. However, higher derivative terms are non-zero for the $n$-vortex solution ($n>1$, still with $a=0$). Barring a miraculous cancellation, the $n$-vortex solution will have classical corrections from the higher derivative terms. Another solution to consider is that with $n=a$, or $l=0$ from the BPS condition (\[BPS\_condition\]). Then noting that $D_zT = \frac{(l+(n-a))}{2z}T=\frac{l}{z}T$, for $l=0$ both $D_{{ \overline{z} }}T = D_zT = 0$, naïvely implying that all higher derivative terms vanish and $n=a$ is an exact solution. Recall, however, that the solution with a wound gauge field requires some care in taking the limit, and we can only take $l\to0$ with $u\to\infty$ and $lu\to\infty$, implying $\lim_{l\to0}D_zT \ne 0$. Therefore corrections due to higher derivative terms cannot be ignored in the case where there is a flux associated with each winding, *i.e.* $n=a$. In conclusion, we do not know how the degeneracy will eventually be lifted. Lower Dimensional Brane Anti-Brane Systems ========================================== Lower dimensional brane anti-brane pairs can be constructed in a straightforward manner by applying T-duality to the action (\[action\]). In these systems, the brane and anti-brane can be separated because under T-duality components of $A^-$ transform into the relative separation of the pair. We follow closely the procedure of [@Myers:1999ps]. Because under T-duality, both the dilaton transforms and there is mixing between the metric and the Kalb-Ramond $B$-field, it is necessary to include these in our action (\[action\]). The T-duality properties of the various fields in the action are well known; the gauge fields in the T-dual directions transform into the adjoint scalars, the metric and Kalb-Ramond field mix, the string coupling scales. Being an open string scalar state, the tachyon is inert under T-duality. Under T-duality in directions labeled by uppercase Latin indices, (lowercase Latin indices labeling unaffected directions on the brane), the fields transform as [@Myers:1999ps] $$\begin{aligned} T &\to T, &A_a &\to A_a, &A_I &\to \frac{\Phi^I}{2\pi\ap},\\ E_{\mu\nu} &\equiv g_{\mu\nu} + B_{\mu\nu}, &e^{2\phi} &\to e^{2\phi}\det E^{IJ}, &E_{IJ} &\to E^{IJ}\\ E_{ab} &\to E_{ab}-E_{aI}E^{IJ}E_{Jb}, &E_{aI} &\to E_{aK}E^{KI}, &E_{Jb} &\to -E^{JK}E_{Kb},\end{aligned}$$ where $E^{IJ}$ is the matrix inverse to $E_{IJ}$. The result of T-dualing $9-p$ dimensions can be written most simply by defining the pull-back in normal coordinates as: $$\begin{aligned} &P[E_{ab}]^{1,2} \equiv E_{ab} + E_{I\{a}\partial_{b\}} \Phi^{I\;1,2} + E_{IJ}\left(\partial_a\Phi^I\partial_b\Phi^J\right)^{1,2}, &P[E_{aI}]^{1,2} &\equiv E_{aI} + E_{JI}\partial_a\Phi^{J\;1,2}.\end{aligned}$$ Care must be taken because there are two sets of scalars which describe the position of each brane; they are denoted herein by $\Phi^{I\;1,2}$ and their difference as $\varphi^{I} \equiv \Phi^{I\;1} - \Phi^{I\;2}$ which is the scalar representing the $(\DD)_p$ separation. In calculating the pull-back of any quantity only the indices corresponding to directions along the brane are affected. After T-dualing the fields in (\[action\]) as above and performing manipulations similar to those in [@Myers:1999ps], we obtain the improved action for a D$p$ brane anti-brane pair: $$\begin{aligned} \nonumber S_{(\DD)_p} = -\tau_p\int d^{p+1}x\;e^{-2\pi\ap T{ \overline{T} }}& \left[ \sqrt{-\det[\G_1]}\operatorname{\EuScript{F}}(\X_1+\sqrt \Y_1)\operatorname{\EuScript{F}}(\X_1-\sqrt \Y_1)\right.\\&+ \left.\sqrt{-\det[\G_2]}\operatorname{\EuScript{F}}(\X_2+\sqrt \Y_2)\operatorname{\EuScript{F}}(\X_2-\sqrt \Y_2) \right]\label{T-dual_action}\end{aligned}$$ where now the effective metric contains the spacetime metric pulled-back to the brane worldvolume (and includes any non-zero NS-NS B field) $$\begin{aligned} &\G^{1,2}_{ab} \equiv P[E_{ab}]^{1,2}+2\pi\ap F^{1,2}_{ab},\end{aligned}$$ and the covariant derivative dependence of $\X$ and $\Y$ in (\[action\]) leads to $\Phi$ dependence in the T-dual action. The complete expressions for $\X$ and $\Y$ are $$\begin{aligned} \X_{1,2} &= 2\pi\ap^2 \left[\begin{array}{r} \G_{1,2}^{\{ab\}}D_aTD_b{ \overline{T} } + \frac1{(2\pi\ap)^2}\varphi^{I}\varphi^{J}T{ \overline{T} } \left(E_{\{IJ\}} -\G_{1,2}^{ab}P[E_{\{Ia}E_{bJ\}}]^{1,2}\right)\\ +\frac i{2\pi\ap}\left(\G_{1,2}^{ab}P[E_{bI}]^{1,2} -\G_{1,2}^{ba}P[E_{Ib}]^{1,2}\right) \left(TD_a{ \overline{T} }-{ \overline{T} }D_a T\right)\varphi^I \end{array}\right]\\ \Y_{1,2} &= \left(2\pi\ap^2\right)^2 \left|\begin{array}{r} \G_{1,2}^{ab}D_aTD_bT +\frac i{2\pi\ap}\G_{1,2}^{ab}\left(P[E_{bI}]^{1,2}D_aT -P[E_{Ia}]^{1,2}D_bT\right)T\varphi^{I}\\ - \frac1{(2\pi\ap)^2}\varphi^{I}\varphi^{J}T^2 \left(E_{IJ}-\G_{1,2}^{ab}P[E_{Ia}E_{bJ}]^{1,2}\right) \end{array}\right|^2.\end{aligned}$$ These expressions simplify considerably in Minkowski spacetime when $B=0$ and $A^{1,2}=0$: $$\begin{aligned} \G_{ab}^{1,2} &= \eta_{ab} + \delta_{IJ}(\partial_a\Phi^I\partial_b\Phi^J)^{1,2},\\ \X_{1,2} &\xrightarrow[B=0,\;A^{1,2}=0]{g=\eta}\; 2\pi\ap^2\left[\G_{1,2}^{ab}\partial_aT\partial_b{ \overline{T} } + \frac1{(2\pi\ap)^2}\varphi^{I}\varphi^{J}T{ \overline{T} } \left(\delta_{IJ} - \G_{1,2}^{ab} \partial_a\Phi^{1,2}_I\partial_b\Phi^{1,2}_J\right)\right],\\ \Y_{1,2} &\xrightarrow[B=0,\;A^{1,2}=0]{g=\eta}\; \left(2\pi\ap^2\right)^2\left|\G_{1,2}^{ab}\partial_aT\partial_bT - \frac1{(2\pi\ap)^2}\varphi^{I}\varphi^{J}T^2 \left(\delta_{IJ} - \G_{1,2}^{ab} \partial_a\Phi^{1,2}_I\partial_b\Phi^{1,2}_J\right) \right|^2.\end{aligned}$$ It is clear that the action contains relative velocity dependent terms. It would be interesting to study the implications of the velocity dependence of this action to density perturbations in brane inflationary models. Some important properties of the separation dependent tachyon potential can be verified. The potential is equal to that part of the Lagrangian which is independent of gauge fields and derivatives, $$\begin{aligned} &V(T,\varphi) = 2\tau_p e^{-2\pi\ap T{ \overline{T} }} \operatorname{\EuScript{F}}\left(\frac1\pi|\varphi|^2T{ \overline{T} }\right), &|\varphi|^2 &\equiv E_{IJ}\varphi^I\varphi^J,\end{aligned}$$ which gives as the position dependent mass of the tachyon $$\begin{aligned} m_T^2 &= \frac1{2\ap}\left(\frac{|\varphi|^2}{2\pi^2\ap} - \frac1{2\ln2}\right).\end{aligned}$$ Apart from the discrepancy by $2\ln2$ which appears in the BSFT calculations of the tachyon mass, this is consistent with the familiar result that as a parallel $Dp$-brane and $Dp$-anti-brane are moved toward each other, the lowest open string scalar mode becomes tachyonic at separations $|\varphi|^2<2\pi^2\ap$ [@Banks:1995ch]. We see that the separated $(\DD)_p$ system, although classically stable, is quantum mechanically unstable for $|\varphi|^2 > \frac{2\pi^2\ap}{2\ln2}$, with a tunneling barrier which increases with their separation, as in Figure \[figure\_potential\]. The system remains unstable at the critical separation $|\varphi_c|^2 \equiv \frac{2\pi^2\ap}{2\ln2}$ (the dotted blue line in Figure \[figure\_potential\]), since the $|T|^4$ term in the potential has a negative coefficient there. This potential was first written down by Hashimoto [@Hashimoto:2002xt] assuming a linear tachyon profile; here we have justified its form for arbitrary $T$, which allows us to calculate the instanton “bounce”, which is spherically symmetric in $p+1$ dimensional Euclidean space. That the separated $(\DD)_p$ system will annihilate via quantum mechanical tunnelling has been studied in the literature [@Callan:1998kz; @Hashimoto:2002xt]. Here we shall use the above effective action to find the decay rate and check the validity of the thin wall approximation. For fixed brane separation (that is, constant $|\varphi|$, a very good approximation in the slow-roll phase during the inflationary epoch in the early universe), we calculate the decay rate. We aim to do so including the contribution from all kinetic terms. The calculation is tractable since the tachyon decays in one direction in field space only, $T = { \overline{T} }$. We set the value of the tachyon potential at the false vacuum, $T=0$ to be zero, and the resulting Euclidean Lagrangian for the bounce [@Coleman:1985] becomes $$\begin{aligned} &\mathcal L_E = 2\tau_p \sqrt{g_E}\;\left[e^{-2\pi\ap T^2} \operatorname{\EuScript{F}}(\frac1\pi|\varphi|^2T^2) \operatorname{\EuScript{F}}(4\pi\ap^2\partial_\mu T\partial^\mu T) - 1\right], &\tau_p = \frac1{(2\pi)^pg_s\ap^{\frac{p+1}2}}.\end{aligned}$$ The tunneling rate can be computed numerically by the standard instanton methods, where the probability of tunneling is $$\begin{aligned} \mathcal{P} \sim K(\varphi) e^{-S_E(\varphi)},\end{aligned}$$ $S_E(\varphi)$ being the Euclidean action of the instanton and the factor $K(\varphi)$ is due to both the quantum fluctuations about the instanton transition and to solutions of higher action which shall in general depend on the separation, $\varphi$. Calculating the “bounce” solution and integrating it numerically gives, to a good approximation, $$\begin{aligned} \label{instanton} &S_E(\varphi) \simeq 4\pi c_1c_2^{p+1}\left[\frac{p^p}{(p+1)g_s} \frac{2\pi^{\frac{p+1}2}}{\Gamma\left(\frac{p+1}2\right)}\right] \left(\frac{|\varphi|-|\varphi_c|}{\sqrt\ap}\right)^{\frac{p+1}2}, &\begin{array}{l} c_1 \sim 1.5,\\c_2 \sim 0.29, \end{array}\end{aligned}$$ when $|\varphi|>|\varphi_c|$. We have expressed $S_E$ in this form to most easily compare to the expression for the thin wall approximation [@Coleman:1985], $$\begin{aligned} S_E(\varphi) \simeq \left[\frac{p^p}{(p+1)} \frac{2\pi^{\frac{p+1}2}}{\Gamma\left(\frac{p+1}2\right)}\right] \left(\frac{S_1}{\epsilon_p}\right)^{p+1}\epsilon_p\end{aligned}$$ where $S_1$ is the action for the one-dimensional instanton and $\epsilon_p = 2\tau_p$. This imples that the thin wall bounce has the form $$\begin{aligned} \frac{S_1}{\sqrt{\ap}\epsilon_p} = 2\pi c_2\left(\frac{|\varphi|-|\varphi_c|}{\sqrt\ap}\right)^{\hf}.\end{aligned}$$ In the thin wall approximation $c_1=1$, and comparison with the numerical result shows that $c_2 \simeq 0.29$. We expect the thin wall approximation to be valid when $\varphi$ becomes large. Note that $S_1$ differs from that in [@Hashimoto:2002xt] (in which $S_1$ is linear in $\varphi$) because that calculation was performed with truncated kinetic and potential terms. Classically, for large enough separation, when $m_T^2>0$, the ground state is $T=0$, and $V(0,\varphi)=1$. This implies that there is no force in the $(\DD)_p$ system. However, since the system is non-supersymmetric, quantum corrections are clearly present. It is known that the one-loop open string contribution is dual to the closed string exchange. For large separation, this is dominated by the exchanges of the graviton, dilaton and RR field $C_{p+1}$ between the D$p$ and the anti D$p$-brane and has been calculated. These one-loop open string corrections can be included by inserting the classical closed string background produced by a D$p$ and a ${ \overline{\text{D}} }p$ brane into the $(\DD)_p$ action. The supergravity solution is well known [@Horowitz:1991cd]. $$\begin{aligned} ds^2 &= {h(r)^{\hf}}\left(-dt^2+\sum_{i=1}^pdx^idx^i\right) + {h(r)^{-\frac32-\frac{5-p}{7-p}}}dr^2 + r^2{h(r)^{\hf-\frac{5-p}{7-p}}}d\Omega_{8-p}^2,\\ e^{-2\phi} &= g_s^{-2}{h(r)^{-\frac{p-3}2}}, \quad\quad\quad h(r) = 1 - \frac{g_s\beta}{r^{7-p}}\\ (C_{p+1})_{a_1\ldots a_{p+1}} &= \frac{\beta}{r^{7-p}} \epsilon_{a_1\ldots a_{p+1}}, \quad\quad\quad \beta \equiv (4\pi)^{\frac{5-p}2}\ap^{\frac{7-p}2} \Gamma\left(\frac{7-p}2\right).\end{aligned}$$ This classical closed string background of a brane shall be “felt” by the anti-brane, so we insert this into that part of the action corresponding to an anti-brane and the similar background into the brane action. On the brane worldvolumes the separation, $r$, is represented by the scalar field $|\varphi|$. The result of performing these steps is that when $|\varphi|^2\gg\ap$, the total tension of the system is renormalized: $$\begin{aligned} S &= S_{\DD}(\varphi) + S_{\text{CS}}(\varphi),\\ \tau_p &\to \tau_p(\varphi) = \tau_p\Bigg( \overbrace{1-\frac{g_s\beta}{|\varphi|^{7-p}}}^{\text{NS-NS}} \underbrace{-\frac{g_s\beta}{|\varphi|^{7-p}}}_{\text{RR}}\Bigg).\end{aligned}$$ Clearly for a brane-brane system, the sign of the RR contribution is reversed and the tension is unrenormalized. The renormalized tension then gives a potential for the scalar representing the separation, and we see there is an attractive force between the brane and anti-brane. When the brane separation decreases, massive closed string modes start to contribute to $\tau_p(\varphi)$. Their contributions are easy to include, except when the brane separation becomes so small that $m_T^2$ becomes negative. When the tachyon appears, $\tau_p(\varphi)$ becomes complex. Fortunately, $\tau_p(\varphi)$ is expected to remain finite [@Garcia-Bellido:2001ky] and tachyon rolling happens so fast that the precise form of $\tau_p(\varphi)$ at short distance becomes phenomenologically unimportant [@Shiu:2002xp]. Conclusion ========== In this paper, we present a fully covariant $\DD$ action (Eq.(\[action\])) based on boundary superstring field theory. The kinetic term has some rather novel features. Its form is almost completely dictated by BSFT and symmetry properties of the system. It is quite amazing that exact multi-vortex multi-anti-vortex solutions (with arbitrary positions and arbitrary constant velocities) can be easily written down (Eq.(\[general\_solution\])). The simplicity of these analytic solutions may be very useful in the study of the production of vortices. In the early universe in brane world scenarios, the production of such vortices corresponds to the production of cosmic strings towards the end of the inflationary epoch. The solitonic solution has a large peculiar degeneracy: the energy and the RR charge of the solutions depend only on the vorticities and not on the “magnetic” flux that may or may not be present inside the vortices. Further improvement on the $\DD$ action should lift this degeneracy. However we are unable to answer the question that we endeavor to address: as a soliton in the exact theory, whether the D$p$-brane has a “magnetic” flux in its core. If the degeneracy is lifted at the classical level, one should simply go back to the path integral expression (\[definition\_S\]) for the $\DD$ action, which is supposedly classically exact, to reexamine the solitonic solutions. The inclusion of the coupling of the gauge field to the tachyon in the $(\DD)_p$ action allows us, via T-duality, to consider the situation when the brane and the anti-brane are separated. This action is suitable for the study of the inflationary scenario in the brane world. 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--- author: - 'J.-B. Delisle' bibliography: - 'laplace.bib' title: | Analytical model of multi-planetary resonant chains\ and constraints on migration scenarios --- Introduction {#sec:introduction} ============ Mean motion resonances (MMR) between two planets are a natural outcome of the convergent migration of planets in a gas-disk [e.g., @weidenschilling_orbital_1985]. The planets initially form farther away from each other, and planet-disk interactions induce a migration of the planets. The period ratio between the planets decreases until they get captured in a MMR. The planets then continue to migrate whilst maintaining their period ratio at a rational value (2/1, 3/2, etc.). The eccentricities increase due to the resonant interactions, until they reach an equilibrium between the migration torque and the eccentricity damping exerted by the disk. The argument of the resonance, which is a combination of the mean longitudes of the two planets, enters into libration (oscillations around an equilibrium value). For systems of three and more planets, once a pair of planets has been captured in a MMR, the other planets might also join this couple to form a chain of resonances. Each time a planet gets captured in the chain, it enters into a MMR (and thus maintains a constant and rational period ratio) with each of the other planets of the chain. The eccentricities of the planets and the resonant arguments of each pair find a new equilibrium. Such multi-planetary resonant chains are expected from simulations of planet migration [e.g., @cresswell_evolution_2006]. Recently, @mills_resonant_2016 showed that the four planets in the Kepler-223 system are in a 3:4:6:8 resonant chain (period ratios of 4/3, 3/2, and 4/3 between consecutive pairs of planets). Using transit timing variations (TTVs), the authors observed that the Laplace angles of the system are librating with small amplitudes. The Laplace angles are combinations of the mean longitudes of three planets in the chain, and the observation of their libration is evidence that the system is indeed captured in the resonant chain. Using numerical simulations, @mills_resonant_2016 showed that the observed orbital configuration is very well reproduced by a smooth convergent migration of the planets. In this article, I present an analytical model of resonant chains. Analytical models have already been proposed, in particular to study the dynamics of the Laplace resonance (1:2:4 chain) between Io, Europa, and Ganymede [e.g., @henrard_orbital_1983]. However, while several numerical studies have been dedicated to the capture of planets in various resonant chains [e.g., @cresswell_evolution_2006; @papaloizou_dynamics_2010; @libert_trapping_2011; @papaloizou_consequences_2016], general analytical models have not yet been proposed. Recently, @papaloizou_three_2015 proposed a semi-analytical model of three-planet resonances taking into account only the interactions between consecutive planets in the chain, with a particular focus on the system [12:15:20 resonant chain, see also @steffen_transit_2012; @gozdziewski_laplace_2016]. This model is very similar to the studies of the Laplace resonance between the Galilean moons, but is not well suited in the general case. For instance, four-planet (or more) resonances are not considered. Moreover, for some three-planet resonances, the interactions between non-consecutive planets cannot be neglected. For instance, in a 3:4:6 resonant chain, each planet is locked in a first-order resonance with each of the other planets. In particular, the innermost and outermost planets are involved in a 2/1 MMR that strongly influences the dynamics of the system. I describe here a general model of resonant chains, with any number of planets, valid for any resonance order. I particularly focus on finding the equilibrium configurations (eccentricities, resonant arguments, etc.) around which a resonant system should librate. While a real system may be observed with significant amplitude of libration around the equilibrium, or could even have some angles circulating, the position of the equilibria still provides useful insights into the dynamics of the system. In Sect. \[sec:model\], I describe this analytical model, and the method I use to find the equilibrium configurations. In Sect. \[sec:application\], I apply the model to . I show that six equilibrium configurations exist for this resonant chain, and that the system is observed to be librating around one of them. I also show that knowing the current configuration of the system allows for interesting constraints to be put on its migration scenario, and in particular on the order in which the planets have been captured in the chain. Model {#sec:model} ===== I consider a planetary system with $n$ planets (which I denote with indices $1,...,n$ from the innermost to the outermost) orbiting around a star (index 0). I assume that the system is coplanar and is locked in a chain of resonances. In such a resonant chain, each pair of planets is locked in a MMR. For two planets $i<j$, I denote by $k_{j,i}/k_{i,j}$ the resonant ratio, such that $$k_{j,i} n_j - k_{i,j} n_i \approx 0,$$ where $n_i$ ($n_j$) is the mean motion of planet $i$ ($j$). I also introduce the degree of the resonance between planet $i$ and planet $j$ $$q_{i,j} = k_{j,i} - k_{i,j}.$$ At low eccentricities, resonances of a lower degree have a stronger influence on the dynamics of the system. In order to study the dynamics of these resonant chains, I generalize to $n$ planets the method developped in the case of two-planet resonances [@delisle_dissipation_2012; @delisle_resonance_2014]. The Hamiltonian of the system takes the form [@laskar_analytical_1991] $$\label{eq:hamposvel} \H = -\sum_{i=1}^n \G\frac{m_0m_i}{2 a_i} + \sum_{1\leq i<j\leq n} \left(-\G\frac{m_i m_j}{||\vec{r}_i-\vec{r}_j||} + \frac{\vec{\tilde{r}}_i.\vec{\tilde{r}}_j}{m_0} \right),$$ where $\G$ is the gravitational constant, $m_i$ is the mass of body $i$, $a_i$ is the semi-major axis, $\vec{r}_i$ the position vector, and $\vec{\tilde{r}}_i$ the canonically conjugated momentum of planet $i$ [in astrocentric coordinates, see @laskar_analytical_1991]. The first sum on the right-hand side of Eq. (\[eq:hamposvel\]) is the Keplerian part of the Hamiltonian (planet-star interactions), while the second sum is the perturbative part (planet-planet interactions). In the coplanar case (which I assume here) the system has $2 n$ degrees of freedom (DOF), with 2 DOF (4 coordinates) associated to each planet. As for two-planet resonances [e.g., @delisle_dissipation_2012], the number of DOF can be reduced by using the conservation of the total angular momentum (1 DOF), and by averaging over the fast angles (1 DOF). Therefore, the problem can be reduced to $2(n-1)$ DOF. Even with these reductions, the phase space is still very complex, especially for systems of many planets such as (chain of 4 planets, 6 DOF), and the problem is, in most cases, non-integrable. In this study, I focus on finding the fixed point of the averaged problem, which provides useful insight into the dynamics of the system, and especially into the values around which the angles of a resonant system should librate. The method described in the following is a generalization of the method presented in @delisle_dissipation_2012 which focuses on finding the fixed points for two-planet MMR. I denote by $\lambda_i$ and $\varpi_i$ the mean longitude and longitude of periastron of planet $i$ (in astrocentric coordinates), respectively. The actions canonically conjugated to the angles $\lambda_i$ and $-\varpi_i$ are the circular angular momentum $\Lambda_i$ and the angular momentum deficit [AMD, see @laskar_spacing_2000] $D_i$, respectively. These actions are defined as follows $$\begin{aligned} \Lambda_i &=& \beta_i\sqrt{\mu_i a_i},\\ D_i &=& \Lambda_i-G_i = \Lambda_i \left(1-\sqrt{1-e_i^2}\right),\end{aligned}$$ where $G_i=\Lambda_i\sqrt{1-e_i^2}$ is the angular momentum of planet $i$, $\beta_i = m_i m_0/(m_0+m_i)$, $\mu_i = \G (m_0+m_i)$. At low eccentricities the deficit of angular momentum $D_i$ is proportional to $e_i^2$. The Hamiltonian (Eq. (\[eq:hamposvel\])) can be expressed using these action-angle coordinates $$\label{eq:hamLaD} \H = - \sum_{i=1}^n \frac{\mu_i^2\beta_i^3}{2\Lambda_i^2} + \sum_{1\leq i<j\leq n} \H_{i,j}(\Lambda_i,\Lambda_j,D_i,D_j,\lambda_i,\lambda_j,\varpi_i,\varpi_j),$$ where the first sum is the Keplerian part, which depends only on $\Lambda_i$ (or equivalently $a_i$), and $\H_{i,j}$ is the perturbation between the planets $i$ and $j$, which depends on the eight coordinates associated to $i$ and $j$. I follow the method described in @laskar_stability_1995 to compute $H_{i,j}$ as a power series of the eccentricities (or equivalently of $\sqrt{D_i}$ and $\sqrt{D_j}$), and a Fourrier series of the angles, where the coefficients are functions of $\Lambda_i$ and $\Lambda_j$ (i.e., of the semi-major axes). For a system that is close to the resonance or resonant, the semi-major axes remain close to the nominal resonant values (Kepler’s third law) $$\label{eq:nominalsma} \frac{a_i}{a_j} \approx \frac{a_{i,0}}{a_{j,0}} = \left(\frac{k_{i,j}}{k_{j,i}}\right)^{2/3} \left(\frac{\mu_i}{\mu_j}\right)^{1/3}.$$ I introduce $$\Delta\Lambda_i = \Lambda_i - \Lambda_{i,0},$$ where $$\Lambda_{i,0} = \beta_i\sqrt{\mu_i a_{i,0}},$$ and expand the Keplerian part at degree 2, and the perturbative part at degree 0 in $\Delta\Lambda_i$ $$\begin{aligned} \label{eq:hamdLaD} \H &=& \sum_{i=1}^n n_{i,0} \Delta\Lambda_i - \frac{3}{2}\frac{n_{i,0}}{\Lambda_{i,0}}\Delta\Lambda_i^2\nonumber\\ &+& \sum_{1\leq i<j\leq n} \H_{i,j}(D_i,D_j,\lambda_i,\lambda_j,\varpi_i,\varpi_j),\end{aligned}$$ where $n_{i,0}$ is the nominal mean motion of planet $i$, such that $$\frac{n_{i,0}}{n_{j,0}} = \frac{k_{j,i}}{k_{i,j}}.$$ The perturbative part does not depend on $\Lambda_i$ anymore, but is simply evaluated at $\Lambda_{i,0}$. In order to perform the reductions associated to the conservation of angular momentum and to the averaging, I first change the system of coordinates. For the sake of readability, I present the general case (with any number of planets, in any resonance of any degree) in Appendix \[sec:generalchgcoord\], and take here the example of a system of four planets in a 3:4:6:8 resonant chain (as is the case for ). I introduce new canonically conjugated angles and actions as follows (see Eqs. (\[eq:angles\]) and (\[eq:actions\])) $$\begin{aligned} \label{eq:k223angacts} \phi_1 = \lambda_1 + \lambda_3 - 2 \lambda_2, &\qquad& L_1 = \Lambda_1,\nonumber\\ \phi_2 = \lambda_2 + 2\lambda_4 - 3 \lambda_3, &\qquad& L_2 = 2 \Lambda_1 + \Lambda_2,\nonumber\\ \phi_3 = \lambda_3 - \lambda_4, &\qquad& \Gamma = 8 \Lambda_1 + 6\Lambda_2 + 4\Lambda_3+ 3\Lambda_4,\nonumber\\ \phi_4 = 4 \lambda_4 - 3 \lambda_3, &\qquad& G = G_1 + G_2 + G_3 + G_4,\nonumber\\ \sigma_1 = 4 \lambda_4 - 3 \lambda_3 - \varpi_1, &\qquad& D_1,\nonumber\\ \sigma_2 = 4 \lambda_4 - 3 \lambda_3 - \varpi_2, &\qquad& D_2,\nonumber\\ \sigma_3 = 4 \lambda_4 - 3 \lambda_3 - \varpi_3, &\qquad& D_3,\nonumber\\ \sigma_4 = 4 \lambda_4 - 3 \lambda_3 - \varpi_4, &\qquad& D_4.\end{aligned}$$ $G$ is the total angular momentum, and is a conserved quantity; Its canonically conjugated angle ($\phi_4$) does not appear in the Hamiltonian. The angle $\phi_3$ is the only fast angle, and the averaging of the Hamiltonian is done over this angle. Therefore, its conjugated action ($\Gamma$) is constant in the average problem. The averaging is simply done by discarding all terms that depends on $\phi_3$ in the Fourrier expansion of the Hamiltonian.[^1] The angle $\phi_1$ is the argument of the Laplace resonance between the three innermost planets. The angle $\phi_2$ is the argument of the Laplace resonance between the three outermost planets. The two outer planets (3 and 4) may seem to play an important role in Eq. (\[eq:k223angacts\]), but this is only due to the arbitrary choice of canonical coordinates (many other choices are possible). Any two-planet resonant angle can be expressed as a combination of the angles of Eq. (\[eq:k223angacts\]). The arguments of the 4/3 resonance between the two outermost planets are $\sigma_3$ and $\sigma_4$. The arguments of the 3/2 resonance between planets 2 and 3 are $\sigma_2 - 2 \phi_2 = 3\lambda_3 - 2\lambda_2 - \varpi_2$ and $\sigma_3 - 2 \phi_2 = 3\lambda_3 - 2\lambda_2 - \varpi_3$. The arguments of the 4/3 resonance between planets 1 and 2 are $\sigma_1 - 2 \phi_2 - 3 \phi_1 = 4\lambda_2 - 3\lambda_1 - \varpi_1$ and $\sigma_2 - 2 \phi_2 - 3 \phi_1 = 4\lambda_2 - 3\lambda_1 - \varpi_2$. Arguments of resonances between non-consecutive pairs can also be expressed in the same way. For instance, for the 2/1 resonance between planets 1 and 3, the arguments are $\sigma_1 - 2 \phi_2 - \phi_1 = 2\lambda_3 - \lambda_1 - \varpi_1$ and $\sigma_3 - 2 \phi_2 - \phi_1 = 2\lambda_3 - \lambda_1 - \varpi_3$. For a system of $n$ planets captured in a resonant chain, $\phi_i$ ($i\leq n-2$) and $\sigma_i$ ($i\leq n$) (and all their linear combinations) librate around equilibrium values. All the actions also oscillate around equilibria. These equilibrium values correspond to stable fixed points of the average problem. For this example, I expand the perturbative part at first order in eccentricities ($\sqrt{D_i}$), and obtain an expression of the form $$\begin{aligned} \label{eq:hamdLaDk223} \H &=& -\frac{3}{2}\sum_{i=1}^3 \frac{n_{i,0}}{\Lambda_{i,0}}\Delta\Lambda_i^2\nonumber\\ &+& C_{1,2} \sqrt{D_1} \cos(\sigma_1 - 2 \phi_2 - 3 \phi_1)\nonumber\\ &+& C_{2,1} \sqrt{D_2} \cos(\sigma_2 - 2 \phi_2 - 3 \phi_1)\nonumber\\ &+& C_{1,3} \sqrt{D_1} \cos(\sigma_1 - 2 \phi_2 - \phi_1)\nonumber\\ &+& C_{3,1} \sqrt{D_3} \cos(\sigma_3 - 2 \phi_2 - \phi_1)\nonumber\\ &+& C_{2,3} \sqrt{D_2} \cos(\sigma_2-2\phi_2) + C_{3,2} \sqrt{D_3} \cos(\sigma_3-2\phi_2)\nonumber\\ &+& C_{2,4} \sqrt{D_2} \cos(\sigma_2-\phi_2) + C_{4,2} \sqrt{D_4} \cos(\sigma_4-\phi_2)\nonumber\\ &+& C_{3,4} \sqrt{D_3} \cos(\sigma_3) + C_{4,3} \sqrt{D_4} \cos(\sigma_4),\end{aligned}$$ where the first term of Eq. (\[eq:hamdLaD\]) vanishes (because $\Gamma$ is constant), and $C_{i,j}$ are constant coefficients that depend on the masses and nominal semi-major axes ($a_{i,0}$). I provide explicit formulas for the case of the 3:4:6:8 resonant chain in Appendix \[sec:coef346\]. Since the Hamiltonian (Eq. (\[eq:hamdLaDk223\])) is developed at first order in eccentricities, only first-order resonances appear. In particular, the 3/8 resonance between planets 1 and 4 is neglected since it would only appear at order 5 in eccentricities. In order to use a consistent (canonical) set of coordinates, $\Delta \Lambda_i$ should be replaced in Eq. (\[eq:hamdLaDk223\]) by $$\begin{aligned} \Delta\Lambda_1 &=& \Delta L_1,\nonumber\\ \Delta\Lambda_2 &=& \Delta L_2 - 2\Delta L_1,\nonumber\\ \Delta\Lambda_3 &=& \Delta L_1 - 3(\Delta L_2+\epsilon),\nonumber\\ \Delta\Lambda_3 &=& 2 \Delta L_2 + 4\epsilon,\end{aligned}$$ where $$\epsilon = D - \delta = \sum_{i=1}^n \Delta\Lambda_{i},$$ which measures the distance of the system to the exact resonance, where$$D = \sum_{i=1}^n D_i,$$ is the total deficit of angular momentum, and $$\delta = \sum_{i=1}^n \Lambda_{i,0} - G$$ is the nominal total deficit of angular momentum (at exact resonance). Since $\Lambda_{i,0}$ and $G$ are constants, $\delta$ is also a conserved quantity and can be used as a parameter instead of $G$. The other parameter $\Gamma$ does not appear explicitly in the Hamiltonian, but is hidden in the values of $\Lambda_{i,0}$, $a_{i,0}$, and $n_{i,0}$. Indeed, the nominal semi-major axis ratios are fixed at the resonant values (Eq. (\[eq:nominalsma\])), but $\Gamma$ sets the global scale of the system ($\Gamma = 8 \Lambda_{1,0} + 6\Lambda_{2,0} + 4\Lambda_{3,0} + 3\Lambda_{4,0}$, for a 3:4:6:8 chain). The value of $\Gamma$ does not influence the dynamics of the system appart from changing the scales of distance, time, and energy [@delisle_dissipation_2012]. Therefore, one only need to vary the value of $\delta/\Gamma$ to study the evolution of the phase space (in particular the positions of fixed points). For a given value of $\delta/\Gamma$, the fixed points are found by solving the following system of equations $$\begin{aligned} \label{eq:Hamjaceqs} \dot{\sigma}_i &=& \frac{\partial\H}{\partial D_i} = 0 \qquad (i\leq n),\nonumber\\ \dot{D}_i &=& -\frac{\partial\H}{\partial \sigma_i} = 0 \qquad (i\leq n),\nonumber\\ \dot{\phi}_i &=& \frac{\partial\H}{\partial L_i} = 0 \qquad (i\leq n-2),\nonumber\\ \dot{L}_i &=& -\frac{\partial\H}{\partial \phi_i} = 0 \qquad (i\leq n-2).\end{aligned}$$ This is a system of $4(n-1)$ equations, with $4(n-1)$ unknowns ($2(n-1)$ DOF),which in general possesses a finite number of solutions. These solutions can correspond to elliptical (stable) fixed points or hyperbolic (unstable) ones. To assess the stability of fixed points, I compute the eigenvalues of the linearized equations of motions around the fixed point. Stable fixed points have purely imaginary eigenvalues, while the eigenvalues around unstable fixed points have a non-zero real part. Application to Kepler-223 {#sec:application} ========================= In this section, I apply my model to the four planets orbiting in a 3:4:6:8 resonant chain around . In Sect. \[sec:equilibrium\], I focus on the comparison between the positions of the stable fixed points and the observed configuration of the system (values of resonant angles, eccentricities). In Sect. \[sec:constraints\], I derive constraints on the order in which the planets have been captured in the resonant chain, from the observation of the equilibrium around which the system is currently librating. Equilibrium configurations {#sec:equilibrium} -------------------------- {width="\linewidth"} ![Comparison of the observed libration amplitudes of the Laplace angles in the system [gray rectangle, values taken from @mills_resonant_2016], with the positions of the (six) fixed points as determined from the analytical model. I use the same colors as in Fig. \[fig:I\]. The observations are consistent with a libration around the red fixed point.[]{data-label="fig:II"}](phiij){width="\linewidth"} I follow the method described in Sect. \[sec:model\], and expand the Hamiltonian at first order in eccentricities (Eq. (\[eq:hamdLaDk223\])), to solve for the positions of the fixed points (Eq. (\[eq:Hamjaceqs\])) as a function of the parameter $\delta/\Gamma$ (see Appendix \[sec:fixedpoints\] for more details). I only consider here the stable (elliptical) fixed points that correspond to the libration in resonance. In the case of , I find six families of stable fixed points (parameterized by $\delta/\Gamma$), corresponding to six possible areas of libration for the system. I show in Fig. \[fig:I\] the positions of these fixed points, and compare them with the observed values [taken from @mills_resonant_2016]. As shown by @mills_resonant_2016, the TTVs constrain the values of $\phi_1$ and $\phi_2$ very well, while the eccentricities are only roughly determined. The two-planet resonant angles $\sigma_i$ (which depend on the longitudes of periastron) are not well constrained [@mills_resonant_2016], so the theoretical values cannot be compared to the observations. As $\delta/\Gamma$ increases, the eccentricities increase, but the angles ($\phi_i$, $\sigma_i$) remain constant. Since the best observational constraints are on the values of $\phi_1$ and $\phi_2$, I compare in Fig. \[fig:II\] the observed values of these angles, with the six possible equilibrium values (which are independent of $\delta/\Gamma$). The observations correspond very well to a libration of the system around one of the six configurations (the red one). This result confirms that the system is captured in the resonant chain, and that my model is correct in first approximation. It is interesting to wonder why the system was captured around this particular configuration and not one of the five others. This might simply be by chance (probability of 1/6 for each configuration), but might also greatly depend on the migration scenario. Observing around which equilibrium the resonant chain is librating could provide interesting constraints on the migration undergone by the planets. Constraints on migration scenarios {#sec:constraints} ---------------------------------- [c|cccccc|c]{} order & ABC & ACB & BAC & BCA & CAB & CBA & mean\ \# & 1543 & 516 & 1343 & 980 & 506 & 870 &\ ------------------------------------------------------------------------ & & & & & & &\ green & $32.0_{-1.2}^{+1.2}$ & $14.5_{-1.6}^{+1.7}$ & $52.3_{-1.4}^{+1.4}$ & $11.5_{-1.0}^{+1.1}$ & $ 9.1_{-1.3}^{+1.5}$ & $22.9_{-1.4}^{+1.5}$ & 23.7\ blue & $ 3.3_{-0.5}^{+0.5}$ & $17.6_{-1.7}^{+1.8}$ & $ 4.8_{-0.6}^{+0.7}$ & $18.5_{-1.3}^{+1.3}$ & $28.1_{-2.0}^{+2.1}$ & $31.1_{-1.6}^{+1.6}$ & 17.2\ orange & $39.0_{-1.3}^{+1.3}$ & $18.4_{-1.7}^{+1.9}$ & $ 5.7_{-0.6}^{+0.7}$ & $ 4.3_{-0.6}^{+0.7}$ & $11.9_{-1.5}^{+1.6}$ & $ 1.7_{-0.4}^{+0.6}$ & 13.5\ black & $ 3.4_{-0.5}^{+0.5}$ & $18.0_{-1.7}^{+1.9}$ & $ 1.4_{-0.3}^{+0.4}$ & $ 1.7_{-0.4}^{+0.5}$ & $16.6_{-1.7}^{+1.8}$ & $ 2.9_{-0.6}^{+0.7}$ & 7.3\ cyan & $18.3_{-1.0}^{+1.0}$ & $ 8.3_{-1.2}^{+1.4}$ & $24.9_{-1.2}^{+1.2}$ & $11.9_{-1.0}^{+1.1}$ & $ 8.3_{-1.2}^{+1.4}$ & $ 6.3_{-0.8}^{+0.9}$ & 13.0\ \[tab:I\] ![Example of a simulation that successfully reproduces the observed configuration of the system. The scenario of capture is of type BCA (see Sect. \[sec:constraints\]). I plot the period ratio between each consecutive pair of planets (*top*), the planets’ eccentricities (*middle*), and the two Laplace angles (*bottom*). The two horizontal black lines in the *bottom* plot represent the equilibrium values of the Laplace angles expected from the analytical model (see Sect. \[sec:equilibrium\]).[]{data-label="fig:III"}](k223exmig){width="\linewidth"} As explained in Sect. \[sec:introduction\], when two planets are captured in a MMR, their eccentricities increase until they reach an equilibrium [e.g., @delisle_dissipation_2012], their period ratio remains locked at the resonant value, and the arguments of the resonance librate. For a resonance between two planets, and at low eccentricities, the equilibrium configuration is unique. When additional planets join the chain, the system evolves toward a new equilibrium (eccentricities, two-planet resonant angles, Laplace angles). However, when more than two planets are involved in the chain, there can be more than one equilibrium (see Sect. \[sec:equilibrium\]), and the probability of capture around each of the new equilibria is not necessarily equal. Moreover, this probability might depend on the order in which the planets are captured in the chain. Indeed, the planets that are already captured in resonance have their eccentricities excited, and the angles associated to the already formed resonances are librating around an equilibrium, while planets that are not yet captured should have lower eccentricities, and their mean longitudes should be randomly distributed. Therefore, the initial conditions (angles, eccentricities) at the moment of the capture of a planet in the chain greatly depend on which planets are already in the chain. In this section, I investigate how the order in which the planets are captured in the resonant chain influences the probability of capture around each of the six equilibria found in the case of (see Sec. \[sec:equilibrium\]). This problem is very complex (the phase space has 6 DOF). In particular, each time an additional planet joins the chain, the system might cross one or several separatrices before being captured around one of the new equilibria. I do not attempt to treat this problem analytically, but rather numerically estimate the probabilities of capture by running 6000 N-body simulations including prescriptions for the migration, with varying initial conditions. I use the same integrator as in @delisle_stability_2015, and set constant timescales for the migration torque and the eccentricity damping of each planet. The migration timescales ($T_{\text{mig.},i}$) are set to $10^{6}$, $3\times 10^{5}$, $2\times 10^{5}$, and $1.5\times 10^{5}$ yr (from the innermost to the outermost planet). The eccentricity damping timescales ($T_{\text{ecc.},i}$) are set such that for each planet $T_{\text{mig.},i}/T_{\text{ecc.},i} = 50$. The planets start with circular and coplanar orbits. The innermost planet is at 1 AU, and $\lambda_1 = 0$, and the other planets’ semi-major axes and mean longitudes are randomly drawn. The mean longitudes are drawn from a uniform distribution between 0 and $2\pi$. The semi-major axis ratio $a_2/a_1$ is uniformly drawn in the range $[1.23, 1.3]$, $a_3/a_2$ in the range $[1.32, 1.38]$, and $a_4/a_3$ in the range $[1.23, 1.28]$. This allows the planet pairs to be captured in MMR in any possible order. The orbits are integrated for $5\times 10^4$ yr. Among the 6000 simulations, 5758 are captured in the 3:4:6:8 resonant chain, while 242 are captured in other chains (or not captured). All of the 5758 captured simulations are observed to librate around one of the six equilibria predicted by my analytical model. This confirms that this first-order model is correctly describing the dynamics of the resonant chain. For each simulation, I check in which order the planets were captured, and around which equilibrium the system ended. The results are shown in Table \[tab:I\]. I use ‘A’ to denote the capture of the two innermost planets (1,2) in the 4/3 MMR, ‘B’ the capture of the intermediate ones (2,3) in the 3/2 MMR, and ‘C’ the capture of the outermost planets (3,4) in the 4/3 MMR. Thus, ABC means that the planets were captured in the chain from the innermost to the outermost one, and so on. For each possible order of capture (ordering of A, B, and C), I compute the percentage of capture around each equilibrium. If the captures were equally probable, the percentages would all be of about $16.7\%$ (1/6). This is clearly not the case for . For instance, the black equilibrium is difficult to reach (low probability) except in the cases ACB and CAB (see Table \[tab:I\]). This means that the two innermost planets and the two outermost ones must first be captured in two independent two-planet resonances, and then the two pairs join to form the four-planet resonant chain. Since the system is currently observed around the red equilibrium (see Figs. \[fig:I\] and \[fig:II\]), it is interesting to look at the corresponding capture probabilities. The most favorable case is BCA, with a probability of $52\%$ to capture the system around the observed equilibrium (Table \[tab:I\]). This case corresponds to a capture of planets 2 and 3 in the 3/2 MMR, then planet 4 joins the chain (2:3:4), and finally planet 1 is captured to form the 3:4:6:8 chain. Figure \[fig:III\] shows an example of a simulation following this scenario, and reproducing the observed configuration of the system. Conversely, in the case ABC (capture from the innermost to the outermost planet), the probability to reproduce the observed configuration is only $4.1\%$. The mean capture probability around the observed configuration (assuming equal probability for each capture order), is the highest value ($25.2\%$). It is thus not surprising to observe the system in this configuration, rather than in one of the five others. Discussion {#sec:discussion} ========== In this article I describe an analytical model of resonant chains. The model is valid for any number of planets involved in the chain, and for any resonances (of any order). In particular, I use it to determine the equilibrium configurations around which a resonant chain librates. I show that contrarily to two-planet MMR, multiple equilibria may exist, even at low eccentricities, when three or more planets are involved. I specifically study the case of the four planets around which have been confirmed to be captured in a 3:4:6:8 resonant chain [using TTVs, see @mills_resonant_2016]. Using the analytical model expanded at first order in eccentricities, I show that six equilibrium configurations exist for this system, and the planets might have been captured around any of these six equilibria. However, the capture probabilities are not the same for each equilibrium, and depend on the order in which the planets have been captured in the chain. Using N-body integrations including migration prescriptions, I show that the scenario the most capable of reproducing the observed configuration of the system is to first capture the intermediate planets (2 and 3) in the 3/2 MMR, then to capture the outermost planet (4) to form a 2:3:4 chain between the three outermost planets, and finally to capture the innermost planet (1). This scenario of capture reproduces the observed configuration in $52\%$ of the simulations, while capturing the planets from the innermost to the outermost reproduces the observed configuration for only $4.1\%$ of the simulations. It should be noted that several hypotheses are made to compute these statistics. The planets are initially outside the resonances, with period ratios slightly higher than the resonant values. The migration and eccentricity damping timescales are fixed for each planet, such that the period ratio between each pair decreases (convergent migration). The planets are initially on circular and coplanar orbits. I only vary the initial semi-major axes and initial mean longitudes of the planets. The statistics would probably slightly change with a different set-up for the simulations, but this would not change the two main results of this study: 1. Resonant chains can be captured around several equilibrium configurations (six for ), 2. observing a system around one of the possible equilibria provides useful constraints on the scenario of formation and migration of the planets. Other properties of the resonant chain might provide useful additional constraints on such a scenario. For instance, in the case of , the two inner planets, as well as the two outer ones, are in a compact 4/3 resonance. If these planets formed with much wider separations, they must have avoided permanent capture into the 2/1 and 3/2 resonances to reach the currently observed 4/3 resonance. However, determining a complete scenario for the formation of the system is a highly degenerated problem and is beyond the scope of this article. I thank the anonymous referee for his/her useful comments. I acknowledge financial support from the Swiss National Science Foundation (SNSF). This work has, in part, been carried out within the framework of the National Centre for Competence in Research PlanetS supported by SNSF. Change of coordinates in the general case {#sec:generalchgcoord} ========================================= In this Appendix I describe the change of coordinates corresponding to Eq. (\[eq:k223angacts\]) in the general case. I introduce the angles $$\begin{aligned} \label{eq:angles} \phi_i &=& \frac{1}{c_i} \left( \frac{k_{i,i+1}}{q_{i,i+1}}\lambda_i +\frac{k_{i+2,i+1}}{q_{i+1,i+2}}\lambda_{i+2} \right.\nonumber\\ &&\phantom{\frac{1}{c_i} (} \left. -\left(\frac{k_{i+1,i}}{q_{i,i+1}}+\frac{k_{i+1,i+2}}{q_{i+1,i+2}} \right)\lambda_{i+1} \right) \qquad (i \leq n-2),\nonumber\\ \phi_{n-1} &=& \lambda_{n-1} - \lambda_n,\nonumber\\ \phi_{n} &=& \frac{k_{n,n-1}\lambda_n - k_{n-1,n}\lambda_{n-1}}{q_{n-1,n}},\nonumber\\ \sigma_i &=& \phi_n - \varpi_i,\end{aligned}$$ where the coefficients $c_i$ are renormalizing factors given by Eq. (\[eq:ci\]). The actions canonically conjugated to $\sigma_i$ are the AMD of the planets ($D_i$), while the actions canonically conjugated to $\phi_i$ are $$\begin{aligned} \label{eq:actions} L_i &=& \sum_{j=1}^i c_i l_{i,j} \Lambda_j \qquad (i\leq n-2),\nonumber\\ \Gamma &=& \frac{k_{n-1,n}}{q_{n-1,n}}\left( \Lambda_n + \frac{k_{n,n-1}}{k_{n-1,n}} \left( \Lambda_{n-1} + \frac{k_{n-1,n-2}}{k_{n-2,n-1}} \left( ... + \frac{k_{2,1}}{k_{1,2}}\Lambda_1 \right)\right)\right),\nonumber\\ G &=& \sum_{i=1}^n \Lambda_i - D_i = \sum_{i=1}^n G_i,\end{aligned}$$ where $$\label{eq:lij} l_{i,j} = \frac{1}{k_{i,i+1}} \sum_{r=j}^i q_{r,r+1} \prod_{s=r+1}^{i}\frac{k_{s+1,s}}{k_{s-1,s}} \qquad (j\leq i \leq n-2).$$ It can be shown that the Hamiltonian does not depend on the angle $\phi_n$, and the conjugated action $G$ (total angular momentum of the system) is thus conserved (as expected). Since $\phi_{n-1}$ is the only fast angle in the new system of coordinates, the averaging is done over this angle, and $\Gamma$ (conjugate of $\phi_{n-1}$) is thus constant in the averaged problem. The remaining $2(n-1)$ DOF (once $G$ and $\Gamma$ are fixed), are thus defined by the angles $\sigma_i$ ($i\leq n$) and $\phi_i$ ($i\leq n-2$), and their conjugated actions $D_i$ ($i\leq n$) and $L_i$ ($i\leq n-2$). The angles $\phi_i$ ($i\leq n-2$) are the Laplace resonant angles between each group of three consecutive planets. The angles $\sigma_{n-1}$ and $\sigma_{n}$ are the classical two-planet resonant angles between the two outermost planets, and any two-planet (not necessarily consecutive) resonant angle can be expressed as a combination of the angles $\sigma_i$ ($i\leq n$) and $\phi_i$ ($i\leq n-2$). In order to express the Hamiltonian in these new coordinates, one first needs to invert the change of coordinates (i.e., express $\Lambda_i$, $\lambda_i$, etc. as functions of $L_i$, $\phi_i$, etc.). For the angles, the inverse transformation reads $$\begin{aligned} \label{eq:invangles} \varpi_i &=& \phi_n - \sigma_i,\nonumber\\ \lambda_i &=& \phi_n + \frac{k_{n-1,n}}{q_{n-1,n}}\prod_{j=i}^{n-1} \frac{k_{j+1,j}}{k_{j,j+1}} \phi_{n-1} + \sum_{j=i}^{n-2} c_j l_{j,i} \phi_j.\end{aligned}$$ Then, the Fourier expansion of the perturbative part of the averaged Hamiltonian exhibits angles of the form $$d (k_{j,i} \lambda_j - k_{i,j} \lambda_i) + p \varpi_i + (d q_{i,j} - p) \varpi_j$$ Since all these angles should be expressed as combinations of $\sigma_i$ ($i \leq n$) and $\phi_i$ ($i\leq n-2$), one have to make sure that only integers appear in the combinations. Indeed, if one of the coefficients is a fraction, then the corresponding angle will not be $2\pi$-periodic. For instance, if one of the angles appearing in the Fourrier series is $3/2 \phi_1 + ...$, then $\phi_1 = 0$ is not equivalent to $\phi_1 = 2\pi$. One should also make the coefficients as small as possible, to avoid introducing an unnecessary periodicity in the Hamiltonian (and thus an artificial duplication of fixed points). I thus choose the renormalizing factors $c_i$ to obtain the smallest possible integers in the Fourier expansion of the perturbative part. $$\label{eq:ci} c_i = \frac{\operatorname{lcm}_{r\leq i, r<s}(D_{i,r,s})}{\gcd_{r\leq i, r<s}(N_{i,r,s})} \qquad (i \leq n-2),$$ where $$\frac{N_{i,r,s}}{D_{i,r,s}} = k_{s,r} l_{i,s} - k_{r,s} l_{i,r} \qquad (l_{i,s} = 0 \text{ for } s>i)$$ is the fraction reduced to its simplest form. The Hamiltonian is expanded in power series of the eccentricities, and truncated at a given degree $d_\text{max}$. Only the combinations for which the degree $q_{r,s} = k_{s,r}-k_{r,s} \leq d_\text{max}$ appear in the truncated Hamiltonian. Therefore, only these combinations should be considered in the computation of the coefficient $c_i$ in Eq. (\[eq:ci\]). For the actions, the inverse change of coordinates reads $$\begin{aligned} \label{eq:invacts} \Lambda_i &=& \frac{k_{i,i+1}}{c_i q_{i,i+1}} L_i + \frac{1}{c_{i-2}}\frac{k_{i,i-1}}{q_{i-1,i}} L_{i-2}\nonumber\\ && - \frac{1}{c_{i-1}}\left(\frac{k_{i,i-1}}{q_{i-1,i}} + \frac{k_{i,i+1}}{q_{i,i+1}}\right) L_{i-1} \qquad (i\leq n-2),\nonumber\\ \Lambda_{n-1} &=& \Gamma - \frac{k_{n-1,n}}{q_{n-1,n}} (G+D) + \frac{1}{c_{n-3}}\frac{k_{n-1,n-2}}{q_{n-2,n-1}} L_{n-3}\nonumber\\ && - \frac{1}{c_{n-2}}\left(\frac{k_{n-1,n-2}}{q_{n-2,n-1}} + \frac{k_{n-1,n}}{q_{n-1,n}}\right) L_{n-2},\nonumber\\ \Lambda_n &=& \frac{k_{n,n-1}}{q_{n-1,n}} (G+D)- \Gamma + \frac{1}{c_{n-2}}\frac{k_{n,n-1}}{q_{n-1,n}} L_{n-2},\end{aligned}$$ where $$D = \sum_{i=1}^n D_i = \sum_{i=1}^n \Lambda_i - G$$ is the total deficit of angular momentum. I introduce $$\delta = \sum_{i=1}^n \Lambda_{i,0} - G,$$ which is the nominal total deficit of angular momentum (at exact resonance). Since $\Lambda_{i,0}$ and $G$ are constants, $\delta$ is also a conserved quantity. I additionally define $$\epsilon = D - \delta = \sum_{i=1}^n \Delta\Lambda_{i},$$ which measures the distance of the system to the exact resonance, such that $$\begin{aligned} \label{eq:invactionsbis} \Delta\Lambda_i &=& \frac{k_{i,i+1}}{c_i q_{i,i+1}} \Delta L_i + \frac{1}{c_{i-2}}\frac{k_{i,i-1}}{q_{i-1,i}} \Delta L_{i-2}\nonumber\\ && - \frac{1}{c_{i-1}}\left(\frac{k_{i,i-1}}{q_{i-1,i}} + \frac{k_{i,i+1}}{q_{i,i+1}}\right) \Delta L_{i-1} \qquad (i\leq n-2),\nonumber\\ \Delta \Lambda_{n-1} &=& - \frac{k_{n-1,n}}{q_{n-1,n}} \epsilon + \frac{1}{c_{n-3}}\frac{k_{n-1,n-2}}{q_{n-2,n-1}} \Delta L_{n-3}\nonumber\\ && - \frac{1}{c_{n-2}}\left(\frac{k_{n-1,n-2}}{q_{n-2,n-1}} + \frac{k_{n-1,n}}{q_{n-1,n}}\right) \Delta L_{n-2},\nonumber\\ \Delta \Lambda_n &=& \frac{k_{n,n-1}}{q_{n-1,n}} \epsilon + \frac{1}{c_{n-2}}\frac{k_{n,n-1}}{q_{n-1,n}} \Delta L_{n-2},\end{aligned}$$ with $\Delta L_i = L_i - \sum_{j\leq i} c_i l_{i,j}\Lambda_{j,0}$. Coefficients of the Hamiltonian for a 3:4:6:8 resonant chain {#sec:coef346} ============================================================ In this appendix, I provide the expressions of the coefficients $C_{i,j}$ that appear in the Hamiltonian of a 3:4:6:8 resonant chain at first order in eccentricities (see Eq. \[eq:hamdLaDk223\]). I follow the method described in @laskar_stability_1995. Each coefficient is a sum of two components, coming from the direct and indirect part of the perturbation $$\begin{aligned} C_{i,j} &=& -\G \frac{m_i m_j}{a_{j,0}} \sqrt{\frac{2}{\Lambda_{i,0}}}\left( C_{i,j}^\text{dir} - \frac{\G m_0}{\sqrt{\mu_i\mu_j\alpha_{i,j}}} C_{i,j}^\text{ind} \right),\nonumber\\ C_{j,i} &=& -\G \frac{m_i m_j}{a_{j,0}} \sqrt{\frac{2}{\Lambda_{j,0}}}\left( C_{j,i}^\text{dir} - \frac{\G m_0}{\sqrt{\mu_i\mu_j\alpha_{i,j}}} C_{j,i}^\text{ind} \right),\end{aligned}$$ where $i<j$, $\alpha_{i,j} = a_{i,0}/a_{j,0}$, and $$\begin{aligned} C_{i,j}^\text{dir} &=& \frac{2}{3} \alpha_{i,j}^{-1} b_{3/2}^{(1)} - b_{3/2}^{(0)} + \frac{7}{6} \alpha_{i,j} b_{3/2}^{(1)} - \frac{5}{2} \alpha_{i,j}^{2} b_{3/2}^{(0)} + \frac{5}{3} \alpha_{i,j}^{3} b_{3/2}^{(1)}\nonumber\\ &\approx&-1.190494,\nonumber\\ C_{j,i}^\text{dir} &=& -b_{3/2}^{(1)} + \frac{5}{2} \alpha_{i,j} b_{3/2}^{(0)} - \frac{3}{2} \alpha_{i,j}^{2} b_{3/2}^{(1)}\nonumber\\ &\approx&1.688311,\nonumber\\ C_{i,j}^\text{ind} &=& 0,\nonumber\\ C_{j,i}^\text{ind} &=& 1,\end{aligned}$$ for a 2/1 resonance between $i$ and $j$ (planets pairs 1,3 and 2,4), $$\begin{aligned} C_{i,j}^\text{dir} &=& \frac{4}{5} \alpha_{i,j}^{-2} b_{3/2}^{(1)} - \frac{6}{5} \alpha_{i,j}^{-1} b_{3/2}^{(0)} + \frac{31}{30} b_{3/2}^{(1)} - \frac{11}{10} \alpha_{i,j} b_{3/2}^{(0)}\nonumber\\ &&+ \frac{23}{15} \alpha_{i,j}^{2} b_{3/2}^{(1)} - \frac{16}{5} \alpha_{i,j}^{3} b_{3/2}^{(0)} + \frac{32}{15} \alpha_{i,j}^{4} b_{3/2}^{(1)}\nonumber\\ &\approx&-2.025223,\nonumber\\ C_{j,i}^\text{dir} &=& -\alpha_{i,j}^{-1} b_{3/2}^{(1)} + \frac{3}{2} b_{3/2}^{(0)} - \frac{3}{2} \alpha_{i,j} b_{3/2}^{(1)} + 3 \alpha_{i,j}^{2} b_{3/2}^{(0)} - 2 \alpha_{i,j}^{3} b_{3/2}^{(1)}\nonumber\\ &\approx&2.484005,\nonumber\\ C_{i,j}^\text{ind} &=& 0,\nonumber\\ C_{j,i}^\text{ind} &=& 0,\end{aligned}$$ for a 3/2 resonance (planets pair 2,3), and $$\begin{aligned} C_{i,j}^\text{dir} &=& \frac{32}{35} \alpha_{i,j}^{-3} b_{3/2}^{(1)} - \frac{48}{35} \alpha_{i,j}^{-2} b_{3/2}^{(0)} + \frac{36}{35} \alpha_{i,j}^{-1} b_{3/2}^{(1)} - \frac{36}{35} b_{3/2}^{(0)}\nonumber\\ &&+ \frac{17}{14} \alpha_{i,j} b_{3/2}^{(1)} - \frac{93}{70} \alpha_{i,j}^{2} b_{3/2}^{(0)} + \frac{64}{35} \alpha_{i,j}^{3} b_{3/2}^{(1)}\nonumber\\ &&- \frac{132}{35} \alpha_{i,j}^{4} b_{3/2}^{(0)} + \frac{88}{35} \alpha_{i,j}^{5} b_{3/2}^{(1)}\nonumber\\ &\approx&-2.840432,\nonumber\\ C_{j,i}^\text{dir} &=& -\frac{16}{15} \alpha_{i,j}^{-2} b_{3/2}^{(1)} + \frac{8}{5} \alpha_{i,j}^{-1} b_{3/2}^{(0)} - \frac{19}{15} b_{3/2}^{(1)} + \frac{13}{10} \alpha_{i,j} b_{3/2}^{(0)}\nonumber\\ &&- \frac{53}{30} \alpha_{i,j}^{2} b_{3/2}^{(1)} + \frac{18}{5} \alpha_{i,j}^{3} b_{3/2}^{(0)} - \frac{12}{5} \alpha_{i,j}^{4} b_{3/2}^{(1)}\nonumber\\ &\approx&3.283257,\nonumber\\ C_{i,j}^\text{ind} &=& 0,\nonumber\\ C_{j,i}^\text{ind} &=& 0,\end{aligned}$$ for a 4/3 resonance (planets pairs 1,2 and 3,4). The coefficients $b_{3/2}^{(0)}$ and $b_{3/2}^{(1)}$ are the Laplace coefficients [e.g., @laskar_stability_1995], evaluated at $\alpha_{i,j}$. Fixed points at first order {#sec:fixedpoints} =========================== In this appendix I describe in more detail how to determine the position of the fixed points of the averaged problem, at first order in eccentricities. In this case, the Hamiltonian takes the form $$\H = \H_0(\epsilon, \Delta L_i) + \sum_{i,j\neq i} C_{i,j} \sqrt{D_i} \cos(\sigma_i + \vec{p}_{i,j}.\vec\phi),$$ where $\vec{p}_{j,i} = \vec{p}_{i,j}$ are vectors of $n-2$ known integer coefficients (which depend on the considered resonances), and $\vec\phi$ is the vector of $\phi_i$ ($i \leq n-2$). The fixed points of the averaged problem are solutions of the following set of equations (see Eq.(\[eq:Hamjaceqs\])) $$\begin{aligned} 0 &=& \frac{\partial\H}{\partial D_i} = \frac{\partial\H_0}{\partial\epsilon} + \frac{1}{2\sqrt{D_i}} \sum_{j\neq i} C_{i,j} \cos(\sigma_i + \vec{p}_{i,j}.\vec\phi)\\ 0 &=& -\frac{\partial\H}{\partial\sigma_i} = \sqrt{D_i} \sum_{j\neq i} C_{i,j} \sin(\sigma_i + \vec{p}_{i,j}.\vec\phi)\\ 0 &=& \frac{\partial\H}{\partial\Delta L_i} = \frac{\partial\H_0}{\partial\Delta L_i}\\ 0 &=& -\frac{\partial\H}{\partial\vec{\phi}} = \sum_{i,j\neq i} \vec{p}_{i,j} C_{i,j} \sqrt{D_i} \sin(\sigma_i + \vec{p}_{i,j}.\vec\phi),\end{aligned}$$ from which I deduce $$\begin{aligned} \label{eq:solDsig} \sqrt{D_i} \expo{\im \sigma_i} &=& - \frac{1}{2}\left(\frac{\partial\H_0}{\partial\epsilon}\right)^{-1} \sum_{j\neq i} C_{i,j} \expo{-\im \vec{p}_{i,j}.\vec\phi},\\ \label{eq:solphi} 0 &=& \sum_{i,j\neq i,r\neq i} \vec{p}_{i,j} C_{i,j} C_{i,r} \sin\left((\vec{p}_{i,j}-\vec{p}_{i,r}).\vec\phi\right).\end{aligned}$$ The parameter $\delta/\Gamma$ only appears in these equations through the value of $\frac{\partial\H_0}{\partial\epsilon}$. Therefore, at first order in eccentricities, all the angles, as well as the eccentricity ratios are independent of $\delta/\Gamma$. The value of this parameter only changes a factor common to all eccentricities (see Eq. (\[eq:solDsig\])). Equation (\[eq:solphi\]) provides a set of $n-2$ equations on the $n-2$ angles $\phi_i$. For instance, for a 3:4:6:8 resonant chain such as , I obtain (see Eqs. (\[eq:hamdLaDk223\]) and (\[eq:solphi\])) $$\begin{aligned} \label{eq:solphik223} 0 &=& 2C_{1,2}C_{1,3}\sin(2\phi_1)\nonumber\\ && +3C_{2,1}C_{2,3}\sin(3\phi_1) + 3C_{2,1}C_{2,4}\sin(3\phi_1+\phi_2)\nonumber\\ && + C_{3,1}C_{3,2}\sin(\phi_1) + C_{3,1}C_{3,4}\sin(\phi_1+2\phi_2),\nonumber\\ 0 &=& C_{2,1}C_{2,4}\sin(3\phi_1+\phi_2) + C_{2,3}C_{2,4}\sin(\phi_2)\nonumber\\ && + 2C_{3,1}C_{3,4}\sin(\phi_1+2\phi_2) + 2C_{3,2}C_{3,4}\sin(2\phi_2)\nonumber\\ && + C_{4,2}C_{4,3}\sin(\phi_2).\end{aligned}$$ There are trivial solutions at $0$ and $\pi$, but other (asymmetric) solutions might exist. The existence of asymmetric solutions is due to the influence of first-order resonances between non-consecutive pairs. In the case where only consecutive pairs are involved in resonances, Eq. (\[eq:solphi\]) simplifies, and it can be shown that only symmetric solutions exist. The number of solutions of a highly non-linear set of equations such as Eq. (\[eq:solphik223\]) is not easily predicted. Moreover, among those solutions, some correspond to elliptical (stable) fixed points, and the others to hyperbolic (unstable) fixed points. In the case of , I solve for the position of fixed points numerically, and only consider the stable fixed points. I find six possible stable solutions. Once a solution is found for $\phi_i$ ($i \leq n-2$), the angles $\sigma_i$, and the eccentricity ratios can easily be deduced from Eq. (\[eq:solDsig\]) $$\label{eq:solesig} e_i \expo{\im \sigma_i} \approx e_0 \sum_{j\neq i} \frac{m_j}{m_0} C'_{i,j} \expo{-\im \vec{p}_{i,j}.\vec\phi},\\$$ with $$\begin{aligned} C'_{i,j} &=& - \frac{a_{n,0}^{3/2}}{a_{\max(i,j),0} \sqrt{a_{i,0}}} \left( C_{i,j}^\text{dir} - \frac{C_{i,j}^\text{ind}}{\sqrt{\alpha_{i,j}}}\right),\nonumber\\ e_0 &=& - n_{n,0} \left(\frac{\partial\H_0}{\partial\epsilon}\right)^{-1}.\end{aligned}$$ For a 3:4:6:8 resonant chain such as , I obtain $$\begin{aligned} \frac{e_1}{e_0} \expo{\im (4\lambda_2 - 3\lambda_1 - \varpi_1)} &\approx& 6.2526 \frac{m_2}{m_0} + 1.9999 \frac{m_3}{m_0} \expo{-\im 2\phi_1},\nonumber\\ \frac{e_2}{e_0} \expo{\im (3\lambda_3 - 2\lambda_2 - \varpi_2)} &\approx& - 6.5665 \frac{m_1}{m_0} \expo{\im 3\phi_1} +3.0911 \frac{m_3}{m_0}\nonumber\\ &&+1.4999 \frac{m_4}{m_0} \expo{-\im \phi_2},\nonumber\\ \frac{e_3}{e_0} \expo{\im (4 \lambda_4 - 3 \lambda_3 - \varpi_3)} &\approx& -0.5712 \frac{m_1}{m_0} \expo{\im (\phi_1+2\phi_2)} - 3.3120 \frac{m_2}{m_0} \expo{\im 2\phi_2}\nonumber\\ &&+3.1263 \frac{m_4}{m_0} ,\nonumber\\ \frac{e_4}{e_0} \expo{\im (4 \lambda_4 - 3 \lambda_3 - \varpi_4)} &\approx& -0.4284 \frac{m_2}{m_0} \expo{\im \phi_2} - 3.2833 \frac{m_3}{m_0}.\end{aligned}$$ [^1]: I restrict this study to first order in the planet-star mass ratio, which means that three-planet terms (of order two in the mass) are neglected.

--- abstract: 'We introduce the Connection Scan Algorithm (CSA) to efficiently answer queries to timetable information systems. The input consists, in the simplest setting, of a source position and a desired target position. The output consist is a sequence of vehicles such as trains or buses that a traveler should take to get from the source to the target. We study several problem variations such as the earliest arrival and profile problems. We present algorithm variants that only optimize the arrival time or additionally optimize the number of transfers in the Pareto sense. An advantage of CSA is that is can easily adjust to changes in the timetable, allowing the easy incorporation of known vehicle delays. We additionally introduce the Minimum Expected Arrival Time (MEAT) problem to handle possible, uncertain, future vehicle delays. We present a solution to the MEAT problem that is based upon CSA. Finally, we extend CSA using the multilevel overlay paradigm to answer complex queries on nation-wide integrated timetables with trains and buses.' author: - | Julian Dibbelt, Thomas Pajor, Ben Strasser, Dorothea Wagner\ Karlsruhe Institute of Technology (KIT), Germany\ work done while at KIT\ `algo@dibbelt.de`    `thomas@tpajor.com`\ `strasser@kit.edu`    `dorothea.wagner@kit.edu` date: March 2017 title: 'Connection Scan Algorithm[^1]' --- Introduction ============ We study the problem of efficiently answering queries to timetable information systems. Efficient algorithms are needed as the foundation of complex web services such as the Google Transit or bahn.de - the German national railroad company’s website. To use these websites, the user enters his desired departure stop, arrival stop and a vague moment in time and the system should compute a journey telling the user when to take which train. In practice, trains do not adhere perfectly to the timetable and therefore it is necessary to be able to quickly adjust the scheduled timetable to the actual situation or account in advance for possible delays. At its core, the studied problem setting consists of the classical shortest path problem. This problem is usually solved using Dijkstra’s algorithm [@d-ntpcg-59] which is build around a priority queue. Algorithmic solutions that reduce timetable information systems to variation of the shortest path problem that are solved with extensions of Dijkstra’s algorithm are therefore common. The time-dependent and time-expanded graph [@pswz-emtip-08] approaches are prominent examples. In this work, we present an alternative approach to the problem, namely the *Connection Scan Algorithm* (CSA). The core idea consists of doing away with the priority queue and replacing it with a list of trains sorted by departure time. Contrary to most competitors, CSA is therefore not build upon Dijkstra’s algorithm. The resulting algorithm is comparatively simple because the complexity inherent to the queue is missing. Further, Dijkstra’s algorithm spends most of its execution time within queue operations. Our approach replaces these with faster more elementary operations on arrays. The resulting algorithm is therefore also able of achieving low query running times. A further advantage of our approach is that the data structure consists primarily of an array of trains sorted by departure time. Maintaining a sorted array is easy even when train schedules change. Modern timetable information systems do not only optimize the arrival time. A common approach consists of optimizing several criteria in the Pareto sense [@mswz-tima-07; @dms-mcspt-08; @bm-somcs-09]. The practicality of this approach was shown by [@mw-pspof-01]. The most common second criterion is the number of transfers. Another often requested criterion is the price [@ms-pltcm-06] but we omit this criterion from our study because of very complex realworld pricing schemes. A further commonly considered problem variant consists of profile queries. In this variant the input does not contain a departure time. Instead, the output should contain all optimal journeys between two stops for all possible departure times. As further problem variant, we propose and study the minimum expected arrival time (MEAT) problem setting to compute delay-robust journeys. CSA is very fast as it does not possess a heavyweight preprocessing step. This makes the algorithm comparatively simple but it also makes the running time inherently dependent on the timetable’s size. For very large instances this can be a problem. We therefore study an algorithmic extension called Connection Scan Accelerated (CSAccel) which combines a multilevel overlay approach [@sww-daola-99; @hsw-emlog-08; @dgpw-crprn-13] with CSA. #### Related Work. Finding routes in transportation networks is the focus of many research projects and thus many publications on this subject exist. The published papers can be roughly divided into two categories depending on whether the studied network is timetable-based. As our research focuses on timetable routing, we restrict our exposition to it and refer to a recent survey [@bdgmpsww-rptn-16] for other routing problems. Some techniques are processing-based and have an expensive and slow startup phase. The advantage of preprocessing is, that it decreases query running times. A major problem with preprocessing-based techniques is that the preprocessing needs to be rerun each time that the timetable changes. We start by proving an overview over techniques without preprocessing and afterwards describe the preprocessing-based techniques. The traditional approach consists of extending Dijkstra’s algorithm. Two common methods exist and are called the time-dependent and time-expanded graph models [@pswz-emtip-08]. In [@dkp-pcbcp-12] the time-dependent model has been refined by coloring graph elements. The authors further introduce SPCS, an efficient algorithm to answer earliest arrival profile queries. A parallel version called PSPCS is also introduced. We experimentally compare CSA to SPCS, to the colored time-dependent model and the basic time-expanded model. Another interesting preprocessing-less technique is called RAPTOR and was introduced in [@dpw-rbptr-14]. Just as CSA it does not employ a priority queue and therefore is not based on Dijkstra’s algorithm. It inherently supports optimizing the number of transfers in the Pareto-sense in addition to the arrival time. A profile extension called rRAPTOR also exists. We experimentally compare CSA with RAPTOR and rRAPTOR. Adjusting the time-dependent and time-expanded graphs to account for realtime delays is conceptually straightforward but the details are non-trivial and difficult as the studies of [@ms-etipd-09] and [@cddfgpz-egbmd-14] show. In [@bgm-fdsut-10] SUBITO was introduced. This is an acceleration of Dijkstra’s algorithm applied to the time-dependent graph model. It works using lower bounds on the travel time between stops to prune the search. As slowing down trains does not invalidate the lower bounds, most realworld train delays can be incorporated. However, CSA supports more flexible timetable updates. For example, contrary to SUBITO CSA supports the efficient insertion of connections between stops that were previously not directly connected. In [@w-tbptr-15] trip-based routing (TB) was introduced. It works by computing all possible transfers between trains in a preprocessing step. The preprocessing running times are still well below those of other preprocessing-based techniques but non-negligible. Unfortunately, the achieved query speedup lacks behind techniques with more extensive preprocessing. In [@w-tbptr-16] the technique was extended with a significantly more heavy-weight preprocessing algorithm that stores a large amount of trees to achieve higher speedups. Many more preprocessing-based techniques exist. For example, in [@g-ctnrt-10] Contraction Hierarchy, a very successful technique for road routing, was adapted for timetable-based routing. In [@ddpw-ptl-15], Hub-labeling, another successful technique for roads, was also adapted for timetable-based routing. Another labeling-based approach was proposed in [@wlyxz-erppt-15]. In addition to SUBITO, [@bgm-fdsut-10] introduces $k$-flags. $k$-flags is an adaptation of Arc-Flags [@l-aefea-04], a further successful technique for roads, to timetables. Another well-known preprocessing-based technique is called Transfer Patterns (TP). It was introduced in [@bceghrv-frvlp-10] and was refined since then over the course of several papers. In [@bs-fbspt-14] the authors combined frequency-based compression with routing and used it to decrease the TP preprocessing running times. In [@bhs-stp-16] TP was combined with a bilevel overlay approach to further decrease preprocessing running times. CSAccel is not the first technique to combine multilevel routing with timetables. This was already done in [@swz-umlgt-02]. We postpone giving an overview over the existing papers related to the MEAT problem until Section \[sec:related-work-meat\], as the details of the MEAT problem are described in Section \[sec:MEAT\]. #### Previous publications. This paper is the aggregated journal version of three conference papers. In [@dpsw-isftr-13], we introduced CSA and the very basic MEAT problem. In [@sw-csa-13], we first described CSAccel. In [@dsw-drjtn-14], we present a more in-depth description and evaluation of the MEAT problem setting. #### Contribution. We describe the Connection Scan family of algorithms (CSA) to solve various routing problems in timetable-based networks. We describe profile and non-profile variants. Algorithm variants are described that optimize arrival time and optionally the number of transfers in the Pareto sense. We further describe Connection Scan Accelerated (CSAccel) a combination of CSA with multilevel overlay techniques. Finally, we define the Minimum Expected Arrival Time (MEAT) problem and describe how it can be solved using CSA. All algorithm descriptions are accompanied by an in-depth experimental analysis and experimental comparison with relevant related work. #### Outline. Our paper is organized into five sections. The first section contains the preliminaries. It consists of the formal timetable definition and precisely states nearly all problem settings considered in the following sections. The second section describes the basic CSA without profiles. The third section extends CSA to profiles. In the fourth section, we describe CSAccel, a multilevel extension of CSA. The fifth section formalizes the MEAT problem and describes how it can be solved within the CSA framework. The final section is a conclusion. Preliminaries ============= We describe the Connection Scan algorithm in terms of train networks. Fortunately, many other transportation networks exist with the same timetable-based structure. Flight, ship, and bus networks are examples thereof. We could therefore formulate our work in more abstract terms such as vehicles. However, to avoid an unnecessary clumsy language, we refrain from it, and just refer to every vehicle as train. Timetable Formalization ----------------------- In this section, we formalize the notion of timetable, which is part of the input of nearly every algorithm presented in this paper. We are not the first to present a formalization. However, even though many previous works exist, they differ when it comes to notation and details. We therefore explain our terminology and the model used in our work in detail to avoid confusion. A timetable encodes what trains exist, when they drive, where they drive, and how travelers can transfer between trains. Especially, the details of the last part — changing trains — vary significantly across related work. Unfortunately, unlike one intuitively might expect, these details impact the algorithm design and can have a huge impact on the running time behavior. Further, these details can make a timetable description verbose. Therefore, we first describe the entities not related to transfers, give examples for these, and only afterwards describe the transfer details. A *timetable* is a quadruple $(\mathcal{S},\mathcal{C},\mathcal{T},\mathcal{F})$ of stops $\mathcal{S}$, connections $\mathcal{C}$, trips $\mathcal{T}$, and footpaths $\mathcal{F}$. The footpaths are used to model transfers. We therefore postpone their description until we describe transfers. A *stop* is a position outside of a train where a traveler can stand. At a stop, trains can halt and passengers can enter or leave trains. A *trip* is a scheduled train. A *connection* is a train that drives from one stop to another stop without intermediate halt. Formally, a connection $c$ is a five tuple $(c_{\ensuremath{\mathrm{dep\_stop}}},c_{\ensuremath{\mathrm{arr\_stop}}},c_{\ensuremath{\mathrm{dep\_time}}},c_{\ensuremath{\mathrm{arr\_time}}},c_{\ensuremath{\mathrm{trip}}})$ We refer to these attributes as $c$’s *departure stop*, *arrival stop*, *departure time*, *arrival time*, and *trip*, respectively. We require from every connection $c$ that $c_{\ensuremath{\mathrm{dep\_stop}}}\neq c_{\ensuremath{\mathrm{arr\_stop}}}$ and $c_{\ensuremath{\mathrm{dep\_time}}}<c_{\ensuremath{\mathrm{arr\_time}}}$. All connections with the same trip form a set. We require that this set can be ordered into a sequence $c^1,c^2\ldots c^k$ such that $c^i_{\ensuremath{\mathrm{arr\_stop}}}=c^{i+1}_{\ensuremath{\mathrm{dep\_stop}}}$ and $c^i_{\ensuremath{\mathrm{arr\_time}}}<c^{i+1}_{\ensuremath{\mathrm{dep\_time}}}$ for every $i$. In a slight abuse of notation, we sometimes identify a trip with its corresponding sequence of connections. #### Examples. Examples for stops are the train main stations, such as “Karlsruhe Hbf”. Other examples include subway or tram stations. Trips include high speed trains, subway trains, trams, buses, ferries, and more. An example for a trip is the “ICE 104” from Basel to Amsterdam that departs at 15:13 on the 2-nd of August 2016. Note, that the description “ICE 104” without the departure time does not uniquely identify a trip as such a train exists on every day of August 2016. In our model, there is a trip for every day, even though these trips share the same sequence stop and the operator refers to all trains by the same name. Pick one of the “ICE 104” trips and name it $x$. The first three stops at which $x$ halts are Basel, Freiburg, and Offenburg. There is a connection with departure stop Basel, arrival stop Freiburg, and trip $x$. There further is a connection with departure stop Freiburg, arrival stop Offenburg, and trip $x$. However, there is no connection with departure stop Basel, arrival stop Offenburg, and trip $x$, as we require that the train of a connection does not halt at an intermediate stop. #### Transfers. A traveler standing at stop $s$ at the time point $\tau$ can be described using a pair $(s,\tau)$. To lighten our notation, we denote these pairs as $s@\tau$. Denote by $P$ the infinit set of these pairs. A *transfer model* is a relation on $P$, which we denote using the $\rightarrow$ symbol. A traveler sitting in an incoming connection $c$, wishing to transfer to an outgoing connection $c'$ of another trip, can do so by definition if and only if $c_{\ensuremath{\mathrm{arr\_stop}}}@c_{\ensuremath{\mathrm{arr\_time}}}\rightarrow c'_{\ensuremath{\mathrm{dep\_stop}}}@c'_{\ensuremath{\mathrm{dep\_time}}}$ holds. Many transfer models exist and the details vary significantly across the literature. Unfortunately, there is no consent on what the best model is. In the following, we focus our description on the model used in our work, which is based upon footpaths. We also briefly discuss the differences to other models. A *footpath* $f$ is a triple $(f_{\ensuremath{\mathrm{dep\_stop}}},f_{\ensuremath{\mathrm{arr\_stop}}},f_{\ensuremath{\mathrm{dur}}})$, which we refer to as $f$’s *departure stop*, $f$’s *arrival stop*, and $f$’s *duration*. We require all footpath durations to be positive, i.e., $f_{\ensuremath{\mathrm{dur}}}> 0$. The set of footpaths $\mathcal{F}$ is the last element of the quadruple that characterizes timetables. These footpaths can be viewed as weighted, directed *footpath graph* $G_\mathcal{F}=(\mathcal{S},\mathcal{F})$, where the stops are the nodes, the footpaths the arcs, and the duration the weights. We define the transfer relation as follows: $a@\tau_a \rightarrow b@\tau_b$ holds, if and only if there is a path from $a$ to $b$ whose length is at most $\tau_b - \tau_a$. Having a large connected footpath graph makes the considered problems significantly harder than having only loosely connected components. Following [@dpw-rbptr-14], we therefore introduce two restrictions on the footpath graph. It must be transitively closed and fulfill the triangle inequality. Transitively closed means that if there is an edge $ab$ and an edge $bc$, then there is an edge $ac$. The triangle inequality further requires that $ab_{\ensuremath{\mathrm{dur}}}+ bc_{\ensuremath{\mathrm{dur}}}\ge ac_{\ensuremath{\mathrm{dur}}}$. From these two properties one can show that if there is a path from $a$ to $b$, then there is a shortest $ab$-path with a single edge. The transfer relation in this special case therefore boils down to $$(a@\tau_a \rightarrow b@\tau_b) \iff \exists f \in F : \tau_b - \tau_a \ge f_{\ensuremath{\mathrm{dur}}}\text{ and } a=f_{\ensuremath{\mathrm{dep\_stop}}}\text{ and }b=f_{\ensuremath{\mathrm{arr\_stop}}}$$ which allows us to limit our searches to single-edge paths. These restrictions come at a price. In each connected component there is a quadratic number of edges because of the transitive closure. As a quadratic memory consumption is prohibitive in practice, we can therefore have no large components. Our footpath-based transfer model is transitive, i.e., if $a@\tau_a \rightarrow b@\tau_b$ and $b@\tau_b \rightarrow c@\tau_c$ then $a@\tau_a \rightarrow c@\tau_c$. We exploit this property in our algorithms. While transfer model transitivity sounds like a very reasonable and desirable property, there is a common class of competitor transfer models that do not have it. They are similar to our model, except that instead of requiring transitive closure and triangle inequality, they limit the maximum path length by some constant $m$. It possible that one can walk within time $m$ from $a$ to $b$ and within time $m$ from $b$ to $c$ but require longer than time $m$ to get from $a$ to $c$, which demonstrates that transitivity breaks. The missing transitivity is the main reason why we chose a different model. An interesting special case are *loops* in the footpath graph. Note that without a loop at a stop $s$, a traveler cannot exit at $s$ and enter another train at $s$. In practice, all stops have therefore loops. The duration of the loop footpath at stop $s$ is called the change time[^2] $s^{\ensuremath{\mathrm{change}}}$. Some competitor works even assume that there are no footpaths beside these loops, which is a significant restriction compared to our model. Footpaths that are not loops are *interstop footpaths*. Our transfer model is in general not reflexive, i.e., it is possible that there are stops $s$ and time points $\tau$ such that $s@\tau\not\rightarrow s@\tau$. However, one can study the special case of reflexive transfer models. This requirement translates to every stop having a change time of 0. The London benchmark instance of [@dpw-rbptr-14], which we also use, has this additional property. #### Examples. In our Germany instance, the Karlsruhe main station is modeled as two stops. There is a stop that represents the main tracks used by the long distance trains. Further, there is a stop that represents the tracks where the local trams halt. Both are connected using a footpath per direction. Further, both stops have loop footpaths. The loop of the main track stop has a duration of 5min and the loop of the local tram stop has a duration of 4min. The footpaths between the two stops have a duration of 6min. Transferring between local trams is therefore possible within 4min. To transfer between long distance trains, the traveler needs 5min. Finally, to transfer from tram to long distance train 6min are needed. Other main stations can be modeled using more stops. For example many stations have an additional stop per subway line. Within cities, it can make sense to insert footpaths between neighboring tram stops. However, one has to be careful to not create large connected components in the footpath graph by doing so. It is also possible to model stations in greater detail using a stop per platform. The London instance uses this approach. This approach gives more precise transfer times at the expense of more stops. Journeys -------- A journey describes how a passenger can travel through a timetable network. They are composed of legs, which are pairs of connections $(l^i_{\ensuremath{\mathrm{enter}}},l^i_{\ensuremath{\mathrm{exit}}})$ within the same trip. $l^i_{\ensuremath{\mathrm{enter}}}$ must appear before $l^i_{\ensuremath{\mathrm{exit}}}$ in the trip. Formally, a journey consists of alternating sequence of legs and footpaths $f^0,l^0,f^1,l^1\ldots f^{k-1},l^{k-1},f^k$. A journey must start and end with a footpath. All intermediate transfers must be feasible according to the transfer model, i.e., for all $i$, $(l^{i-1}_{\ensuremath{\mathrm{exit}}})_{\ensuremath{\mathrm{arr\_stop}}}@ (l^{i-1}_{\ensuremath{\mathrm{exit}}})_{\ensuremath{\mathrm{arr\_time}}}\rightarrow (l^i_{\ensuremath{\mathrm{enter}}})_{\ensuremath{\mathrm{dep\_stop}}}@ (l^i_{\ensuremath{\mathrm{enter}}})_{\ensuremath{\mathrm{dep\_time}}}$ must hold. We refer to $f^0$ as *initial footpath* and to $f^k$ as *final footpath*. The remaining footpaths are called *transfer footpaths*. Further, for a journey $j$ we refer to $f^0_{\ensuremath{\mathrm{dep\_stop}}}$ as *$j$’s departure stop*, to $f^k_{\ensuremath{\mathrm{arr\_stop}}}$ as the *$j$’s arrival stop*, to $(l^0_{\ensuremath{\mathrm{enter}}}){\ensuremath{\mathrm{dep\_time}}}- f^0_{\ensuremath{\mathrm{dur}}}$ as *$j$’s departure time*, to *$(l^{k-1}_{\ensuremath{\mathrm{exit}}}){\ensuremath{\mathrm{arr\_time}}}+ f^k_{\ensuremath{\mathrm{dur}}}$ as $j$’s arrival time*, and to $k$ as *$j$’s number of legs*. We also use $j_{\ensuremath{\mathrm{leg}}}$ to refer to the number of legs, i.e., $k$. Finally, we refer to $j_{\ensuremath{\mathrm{arr\_time}}}-j_{\ensuremath{\mathrm{dep\_time}}}$ as *$j$’s travel time*. Formally, journeys are allowed to consist of a single footpath and no leg. However, we forbid this special case in certain problem settings to avoid unnecessary, simple but cumbersome special cases in our algorithms. A journey $j$ that is missing its initial footpath, i.e., a sequence $l^0,f^1,l^1\ldots f^{k-1},l^{k-1},f^k$ is called a *partial journey*. We say that $j$ departs in the connection $l^0_{\ensuremath{\mathrm{enter}}}$. Note, that the number of legs and the number of transfers differ slightly. For every journey with at least one leg, the number of transfers is $j_{\ensuremath{\mathrm{leg}}}-1$. The numbers are therefore essentially the same, except for a subtle difference. A journey without leg has 0 legs but also has 0 transfers and not -1 transfers. Counting legs eliminates some special cases in our algorithms and avoids some -1/+1-operations. Hence, for simplicity, we count legs. Considered Problem Settings --------------------------- In this section, we describe most problem settings studied in this paper. Several of these problems are defined in terms of Pareto-optimization. We therefore first recapitulate the definition of Pareto-optimal and domination and then state the problems considered in our paper. Section \[sec:MEAT\] introduces another problem setting called Minimum Expected Arrival Time problem. As its details are more involved, we introduce the problem setting in its own section. A tuple $x$ *dominates* a tuple $y$ if there is no component in which $y$ is strictly smaller than $x$ and there is component in which $x$ is strictly smaller than $y$. Pareto-optimal is defined in terms of domination. Denote by $P$ a multi-set of $n$-dimensional tuple with scalar components. A tuple $x$ is *Pareto-optimal* with respect to $P$, if no other tuple $y\in P$ exists, such that $y$ dominates $x$. In our setting, the tuples are journey attributes such as a journey’s travel time. $P$ is the set of attribute-tuples of all journeys. The easiest problem, that we consider, asks when a traveler will arrive the earliest possible. Formally, it can be stated as follows: [problemtitle[Earliest Arrival Problem]{}]{} [probleminput[timetable, source stop $s$, target stop $t$, source time $\tau$]{}]{} [problemoutput[The minimum arrival time over all journeys that depart after $\tau$ at $s$ and arrive at $t$]{}]{} While simple, the earliest arrival problem has several downsides. For one, a traveler often does not have a fixed departure time, but is flexible and has a range of possible departure times. One can resolve this issue by iteratively solving the earliest arrival problem with varying source times. Fortunately, we can do better and therefore formalize the aggregated problem as follows: [problemtitle[Earliest Arrival Profile Problem]{}]{} [probleminput[timetable, source stop $s$, target stop $t$, minimum departure time $\tau_s$, maximum arrival time $\tau_t$]{}]{} problemoutput The set of all $(j_{\ensuremath{\mathrm{dep\_time}}},j_{\ensuremath{\mathrm{arr\_time}}})$ over journeys $j$ such that - $j$ departs not before $\tau_s$ at $s$, - $j$ arrives not after $\tau_t$ at $t$, - the pair $(-j_{\ensuremath{\mathrm{dep\_time}}},j_{\ensuremath{\mathrm{arr\_time}}})$ is Pareto-optimal among all journeys, and - $j$ contains at least one leg. ![Profile function that maps the departure times at a stop $s$ onto the arrival times at stop $t$. The black dots represent the solution to the earliest arrival profile problem. Only the black part needs to be computed. The gray part is excluded by the minimum departure time or maximum arrival time.[]{data-label="fig:profile-plot"}](profile-plot) The result of the profile problem can be represented using a plot such as the one in Figure \[fig:profile-plot\]. The result is a compact representation of the functions that maps a departure time at $s$ onto the earliest arrival time at $t$. We refer to this function as *profile function*. Formulated differently, the profile problem asks to simultaneously solve the earliest arrival problem for all source times. We require $j$ to have at least one leg, to be able to guarantee that the profile function is a step function. Dropping this restriction, can break this property if $s$ and $t$ are connected via a footpath $f$. At least in our setting, handling such a situation is trivial but requires special case handling in our algorithm. To simplify our descriptions and to focus on the algorithmically interesting aspects, we decided to forbid journeys without leg. ![Example for an “optimal” journey that visits a stop twice. Circles depict stops, arrows depict connections and are annotated with their departure and arrival times. The journey $A{\rightsquigarrow}B{\rightsquigarrow}D{\rightsquigarrow}E{\rightsquigarrow}B{\rightsquigarrow}C$ visits stop $B$ twice and has a minimum arrival time. The journey $A{\rightsquigarrow}B{\rightsquigarrow}C$ has the same arrival time but uses fewer legs.[]{data-label="fig:loop-journey"}](loop-journey) An issue common with the earliest arrival problem and with its profile counterpart is that solely optimizing arrival time can lead to very absurd but “optimal” journeys. For example, Figure \[fig:loop-journey\] depicts a journey that is “optimal” with respect to its arrival time but visits a stop twice. Similarly, it “optimal” journeys exist that enter a trip multiple times. When computing earliest arrival journeys and not just their arrival time, one therefore usually also require that the journeys visit no stop or trip twice. A simple solution to this problem consists of picking among all journeys with a minimum arrival time one that minimizes the number legs. This implies that no stop or trip is used twice. We say that the first optimization criterion is arrival time and the second criterion is the number of legs. This slight change is enough to guarantee that no stop is visited twice. While this small change solves many transfer-related problems, some remain. Suppose, for example that there are two journeys whose arrival times differ by one second but the earlier one needs significantly more legs. In this case one would like to pick the journey that arrives slightly later. This problem can be mitigated by rounding the arrival times at the target stop. However, in many application one wants to find both journeys. We therefore also consider the following problem setting. [problemtitle[Pareto Profile Problem]{}]{} [probleminput[timetable, source stop $s$, target stop $t$, minimum departure time $\tau_s$, maximum arrival time $\tau_t$, maximum number of legs $\max_{\ensuremath{\mathrm{leg}}}$]{}]{} problemoutput The set of all $(j_{\ensuremath{\mathrm{dep\_time}}},j_{\ensuremath{\mathrm{arr\_time}}},j_{\ensuremath{\mathrm{leg}}})$ over journeys $j$ such that - $j$ departs not before $\tau_s$ at $s$, - $j$ arrives not after $\tau_t$ at $t$, - $j$ has at most $\max_{\ensuremath{\mathrm{leg}}}$ legs, - the pair $(-j_{\ensuremath{\mathrm{dep\_time}}},j_{\ensuremath{\mathrm{arr\_time}}},j_{\ensuremath{\mathrm{leg}}})$ is Pareto-optimal among all journeys, and - $j$ contains at least one leg. Besides the profile problem setting, we also consider *range problem* variants. In these, we set $\tau_t$ to $\tau_s + 2\cdot (x-\tau_s)$, where $x$ is the earliest arrival time. Formulated differently, we are only interested in journeys that are at most two times as long as possible. The solution to the range problems is a subset of the solution to the profile problems. The range problems can therefore often be solved faster. Fortunately, travelers usually do not want to arrive significantly later than the earliest arrival time. The solution to the range problem thus often consists of the journeys that actually interest a traveler. The range problem special cases are therefore of high practical relevance. Beside determining the attributes of optimal journeys, i.e., departure time, arrival time, and number of legs, we also consider the problem of computing corresponding journeys in Sections \[sec:ea-extraction\] and \[sec:extraction\]. Note that optimal journeys are usually not unique. There usually are multiple journeys for a combination of departure time, arrival time, and number of legs. We regard all of them as being equal and only extract one of them. Extracting all journeys for a combination is a different problem setting. Earliest Arrival Connection Scan {#sec:base-csa} ================================ In this section, we describe the earliest arrival Connection Scan variant. It assumes that the connections are stored as array of quintuples that are sorted by departure time. Further, the footpaths must be stored in a data structure that allows an efficient iteration over the incoming and outgoing footpaths of a stop, such as for example an adjacency array. Similar to Dijkstra’s algorithm, CSA maintains a tentative arrival time array, that stores for each stop the earliest known arrival time. A connection is called *reachable* if there is a way for the traveler to sit in the connection. Contrary to Dijkstra’s algorithm, ours does not employ a priority queue. Instead, it iterates over all connections increasing by departure time. The algorithm tests for every connection whether it is reachable. For each reachable connection, the algorithm adjusts the tentative arrival times of the stops reachable by foot from the connection’s arrival stop. After the execution of our algorithm, the output is $t$’s tentative arrival time. Contrary to most adaptations of Dijkstra’s algorithm, our algorithm touches more connections. But the work required per connection does not involve a priority queue operation and is therefore significantly faster. Our algorithm maintains two arrays $S$ and $T$. The array $S$ stores for every stop the tentative arrival time. The array $T$ stores for every trip a bit indicating whether the traveler was able to reach any of the connections in the trip. Testing whether a connection $c$ is reachable boils down to testing, whether $S[c_{\ensuremath{\mathrm{dep\_stop}}}]\le c_{\ensuremath{\mathrm{dep\_time}}}$ or $T[c_{\ensuremath{\mathrm{trip}}}]$ is set. To adjust the tentative arrival times, our algorithm relaxes all footpaths outgoing from $c_{\ensuremath{\mathrm{arr\_stop}}}$. The algorithm is described in pseudo-code form in Figure \[fig:unoptimized-earliest-arrival-connection-scan\]. Optimizations ------------- Find first connection $c^0$ departing not before $\tau$ using a binary search In this subsection, we describe three optimizations to the earliest arrival Connection Scan algorithm. Figure \[alg:optimized-earliest-arrival-connection-scan\] presents pseudo-code that incorporates all three optimizations. In the following, $c$ always denotes the connection currently being processed. #### Stopping criterion. We can abort the execution of the algorithm as soon as $S[t] \le c_{\ensuremath{\mathrm{dep\_time}}}$. This is correct because processing a connection $c$ never assigns a value below $c_{\ensuremath{\mathrm{dep\_time}}}$ to any tentative arrival time. Further, as we process the connections increasing by $c_{\ensuremath{\mathrm{dep\_time}}}$, it follows that $S[t]$ will not be changed by our algorithm after the inequality holds. #### Starting criterion. No connection departing before the source time $\tau$ is reachable, as for every journey $j$, $\tau \le j_{\ensuremath{\mathrm{dep\_time}}}< j_{\ensuremath{\mathrm{arr\_time}}}$ must hold. The proposed optimization exploits this. It runs a binary search to determine the first connection $c^0$ departing no later than $\tau$. The iteration is started from $c^0$ instead of the first connection in the timetable. #### Limited Walking. If $S[c_{\ensuremath{\mathrm{arr\_stop}}}]$ cannot be improved even with an instant transfer, i.e., $S[c_{\ensuremath{\mathrm{arr\_stop}}}]\le c_{\ensuremath{\mathrm{arr\_time}}}$ holds, then no tentative arrival time can be improved. The optimization consist of not iterating over the outgoing footpaths of $c_{\ensuremath{\mathrm{arr\_stop}}}$ in this case. The correctness of this optimization relies on the transitivity of the transfer model. Denote by $y=c_{\ensuremath{\mathrm{arr\_stop}}}$. As $S[y]\neq\infty$ a journey $j$ ending at $y$ has already been found. Denote by $f^{xy}$ the last footpath of $j$ departing at $x$. Note, that it is possible that $x=y$ and that $f^{xy}$ is a loop. For every outgoing footpath $f^{yz}$ of $y$ to some stop $z$, there exists a footpath $f^{xz}$ from $x$ to $z$ such that $f^{xz}_{\ensuremath{\mathrm{dur}}}\le f^{xy}_{\ensuremath{\mathrm{dur}}}+ f^{yz}_{\ensuremath{\mathrm{dur}}}$. We can replace the last footpath of $j$ by $f^{xz}$ and have obtained a journey arriving at $z$ no later than the journey involving $c$. As this argumentation works for all outgoing footpaths, no tentative arrival time can be improved. Iterating over the outgoing footpaths is therefore superfluous. The optimization is thus correct. ![Counterexample for the correctness of limited walking optimization in combination with a maximum path length transfer model. The example includes three stops $A$, $B$, and $C$, two connections $c^A$, and $c^B$ with annotated departure and arrival times, and walking radii of 10min. The light gray area is reachable by foot from $A$. The dark grey area is reachable by foot from $B$ but not from $A$.[]{data-label="fig:limited-walking-max-path-counter-example"}](limited-walking-max-path-counter-example) Note that the limited walking optimization crucially depends on the transitivity of the transfer model. For example, it does not hold for a transfer model with a maximum path length. Consider the example depicted in Figure \[fig:limited-walking-max-path-counter-example\]. Assume that both $c^A$ and $c^B$ are reachable connections. When processing the connection $c^A$ arriving at $A$ the tentative arrival time at $B$ is set to 8:07. However, the tentative arrival time at $C$ remains $\infty$ as the path is too long. Because the tentative arrival time at $B$ is smaller than 9:00 the limited walking optimization activates when processing $c^B$. The tentative arrival time at $C$ therefore remains at $\infty$, which is clearly incorrect. Journey Extraction {#sec:ea-extraction} ------------------ The algorithm described in the previous section only computes the earliest arrival time. In this section, we describe how to compute an earliest arrival journey in a post processing step. Our algorithm guarantees that the extracted journey visits no stop nor trip twice. The algorithm comes in two variants. The first variant augments the data structures used during the Connection Scan with additional *journey pointers* that can be used to reconstruct a journey. The second variant leaves the earliest arrival scan untouched but needs to perform more complex tasks to reconstruct an earliest arrival journey. The trade-off between the two variants is that the former is conceptually slightly more straight-forward and therefore easier to implement. Further, the former has a lower extraction running time, which comes at the cost of a higher scan running time. Finally, the later requires additional data structures, which must be computed in a fast preprocessing step. If only a journey towards one target stop should be extracted, then the later variant is faster. If journeys from one source stop to many target stops should be extracted, the former can be faster. ### With Journey Pointers $j\gets\{\}$ Prepend $j$ with the footpath from $s$ to $t$ Output $j$ Our algorithm, which is illustrated in Figure \[fig:unoptimized-earliest-arrival-connection-scan-with-journey-pointer\], stores for every stop $x$ a triple $J[x]$ of final enter connection, final exit connection, and final footpath of an earliest arrival journey towards $x$. We refer to this triple as *journey pointer*. If no optimal journey exists, then the journey pointer is set to an invalid value. A journey $j$ from a source stop $s$ to a target stop $t$ can be constructed backwards. Initially $j$ is empty. If $t$ has a valid journey pointer, we the prepend $t$’s journey pointer to to $j$. Further, we set $t$ to the departure stop of the journey pointer’s enter connection and iterate. If $t$ does not have a valid journey pointer, we prepend $j$ with a footpath from $s$ to $t$ and the journey extraction terminates. #### Journey Pointer Construction. When a tentative arrival time is modified, our algorithm stores a corresponding journey pointer. To this end, our algorithm must determine the three elements of the triple. Determining the exit connection and the final footpath is easy. These are the values denoted by the variables $c$ and $f$ in the code depicted Figure \[fig:unoptimized-earliest-arrival-connection-scan-with-journey-pointer\]. Computing the enter connection is more difficult. We replace the bit array $T$ in the base algorithm by an array that contains for every trip a connection ID. This connection ID indicates the earliest connection reachable in a trip. It may be invalid, if no connection was reached. The ID being valid corresponds to the bit being set in the base algorithm. We set an ID when a trip is first reached. It remains to show that the extracted journey does not visit a stop or trip twice. A trip cannot be visited twice by the extracted journey because $T$ is set to the first connection reachable in a trip. A stop cannot be visited twice as our algorithms stores at each stop the first journey pointer found towards it. Fortunately, a journey pointer leading to a journey with a loop cannot be the first. ### Without Journey Pointers A journey can be extracted without storing journey pointers. However, additional data structures are necessary. #### Additional Datastructures. Our algorithm needs to enumerate the connections in a trip that precede a given connection. We therefore construct an adjacency array that maps a trip ID onto the IDs of the connections in the trip. The connections are sorted by position in the trip. Our algorithm can thus enumerate all connections in a trip rapidly and stop once the given connection is found. Further, our algorithm needs to enumerate the connections arriving at a given stop at a given timepoint. We therefore construct a second adjacency array that maps a stop ID onto the IDs of the connections arriving at the stop. The connections are sorted by arrival time. We can use a binary search to efficiently enumerate all requested connections. #### Extraction. Our algorithm works similar to the one using journey pointers. However, the journey pointer is generated on the fly. We therefore need a subroutine to determine a triple of enter connection $c^{\ensuremath{\mathrm{enter}}}$, exit connection $c^{\ensuremath{\mathrm{exit}}}$, and final footpath $f$. We start by constructing a set of candidates for $c^{\ensuremath{\mathrm{exit}}}$. This set is then pruned. Finally, our algorithm iterates over the candidates and tries to find a corresponding $c^{\ensuremath{\mathrm{enter}}}$. To generate the candidate set our algorithm enumerates all incoming footpaths $f$ of the stop $t$. For every $f$, all connections arriving at $f_{\ensuremath{\mathrm{dep\_stop}}}$ at $S[t]-f_{\ensuremath{\mathrm{dur}}}$ are added to the candidate set. A candidate can only be a valid $c^{\ensuremath{\mathrm{exit}}}$ if it is reachable. If it is reachable then the trip bit must be set. We can therefore prune all candidate connections $c$ for which $T[c_{\ensuremath{\mathrm{trip}}}]$ is false. Note, that the bit being set does not imply that the candidate is reachable. It is also possible that only a later connection in the same trip is reachable. Finally, our algorithm iterates over the remaining candidates $c$. For each candidate, it enumerates all connections $x$ in $c_{\ensuremath{\mathrm{trip}}}$ not after $c$. It then checks whether $S[x_{\ensuremath{\mathrm{dep\_stop}}}]\le S[x_{\ensuremath{\mathrm{dep\_time}}}]$. If it holds, then $x$ is a valid enter connection, $c$ a valid exit connection and $f$ a valid final footpath. Further, as $x$ is the first connection in the trip, the extracted journey cannot visit a trip twice. As our approach constructs a journey for the earliest timepoint where $t$ is reachable, we can guarantee that the extracted journey does not visit a stop twice. If no journey pointer can be generated, then $t$ was reached by foot from $s$. This corresponds to $J[t]$ being invalid in the algorithm of Figure \[fig:unoptimized-earliest-arrival-connection-scan\]. Experiments ----------- We experimentally evaluated the earliest arrival Connection Scan algorithm and compare it with competing algorithms. Beside, only measuring the query running times, we also report how much time is needed to setup the data structures. The setup time is an upper bound to the time needed to update a timetable. The section is structured as follows: We first describe the machines on which we run our experiments. We then describe the test instances and how we generate our test queries. Afterwards, we report the running times needed by the Connection Scan algorithm. Finally, we compare the achieved running times with related work. ### Experimental Setup {#sec:experimental-setup} #### Machine. Unless specified otherwise, we ran all experiments on a single pinned thread of an Intel Xeon E5-1630v3, with 10 MiB of L3 cache and 128GiB of DDR4-2133MHz. This is a CPU with Haswell architecture. Some experiments were executed on an older dual 8-core Intel Xeon E5-2670, with 20 MiB of L3 cache and 64GiB of DDR3-1600 RAM, a CPU with Sandy Bridge architecture. Hyperthreading was deactivated in all experiments. Our implementation is written in C++ and is compiled using g++ 4.8.4 with the optimization flags `-O3 -march=native`. #### Instances. Instance Stops Connections Trips Routes Interstop Footpaths ---------- -------- ------------- --------- -------- --------------------- Germany 252374 46218148 2395656 248261 103535 London 20843 4850431 125537 2135 45652 : Instance sizes.[]{data-label="tab:instances"} We performed our experiments on two main benchmark instances. Table \[tab:instances\] reports the sizes. The first instance is based on the data of [bahn.de](bahn.de) during winter 2011/2012. The data was provided to use by Deutsche Bahn (DB), the German national railway company. We thank DB for making this data accessible to us for research purposes. The data contains European long distance trains, German local trains, and many buses inside of Germany. The data includes vehicles of local operators beside DB. The raw data contains for every vehicle a day of operation. Unfortunately, no day exists every local operator operates. The planning horizon of some operators ends before the reported data of other operators begins. To avoid holes in our timetable, we therefore extract all trips regardless of their day of operation and assume that they depart within the first day. Our extracted instance contains therefore more connections per day than the instance in productive use. Further, to support night trains, we consider two successive identical days. The raw data contains footpaths. We did not generate additional ones based upon geographic positions but did add footpaths to make the graph transitively closed. We removed data errors such as exactly duplicated trips, vehicles driving at more than 300 km/h or footpaths at more than 50 km/h. The second instance is based on open data made available by Transport for London (TfL). The raw input data is available in the London data store[^3]. We thank TfL for making this data openly available. The data includes tube (subway), bus, tram, Dockland Light Rail (DLR). The data corresponds to a Tuesday of the periodic summer schedule of 2011. In contrast to the Germany instance, the London instance thus only contains data for a single day. Stops correspond to platforms in this data set. As a consequence all change times are zero, i.e., the transfer model is reflexive. This data set is the main instance used in [@dpw-rbptr-12], one of our main competitor algorithms. We removed some obvious data errors from the data. The instance sizes we report are therefore slightly smaller than in [@dpw-rbptr-12]. #### Test Query Generation. To evaluate our algorithms, we generate random test queries. The source and target stops are chosen uniformly at random. The source time is chosen uniformly at random within the first 24 hours. Unless noted otherwise all reported running times are averaged over $10^4$ queries. ### Earliest Arrival Connection Scan {#earliest-arrival-connection-scan} We experimentally evaluated the earliest arrival Connection Scan algorithm and report the average running time in Table \[tab:ea-time\]. We successively activate the proposed optimizations. Further, we evaluate the running time of the journey extraction without journey pointers. The start and stop criteria drastically reduce the running times. The explanation is that significantly fewer connections have to be scanned. On the London instance the speedup is 15 times whereas the speedup on the Germany instance is “only” 5. This is due to the differences in journey lengths. In London a traveler needs on average less time to traverse the whole network than in Germany. The stop criterion therefore activates sooner reducing the number of scanned connections. The limited walking optimization further reduces running times by 1.5 to 2.0 times. Finally, we report the running time needed to perform a journey extraction in addition to the earliest arrival Connection Scan. As we only extract a single journey per scan, we use the extraction process that does not store journey pointers. The extraction process is very faster compared to the scan. On the Germany instance, it only need about 1.2ms and on the London instance 0.1ms. Instance Sort \[s\] Journey \[s\] ---------- ------------ --------------- Germany 3.56 6.15 London 0.35 0.39 : Datastructure construction running time averaged over $100$ runs. “Sort” is the time needed to sort the connection array by departure time. “Journey” is the additional time needed to construct the journey extraction data structures. []{data-label="tab:ea-construction"} #### Datastructure Construction. In Table Table \[tab:ea-construction\], we report the running time needed to sort the connection array and the running time needed to construct the journey extraction data structures. To avoid accelerating the sort algorithm by providing it with nearly sorted data, we randomly permute the array before sorting it. We use GCC’s std::sort implementation. If the timetable significantly changes, then these two steps need to be rerun. If the changes are only small, then it is probably faster to patch the existing data structures. In practice, when delays occurs, the operator needs to simulate how the delay propagates through the network. This propagation is in practice probably slower, than the few seconds needed to construct the data structures needed by our algorithms. #### Comparison with Related Work. In Table \[tab:ea-related-work\], we compare our algorithms with related work. The employed implementations are based upon the code of [@dpw-rbptr-14]. All competitors are run with stopping criterion active. We compare the Connection Scan algorithm’s running times against three extensions of Dijkstra’s algorithm and RAPTOR. The first extension is based on a time-expanded graph model. The second uses a time-dependent graph model. We refer to [@pswz-emtip-08] for a detailed exposition of these models. The third uses an optimized time-dependent graph model, proposed in [@dkp-pcbcp-12], that merges nodes based on colored timetable elements. Finally, we compare against RAPTOR [@dpw-rbptr-14], an algorithm that does not employ a graph based model. Instead, it operates directly on the timetable, similarly to the Connection Scan algorithm. We experimentally compare the performance of the algorithms with respect to the earliest arrival time problem. However, RAPTOR does not fit precisely into this category. It is designed in a way that inherently optimizes the number of transfers in the Pareto-sense. It can and must thus solve a more general problem. It does not benefit from restricting the problem setting. We therefore report its running times alongside the other earliest arrival time algorithms. Table \[tab:ea-related-work\] shows that the non-graph based algorithms clearly dominate the base versions of the time-dependent and time-expanded extensions of Dijkstra’s algorithm. The time-dependent extension can be engineered to be about a factor of 2 faster than RAPTOR. The Connection Scan algorithm is faster than all of the competitors. #### Section Conclusions. CSA enables answering earliest arrival time queries in mere milliseconds. A corresponding earliest arrival journey can be extracted afterwards in a nearly negligible amount of addition query running time. Even on the large Germany network with integrated local transit average query running times below 50ms are possible. The data structures can be constructed in less than 10 seconds even for the large Germany instance. This enables an easy straightforward and fast integration of realtime train delays. Profile Connection Scan ======================= The Connection Scan algorithm can be extended to solve the profile problem variants. The algorithm is very flexible and, compared with many other algorithms to solve the profile problem, comparatively easy. We first present the algorithm on a very high level in the form of an abstract framework. Afterwards, we illustrate how this framework can be used to solve the various profile problem variants. We start with a very restricted problem setting to simplify the exposition. We then extend the algorithm, iteratively dropping these restrictions. The initial simplifications are: - The time horizon is unbounded, i.e., there is no minimum departure nor maximum arrival time in the input. - We solve the all-to-one problem, i.e., there the input contains only a target stop and the profile functions from every stop to this target should be computed. - We assume that there are no interstop footpaths, i.e., there are only change times, i.e., there are only loops in the footpath graph. - We solve the earliest arrival profile problem, i.e., we do not optimize the number of transfers. Framework --------- Figure \[alg:profile-connection-scan-framework\] depicts the high level framework of all Connection Scan based profile algorithms. Understanding this structure is crucial to understand any of the algorithms. At its core the algorithm uses dynamic programming. It constructs journeys from late to early and exploits that an early journey can only have later journeys as subjourneys. Further, it exploits the observation that a traveler sitting in a connection only has three options to continue his journey. The three options to continue his journey are: - The traveler can exit the train and, if there is a footpath to the target, walk there, or - he can remain seated reaching the next connection in the trip, if there is a next connection, or - he can exit the train and use a footpath towards some other stop and enter another train. The two ways how a traveler can have reached a connection $c$ are: - He can have been sitting in the train, i.e, he reached a connection before $c$ in the same trip, or - or he entered the train at $c_{\ensuremath{\mathrm{dep\_stop}}}$ proceeded by a footpath. The algorithm scans the connections decreasing by departure time. We say that the algorithm iteratively *scans* the connections. In the following, we always use the letter $c$ to indicating the connection currently being scanned. The algorithm stores at each stop $x$ a profile from $x$ to the target $t$ and at each trip the earliest arrival time over all partial journeys departing in a connection of the trip. The algorithm’s structure is depicted in the pseudo-code of Figure \[alg:profile-connection-scan-framework\] which mirrors this high level description very closely. Earliest Arrival Profile Algorithm without Interstop Footpaths -------------------------------------------------------------- Figure \[alg:profile-connection-scan-framework\] contains the pseudo-code of the basic Connection Scan profile framework. In this section we describe, how to instantiate this framework to obtain an algorithm to solve the earliest arrival profile algorithm. The pseudo-code of the instantiated algorithm is depicted in Figure \[alg:earliest-arrival-profile-connection-scan\]. We start our description by describing how the stop data structure $S$ and the trip data structure $T$ are implemented. Afterwards, we describe the operations that modify $S$ and $T$. For every trip, our algorithm stores one integer, i.e., $T$ is an array of integers whose size is the number of trips. This number represents the earliest arrival time for the partial journey departing in the earliest scanned connection of the corresponding trip. For every stop, we store a profile function. A function is stored as sorted array of pairs of departure and arrival times. This means that $S$ is an array whose size is the number of stops. The elements of $S$ are arrays with a dynamic size. The elements of these inner arrays are pairs of departure and arrival times. After the execution of the algorithm, $S[x]$ contains the $xt$-profile. We initialize all elements of $T$ with $\infty$ and all elements of $S$ with a singleton array containing a $(\infty,\infty)$-pair. This algorithm state encodes that all travel times are $\infty$, i.e., the traveler cannot get anywhere. This would also be the correct solution, if the timetable contained no connections. When scanning the connection $c$, we modify $S$ and $T$ to account for all journeys that use $c$. One can thus view our approach as maintaining profiles corresponding to the timetable consisting of only the latest connections. We start with no connection and iteratively add connections. The scanned connection $c$ is the connection currently being added. Scanning $c$ consists of two parts. First, $\tau_c$ must be computed and then $\tau_c$ must be integrated into $T$ and $S$. Computing $\tau_c$ consists of the already mentioned three subcases and the integration has two subcases. Luckily, most of these cases are trivial in the simplified problem variant considered here. Computing the arrival time at the target $\tau_1$ is trivial: Either $c$ arrives at the target stop, in which case the arrival time is $c_{\ensuremath{\mathrm{arr\_time}}}+c_{\ensuremath{\mathrm{arr\_stop}}}^{\ensuremath{\mathrm{change}}}$, or the target is unreachable, as there are no interstop footpaths. If the traveler remains seated, then his arrival time will be the same, as the arrival time if he was sitting in the next connection of the trip. This arrival time is stored in $T[c_{\ensuremath{\mathrm{trip}}}]$. Incorporating $\tau_c$ into the trip data structure is also trivial, it consists of a single assignment: $T[c_{\ensuremath{\mathrm{trip}}}]\leftarrow \tau_c$. Slightly more complex are the incorporation of $\tau_c$ into the profile and the efficient computation of $\tau_3$. Incorporating $\tau_c$ consists of adding the pair $p=(c_{\ensuremath{\mathrm{dep\_stop}}}-c_{\ensuremath{\mathrm{dep\_stop}}}^{\ensuremath{\mathrm{change}}},\tau_c)$ into the array if it is non-dominated. Because the connections are scanned decreasing by departure time, there cannot be pair with an earlier departure time. However, there can be a pair with the same departure time. It is therefore sufficient for the domination test to look at the earliest pair $q$ already in the array. If $p$ is not dominated by $q$, we either add $p$ or replace $q$, depending on whether the departure times are equal. Evaluating the function is done by finding the pair $p$ in the array $S[c_{\ensuremath{\mathrm{arr\_stop}}}]$ with the earliest departure time no earlier than $c_{\ensuremath{\mathrm{arr\_time}}}$. The arrival time of $p$ is $\tau_3$. As the array is sorted, the evaluation can be done in logarithmic running time using a binary search. However, as $c_{\ensuremath{\mathrm{arr\_time}}}-c_{\ensuremath{\mathrm{dep\_time}}}$ is usually small in practice, the requested pair is usually near the beginning of the array. A sequential search is therefore faster in practice. Optimizations ------------- Several optimizations exist for the Connection Scan profile algorithm. The first optimization, that we describe exploits a hardware feature called prefetching. The next three optimizations exploit that in most cases we do not want to compute journeys from every stop to the target. They exploit additional information in the input such as the source stop to accelerate the computation. #### Memory Prefetching. The Connection Scan profile algorithm can be slightly accelerated by using processor memory prefetch instructions. Modern processors are capable of detecting simple memory access patterns and to fetch data sufficiently early to hide memory access latency. The sequential scan over the connection array is an example of such a simple memory access pattern. However, detecting the stop profile access is more complex. When scanning the $c$-th connection, we therefore execute prefetch instructions for the stop profiles $S[{(c-4)}_{\ensuremath{\mathrm{dep\_stop}}}]$, and $S[{(c-4)}_{\ensuremath{\mathrm{arr\_stop}}}]$ and the trip arrival time $S[{(c-4)}_{\ensuremath{\mathrm{dep\_stop}}}]$. These instructions help hide memory latency by overlapping the processing of connection $c$ with the memory fetching of the four connections $c-4$, $c-3$, $c-2$, and $c-1$. #### Bounded Time-Horizon. The minimum departure time $\tau_s$ and maximum arrival time $\tau_t$ can exploited by only scanning connections $c$ with $\tau_s \le c_{\ensuremath{\mathrm{dep\_time}}}\le \tau_t$. The earliest connection can be determined using a binary search. To determine the latest connection, a binary search can be used. However, it is also a byproduct of the next optimization. #### Scanning only Reachable Trips. The source stop and source times can be exploited by running a non-profile earliest arrival scan before the profile scan. The objective of this initial scan is to determine, which trips are reachable. If a trip is not reachable, then no connection in it can be reachable. We do not have to scan non-reachable connections as they cannot influence the profile at the source stop. We can thus skip connections for which the trip bit is not set. An efficient implementation starts by finding the first connection not before $\tau_s$ using a binary search. It then performs the earliest arrival scan increasing by departure time until a connection departing after $\tau_t$ is encountered. The same connections are then scanned in the reverse order in the profile scan. #### Source Domination. The source stop can be exploited in another way. In the profile framework depicted in Figure \[alg:profile-connection-scan-framework\], scanning a connection consists of two parts. The first part determines the arrival time when sitting in the connection $\tau_c$. The second part incorporates $\tau_c$ into the data structures. Consider the pair $p=(c_{\ensuremath{\mathrm{dep\_time}}},\tau_c)$. If $p$ is dominated by the pairs in the profile of the source stop, then the second part can be skipped. This optimization is correct because every journey starting at the source stop and using $c$ would be dominated. It remains to describe, how to efficiently implement the domination test. For the test, we need to know the arrival time of the earliest pair $q$ in the profile of the source stop such that $q_{\ensuremath{\mathrm{dep\_time}}}\ge c_{\ensuremath{\mathrm{dep\_time}}}$. This information can be obtained by evaluating the source stop’s profile. However, as the connections are scanned decreasing by departure time, we can do better by maintaining a pointer to the relevant pair in the source stop’s profile. When scanning a connection our algorithm first decreases the pointer if necessary and then looks up the arrival time. As the pointer can only be decreased as often as there are pairs in the source stop’s profile, we can bound the running time needed to perform these evaluations by the size of the source stop’s profile. Interstop Footpaths {#sec:interstop-footpath} ------------------- In this section, we expand the profile algorithm to handle interstop footpaths. Initial and transfer footpaths are handled in the same way, but a different strategy is needed for final footpaths. We start our description with the later, as the idea is simpler. The pseudo-code for this algorithm variant is presented in Figure \[alg:profile-connection-scan-with-footpaths\]. #### Final Footpaths. Handling final footpaths consists of modifying the computation of $\tau_1$ in the framework of Figure \[alg:profile-connection-scan-framework\]. In the base algorithm, the traveler can only arrive at the target by train. In the extended version, he can also walk at the end. For this extension, we add a new array of integers $S$. It stores for every stop the walking distance to the target or $\infty$, if walking is not possible. Computing $\tau_1$ for a connection $c$ can be done in constant time by evaluation $c_{\ensuremath{\mathrm{arr\_time}}}+ D[c_{\ensuremath{\mathrm{arr\_stop}}}]$. For efficiently reasons, we do not reset all elements of $D$ for each query. Instead, we initialize all elements of $D$ to $\infty$ during the algorithm setup. We do this initialization only once. Each query begins by iterating over the incoming footpaths of the target stop. It sets $D$ to the appropriate values for all stops from which the traveler can transfer to the target. After the profile computation, our algorithms iterates a second time over the same footpaths to reset all values of $D$ to $\infty$. #### Transfer and Initial Footpaths. Our algorithm handles transfer and initial footpaths by iterating over the incoming footpaths $f$ of $c_{\ensuremath{\mathrm{dep\_stop}}}$ when incorporating $\tau_c$ into the profiles. It inserts a pair $p=(c_{\ensuremath{\mathrm{dep\_time}}}-f_{\ensuremath{\mathrm{dur}}},\tau_c)$ into the profile of the stop $f_{\ensuremath{\mathrm{dep\_stop}}}$, if $p$ is not dominated in $f_{\ensuremath{\mathrm{dep\_stop}}}$’s profile. Unfortunately, we can no longer guarantee, that the departure time of $p$ will be the earliest in each profile. A slightly more complex insertion algorithm is therefore needed: Our algorithm temporarily removes pairs departing before the new pair. It then inserts $p$, if non-dominated, and then reinserts all previously removed pairs that are not dominated by $p$. #### Limited Walking If the number of interstop footpaths is large, handling transfer and initial footpaths can be computationally expensive. Especially, the iteration over the incoming footpaths of $c_{\ensuremath{\mathrm{dep\_stop}}}$ can be costly. Fortunately, the limited walking optimization can be adapted and can drastically reduce running time on some instances. The idea is as follows: If the pair $(c_{\ensuremath{\mathrm{dep\_time}}},\tau_c)$ is dominated in the profile of $c_{\ensuremath{\mathrm{dep\_stop}}}$, then all pairs computed when scanning $c$ are dominated. The correctness argument is essentially the same as for the non-profile algorithm. One can prefix the journey of the dominating pair with each footpath and obtain at each stop a pair that would dominate each of the pairs created during the scanning of $c$. We thus do not need to generate them as they would be dominated anyway, i.e., we do not need to iterate over the incoming footpaths. #### Different Set of Footpaths for Initial and Final Footpaths. In our proposed transfer model, we only have one type of footpaths. However, many applications have an extended set of footpaths for the initial and final footpaths. In some applications the traveler can walk for a longer amount of time at the beginning or at the end of his journey than when changing trains. Further, some applications have source and target locations that are not stops but might, for example, be city districts. Luckily, our algorithm can easily be extended to handle these cases. Final footpaths can be handled by iterating over the extended footpath set during the initialization of $D$. Handling initial footpaths is slightly more complex. Denote by $s$ the source location, for which the profile should be computed. In a first step, we create a set $X$ of pairs that may contain dominated entries. After removing the dominated entries, the profile of $s$ is obtained. Our algorithm starts by iterating over all outgoing extended footpaths $f$ of $s$. For every pair $(d,a)$ in the profile of $f_{\ensuremath{\mathrm{arr\_stop}}}$, there is a $(d-f_{\ensuremath{\mathrm{dur}}},a)$ pair in $X$. After removing dominated pairs from $X$, the profile of $s$ is obtained. It is possible to generate the set of extended footpaths using Dijkstra’s algorithm on the fly. We can therefore drop the requirement that the set of extended footpaths must be transitively closed. This allows us to have very long initial and final footpaths. Unfortunately, the restrictions still apply for transfer footpaths. Optimizing the Number of Legs ----------------------------- In the previous section, we presented the basic Connection Scan profile algorithm and extended it to a footpath-based transfer model. In this section, we further extend it to optimize the number of legs beside the arrival time. We present three ways to perform this optimization. The first and easiest approach optimizes the number of legs as a secondary criterion. The second approach is a refinement of the first that heuristically mitigates some of its problems. Finally, we present as third approach an extension that optimizes the number of legs and the arrival time in the Pareto-sense. The overhead of the first two approaches over the basic algorithm is negligible. Unfortunately, the optimization in the Pareto-sense adds a significant overhead. We therefore recommend to the reader to first try the first two approaches and only use the third if it is really necessary for the particular application at hand. Our algorithm optimizes the number of legs by counting the number of times a traveler exits a train. As there is an exit per leg, the number of exits and the number of legs coincide. The exit counter is increased each time that a profile is evaluated, i.e., during the computation of $\tau_3$ in the framework. #### Number of Legs as Secondary Criterion. Optimizing the number of legs as secondary criterion, i.e., computing a journey with a minimum number of legs among all journeys with a minimum arrival time, is surprisingly easy. Denote by $\epsilon$ a negligibly small time value, i.e., think of $\epsilon$ as one millisecond. The modification of our algorithm consists of increasing $\tau_3$ by $\epsilon$ after each profile evaluation, i.e., the modification consists of inserting a single addition compared to the base algorithm. If two journeys have different arrival times, then the earlier journey is chosen. If the arrival times are equal, the number of $\epsilon$s added determines which journey is chosen. As an $\epsilon$ is added each time that the travelers exits a train, the number of $\epsilon$s corresponds to the number of legs. The number of legs is thus optimized as secondary criterion. In a real implementation, we multiply all departure and arrival times in the timetable with a small constant, such as for example $2^5$. Timestamps, even with second resolution, usually require significantly fewer than 32 bits. For example, to encode all seconds within a year, 25 bits are enough. We can therefore encode the modified timestamps using 32 bit integers. The value of $\epsilon$ is set to 1. The modifications to the algorithm depicted in Figure \[alg:profile-connection-scan-with-footpaths\] adding a “+1” in line 7 and perform the scaling using two bit shift operations between the lines 4 and 5. Stated differently, we encode the number of legs in the lower 5 bits of a timestamp. The higher 27 bits encode the arrival time. As an integer comparison only compares the lower bits if the higher bits are equal, we obtain the desired effect, that the journeys are tie-broken using the number of legs. #### Rounding the Arrival Times. Optimizing the number of legs as secondary criterion, eliminates the most problematic earliest arrival journeys, such as those visiting a stop several times or those entering a trip multiple times. However, a journey that arrives at $8{:}02$ with 10 legs is still preferred over a journey with 2 legs arriving at $8{:}03$. While the former arrives earlier, most travelers prefer the later. This problem can be avoid by optimizing the number of legs in the Pareto-sense. Fortunately, a simpler partial solution to the problem exists that might be good enough for some applications. The idea consists of rounding the value of $\tau_1$ in the framework of Figure \[alg:profile-connection-scan-framework\]. If $\tau_1$ is rounded down the lowest multiple of say 5 minutes, then both journeys are equal with respect to arrival time and therefore the journey with 2 legs is chosen. Rounding down to multiple of 5 minutes divides a day into 288 time buckets. Journeys arriving within one bucket are regarded as arriving at the same time and thus one with a minimum number of legs is picked. This avoids many problematic journeys, but it is only a partial solution as the problem remains at the time bucket borders. Further, the trick has no effect, if the difference in journey arrival times is larger than the bucket size. Notice, that we are only rounding the arrival times at the target stop. We do not round the departure or arrival times of intermediate connections. This trick therefore does not modify the transfer model. A problem with this trick is that the profiles contain rounded arrival times. However, we want to display the non-rounded arrival times to the user. Further, there will only be one journey per bucket. Fortunately, these problems can be solved by permuting some bits in the timestamps. ![Encoding used to represent timestamps. The numbers represent the bit-offsets within a 32 bit integer of the three data items.[]{data-label="fig:bit-encoding"}](bit-shuffle) Suppose that, we want to use 5 bits to encode the number of legs. Further, assume that we round the arrival times down to $2^8=256$. With seconds resolution that corresponds to rounding down to multiples of $\approx$4.2 minutes. The idea consists of not encoding the number of legs in the lowest bits of a timestamp. Instead, we use bits in the middle. The lowest 8 bits are the lower bits of the arrival time. The next higher 5 bits are the number of legs. The remaining bits encode the higher bits of the arrival time. Figure \[fig:bit-encoding\] illustrates the layout. The effect of this modification is that our algorithm now optimizes three criteria. These are: 1. The rounded arrival time, 2. the number of legs, and 3. the exact arrival time. Criteria 2 and 3 are used as second and third criteria, i.e., they are tie-breakers. The exact arrival times can easily be reconstructed from this encoding. Further, assume that there are two journeys that arrive within the same bucket and have the same number of legs but have different arrival times. In the base version only one would be found. Using the refined algorithm both are found as they are not identical with respect to the third criterion. Unfortunately, as already mentioned this trick mitigates but does not resolve the problem of trading many addition transfers for a tiny improvement in arrival time. However, for certain applications this trick reduces the number of problematic cases to a sufficiently small amount. The main advantage of this trick is that it is significantly easier to implement than the more complex solution described in the next paragraph. Further, the incurred overhead is comparatively low. #### Pareto Optimization. The number of legs and the arrival times can be optimized in the Pareto-sense. For a fixed target $t$, we want to compute for every source stop $s$, every source time $\tau_s$, and every number of legs $\ell$, the earliest arrival time $\tau_t$ over all journeys from $s$ to $t$ not departing before $\tau_s$ with at most $\ell$ legs. To simplify this problem slightly, we bound $\ell$ by ${\ensuremath{\mathrm{leg}}}_{\max}$ which is a constant in the algorithm. We usually set ${\ensuremath{\mathrm{leg}}}_{\max}$ to 8 or a similarly large value, exploiting that travelers in practice do not care about journeys with too many legs. We modify our algorithm by replacing all arrival times by constant-sized vectors. ${\ensuremath{\mathrm{leg}}}_{\max}$ is the dimension of the vectors. We denote the elements of a vector $A$ as $A[1],A[2]\ldots A[{\ensuremath{\mathrm{leg}}}_{\max}]$. The element $A[\ell]$ is the arrival time at the target, if the journey has at most $\ell$ legs. We define two operations that modify these vectors. The first is the *component wise minimum*, i.e., the result of the minimum operation of two vector $A$ and $B$ is a vector $C$ such that $C[i]=\min\{A[i],B[i]\}$ for all indices $i$. The second operation is the *shift* operation, which is defined as follows: Shifting $A$ yields a vector $B$ such that $B[1]=\infty$ and $B[i]=A[i-1]$ for all other indices $i$. The interpretation of the minimum operation consists of taking the best of two options. Further, the shift operation can be interpreted as increasing the number of legs. All $\tau$-variables in the framework from Figure \[alg:profile-connection-scan-framework\] become vectors. The trip data structure $T$ becomes an array of vectors. The profile data structure $S$ becomes an array of dynamic-sized arrays of pairs of an integer and a vector. The walking distance to the target $D$ remains an array of integers. It is possible that a vector $A$ dominates another vector $B$ in one component, for example $A[1]<B[1]$, but $B$ dominates $A$ in another component, for example $A[2]>B[2]$. For this reason, the vector insertion must be modified. If all components of the new vector are dominated, then the profile is not modified. Otherwise, we insert the minimum of the new vector and the minimum of the earliest vector already in the profile. Two successive pairs can have the same arrival time with respect to certain but not all values of $\ell$ but different departure times. In the base algorithm the profiles are initialized with a sentinel $(\infty,\infty)$ pair. The arrival time of this pair is a vector in the extended algorithm, i.e., the new sentinel is $(\infty, (\infty,\infty\ldots \infty))$. The computation of $\tau_1$ starts analogous to the non-Pareto case. Our algorithm starts by computing the walking time $x$ to the target. Afterwards $x$ is converted to a vector $A$ by setting $A[i]=x$ for all indices $i$. The operation of setting all components of a vector to one value is sometimes called *broadcast*. In Figure \[alg:pareto-profile-connection-scan\] we present the profile Pareto algorithm in pseudo-code form. To simplify its exposition, we omit interstop footpaths. Fortunately, they can be incorporated in the same way as already described in Section \[sec:interstop-footpath\] and depicted in Figure \[alg:profile-connection-scan-with-footpaths\]. ![ Example timetable. The circles are stops and the arrows are connections annotated by their departure and arrival times. All connections are part of different trips. There are 4 journeys from $s$ to $t$ with a varying number of legs: $s@5{\rightarrow}t@14$, $s@7{\rightarrow}z{\rightarrow}t@12$, $s@6{\rightarrow}x{\rightarrow}t@13$, and $s@6{\rightarrow}x{\rightarrow}y{\rightarrow}t@11$ []{data-label="fig:pareto-example"}](pareto-example.pdf) #### Example. Consider the example timetable depicted in Figure \[fig:pareto-example\]. We describe how the profile of $s$ evolves during the execution of our algorithm. We set the target stop to $t$ and ${\ensuremath{\mathrm{leg}}}_{\max}$ to 3. The profile is a dynamic array of pairs of departure time and arrival time vectors. Initially it only contains an infinity sentinel, i.e., initially we have $S[s]=\{(\infty,(\infty,\infty,\infty))\}$. The profile $S[s]$ is changed for the first time, when the connection from $s$ to $z$ is scanned. The value of $\tau_c$ is $(\infty,12,12)$. As there is no way to reach $t$ with at most 1 leg, the first component $\tau_c[1]$ is $\infty$. $\tau_c[2]$ is $12$ as the target can be reached at 12 with 2 legs. Further, $\tau_c[3]$ is $12$ also as the target can be reached at 12 with at most 3 legs. Notice that $\tau_c[3]$ is $12$ even though that the corresponding journey only contains 2 legs. $\tau_c$ is better in two components than the earliest vector in the profile, which is the $(\infty,\infty,\infty)$ sentinel. The algorithm therefore inserts a new pair, namely $(7,(\infty,12,12))$ into the profile $S[s]$. The profile $S[s]$ after the scan is $\{(7,(\infty,12,12)), (\infty,(\infty,\infty,\infty))\}$. The profile $S[s]$ is changed for the second time, when the connection from $s$ to $x$ is scanned. The value of $\tau_c$ is $(\infty,13,11)$. $\tau_c[1]$ is $\infty$ as $t$ cannot be reached without transfer. $\tau_c[2]$ is $13$ because the journey $s@6{\rightarrow}x{\rightarrow}t@13$ contains 2 journeys. Further, $\tau_c[3]$ is $10$ because the journey $s@6{\rightarrow}x{\rightarrow}y{\rightarrow}t@11$ with 3 legs exists. As the later has more than 2 legs, we have that $\tau_c[2]\neq 11$. $\tau_c$ is better in at least one component than the earliest vector in the profile, i.e., $(\infty,12,12)$. However, it is not better in every component. The algorithm therefore computes the minimum $\min\{(\infty,13,11),(\infty,12,12)\} = (\infty,12,11)$. The pair $(6,(\infty,12,11))$ is added to the profile $S[s]$. The resulting profile has the value $\{(6,(\infty,12,10)), (7,(\infty,12,12)), (\infty,(\infty,\infty,\infty))\}$. The last time that the profile $S[s]$ might be change is when the connection from $s$ to $t$ is scanned. The value of $\tau_c$ is $(14,14,14)$. However, $\tau_c$ is not better in any component than the earliest vector in the profile, i.e., $(\infty,12,10)$. No pair is thus added. After the execution of the algorithm the profile $S[s]$ is $\{(6,(\infty,12,10)), (7,(\infty,12,12)), (\infty,(\infty,\infty,\infty))\}$. To determine the arrival time for a source time $\tau_s$ and maximum number of legs $\ell$, find the earliest pair with a departure time no earlier than $\tau_s$. The $\ell$-th component of the corresponding arrival time vector contains the answer. For $\tau_s = 6.5$ and $\ell = 3$, we therefore first look up the first pair with a departure time after $6$. This is $(7,(\infty,12,12))$. The $\ell$-th, i.e., third, component is 12. The traveler can thus arrive at 12. #### Earliest Arrival Time. In some cases, one is more interested in the minimum arrival time over all journeys than in the minimum arrival time over all journeys with at most ${\ensuremath{\mathrm{leg}}}_{\max}$ legs. This can be implemented using a small change in the definition of the shift operation. The result of the modified shift of a vector $A$ is a vector $B$ such that $B[1]=\infty$, $B[{\ensuremath{\mathrm{leg}}}_{\max}] = \min\{A[{\ensuremath{\mathrm{leg}}}_{\max}-1], A[{\ensuremath{\mathrm{leg}}}_{\max}]\}$, and $B[i] = A[i-1]$ for all other indices $i$. With this modification, the ${\ensuremath{\mathrm{leg}}}_{\max}$-th vector component contains the earliest arrival times over all journeys. #### SIMD. All vectors operations, i.e., component-wise minimum, component shifting, and broadcasting a value to all components, can be implemented using SIMD operations on all common processor architectures. This includes x86 processors with the SSE and AVX2 instruction sets. One SSE vector has 4 components with 32 bit integers. Concatenating two vectors, yields an efficient implementation for ${\ensuremath{\mathrm{leg}}}_{\max}=8$. Alternatively, AVX2 vectors have 8 components with 32 bit integers. One AVX2 vector is therefore large enough. Journey Extraction {#sec:extraction} ------------------ In the previous section, we introduced an algorithm to compute profiles. In this section, we describe how to extract corresponding journeys in a post processing step. Similar to the extraction process for the earliest arrival time Connection Scan algorithm, the extraction comes in two variants. The first conceptually simpler approach consists of storing journey pointers. The second approach computes the journey pointers on the fly during the extraction. The input consists of a source stop $s$ and source time $\tau_s$. The output consists of an earliest arrival journey towards the target stop for which the profile was computed. If transfers are optimized in the Pareto-sense, then the input contains additionally a maximum number of legs $\ell$. Several journeys can exist that are identical with respect to all considered criteria, i.e., they depart at the same source stop at the same source time and arrive at the same target stop at the same target time and have the same number of transfers. We only consider the problem setting of extracting one of these journeys. Our algorithms guarantee that the extracted journey visits no stop or trip twice even when the number of legs is not optimized. ### Journey Pointers In the base profile algorithm, the pairs $(d,a)$ contain two pieces of information namely a departure time $d$ and an arrival time $a$. We extend the pairs with two connection IDs $l^{\ensuremath{\mathrm{enter}}}$, $l^{\ensuremath{\mathrm{exit}}}$, turning the pairs into quadruples $(d,a,l^{\ensuremath{\mathrm{enter}}},l^{\ensuremath{\mathrm{exit}}})$. The meaning of such a quadruple is that there is an optimal journey $j$ that arrives at the target stop at time $a$ and departs at time $d$. The extracted journey $j$ starts with a footpath towards $l^{\ensuremath{\mathrm{enter}}}_{\ensuremath{\mathrm{dep\_stop}}}$. $j$ leaves the stop using the connection $l^{\ensuremath{\mathrm{enter}}}$. The traveler exits the train at the end of the connection $l^{\ensuremath{\mathrm{exit}}}$. These quadruples can be used to iteratively extract an optimal journey. The extraction starts by computing the time needed to directly transfer to the target. Doing this trivial without interstop footpaths. With footpaths, we use the $D$ array of the base profile algorithm. In the next step, our algorithm determines the first quadruple $p$ after $\tau_s$ in the profile $P[s]$ of the source stop $s$. If directly transferring to the target is faster, then the journey consists of a single footpath and there is nothing left to do. Otherwise, $p$ contains the first leg of an optimal journey. The algorithm then sets $s$ to $l^{\ensuremath{\mathrm{exit}}}_{\ensuremath{\mathrm{arr\_stop}}}$ and $\tau_s$ to $l^{\ensuremath{\mathrm{exit}}}_{\ensuremath{\mathrm{arr\_time}}}$ and iteratively continues to find the remaining legs of the output journey. It remains to describe how $l^{\ensuremath{\mathrm{enter}}}$ and $l^{\ensuremath{\mathrm{exit}}}$ are determined when inserting the quadruple into the profile during the scan. $l^{\ensuremath{\mathrm{enter}}}$ is the connection being scanned and is therefore already known. To determine $l^{\ensuremath{\mathrm{exit}}}$ efficiently, we extend the trip information $T$ with a connection ID for each trip, i.e., $T$ becomes an array of pairs of arrival times and connection IDs. Each time that the arrival time stored in $T$ is decreased, the algorithm sets the trip’s connection ID to the currently scanned connection. When inserting the quadruple, $c^{\ensuremath{\mathrm{exit}}}$ is the connection ID stored with currently scanned connection’s trip. This approach can be combined with Pareto-optimization by replacing $l^{\ensuremath{\mathrm{enter}}}$, $l^{\ensuremath{\mathrm{exit}}}$, and the trip connection IDs with constant-sized vectors. The input of the algorithm must be extended with the maximum number of desired legs. ### Without Journey Pointers Similarly to the earliest arrival Connection Scan, it is possible to implement a journey extraction without modifying the scan. Our algorithms require enumerating the outgoing connections of a stop ordered by departure time. To efficiently support this operation, we create an auxiliary data structure that consists of an adjacency array that maps a stop $s$ onto the departure time and the ID of all connections $c$ departing at $s$, i.e., onto the connections $c$ for which $c_{\ensuremath{\mathrm{dep\_stop}}}=s$ holds. The outgoing connections are ordered by departure time. Further, our algorithm needs to be able to enumerate all connections in a trip after a given connection. To efficiently support the second operation, we create another auxiliary adjacency array that maps a trip $t$ onto the IDs of the connections $c$ in the trip, i.e., onto the connections $c$ for which $c_{\ensuremath{\mathrm{trip}}}= t$ holds. The connections are ordered by their position in the trip. To enumerate the connections in a trip after a given connection $c$, we enumerate the connections in $c_{\ensuremath{\mathrm{trip}}}$ from late to early and abort the enumeration once $c$ is encountered. Notice, that all auxiliary data structures are independent of the target stop. Further, both data structures can be computed by essentially sorting the connections by various criteria. We can therefore compute the auxiliary data in a fast preprocessing step. Similarly, to the journey pointer approach, our second approach starts by checking, whether directly walking from the source stop $s$ and the source time $\tau_s$ to the target $t$ is optimal. It terminates, if this is the case. Otherwise, our algorithm must compute a pair of valid $l^{\ensuremath{\mathrm{enter}}}$ and $l^{\ensuremath{\mathrm{exit}}}$. In the first approach, these were stored in the pairs which is no longer the case in the second approach. Our algorithm therefore needs to infer the values. It does so by searching for the earliest pair $(d,a)$ after $\tau_s$ in $s$’s profile $P[s]$ using a binary search. We know that there must be a footpath $f$ outgoing from $s$ towards $l^{\ensuremath{\mathrm{enter}}}_{\ensuremath{\mathrm{dep\_stop}}}$ such that $l^{\ensuremath{\mathrm{enter}}}_{\ensuremath{\mathrm{dep\_time}}}= d+f_{\ensuremath{\mathrm{dur}}}$. By iterating over the outgoing footpaths of $s$ and checking this condition, we obtain a set $\{c^1,c^2\ldots c^k\}$ of candidates for $l^{\ensuremath{\mathrm{enter}}}$. We know that there must be an optimal first leg $l$, such that $l^{\ensuremath{\mathrm{enter}}}$ is among the candidates. We can optionally prune the candidate set using the trip arrival times $T[x]$ computed during the profile scan. $T[x]$ is the minimum arrival time over all optimal journeys departing in a connection of trip $x$. We therefore know that if for a candidate $T[c^i_{\ensuremath{\mathrm{trip}}}]>a$ holds, that $c^i$ cannot be $l^{\ensuremath{\mathrm{enter}}}$ and we can therefore remove $c^i$ from the set. For the remaining candidates, we need to look at the connections in their trips. For each potential candidate $c^i$, our algorithm enumerates all connections $c$ in its trip that come after $c^i$, including $c^i$ itself. For each $c$, our algorithm searches for the earliest pair $(d',a')$ in $c_{\ensuremath{\mathrm{arr\_stop}}}$’s profile after $c_{\ensuremath{\mathrm{arr\_time}}}$ using a binary search. If $a=a'$, then we found an optimal first leg and $c$ is the corresponding $l^{\ensuremath{\mathrm{exit}}}$. If we only wish to extract one journey, then our algorithm can discard the remaining candidates. Our algorithm iterates by setting $s$ to $l^{\ensuremath{\mathrm{exit}}}_{\ensuremath{\mathrm{arr\_stop}}}$ and $\tau_s$ to $l^{\ensuremath{\mathrm{exit}}}_{\ensuremath{\mathrm{arr\_time}}}$. To assure that no trip is used twice in a journey, we pick the latest valid $l^{\ensuremath{\mathrm{exit}}}$ in the trip. As we enumerate connections from late to early, the first valid $l^{\ensuremath{\mathrm{exit}}}$ we encounter is automatically the latest. #### Pareto Optimization. The candidate set is computed by finding the first pair $(d,a)$ departing after $\tau_s$. This is correct for the base profile scan algorithm. However, the Pareto-extension can insert several pairs with the same departure time with respect to $\ell$. A modification to the extraction is therefore necessary. Consider for example the example illustrated in Figure \[fig:pareto-example\]. Suppose that the traveler departs at $s$ at 5 and wants to use at most 2 legs. Already the first pair $(6,(\infty,12,10))$ in the profile departs later than 5. However, there is no earliest arrival journey towards $t$ departing at 6 towards $t$ with at most 2 legs. The corresponding journey departs at 7. Indeed, the second pair $(7,(\infty,12,12))$ in the profile has the correct departure time and arrives at the same time. To fix this problem, we slightly modify the algorithm. First we find the earliest pair $p$ departing no earlier than $\tau_s$. In a second step, we iterated over the pairs in the profile from early to late starting at $p$ until we find the last pair $q$ with the same arrive time than $p$ for the requested number of legs. The departure time of $q$ is used to determine the candidate set. Experiments ----------- We use the experimental setup described in Section \[sec:experimental-setup\]. In Table \[tab:ea\_profile\], we report the running times of the earliest arrival Connection Scan profile algorithm. We report the running times for both main instances on both of our test machines. We iteratively activate optimizations to show their impact. Activating range queries also includes not processing unreachable trips. We also report the running time needed to perform the scan and extract for every pair in the source stop’s profile a corresponding earliest arrival journey. The comparison between the two machines is interesting. We expect the newer machine to be faster, as it has a faster processor, a newer architecture, and faster RAM. This expected behavior is also nearly always the observed behavior, except on the Germany instance for non-range queries. The differences in L3 cache sizes explains the effect. The newer machine is better with respect to every criterion except L3 cache. The old machine has 20 MiB while the newer one only has 10 MiB. The London instance is smaller and therefore a larger part of the stop profiles fit into the 10 MiB. If we compute range queries, only parts of the stop profiles are computed. This part is smaller and therefore a greater percentage fits into the L3 cache. The newer machine is therefore faster on range queries and slower on non-range queries. The conclusion is that a sufficiently large cache is necessary for a good Connection Scan profile performance. Activating prefetching decreases the running times. On the newer machine and the London instance the speedup is only about 1.02. However, on the Germany instance the gain is already 1.05. This observation again illustrates that caching effects matter for good performance. On the London instance large parts of the frequently used data structures are never evicted from L3 cache. The gain from prefetching comes therefore mostly from moving data to the lower cache levels. On the Germany instance prefeching moves data from the RAM into L3 cache more often. As the absolute differences in access speeds between L3 cache and RAM are greater than between L2 and L3 cache, the speedup is lower for the London instance. Activating the limited walking optimization further reduces the running times. The speedup is about 1.4 to 1.7, which is roughly comparable to the speedups achieved for the non-profile algorithm variants. Activating source domination further reduces the running times. As source domination prunes pairs from profiles except the source stop, the algorithm solves a more restricted problem setting. Instead of computing the profiles from every stop towards the target stop, it now only computes a single profile from the source stop to the target. Switching to range queries drastically reduces the running times. The specified maximum travel time of twice the minimum travel time allows the algorithm to limit the connections that need to be scanned. On the Germany instance the speedup is about a factor 6. On the London instance the speedup of 11 is higher. These speedups are roughly comparable to the speedups achieved by activating the start and stop criteria in the non-profile earliest arrival algorithm. The reason is that the decrease in scanned connections is roughly comparable. Further, as already observed, a traveler needs less time to traverse London than to traverse Germany. The relative decrease in scanned connections is thus higher on the London instance and as a consequence the achieved speedups are higher. In Table \[tab:pareto\], we report running times of the Connection Scan Pareto profile algorithm. It optimizes the number of legs, the arrival time, and the departure time in the Pareto-sense. The maximum number of legs is set to 8. We use the algorithm variant that computes the earliest arrival time in the 8-th vector component. We iteratively activate our proposed optimizations to demonstrate their effectiveness. We present three SIMD variants. All three use the same memory layout. All use vectors with 256 bits that contain 8 components with a 32-bit timestamp. They differ in what processor instructions are used to operate on the vectors. The first variant uses no special instructions and works with loops with a fixed number of iterations. The second variant uses SSE instructions. SSE registers are 128 bits wide. To process one vector, two SSE instructions are thus required. The third variant uses AVX registers. Luckily, these are 256 bits wide and therefore a single instruction is sufficient. We use integer AVX arithmetic instructions. These were introduced with AVX2, a feature introduced in the Haswell processor architecture. Our AVX code can therefore not run on our older test machine, which does not yet support AVX2. The first optimization that we consider consists of prefetching memory. On the Germany instance without SSE nor AVX, a speedup of 1.16 was achieved. This is significant, considering that no algorithmic changes were performed. Interestingly, the speedup is only 1.02, when comparing the AVX prefetch and AVX non-prefetch running times. It is also interesting that by using AVX compared to the base version a speedup of 1.9 is achievable. Especially, the later is interesting, as we expect SIMD to have the largest benefit in compute-bound algorithms and our previous experiments suggest that the Connection Scan algorithm heavily depends on memory access speeds. One explanation for these two effects is that the AVX code has fewer instructions, making it easier for processor to predict memory access patterns. This would explain why the benefit of prefetching nearly vanishes but running times drastically decrease. This explanation is also consistent with the observation that using AVX is a benefit over SSE as the AVX code requires fewer instructions. The speedups of the limited walking and source domination optimizations are comparable to those observed for the earliest arrival profile algorithm. We refer to discussion of these experiments for an interpretation of the observed effects. The speedup of the range query variant is about 10 on the Germany instance and 17-19 on the London instance. These speedups are larger than those observed for the earliest arrival profile algorithm. The difference is likely due to the Pareto algorithms having a larger overall memory consumption. As a consequence caching effects have a larger impact and therefore a reducing of the memory footprint yields a large relative advantage. #### Comparison with Related Work. In Table \[tab:profile\_compare\], we compare the Connection Scan profile algorithm with two competitor algorithms. The first is Self-Pruning Connection-Setting (SPCS) algorithm [@dkp-pcbcp-12]. It computes profiles that optimize departure and arrival time in the Pareto-sense but does not optimize transfers. The algorithm can be combined with the colored timetable optimization, which was used in our experiments. We therefore refer to the algorithm as SPCS-col in Table \[tab:profile\_compare\]. The second competitor is rRAPTOR [@dpw-rbptr-14]. Similar to the base RAPTOR algorithm, it inherently optimizes transfers in the Pareto-sense. The Connection Scan algorithm (CSA) was run with AVX and limited walking activated. Both rRATPOR and CSA clearly dominate SPCS-col in terms of running time. The difference between CSA and rRATPOR is smaller. CSA is always faster, but on the Germany instance the gap is only up to a factor of 2. On the London instance, there is a speedup of up to 4.7. #### Section Conclusions. Using CSA and by exploiting the full capabilities of modern processors, it is possible to answer Pareto range queries on the large Germany instance in a quarter of a second. It is feasible to construct interactive timetable information systems upon these running times. However, ideally lower running are desirable. For example, spending a quarter of a second per query in a web server severely limits throughput. Fortunately, we were able to achieve these running times without compromising the excellent data structure construction times of the base algorithm. Flexible realtime updates are possible. Connection Scan Accelerated =========================== In the previous sections, we present the Connection Scan family of algorithms. We demonstrate that queries can be answered very quickly on modern hardware. Even Pareto range queries can be answered in well below a second even on the large Germany instance. A significant advantage of the Connection Scan algorithms is that the preprocessing is very lightweight. It mainly consists of sorting the connections, which can be done in very few seconds. This allows us to quickly update the timetable to account for disturbances, such as delayed trains, blocked stops or tracks, or overbooked trains. While, all of these properties make the Connection Scan family of algorithms a good fit for many applications, it is also interesting to investigate whether further gains are achievable by using more heavy-weight preprocessing techniques. Further, even though the achieved running times on the Germany instance are low enough for interactive applications, we expect them to consume a significant amount of resources. Lower running times are therefore very desirable in practice. Investigating the combination of Connection Scan with more heavy-weight preprocessing techniques is therefore the topic of this section. We investigate a multilevel overlay extension to the Connection Scan algorithms, which we call Connection Scan Accelerated (CSAccel). The central ideas are similar to those used in [@sww-daola-99; @hsw-emlog-08; @dgpw-crprn-13]. In several studies, this approach has proven to enable very fast queries in road networks. Compared to Dijkstra’s algorithms, speedups on the order of 1000 are possible. It is therefore reasonable to expect similar speedups on timetable networks. We are not the first to investigate this question. Unfortunately, previous research [@bdgm-atdmc-09; @bgm-fdsut-10] has shown that achieving similar speedups is harder than one would naively expect. Our work is no exception to this observation. Our multilevel extension manages to provide a significant speedup on the Germany instance. However, the speedup lacks far behind of what is achievable in road networks. The core idea of our extension is best illustrated using an example: When planning a journey from Karlsruhe to Stuttgart, do not scan rural bus connections around Hamburg. We use overlays to formalize the concept of rural bus. Our algorithm partitions the stop set into cells. Karlsruhe and Stuttgart are put into the same cell. Hamburg is in a different cell. For every cell, our algorithms computes a subset of *transit connections*. For every pair of connections entering and leaving a cell $z$, there must be a journey with a minimum number of transfers, that only enters or exits trips at transit connections of $z$. For rural buses, usually no such journey exists and thus they are not in the transit connection set. When traveling from Karlsruhe to Stuttgart, our algorithm only looks at the transit connections of Hamburg’s cell and thus skips the rural buses around Hamburg. Following the setup and terminology of [@dgpw-crprn-13], our algorithm works in three phases. In the first phase, called *preprocessing phase*, a multilevel partition of the stop set is computed. In the second phase, called *customization phase*, our algorithm computes overlays for every cell. Finally, in the third phase, called *query phase*, our algorithm computes arrival times and journeys. The second phase uses the results of the first phase. Similarly, the third phase uses the results of the first and the second phases. The preprocessing phase should only use data that rarely changes, such as for example what tracks exist and perhaps what tracks are highly frequented. The idea is that the preprocessing phase does not have to be rerun very often and may therefore be slow. To update the timetable, it should be sufficient to rerun the customization, which should be fast. Our customization phase works with every stop partitioning, as long footpaths do not cross cell boundaries and the stop sets are identical. However, if the timetables used during preprocessing and customization differ too much, then customization and query performance will significantly degrade. multilevel approaches inherently rely on the structure of the network. Small, balanced graph cuts are a necessity. Without these, the achievable speedups crumble. Fortunately, as shown in many studies, road graphs typically have this structure. However, for timetables, the situation is less clear. Indeed, country-wide timetables that consist of many urban centers differ in structure from timetables that consist of a single large urban region. There typically exist small, balanced cuts between cities, however, cutting through a city is significantly more difficult. Many cities contain natural cuts such as rivers or large main roads. This property is exploited to achieve fast shortest path queries in road networks. Unfortunately, in timetable networks, rivers are not necessarily advantageous. Often, several trains or buses lines pass over a single bridge. Cutting through tracks with a high public transit frequency is expensive, in the context of timetables, as we need to weight the cuts by the number vehicles that pass over it. We therefore expect the performance of all multilevel overlay extensions to perform poorer on pure urban instances. This differs from the basic Connection Scan algorithm, whose performance is nearly independent of the timetable structure. Connection Scan algorithms find a journey $j$ with legs $l^1,l^2 \ldots l^k$, if the connections $l^1_{\ensuremath{\mathrm{exit}}}$, $l^1_{\ensuremath{\mathrm{enter}}}$ $\ldots$ $l^k_{\ensuremath{\mathrm{exit}}}$, $l^k_{\ensuremath{\mathrm{enter}}}$ are scanned in the correct order. These are the connections where the traveler transfers, i.e., enters or exits. A connection where the traveler does neither does not have to be scanned. Scanning all connections ordered by departure time fulfills this property for all journeys. This is the core observation exploited by the Connection Scan base algorithms. For a fixed source and target stop it can be sufficient to only scan a subset of the connections. Our algorithm exploits this observation. Our query phase thus works in two phases. In the first phase a sorted connection subset $\mathcal{C_S}$ is assembled. For every pair in the $st$-profile, there must be a journey $j$, such that all transfer connections of $j$ are included in $\mathcal{C_S}$. In the second phase the Connection Scan base algorithms are run restricted to the connections in $\mathcal{C_S}$. Our algorithm computes $\mathcal{C_S}$ by merging arrays of sorted connections. In the base setting every cell has an associated sorted array of transit connections. To compute $\mathcal{C_S}$, one would identify all potentially relevant cells and merge their transit connections. Unfortunately, the number of these cells can be large and merging sorted arrays is a task that requires some running time. We therefore want to reduce the number of arrays merged. We therefore introduce the concept of *long distance connections*. A transit connection of a cell $z$ is a long distance connection of its direct parent cell. On the lowest level, all connections within a cell $z$ are long distance connections of $z$. For every cell, our algorithm stores a sorted array of long distance connections. To assemble $\mathcal{C_S}$, our algorithm merges the long distance connections of all cells that contain the source or target stop or both. If the long distance connections of a cell $z$ are merged into $\mathcal{C_S}$, then also the connections of $z$’s parent are merged. We can exploit this observation to further thin out the long distance connection set. If $c$ is a long distance connection of a cell $z$ and of $z$’s parent cell, then it is sufficient to store $c$ in the parent cell’s array. Further, we can construct the transit connections with the property that, if $c$ is a long distance connection of $z$’s parent, then $c$ is a long distance connection of $z$. A consequence of this is that every connection is contained in at most one thinned out long distance connection set. The memory consumption is therefore linear in the number of connections. To prove that our algorithm is correct, we show that for every Pareto-optimal journey $j$, there exists a Pareto-optimal journey $j'$ that only enters or exists trips in the merged connection subset $\mathcal{C_S}$, such that $j$ and $j'$ have the same departure and arrival time and have the same number of legs. Before formally proving the correctness, we illustrate the employed arguments using an example. ![ Multi level journey example from stop $s$ to stop $t$. []{data-label="fig:multilevel-journey"}](multi-level-journey) Figure \[fig:multilevel-journey\] illustrates a stop set that was recursively partitioned along the straight solid lines. At every level, every cell was partitioned into four parts. The thickness of the lines indicates the level – the thicker the line the higher a level. The solid bent line represents the journey $j$. The colored areas represent transit connections merged into $\mathcal{C_S}$. Red means lowest level, blue is next higher, then orange and green is the highest level. The white dots represent connections, where $j$ crosses cell boundaries. The dotted line represents an alternative subjourney of $j$ within the green bottom-right cell. The journey $j$ consists of a prefix from $s$ to the first boundary connection, several subjourneys that traverse cells, and a suffix from the last boundary connection to $t$. The subjourneys are enclosed by the white dots in Figure \[fig:multilevel-journey\]. The constructed journey $j'$ has the same prefix and suffix and crosses the cell boundaries in the same connections as $j$, i.e., in the white dots. Because the prefixes and the suffixes are equal, the departure and arrival times of $j$ and $j'$ are equal. The subjourneys within a cell can differ. For example it is possible that $j$ uses the solid line, whereas $j'$ uses the dotted line. By construction, we know that for every cell, entry connection, and exit connection, there exists a subjourney with a minimum number of transfers that only enters or exits trips at transit connections in $\mathcal{C_S}$. For every cell $z$ that $j$ traverses, replace the subjourney of $j$ within $z$ with the corresponding minimum transfer journey to obtain $j'$. $j'$ cannot have more transfers than $j$ because otherwise one of the employed subjourneys would not have had a minimum number of transfers. Further, as $j$ was Pareto-optimal, $j'$ cannot have fewer transfers than $j$. $j$ and $j'$ therefore have the same number of transfers. In the following, we describe the details of our multi level extension. The text is organized along the three main phases. It first describes the the preprocessing phase which mostly consists of a graph partitioning problem. Afterwards, the customization phase is described, which primarily consists of computing the transit connections. Next, we explain how to perform the queries, which consists of computing $\mathcal{C_S}$. Finally, we present an experimental evaluation of the algorithm and a comparison with related work. Phase 1: Partitioning the Stop Set ---------------------------------- A $k$-partition of the stop set $V$ divides $V$ into $k$ cells such that every stop is in exactly one cell. We require that stops connected by footpaths must be in the same cell, i.e., footpaths must not cross cell borders. A connection is interior (exterior) to a cell if it departs at a stop inside (outside) the cell. In an $l$-level partition with $k$ children, the stop set is recursively split into $k$ cells over $l$ levels. At the bottom level there are $k^l$ cells. The top level consists of a single cell that contains all stops. The parent $p$ of a cell $z$ is the cell which was split to create $z$. Similarly, $z$ is a child of $p$. The bottom level cells do not have children and the top level cell does not have a parent. The preprocessing step consists of computing an $l$-level partition with $k$ children where $l$ and $k$ are tuning parameters of the algorithm. We perform the partitioning using a graph partitioner. From the timetable, we build an undirected, weighted graph as follows: The stops form the node set of the graph. There is an edge between two nodes if there is a connection or footpath between the corresponding stops. If there is a footpath, we weight the corresponding edge with $\infty$, to assure that it is not cut. Otherwise, the weight of an edge reflects the number of connections between the edge’s endpoints. We partition the graph into $k$ parts using KaHip v1.0c[^4] with 20% imbalance. We recursively repeat this operation $l$ times. We run KaHip using the “strong”-preconfiguration. Unfortunately, the results we get from KaHip vary significantly depending on the random seed given to it. We therefore run KaHip at each level in a loop with varying seeds until for 10 iterations no smaller cut is found. This setup is definitely not the fastest partitioning method. Fortunately, it is fast enough and the obtained cuts are reliably small. Phase 2: Computing Transit Connections -------------------------------------- In this section, we describe how to compute the transit and long distance connections. We start by describing how to compute journeys with a minimum number of transfers. In the next step, we describe how to use this algorithm to compute transit and long distance connections sequentially. Finally, we describe how the customization algorithm can efficiently be parallelized. #### Minimum Number of Transfers. $T[c_t]\gets 0$ To compute overlays, our algorithm needs to quickly compute journeys with a minimum number of transfers between each pair of connection entering and leaving a cell $z$. We implement this using a variant of the earliest arrival Connection Scan profile algorithm with secondary transfer optimization. We run this algorithm on a part of the network restricted to the connection inside of $z$. Contrary to the algorithms described in Section \[sec:base-csa\], the traveler does not start and end at a stop but starts in an entry connection $c_s$ and ends in an exit connection $c_t$. A key observation is that, for a fixed target exit connection $c_t$, the arrival times of all journeys are the same. The algorithm therefore only optimizes the number of transfers. Our algorithm iterates in an outer loop over the exit connections $c_t$ and computes a backward profile for each. In an inner loop, it iterates over all entry connections and evaluates the profile. It extracts a corresponding journey $j$ from $c_s$ to $c_t$. All connections where $j$ exits or enters a trip are marked as transit connections including $c_s$ and $c_t$. In the following, we describe the inner loop of our algorithm in greater detail. In the inner loop, we have a fixed target exit connection $c_t$ and a set of source enter connections $C_s$. For every connection $c_s$ in $C_s$, a minimum transfer journey from $c_s$ to $c_t$ should be computed. The pseudo-code of the algorithm is given in Figure \[alg:profile-min-trans\]. We start our scan with the connection $c_t$, as all connections after it are obviously not reachable. The body of the loop is left mostly unchanged compared to the base algorithm. There is only one major modification: We no longer compute a walk-to-target time $\tau_1$. It would be $\infty$ for every connection except $c_t$, which is not useful. Instead, we introduce a special case for $c_t$ outside of the loop. As all journey end in $c_t$, it does not matter what arrival time we give $c_t$. For simplicity, we use 0. To quickly extract journeys, we use journey pointers. The extraction works analogous to the base algorithm with one modification. To extract the first leg of a journey starting in a connection $c_s$, we need to look at the exit connection stored with $c_s$’s trip. However, this exit connection may be overwritten, if there are several entry connection inside of this trip. We therefore extract the journey directly after processing $c_s$. A trip can contain multiple entry connections, if it leaves and enters a cell multiple times. #### Computing Transit and Long-Distance Connections. We compute transit connections bottom-up, i.e., the transit connections of the lowest level are computed first. To accelerate the computations on the higher levels, we use transit connections of lower levels. A central observation is that for every cell $z$ there is a valid transit connection set $T_z$ that is a subset of the long-distance connection set $L_z$ of $z$. Our algorithm thus works as following: For all levels $l$ from bottom to the top and all cells $z$ in the level $l$, first compute the long-distance connections $L_z$ of $z$, then compute the transit connections $T_z$ of $z$ by restricting the search to the long-distance connections $L_z$. For the lowest level cells, the long-distance connection set contains all interior connections. In a second faster step, we iterate a second time over the levels and cells and thin out the long distance connection sets. #### Parallelization. Significant speedups can be achieved by parallelizing the transit connection computation. There are two levels of granularity on which we can parallelize: (1) we can compute the transit connections of cells on the same level in parallel, and (2) we can compute the journeys for different exit connections within a cell in parallel. The former has the advantage that the data structures of different cells are completely disjoint, minimizing the necessary communication and synchronization. However, the boundary sizes of cells are very skewed because of urban centers. It is therefore difficult to keep all threads fully occupied. The later is more fine-grained and therefore allows us to fully occupy all threads. However, more communication and synchronization is needed. We use a hybrid approach that combines the best properties of both. In a first step, we sort all cells first by level from bottom to top and as a secondary criterion by decreasing boundary size. The obtained list is a topological sorting of the dependencies between the cells. We sort the cells by boundary size, to assure that the more expensive cells, i.e., those with a larger boundary, are processed first. We attach to every cell an atomic counter, that indicates the number of children cells have not yet been computed. If this counter reaches zero then processing can start. The bottom level cells start with a counter of zero. The higher level cells start with the number of children used in the partitioning. We spawn as many threads as the hardware can process simultaneously. Every thread iterates over the list of cells once. If it finds a cell with counter 0, it grabs the cells by atomically increases the counter to prevent other threads from seeing the 0 counter value. The thread then processes the cell and once it is finished decreases the counter of its parent. When a thread reaches the end of the list, it puts itself into a pool of idle threads. The threads that are still processing cells, look at whether this pool is non-empty between processing two target exit connections. If it is non-empty, they extract an idle thread atomically and the thread helps processing the cell. At the end of processing a cell, all threads but the main one are put back into the idle pool. Phase 3: Answering Queries -------------------------- In this section, we describe how to compute the connection subset $\mathcal{C_S}$ and how the query algorithms need to be modified. #### 2-way vs $k$-way. Efficiently computing $\mathcal{C_S}$ is a crucial component of an efficient implementation of our query algorithms. The input consists of several arrays of sorted data that should be merged. Three major strategies exists [@k-taocp-97]. The first consists of iteratively performing a two-way merge to combine pairs of arrays. The other two are direct $k$-way merges. The idea consists of storing a pointer into each array and iteratively determining the smallest element and increasing the corresponding pointer. Determining which element is the smallest is the challenging part. There are two approaches. One can use a binary heap or one can use tournament trees. All three variants have a worst case running time of $O(n \log k)$, where $n$ is the total number of elements. We implemented all three variants and in preliminary experiments on our data set, the iterative two-way merge was the fastest, followed by the binary heap, and the tournament heaps came last. Unfortunately, the iterative two-way merge can only compute $\mathcal{C_S}$ as a whole. The direct $k$-way approach allows us to perform a partial merge, i.e., only merge the first $x$ connections, which is enough for some of our applications. #### Profile Queries. We implement the earliest arrival and Pareto profile algorithms in the straight forward way. First, our algorithm computes $\mathcal{C_S}$ using an iterative two-way merge. In a second step, the Connection Scan base algorithm is applied restricted to $\mathcal{C_S}$. #### Earliest Arrival Queries. Earliest arrival queries have a start and stop criterion. We therefore use a direct $k$-way merge to avoid computing parts of $\mathcal{C_S}$ that will not be scanned. For each of the $k$ arrays, we run a binary search to determine the first connection not before the source time. We then start the $k$-way merge. We run the merging process until the stop criterion aborts the scan. #### Range Queries. For range queries, we use a similar approach. We first perform the $k$ binary search and then start with the $k$-way merge. To determine the reachable trips, we execute a non-profile earliest arrival scan. Once the stop-criterion activates, we continue the merge until all connections departing within the desired range have been computed. We store the output of the merging process into a temporary array. We then run the profile algorithm restricted to the connections in this temporary array. Experiments {#sec:csa_accel_exp} ----------- In this section, we experimentally evaluate CSAccel. We use the experimental setup described in Section \[sec:experimental-setup\]. We start by comparing various multilevel configuration in terms of preprocessing, earliest arrival query, and profile query running time. For one of the best configurations, we present an evaluation of range queries. We conclude with a comparison of experimental results with related work. #### Query Experiments. In Table \[tab:accel\_times\], we experimentally evaluate Connection Scan Accelerated for various configurations. A label X-$c$-$l$ refers to a recursive partitioning of timetable X, over $l$ levels, with $c$ children per level. The number of lowest level cells is $c^l$. We report $c^l$ in the table to give an overview over the granularity of the partition. We report the time needed to compute the multilevel partitioning with KaHip version 1.0c. In preliminary experiments, we also tried using Metis. The partition running times were significantly lower but the customization and query running times were higher. As we focus on the later two values, we therefore refrain from reporting these experiments. Further, we report the customization running times. Both the preprocessing and customization experiments were performed on our older Xeon E5-2670 machine with 16 physical hardware threads. The customization running times are parallelized, whereas the preprocessing running times are sequential. We also report running times for various query variants. The query experiments were run sequentially on the newer Xeon E5-1630v3 machine. We report the average running times for the earliest arrival time, the earliest arrival profile, and the Pareto profile problem settings. Journey extraction was not performed. Range query experiments are reported in Table \[tab:accel\_range\] and discussed later in this section. We activated all optimizations of the base algorithm, i.e., start and stop criteria, source domination, limited walking, and AVX. Beside the query running times, we also report the number of connections in $\mathcal{C_S}$. These are the number of connections that are scanned by the profile algorithms. The earliest arrival algorithm only needs to scan a subset of these connections because of the start and the stop criteria. The preprocessing running times roughly grows with the number of lowest level cells. This is non-surprising, as the number of partitioner invocations follows this trend. The customization running times follow the same general trend and grow with the number of cells. However, having a large number of children helps the customization but hampers the partitioning. The minimum partitioning running times are therefore achieved for Germany-2-9 and London-3-3, which have a low number of children, whereas the customization running times are minimum for Germany-8-3 and London-8-2, i.e., a high number of children. To minimize the number of connections, a recursive bisection strategy with many levels performs best. Scanning fewer connections reduces the running time spent in the Connection Scan algorithm. Query running times are therefore comparatively fast for nested dissection configurations. The only exception to this trend is the earliest arrival running time on London, which is fastest for London-8-3 and London-8-2. The explanation is that the $k$-way merge step dominates the running time. Having more levels results in more arrays to be merged and thus increases the running time of the merge step. London-8-3 and London-8-2 have the fewest levels and therefore the fastest merge steps. Compared to the non-accelerated running times, we observe a significant decrease in running times for every query type on the Germany instance. However, the speedups are significantly less impressive on the London instance. The explanation is that the London instance only has 4850K connections but even for London-2-10 1921K connections have to be scanned. The speedup is therefore very slim. In fact for the earliest arrival time problem, the base algorithm is even faster. The explanation is that it is faster to scan the few additional connections, than to perform the $k$-way merge. It is very surprising that CSAccel is faster in absolute terms on the Germany instance compared to the London instance. There are several reasons for this effect. London has at the time of writing nearly 9M inhabitants. This contrasts with the largest German city Berlin that has only 3.5M inhabitants. As a consequence the London urban transit is larger than any urban transit contained in the Germany instance. Another explanation is the difference in stop modeling. The London instance has a reflexive transfer model with usually one stop per platform. The Germany instance groups nearby platforms into one stop and uses loops in the footpath graph. London is thus modeled in greater detail than Berlin. Having more stops increases computation times. #### Range Queries. In Table \[tab:accel\_range\], we report range query results. We restrict our exposition to Germany-2-12 and London-2-7, as we obtained very good results for these configurations for non-range profile queries. Compared to the profile query running times, we observe significant speedups. These speedups are similar to those observed when comparing profile with range queries in the non-accelerated Connection Scan base algorithm. The speedups are due to cache effects and fewer connections being scanned. [lrrrrrrr]{} & & & &\ (lr)[5-8]{} & \#Stop & \#Conn & Prepro & &\ (lr)[5-6]{} (lr)[7-8]{} Algo & \[K\] & \[M\] & \[min\] & EA & Pareto & EA & Pareto\ RAPTOR [@dpw-rbptr-14] & 252.4 & 46.2 & — & — & 325.8 & — & 4730\ CSA & 252.4 & 46.2 & 0.1 & 44.9 & ${259.2}^\dagger$ & 1246 & 2490\ CSAccel-2-12 & 252.4 & 46.2 & (122)+88 & 6.2 & $24.7^\dagger$ & 47.3 & 78.9\ TP [@bs-fbspt-14] & 248.4 & 13.9 & 22320 & — & 0.3 & — & 5.0\ S-TP [@bhs-stp-16] & 250.0 & 15.0 & 990 & — & 32.0 & — & —\ TB-ST [@w-tbptr-16] & 247.9 & 27.1 & 13878 & — & 0.156 & — & 0.512\ TB [@w-tbptr-15] & 249.7 & 46.1 & 39 & — & 40.8 & — & 301.7\ RAPTOR [@dpw-rbptr-14] & 20.8 & 4.9 & — & — & 6.4 & — & 680\ CSA & 20.8 & 4.9 & $<$ 0.1 & 1.2 & ${10.7}^\dagger$ & 106.9 & 170.2\ CSAccel-2-7 & 20.8 & 4.9 & (4)+27 & 2.6 & $12.0^\dagger$ & 91.7 & 134.4\ PTL [@ddpw-ptl-15] & 20.8 & 5.1 & 54 & 0.0028 & — & 0.074 & —\ Pareto-PTL [@ddpw-ptl-15]& 20.8 & 5.1 & 2958 & — & 0.0266 & — & —\ TB-ST [@w-tbptr-16] & 20.8 & 5.0 & 696 & — & 1.7 & — & 16.1\ TB [@w-tbptr-15] & 20.8 & 5.0 & 6 & — & 1.2 & — & 70.0\ #### Comparison with Related Work. In Table \[tab:accel\_comp\], we compare various algorithms for timetable routing. Some make use of very heavy-weight preprocessing, while others are very lightweight. We compare RAPTOR [@dpw-rbptr-14], our Connection Scan algorithm (CSA), our multilevel extension (CSAccel), public transit labeling (PTL,Pareto-PTL), [@ddpw-ptl-15], Trip-Based routing (TB) [@w-tbptr-15; @w-tbptr-16], and transfer patterns (TP) [@bceghrv-frvlp-10; @bs-fbspt-14; @bhs-stp-16]. Two PTL variants exist: the base version (PTL), and an extension that supports optimizing transfers in the Pareto-sense (Pareto-PTL). There are also two variants of Trip-Based routing: the base variant TB [@w-tbptr-15] and a newer version [@w-tbptr-16] (TB-ST) that precomputes prefix and suffix trees. Transfer patterns were introduced in [@bceghrv-frvlp-10] and overhauled in [@bs-fbspt-14]. We refer to the overhauled version as TP. Another variant called “Scalable Transfer Patterns” was introduced in [@bhs-stp-16]. We refer to it as S-TP. The various papers use different instances that are based upon the same input data. The only exception is S-TP which uses a newer version of the Deutsche Bahn data set. Unfortunately, the papers significantly differ in how they extract a formal timetable from the input. The variations on the London instance are comparatively small and originate from differences in how data errors are repaired. The differences on the Germany instance are more significant. S-TP is based on newer input data than TP and therefore the corresponding numbers differ. The TP instance is based on the same input as the other papers. CSAccel, TB, and TB-ST extract a two day instance. TP and S-TP extract a single day but have days-of-operation flags. Using these flags multiple days discerned. The difference between a two day instance and a one day instance with flags explains the different number of connections between TP and TB. The difference in size between TB-ST and TB originates from a different interpretation. Following our original CSAccel paper, TB extracts all connections regardless of the day of operation. This is done because some local operators do not have a schedule for every day. The downside of this approach is that several variations of the same trip appear simultaneously. For example some trips drive differently on Sundays than on workdays. Fortunately, having more connections will most likely not decrease the running times. The reported numbers of CSAccel and TB are therefore upper bounds. The difference between CSAccel and TB is the result of correcting data errors differently. These differences in instances makes a detailed comparison difficult, if not impossible. We can only confidently compare orders of magnitude between the running times reported in the various papers. We therefore refrain from scaling running times with respect to machines as the numbers are not directly comparable anyway. Further, cache sizes can have a larger impact on the running time than the processor clock speed as demonstrated in Table \[tab:ea\_profile\]. Unfortunately, cache sizes are rarely reported in papers. Scaling by processor clock speed is therefore not meaningful, even if the instances were equal. All reported running times are sequentially. The reason that the preprocessing times seem large, stems from the fact that papers usually report parallelized running times. CSAccel is the only algorithm to split preprocessing into two phases. We therefore report its preprocessing as $(p)+c$ where $p$ is the preprocessing and $c$ the customization running time. Unfortunately, we cannot report numbers for every query type and algorithm. This has various reasons. For RAPTOR, we do not report non-Pareto running times because RAPTOR does not benefit from not optimizing transfers. We report no preprocessing time for RAPTOR, because the original implementation that we use was not tuned for this criteria. For CSA, we report range query running times instead of non-profile Pareto running times. The reason is that we do not know how to implement non-profile Pareto queries in a way that significantly outperforms range queries. Range queries usually compute more journeys because they allow for a flexible departure time. In some sense the problem is therefore harder, However, the latest arrival time is bounded, which makes the problem also somewhat easier. Fortunately, both problems have similar applications and therefore we present the results in the same column. The CSA numbers are marked with a $\dagger$ to illustrate that range queries are computed. PTL’s preprocessing can optionally optimize transfers. This explains the two PTL variants in the table. The authors evaluated the non-transfer variant for earliest arrival time and earliest arrival profiles. Unfortunately, the authors were not able to evaluate PTL on the Germany instance because of legal restrictions. Further, they did not evaluate Pareto-profile queries. The trip-based techniques TB and TB-ST, just as RAPTOR, do not benefit from not optimizing transfers in the Pareto-sense and thus no earliest-arrival-only numbers exist. The transfer patterns techniques TP and S-TP could in theory be implemented in a variant that only optimizes arrival time. This theoretical variant would probably benefit from smaller query graphs but it was, to the best of our knowledge, never implemented and thus we cannot report numbers. Unfortunately, TP was not evaluated on the London instance. #### Discussion of the Germany instance. Ordering the algorithms by preprocessing running times yields: CSA, RAPTOR, TB, CSAccel, S-TP, TB-ST, and finally TP. With the exception of TB-ST and TP, the gaps between each of these techniques are large enough that we can be confident, that the differences are not solely due to differences in experimental setup. Comparing query running times is more difficult because of the various query types. Further, the differences between running times are smaller. It is thus possible that a number is only lower because of a different experimental setup. With respect to non-profile Pareto query running times, the group of fastest algorithms clearly contains TP and TB-ST. The next-slower group contains CSAccel, S-TP, and TB. The slowest group contains CSA and RAPTOR. Meaningfully comparing algorithms within a group requires a more similar experimental setup. Overall, CSAccel strikes a good trade-off between the various criteria. No query running time is above 100ms and preprocessing running times are manageable. #### Discussion of the London instance. On the London instance, only PTL achieves a speedup above a factor of 11 over the CSA baseline. Given the simplicity and near-instant preprocessing running times, this makes CSA a perfect fit for this instance. PTL achieves an interesting performance trade-off when not optimizing transfers. The preprocessing time is slightly below an hour, which is still somewhat manageable. The benefit is that PTL achieves query running times are on the microsecond scale. Unfortunately, when additionally optimizing transfers the preprocessing running time of PTL becomes prohibitively large. Overall, assuming that some form of transfer optimization is required, we recommend using CSA as it is never drastically slower than the alternatives but is simple to implement and can update the timetable almost instantly. #### Section Conclusions. The conclusions we draw from the experiments are mixed and depend on the test instance. On the Germany instance, CSAccel can answer Pareto range queries on average in about 25ms. This is a significant improvement over the 250ms of CSA. Interactive timetable systems with a high throughput can be constructed with an average query running times of 25ms. However, the factor 10 speedup comes at a high cost. The obvious cost is the increased preprocessing time. CSA needs 10 seconds single core to adjust to a completely new timetable. On the other hand, CSAccel requires 2min with 16 cores. Requiring 2min to update the timetable is probably acceptable in practice but far from ideal. Further, CSAccel requires that the new timetable is sufficiently similar to the old one. CSA does not have this restriction. A further cost associated with CSAccel is the significant increase in code and algorithm complexity compared to CSA. Arguably the most important selling point of CSA is its simplicity. It is so simple that not even a heap-based priority queue is needed as a component. The earliest arrival CSA base is arguably even easier than Dijkstra’s algorithm. CSAccel requires solving among other things a graph partitioning problem as subroutine. This is an NP-hard task and the state-of-the-art heuristics alone have a complexity far exceeding that of CSA. Depending on the application, the increase in complexity of CSAccel compared to CSA might even be worse than the increased preprocessing times. However, for applications where query running times of 250ms are prohibitive and realtime updates are needed, CSAccel is still attractive because of the lack of alternatives. None of the other techniques achieves preprocessing running times on the order of only a few minutes and similar query running times. On the urban London network, the decrease in query running time of CSAccel over the CSA baseline is slim. We do not believe that it outweighs the significantly larger preprocessing costs and especially not the significant increase in code complexity. Use CSA in primarily urban networks. An advantage of CSA is that its performance is nearly independent of the timetable structure and mostly depends on its size. On the other hand, the performance of CSAccel is heavily dependent on the timetable structure, as the differences between the test instances shows. When starting a new timetable information system, using CSA until the query running times get prohibitive is a good approach. CSA is easy to implement and therefore not much effort is lost when switching to other approaches. Further, chances are high that the size of your timetables will never reach the prohibitive size. For example, we have not been able to assemble a realistic timetable with only rail-bound vehicles that was large enough. The Germany test instance is only large enough because buses are included. Differences from original CSAccel publication [@sw-csa-13] ---------------------------------------------------------- In this section, we briefly explain where the differences in experimental results between the experimental evaluation presented here and the original conference article [@sw-csa-13] stems from. If you have only read this journal article then you can ignore this section. In [@sw-csa-13] we report 1794.7 seconds to perform a customization using 16 threads on the same test machine on the Germany instance. We improved this to 113.6 seconds, which constitutes an improvement of over a magnitude. This improvement is the combination of three smaller changes. The first and largest improvement is due to an improved parallelization scheme. The cell boundary sizes differ significantly. However, in [@sw-csa-13] we only parallelized over the cells in a level but not not across level and not within a cell. The result was that during large parts of the customization only a single thread was working. The new parallelization approach manages to keep all threads occupied over nearly the whole process. The second improvement is due to using KaHip instead of Metis. The newer KaHip versions achieve smaller cut sizes than Metis. This translates into slightly smaller query running times and drastically lower customization times. The third and smallest improvement is due to implementing the minimum transfer CSA more carefully. Minimum Expected Arrival Time {#sec:MEAT} ============================= The Connection Scan profile framework is very flexible. In the previous sections, we have seen how the timetable can be adjusted to account for known delays. In this section, we want to plan ahead and compute a journey that is robust with respect to unknown, future delays. We do this by computing for every transfer backup journeys. For every transfer in a journey from train $A$ to train $B$, we compute a list of backup trains $C_1,C_2\ldots$ that the traveler can take if he cannot reach train $B$ because $A$ is delayed. If a transfer breaks, then a traveler should take the backup train with the earliest departure time that he can get. ![Delay-Robust journey from Karlsruhe at 9:00 to Berlin.[]{data-label="fig:meat-example"}](ka_be_exp) An example of such a delay-robust journey from Karlsruhe to Berlin is depicted in Figure \[fig:meat-example\]. We refer to the depicted graph as *decision graph*. If no train is delayed, then the traveler should take the train from Karlsruhe to Mannheim, transfer there, and take the direct train to Berlin. The question is what the traveler should do, if he misses his connecting train in Mannheim. The answer is to take a later journey with an additional transfer in Hannover. This additional transfer might also break. The backup journey therefore needs its own backup journey. Fortunately, the second backup goes directly to the target and therefore no further backup trains are necessary. Further, examples can be generated using our proof-of-concept demonstration at <http://meatdemo.iti.kit.edu>. As we expect readers to be unfamiliar with the concept, we highly recommend readers to experiment with the demonstration to get a basic understanding on an intuitive level before reading on. Computing such journeys is a very different setting than computing an earliest arrival journey with respect to a timetable aware of the realtime delay situation. All the algorithm’s decision need to be performed in advance, when the exact delays are not yet known. To be able to do this, we assume that we have an estimation of how likely it is that a train is delayed. We refer to this estimation as *delay model*. One way to obtain such an estimation is to aggregate historic delay data. If a train was never delayed in the past, then we can assume that it is unlikely that it is delayed today. If a train was nearly always delayed, then we definitely need to have a good backup journey. Consider a train $A$ that is part of a very fast journey but no reasonable backup trains exist and $A$ is likely delayed. A risk averse traveler will not want to take $A$ because it is too risky. The algorithmically interesting question consists of identifying risky trains and avoiding them. Neither optimizing the arrival time nor the number of trains achieves this. While developing the Connection Scan algorithms, we have discovered a surprisingly easy way to solve this problem. Consider the Connection Scan earliest arrival profile algorithm. Suppose that the traveler arrives with a train $A$, transfers at a stop $X$, to take a train $B$. We want to find the next best backup $C$ in case that the transfer breaks. To compute this route our, the Connection Scan profile algorithm looks up $B$’s pair in $X$’s profile. What will a traveler do, if he misses $B$? He will wait at $X$ for the next train $C$ heading into the correct direction to depart. Formulated differently, $C$ is the backup train. Computing $C$ is easy. The pair in $X$’s profile after $B$’s pair corresponds to $C$. A problem remains. If no reasonable backup exists, i.e., the transfer is very risky, then the so computed backup $C$ will arrive very late. To solve this issue, we do not store the arrival time of the next train in the profiles. Instead, we store the average over all trains in the profile weighted by the probability of the traveler taking the train. In probability theory, the expected value of a random variable is the average of all possible outcomes weighted by their probability. Following this terminology, we refer to the modified arrival times in our profiles as *expected arrival time*. If the first train $B$ has an early arrival time but no good alternative exists, then the expected arrival time will be large. Minimizing the expected arrival time therefore solves the problem. We refer to the corresponding problem setting as *minimum expected arrival time (MEAT)* problem. The decision graph in Figure \[fig:meat-example\] is tiny. Unfortunately, not all cities are as well connected as Karlsruhe and Berlin. Decision graphs between more remote areas can quickly grow in size and contain backups over numerous layers. We therefore investigate approaches to reduce the graph size and to represent it more compactly. Related Work to MEAT {#sec:related-work-meat} -------------------- There has been a lot of research in the area of train networks and delays. In contrast to our algorithm most of them compute single paths through the network instead of subgraphs containing all backups. To make this distinction clear we refer to such paths as *single-path-journeys*. The authors of [@dms-mcspt-08] define the reliability of a single-path-journey and propose to optimize it in the Pareto-sense with other criteria such as arrival time or the number of transfers. The availability of backups is not considered. The authors of [@bmpvw-rrupt-13], based on delays occurred in the past, search for a single-path-journey that would have provided close to optimal travel times in every of the observed situations. Again, backups do not play a role. The authors of [@gkmss-tprti-11] propose to first compute a set of safe transfers (i.e. those that always work). They then develop algorithms to compute single-path-journeys that arrive before a given latest arrival time and only use safe transfers or at least minimize use of unsafe transfers. The problem with this is approach is that unsafe transfers are avoided at all costs. In the example of Figure \[fig:meat-example\], the direct train from Mannheim to Berlin would be missed because the transfer is unsafe. In [@ghms-rrti-13], a robust primary journey is computed such that for every transfer stop a good backup single-path-journey to the target exists. However, the backups do not have their own backups. The approach optimizes the primary arrival time subject to a maximum backup arrival time. The authors of [@fils-itrar-13] study the correlation between real world public transit schedules in Rom and compare them against the single-path-journeys computed by state-of-the-art route planners based on the scheduled timetable. They observe a significant discrepancy and conclude that one should consider the availability of good backups already at the planning stage. The authors of [@bss-drtpp-13] examine delay-robustness in a different context: Having computed a set of transfer patters on a scheduled timetable in a urban setting, they show that single-path-journeys based on these patterns are still nearly optimal even when introducing delays. The conclusion is that these sets are fairly robust (i.e., the paths in the delayed timetable often use the same or similar patterns). In [@bs-fbgr-14] the authors propose to present to the user a small set of transfer patterns that cover most optimal journeys. They show that in an urban setting few patterns are enough to cover most single-path-journeys. In a different line of work, the authors of [@bgmo-sdplt-11] investigate how a delay-perturbed timetable will evolve over time using stochastic methods. Their study shows that this is a computationally expensive task (running time in the seconds) if the delay model accounts many real-world details. Using a model with such a degree of realism therefore seems unfeasible for delay-robust route planning (requiring query times in the milliseconds). Delay Model {#sec:delay_model} ----------- Every random variable $X$ in this work is denoted by capital letters, is continuous, non-negative, and has a maximum value $\max X$. We denote by $P[X\le x]$ the probability that the random variable is below some constant $x$ and by $E[X]$ the expected value of $X$. A crucial component of any delay-robust routing system is choosing against which types of delays the system should be robust and how to model these delays. This choice has deep implications throughout the whole system. While a too simplistic model does not yield useful routes, a too complicated model makes routing algorithms too inefficient to be useful in interactive timetable information systems. We therefore propose a simple stochastic model. While our model that does not cover every situation and is not delay-robust in every possible scenario, it works well enough to give useful routes with backups. Further, we were not able to construct a proof-of-concept implementation for a more complex model while maintaining reasonable query running times. The central simplification is that we assume that all random variables are independent. Clearly, in reality this is not always the case. However, if delays between many trains interact then the timetable perturbation must be significant. An example of a significant perturbation is a train track that is blocked for an extended period of time. As reaction to such a perturbation even trains in the medium or distant future need to be rescheduled (or arrive at least not on-time). The set of possible outcomes and the associated uncertainty is huge. Accounting for every outcome seems infeasible to us. We argue that if the perturbation is large then we cannot account for all possible recovery scenarios in advance. Instead, the user should replan his journey based on the realtime delay situation. Furthermore, even if we could account for all scenarios, we would still face the problem of explaining every possible outcome to the user, which is a show-stopper in practice. Our model therefore only accounts for small disturbances as we only intend to be robust against these. We believe that assuming independence for small disturbances is a model simplification that is acceptable in practice. Formally, our model contains one random variable $\mathcal{D}_{c}$ per connection $c$. This variable indicates with which delay the train will arrive at $c_{{\ensuremath{\mathrm{arr\_stop}}}}$. We assume that all connections depart on time. This assumption does not induce a significant error because it roughly does not matter whether the incoming or the outgoing train is delayed. Furthermore, we assume that every connection $c$ has a maximum delay, i.e., $\max\mathcal{D}_{c}$ is a finite value. Finally, we assume that all random variables are independent. Delays between trips are independent because if they were not then the perturbation would be large. We can assume that delays within a trip are independent as there nearly never exists an optimal decision graph that uses a trip more than once. We assume that the changing times at stops are encoded in $\mathcal{D}_{c}$. An transfer with a slack time below the regular change time should have a very low success probability but it should not be zero. This way the computed decision graph will also include the very risky transfers that in practice only have a chance of working if the outgoing train departs delayed. However, as the probability is low not much weight is attributed to them. We further assume that inter-stop footpaths are handled by contracting adjacent stops and adjusting the change time. These simplifications allows us to omit footpath and change time handling from the algorithm. Fortunately, for applications that require them they can be incorporated analogously to how they are handled in the earliest arrival profile Connection Scan algorithm. The only remaining modeling issue is to define what distribution the random variables $\mathcal{D}_{c}$ should have. An obvious choice is to estimate a distribution based on historic delay data. However, this has two shortcomings: - it is hard to get access to delay data (we do not have it), and - you need to have records of many days with precisely the same planned schedule. Suppose for example that the user is in the middle of his journey and a significant perturbation occurs. The operator then adjusts the short-term timetable to reflect this and the user wants to reroutes based on this adjusted data. With historic data this often is not possible because this exact recovery scenario may never have occurred in the past and almost certainly not often enough to extrapolate from the historic data. ![\[fig:delay-function\]Plot showing $P[\mathcal{D}_{c}\le x]$ in function of $x$ for $m=5$ and $d=30$.](delay-plot) For these reasons, we propose to use synthetic delay distributions that are only parametrized on the planned timetable. We propose to add to each connection $c$ a synthetic delay variable $\mathcal{D}_{c}$ that depends on the change time $m$ of $c_{{\ensuremath{\mathrm{arr\_stop}}}}$ and on a global[^5] *maximum delay parameter* $d$. We define $\mathcal{D}_{c}$ as follows: $\forall x\in(-\infty,0]:P[\mathcal{D}_{c}\le x]=0$, $\forall x\in(0,m]:P[\mathcal{D}_{c}\le x]=\frac{2x}{6m-3x}$, $\forall x\in(m,m+d]:P[\mathcal{D}_{c}\le x]=\frac{31(x-m)+2d}{30(x-m)+3d}$, and $\forall x\in(m+d,\infty):P[\mathcal{D}_{c}\le x]=1$. The function is illustrated in Figure \[fig:delay-function\] and the rational for its our design is given in the next section. ### Synthetic Delay Distribution There are many methods to come up with formulas for synthetic delays. The lack of any effectively accessible ground truth makes any conclusive experimental evaluation of their quality very difficult. The only real criteria that we have is “intuitively reasonable”. The approach presented here is by no way the final answer to the question of how to design the best synthetic delay distribution. In this section, we describe the rational for our design decisions. We define for every connection $c$ its delay $\mathcal{D}_{c}$ by defining its cumulative distribution function $f_{m,d}(x)$, where $d$ is the maximum delay of $c$ and $m$ the minimum change time at $c_{{\ensuremath{\mathrm{arr\_stop}}}}$. Our delays do not depend on any other parameter than $m$ and $d$. We have the following hard requirements on $f_{m,d}$ resulting from our algorithm: - $f_{m,d}(x)$ is a probability, i.e., $\forall x:0\le f(x)\le1$ - $f_{m,d}(x)$ is a cumulative distribution function and therefore non-decreasing, i.e., $\forall x:f_{m,d}'(x)\ge0$ - $\max\mathcal{D}_{c}$ should be $m+d$, i.e., $\forall x\ge m+d:f(x)=1$ - Our model does not allow for trains that arrive too early, i.e.,$\forall x<0:f(x)=0$ These requirements already completely define what happens outside of $x\in(0,m+d)$. Because of the limitations of current hardware, there are two additional more fuzzy but important requirements: - We need to evaluate $f_{m,d}(x)$ many times. The formula must therefore not be computationally expensive. - Our algorithm computes a lot of $\left(f_{m,d}(x_{1})+a_{1}\right)\cdot\left(f_{m,d}(x_{2})+a_{2}\right)\cdot\left(f_{m,d}(x_{3})+a_{3}\right)\cdots$ chains. The chain length reflects the number of rides in the longest journey considered during the computations. As 64-bit-floating points only have a limited precision, we must make sure that order of magnitude of the various values of $f_{m,d}$ do not differ too much. If they do differ a lot then the less likely journeys have no impact on the overall EAT because their impact is rounded away. Finally there are a couple of soft constraints coming from our intuition: - $f(m)$ is the probability that everything works as scheduled without the slightest delay. In practice this does happen and therefore this should have reasonable high probability. On the other hand a too high $f(m)$ can lead to problems with rounding. We set $f(m)=\frac{2}{3}$ as we believe that it is a good compromise. - We want $f$ to be continuous. - The maximum variation should be at $x=m$, i.e., $f'(m)$ should be the unique local maximum of $f'$. - Initially the function should grow slowly and then once $x=m$ is reached the growth should slow down. This can be formalized as $f''(x)>0$ for $x\in(0,m)$ and $f''(x)<0$ for $x\in(m,m+d)$. We define $f$ using two piece function $f_{1}$ and $f_{2}$. For these pieces we assume $m=5\mbox{min}$ and $d=30\mbox{min}$ and scale them to accommodate for different values, as following: $$f_{m,d}(x)=\begin{cases} 0 & \mbox{ if }x<0\\ f_{1}(\frac{5x}{m}) & \mbox{ if }0\le x\le m\\ f_{2}\left(\frac{30(x-m)}{d}\right) & \mbox{ if }m<x<m+d\\ 1 & \mbox{ if }m+d\le x \end{cases}$$ It remains to define $f_{1}$ and $f_{2}$. We started with a $-1/x$ function and shifted and stretched the function graphs until we ended up with something that looks “intuitively reasonable”. $$\begin{aligned} f_{1}(x) & = & \frac{2x}{3(10-x)}\\ f_{2}(x) & = & \frac{31x+60}{30(x+3)}\end{aligned}$$ The resulting function $f$ fulfills all requirements and is illustrated in Figure \[fig:delay-function\]. To sum up: We define the $f_{m,d}$ as following: $$f_{m,d}(x)=\begin{cases} 0 & \mbox{ if }x<0\\ \frac{2x}{6m-3x} & \mbox{ if }0\le x\le m\\ \frac{31(x-m)+2d}{30(x-m)+3d} & \mbox{ if }m<x<m+d\\ 1 & \mbox{ if }m+d\le x \end{cases}$$ Decision Graphs {#sec:decision_graphs} --------------- In this subsection, we first introduce the notion of safe journey, then formally define decision graphs, and then introduce three problem variants: (i) the unbounded, (ii) the bounded, and (iii) the $\alpha$-bounded MEAT problems. The first two are of more theoretical interest, whereas the third one has the highest practical impact. We prove basic properties of the unbounded and bounded problems and show a relation to the earliest safe arrival problem. ### Formal Definition A *safe $(s,\tau_s,t)$-journey* is a $(s,\tau_s,t)$-journey such for every transfer the time difference between the arrival of the incoming train and the departure of the outgoing train is at least the maximum delay of the incoming train. We denote by $\mathrm{eat}(s,\tau_s,t)$ the arrival time of an optimal earliest arrival journeys and by $\mathrm{esat}(s,\tau_s,t)$ the arrival time of an optimal safe earliest arrival journey. A *$(s,\tau_s,t)$-decision graph* from source stop $s$ to target stop $t$ with the traveler departing at time $\tau_s$ is a directed reflexive-loop-free multi-graph $G=(V,A)$ whose vertices correspond to stops and whose arcs correspond to legs $l$ directed from $l_{{\ensuremath{\mathrm{dep\_stop}}}}$ to $l_{{\ensuremath{\mathrm{arr\_stop}}}}$. There may be several legs between a pair of stops, but they must be of part of different trips and depart at different times. We formalize this as: $\forall l^{1},l^{2}\in A:l_{{\ensuremath{\mathrm{dep\_time}}}}^{1}\neq l_{{\ensuremath{\mathrm{dep\_time}}}}^{2}\vee l_{{\ensuremath{\mathrm{dep\_stop}}}}^{1}\neq l_{{\ensuremath{\mathrm{dep\_stop}}}}^{2}$. We require that the user must be able to reach every leg and must always be able to get to the target. Formally, we require that for every $l\in A$ there exists a $(s,\tau_s,l_{{\ensuremath{\mathrm{dep\_stop}}}})$-journey $j$ with $j_{{\ensuremath{\mathrm{arr\_time}}}}\le l_{{\ensuremath{\mathrm{dep\_time}}}}$ to reach the leg, and a safe $(l_{{\ensuremath{\mathrm{arr\_stop}}}},l_{{\ensuremath{\mathrm{arr\_time}}}}+\max\mathcal{D}_{r},t)$-journey $j'$ to reach the target. To exclude decision graphs with unreachable stops, we require that every stop in $V$ except $s$ and $t$ have non-zero in- and out degree. We first recursively define the *expected arrival time* $e(l)$ (short EAT) of a leg $l\in A$ and define in terms of $e(l)$ the EAT $e(G)$ of the whole decision graph $G$. If $l_{{\ensuremath{\mathrm{arr\_stop}}}}=t$, we define $e(l)=l_{{\ensuremath{\mathrm{arr\_time}}}}+E[\mathcal{D}_{l}]$. Otherwise $e(l)$ is defined in terms of other legs. Denote by $q_{1}\ldots q_{n}$ the sequence of legs in $G$ ordered by departure time, departing at $l_{{\ensuremath{\mathrm{arr\_stop}}}}$ after $l_{{\ensuremath{\mathrm{arr\_time}}}}$, i.e., every leg that the user could reach after $l$ arrives. Denote by $d_{1}\ldots d_{n}$ their departure times and set $d_{0}=l_{{\ensuremath{\mathrm{arr\_time}}}}$. We define $e(l)=\sum_{i\in\{1\ldots n\}}P[d_{i-1}<\mathcal{D}_{l}<d_{i}] \cdot e(q_{i})$, i.e., the average of the EATs of the connecting legs weighted by the transfer probability. Note that this definition is well-defined because $e(l)$ only depends on $e(q)$ of legs with a later departure time, i.e., $l_{{\ensuremath{\mathrm{dep\_time}}}}<q_{{\ensuremath{\mathrm{dep\_time}}}}$. Further notice that $P[\mathcal{D}_{l}<d_{n}]=1$. Otherwise no safe journey to the target would exist invalidating the decision graph. We denote by $G^{\ensuremath{\mathrm{first}}}$ the leg $l\in A$ with minimum $l_{{\ensuremath{\mathrm{dep\_time}}}}$. This is the leg that the user must initially take at $s$. We define the *expected arrival time* $e(G)$ (short EAT) of the decision graph $G$ as $e(G^{\ensuremath{\mathrm{first}}})$. Furthermore, the *latest arrival time* $G_{\max {\ensuremath{\mathrm{arr\_time}}}}$ is the maximum $l_{{\ensuremath{\mathrm{arr\_time}}}}+\max\mathcal{D}_{l}$ over all $l\in A$. Note that by minimizing $G_{\max {\ensuremath{\mathrm{arr\_time}}}}$, we can bound the worst case arrival time giving us some control over the arrival time variance. The *unbounded $(s,\tau_s,t)$-minimum expected arrival time* (short MEAT) problem consists of computing a *$(s,\tau_s,t)$-*decision graph $G$ minimizing $e(G)$. The bounded *$(s,\tau_s,t)$-MEAT* problem consists of computing a *$(s,\tau_s,t)$*-decision graph $G$ minimizing $e(G)$ subject to a minimum $G_{\max {\ensuremath{\mathrm{arr\_time}}}}$. As a compromise between bounded and unbounded, we further define the $\alpha$-bounded MEAT problem: We require that $G_{\max {\ensuremath{\mathrm{arr\_time}}}}-\tau_s\le\alpha\left(\mathrm{esat}\left(s,\tau_s,t\right)-\tau_s\right)$, i.e., the maximum travel time must not be bigger than $\alpha$ times the delay-free optimum. Notice that the bounded and 1-bounded MEAT problems are equivalent. ### Decision Graph Existence There is a *$(s,\tau_s,t)$-*decision graph $G$ iff there exists a safe *$(s,\tau_s,t)$*-journey $j$. If there exists a *$(s,\tau_s,t)$-*decision graph $G$ then by the decision graph definition we know that there exists a safe $(G_{{\ensuremath{\mathrm{arr\_stop}}}}^{\ensuremath{\mathrm{first}}},G_{{\ensuremath{\mathrm{arr\_time}}}}^{\ensuremath{\mathrm{first}}}+\max\mathcal{D}_{G^{\ensuremath{\mathrm{first}}}},t)$-journey $j'$. Prefixing $j'$ with $G^{\ensuremath{\mathrm{first}}}$ yields the required $(s,\tau_s,t)$-journey $j$. Conversly, if there exists a $(s,\tau_s,t)$-journey $j$, we can construct a (non-optimal) $(s,\tau_s,t)$-decision graph $G$ that contains exactly the same legs as $j$. A direct consequence of this lemma is that the minimum $G_{\max {\ensuremath{\mathrm{arr\_time}}}}$ over all *$(s,\tau_s,t)$-*decision graphs $G$ is equal to $\mathrm{esat}(s,\tau_s,t)$. Using this observation we can reduce the bounded MEAT problem to the unbounded MEAT problem. Formally stated: An optimal solution $G$ to the bounded *$(s,\tau_s,t)$-*MEAT problem on timetable $T$ is an optimal solution to the unbounded $(s,\tau_s,t)$-MEAT problem on a timetable $T'$ where $T'$ is obtained by removing all connections $c$ from $T$ with $c_{{\ensuremath{\mathrm{arr\_time}}}}$ above the $\mathrm{esat}(s,\tau_s,t)$. There are two central observations needed for the proof: First, every $(s,\tau_s,t)$-decision graph on timetable $T'$ is a $(s,\tau_s,t)$-decision graph on the strictly larger timetable $T$. Second, every safe *$(s,\tau_s,t)$-*journey in $T'$ is an earliest safe *$(s,\tau_s,t)$-*journey in $T$. Suppose that a *$(s,\tau_s,t)$*-decision graph $G'$ on $T'$ would exist with a suboptimal $G'_{\max {\ensuremath{\mathrm{arr\_time}}}}$ then there would also exist a safe *$(s,\tau_s,t)$-*journey $j'$ in $T'$ with a suboptimal $j_{{\ensuremath{\mathrm{arr\_time}}}}'$, which is not possible by construction of $T'$, which is a contradiction. ![\[fig:infinite-network\]A timetable $T_{p}$ has 4 stops: $s$, $a$, $b$ and $t$. The arrows denote connections. An arrow annotated with its departure time and arrival time. A simple arrow ($\rightarrow$) denotes a single non-repeating connection. A double arrow ($\rightrightarrows$) is repeated every 4 time units, i.e. $1\rightrightarrows2$ is a shorthand for $1+4i\rightarrow2+4i$ for every $i\in\mathbb{N}$. All connections are part of their own trip and have the same delay variable $\mathcal{D}$. We define $P[\mathcal{D}=0]=p$ (with $p\neq0$) and $P[\mathcal{D}<1]=1$.](infinit-timetable) Having shown how to explicitly bound $G_{\max {\ensuremath{\mathrm{arr\_time}}}}$ it is natural to ask what would happen if we dropped this bound and solely minimized $e(G)$. For this we consider the timetable $T_{p}$ with an infinite connection set illustrated and defined in Figure \[fig:infinite-network\]. Notice that $T_{p}$ is constructed such that it does not matter whether the user arrives at $a$ at moments $1+4\mathbb{N}$ or at $b$ at moments $3+4\mathbb{N}$ as the two states are completely symmetric with the stops $a$ and $b$ swapping roles. By exploiting this symmetry we can reduce the set of possibly optimal $(s,0,t)$-decision graphs to 2 elements: the decision graph $G^{1}$ that waits at $a$ and never goes over $b$, and the decision graph $G^{2}$ that oscillates between $a$ and $b$. The corresponding expected arrival times are $e(G^{1})=p\left(2+E[\mathcal{D}]\right)+(1-p)\left(7+E[\mathcal{D}]\right)$ and $e(G^{2})=p\left(2+E[\mathcal{D}]\right)+(1-p)\left(3+e(G^{2})\right)$. The later equation can be resolved to $e(G^{2})=E[\mathcal{D}]-1+\frac{3}{p}$. We can solve $e(G^{1})<e(G^{2})$ in terms of $p$. The result is that $G^{1}$ is better if $p<\frac{\sqrt{43}-4}{9}\approx0.28$. If they are equal then $G^{1}$ and $G^{2}$ are equivalent and otherwise $G^{2}$ is better. This has consequences even for timetables with a finite connection set. One could expect that to compute a decision graph it is sufficient to look at a time-interval proportional to its expected travel time: It seems reasonable that a connection scheduled to occur in ten years would not be relevant for a decision graph departing today with an expected travel time of one hour. However, this intuition is false in the worst case: Consider the finite sub-timetable $T'$ of the periodic timetable $T_{p}$ that encompasses the first ten years (i.e., we “unroll” $T_p$ for ten years). For $p{>}0.28$, an optimal $(s,0,t)$-decision graph will use all connections in $T'$, including the ones in ten years (as $G^2$ would). Fortunately, the bounded MEAT problem does not suffer from this weakness: No connection arriving after $\mathrm{esat}(s,0,t)$ can be relevant. Therefore, even on infinite networks the bounded MEAT problem always admits finite solutions. This property is the main motivation to study the bounded MEAT problem. ### Non-dominated Pairs and Decision Graphs In this section, we only consider decision graphs and journeys arriving at a fixed target stop $t$. All lemmas and definition are therefore with respect to $t$. To simplify our notation, we omit $t$ in this section. We consider, for every connection $c$, the pair $p_c=(c_{\ensuremath{\mathrm{dep\_time}}}, e(G))$ where $G$ is a decision graph that minimizes the expected arrival time, subject to $c$ being the first connection, i.e., $G^{\ensuremath{\mathrm{first}}}_{\ensuremath{\mathrm{enter}}}=c$. Denote by $O$ the outgoing connections of a stop. Every connection has an associated pair, which can be dominated within $O$. This allows us to define when a connection is dominated: It is dominated when its pair is dominated. A leg $l$ is dominated if $l_{\ensuremath{\mathrm{enter}}}$ is dominated. Non-dominated connections have an important role in the computation of optimal decision graphs as the following lemma shows. \[lem:meat-only-non-dominated\] For every source stop $s$ and source time $\tau_s$, if there exists a decision graph, then there exists an optimal decision graph, such that for every leg $l$ of $G$, the entry connection $l^{\ensuremath{\mathrm{enter}}}$ is non-dominated at $l^{\ensuremath{\mathrm{enter}}}_{\ensuremath{\mathrm{dep\_stop}}}$. We know that an optimal decision graph $H$ exists as we required the existence of a decision graph. If $H^{\ensuremath{\mathrm{first}}}_{\ensuremath{\mathrm{enter}}}$ is dominated, then there is another optimal decision graph associated with the dominating connection. Without loose of generality we can therefore assume that, $H^{\ensuremath{\mathrm{first}}}_{\ensuremath{\mathrm{enter}}}$ is non-dominated. Suppose that $H$ contained some other leg $l$ such that $l^{\ensuremath{\mathrm{enter}}}$ is dominated. Further denote by $l'$ an incoming leg from which the traveler might transfer to $l$. $l'$ must exist because $l$ is not the first leg in the decision graph. As $l$ is dominated, removing it and all legs that can only reached via $l$ from $H$ improves $e(l')$, which in terms improves $e(H)$, which is a contradiction to $H$ being optimal. Solving the MEAT Problem ------------------------ The unbounded MEAT problem can be solved to optimality on finite networks, and by extension also the bounded and $\alpha$-bounded MEAT problems. We first describe an algorithm to optimally solve the unbounded MEAT problem. By applying this algorithm to a restricted timetable we solve the bounded and $\alpha$-bounded MEAT problems. ### Solving the Unbounded MEAT problem Our algorithm works in two phases: - Compute the minimum expected arrival times for all connections $c$, - extract a desired $(s,\tau_s,t)$-decision graph. The first phase is a variant of the earliest arrival profile Connection Scan algorithm. The second phase is an extension of the journey extraction algorithm. #### Phase 1: Computing all Expected Arrival Times. Recall the basic Connection Scan profile framework depicted in Figure \[alg:profile-connection-scan-framework\] and especially the earliest arrival time instantiation depicted in Figure \[alg:earliest-arrival-profile-connection-scan\]. We first describe the algorithmic differences to the later and then explain why the proposed algorithm is correct. In the context of this subsection $c$ always refers to the connection being scanned. The first key idea consists of replacing all earliest arrival times with minimum expected arrival times. This works similarly to the profile Pareto-optimization where all earliest arrival times were replaced by vectors. The stop data structure becomes an array of dynamic arrays of pairs of departure time and expected arrival time. The trip data structure becomes an array of expected arrival times. The computation of the expected arrival time when arriving at the target $\tau_1$ is only modified in a minor way: We need to add $E\mathcal{D}_c$, a constant, to the arrival time of the connection. The arrival time when the traveler remains sitting $\tau_2$ is computed in exactly the same way by reading the value of $T[c_{\ensuremath{\mathrm{trip}}}]$. The computation of the arrival time when changing trains $\tau_3$ is significantly modified and is described below. The value of $\tau_c$ is still computed as the minimum of $\tau_1$, $\tau_2$, and $\tau_3$. $\tau_c$ is the minimum expected arrival time over all decision graphs starting in $c$. Formulated differently, $\tau_c$ is the minimum $e(G)$ over all decision graphs such that $G^\mathrm{first}_{\ensuremath{\mathrm{enter}}}= c$. Incorporating $\tau_c$ into the trip data structure $T$ and the stop profiles $S$ works completely analogous to the earliest arrival profile algorithm. The computation of $\tau_3$, i.e., the computation of the arrival time when transferring trains is changed. The reason for this change is that the arriving train $c$ has a random arrival time between $c_{\ensuremath{\mathrm{arr\_time}}}$ and $c_{\ensuremath{\mathrm{arr\_time}}}+ \max \mathcal{D}_c$. Our algorithm starts by determining using a sequential scan all pairs $p^1\ldots p^k$ in the profile $S[c_{\ensuremath{\mathrm{arr\_stop}}}]$ that might be relevant. These are all pairs departing between $c_{\ensuremath{\mathrm{arr\_time}}}$ and $c_{\ensuremath{\mathrm{arr\_time}}}+ \max \mathcal{D}_c$ and the first pair after $c_{\ensuremath{\mathrm{arr\_time}}}+ \max \mathcal{D}_c$. These correspond to all outgoing trains that are worth taking. It then computes $\tau_3$ as the weighted sum over the expected arrival times of all $p^i$. A pair is weighted by the probability of the incoming being delay in such a way that the traveler will take it. Formally this means: $p^1$ is weighted by the probability $P[c_{\ensuremath{\mathrm{arr\_time}}}+ \max \mathcal{D}_c \le p^1_{\ensuremath{\mathrm{dep\_time}}}]$ and all other $p^i$ are weighted by $P[p^{i-1}_{\ensuremath{\mathrm{dep\_time}}}\le c_{\ensuremath{\mathrm{arr\_time}}}+ \max \mathcal{D}_c \le p^i_{\ensuremath{\mathrm{dep\_time}}}]$. Formulated differently, $\tau_2$ is the average over the expected arrival time of the non-dominated outgoing trains weighted by the probability of the traveler transferring to them. The correctness of our algorithm relies on optimal decision graphs not containing any dominated legs as shown in Lemma \[lem:meat-only-non-dominated\]. The domination test in the profile insertion filters dominated pairs and pairs which appear several times. In the later case there are two or more connections that depart at the same time and have the same expected arrival time. In this case, it does not matter which we insert into the decision graph but we may only insert one. Our algorithm picks the connection that appears last in the connection array. It remains to show why our strategy of selecting all outgoing non-dominated connections during the evaluation is optimal. This directly follows from the pairs being ordered. One does not want to skip earlier pairs because they have lower expected arrival times than the later trains. One cannot remove the later trains because it is not guaranteed that the earlier trains can be reached. Connection not in the profile are dominated. From Lemma \[lem:meat-only-non-dominated\] follows that we can ignore dominated connections. #### Phase 2: Extracting Decision Graphs. We extract a $(s,\tau_s,t)$-decision graph $G=(V,A)$ by enumerating all legs in $A$. The stop set $V$ can then be inferred from $A$. At the core, our algorithm uses a min-priority queue that contains connections ordered increasing by their departure time. Initially, we add the earliest connection in the profile of $s$ to the queue. While the queue is not empty we pop the earliest connection $c^{1}$ from it. Denote by $c^{2}\ldots c^{n}$ all subsequent connections in the trip $c_{{\ensuremath{\mathrm{trip}}}}^{1}$. The desired leg $l=(c^{1},c^{i})$ is given by the first $i$ such that $e(c^{1})\neq e(c^{i+1})$ (or $i=n$ if all are equal). We add $l$ to $G$. If $c_{{\ensuremath{\mathrm{arr\_stop}}}}^{i}\neq t$ we add the following connections to the queue: (i) All connections in the profile of $c_{{\ensuremath{\mathrm{arr\_stop}}}}^{i}$ departing between $c_{{\ensuremath{\mathrm{arr\_time}}}}^{i}$ and $c_{{\ensuremath{\mathrm{arr\_time}}}}^{i}+\max\mathcal{D}_{c^{i}}$, and (ii) the first connection in the profile of $c_{{\ensuremath{\mathrm{arr\_stop}}}}^{i}$ departing after $c_{{\ensuremath{\mathrm{arr\_time}}}}^{i}+\max\mathcal{D}_{c^{i}}$. ### Solving the $\alpha$-Bounded MEAT Problem {#sub:Bounded-MEAT-Algo} We assume that the connection set is stored as an array ordered by departure time. To solve the $\alpha$-bounded $(s,\tau_s,t)$-MEAT problem we perform the following steps: (i) run a binary search on the connection set to determine the earliest connection $c^{\ensuremath{\mathrm{first}}}$ departing after $\tau_s$, (ii) run a one-to-one Connection Scan from $s$ to $t$ that assumes all connections $c$ are delayed by $\max \mathcal{D}_c$ to determine $\mathrm{esat}\left(s,\tau_s,t\right)$ (iii) let $\tau_{\mathrm{last}}=\tau_s+\alpha\cdot\left(\mathrm{esat}\left(s,\tau_s,t\right)-\tau_s\right)$ and run a second binary search on the connection set to find the last connection $c^{\mathrm{last}}$ departing before $\tau_{\mathrm{last}}$, (iv) run a one-to-all Connection Scan from $s$ restricted to the connections from $c^{\ensuremath{\mathrm{first}}}$ to $c^{\mathrm{last}}$ to determine all $\mathrm{eat}\left(s,\tau_s,\cdot\right)$, (v) run Phase 1 of the unbounded MEAT algorithm scanning the connections from $c^{\mathrm{last}}$ to $c^{\ensuremath{\mathrm{first}}}$ skipping connections $c$ for which $c_{\ensuremath{\mathrm{arr\_time}}}> \tau_{\mathrm{last}}$ or $\mathrm{eat}(s,\tau_s,c_{{\ensuremath{\mathrm{dep\_stop}}}})\le c_{{\ensuremath{\mathrm{dep\_time}}}}$ does not hold, and finally (vi) run Phase 2 of the unbounded MEAT algorithm, i.e., extract the $(s,\tau_s,t)$-decision graph. Decision Graph Representation {#sec:representation} ----------------------------- In the previous section we described how to compute decision graphs. In practice this is not enough and we must be able to represent the graph in a form that the user can effectively comprehend. The main obstacle here is to prevent the user from being overwhelmed with information. A secondary obstacle is how to actually layout the graph. In this section, we solely focus to reducing the amount of information. The presented drawings were created by hand. In the demonstration, we use GraphViz [@egknw-gdsdg-03]. ### Expanded Decision Graph Representation    The expanded decision graph subdivides each node $v$ into slots $s_{v,1}\ldots s_{v,n}$ that correspond to moments in time that an arc arrives or departs at $v$. The slots in each node are ordered from top to bottom in chronological order. Each arc $(u,v)$ connects the corresponding slots $s_{u,i}$ and $s_{v,j}$. To determine his next train the user has to search for the box corresponding to his current stop and pick the first departure slot after the current moment in time. The arrows guide him to the box corresponding to his next stop. Figure \[fig:meat-expanded\] illustrates this graph drawing style. ### Compact Decision Graph Representation The scheduled arrival time of trains is an information contained in the expanded decision graph that is not strictly necessary. A traveler decides what outgoing train to take when he arrives. At that moment, he can look at any clock to figure out the precise arrival time. The scheduled arrival time recorded in the timetable is not needed for his decision. The compact decision graph exploits this observation by removing the arrival time information from the representation. Each arc $(u,v)$ connects the corresponding departure slot $s_{u,i}$ directly to the stop $v$ instead of a slot. Time slots that only appear as arrival slots are removed. If two outgoing arcs of a node $u$ have the same destination and depart subsequently, they are grouped and only displayed once. The compact decision graph is never larger than the expanded one and most of the time significantly smaller. See Figure \[fig:meat-compact\] for an example of a compact decision graph. ### Relaxed Dominance Decision graphs exist that contain legs that have near to no impact on the EAT. Removing them increases the EAT by only a small amount, resulting in an almost optimal decision graph that can be significantly smaller. To exploit this, we introduce a *relaxation tuning parameter* $\beta$. EATs are regarded as equal if their difference is below $\beta$. Formulated in terms of the framework depicted in Figure \[alg:profile-connection-scan-framework\], we only insert a new pair into the profile $S[x]$, if the expected arrival time of the earliest pair of $S[x]$ is at least $\beta$ time units later than $\tau_c$. ### Displaying only the Relevant Subgraphs In many scenarios, we have a canvas of fixed size. If even the compact relaxed decision graph is too large to fit, we can only draw parts of it. We observe that the decision graph extraction phase does not rely on the actual distributions of the delay variables $\mathcal{D}_{c}$ but only on $\max\mathcal{D}_{c}$. It extracts all connections departing in an interval $I$, plus the first connection directly afterwards. The full decision graph is extracted when $I=[c_{\ensuremath{\mathrm{arr\_stop}}},c_{\ensuremath{\mathrm{arr\_stop}}}+\max \mathcal{D}_c]$. Reducing the size of $I$ reduces the number of legs displayed, while still guaranteeing that backup legs exist. For example a smaller partial decision graph is extracted, if we only follow the connections departing in $I=[c_{\ensuremath{\mathrm{arr\_stop}}},c_{\ensuremath{\mathrm{arr\_stop}}}+\kappa]$ for $\kappa=1/2 \cdot \max \mathcal{D}_c$. Valid values for $\kappa$ are from $0$ to $\max \mathcal{D}_c$. We refer to $\kappa$ as *display window*. Given an upper bound $\gamma$ on the number of arcs in the compact or expanded representation, we use a binary search to determine the maximum display window $\kappa$ and draw the corresponding subgraph. Note that in the worst case the display window has size 0. In this case the decision graph degenerates to a single-path-journey. Experiments {#sec:experiments} ----------- -------------------------------------- ----- ------ ----- ------- ------- ----- ----- ------ ------ ----- ---- ------ ----- (lr)[3-6]{}(lr)[7-10]{}(lr)[11-14]{} Avg 6452 12 98 42 138 12 87 35 26 9 45 19 33% 6209 7 22 10 84 7 22 10 16 7 15 7 66% 7407 13 70 31 162 13 69 31 27 10 40 19 95% 7635 25 349 125 312 24 330 119 66 19 149 57 Max 7805 280 35450 28848 817 173 5540 4703 288 38 1607 366 Avg 5122 12 88 39 116 12 73 31 25 9 39 17 33% 4628 8 26 12 75 8 25 12 16 6 14 7 66% 6026 13 66 31 136 13 64 30 26 10 36 17 95% 6368 24 284 110 249 24 257 100 64 18 123 52 Max 6595 50 12603 6558 685 50 1576 478 240 37 1390 289 Avg 4180 11 66 33 100 11 51 25 24 9 29 15 33% 3845 8 24 12 66 8 23 11 15 6 13 6 66% 4808 13 53 26 115 12 51 25 25 10 30 15 95% 5028 22 178 82 216 22 155 74 61 17 84 42 Max 5159 54 6640 3220 553 54 760 285 196 34 590 183 -------------------------------------- ----- ------ ----- ------- ------- ----- ----- ------ ------ ----- ---- ------ ----- : \[tab:meat-results\]The time (in ms) needed to compute a decision graph and its size. Arcs is the number of arcs in the compact representation. The number of rides corresponds to the number of arcs in the expanded representation. The maximum delay parameter is set to 1h. We report average, maximum and the 33%-, 66%- and 95%-quantiles. --------- ------------ \#Stop 16 991 \#Conn. 55 930 920 \#Trip 3 965 040 --------- ------------ : \[tab:meat-instance\]Instance Size For our experiments, we used on a single core of a Xeon E5-2670 at 2.6 GHz, with 64 GiB of DDR3-1600 RAM, 20 MiB of L3 and 256 KiB of L2 cache. This is the “older” machine used in the experiments of the previous sections. We implemented the algorithm in C++ and compiled it using GCC 4.7.1 with -O3. The timetable is based on the data of [bahn.de](bahn.de) during winter 2011/2012. This the same primary data source as used for the experiments of Section \[sec:csa\_accel\_exp\]. However, we extracted a different formal timetable. We extracted every vehicle except for most buses as we mainly focus on train networks. Not having buses explains the significant instance size difference compared to the Germany instance of the previous sections. Not having buses allows us to get the running times onto a manageable level. Further, it allows us to focus on long-distance trains where delays have a significantly larger impact than in high-frequent inner-city transit. We removed footpaths longer than 10min, connected stops with a distance below 100m, and then contracted stops connected through footpaths adjusting their minimum change times resulting in an instance without footpaths. Not having footpaths again benefits query running times. We pick the largest strongly connected component to make sure that there always exists a journey (assuming enough days are considered). We extract one day of maximum operation (i.e. extract everything regardless of the day of operation and remove exact duplicates). We then replicated this day 30 times to have a timetable spanning about one month of operation. The detailed sizes are in Table \[tab:meat-instance\]. We ran 10000 random queries. Source and target stop are picked uniformly at random. The source time is chosen within the first 24h. We filter queries out that have an minimum delay-free travel time above 24h. Our experimental results are presented in Table \[tab:meat-results\]. The compact representation is smaller by a factor of 2 in terms of arcs than the expanded one. As expected, a larger relaxation parameter gives smaller graphs. Increasing the $\alpha$-bound leads to larger graphs and running times grow. The running times of unbounded queries are proportional to the timespan of the timetable (i.e. 30 days). On the other hand, the running times of bounded queries depend only on the maximum travel time of the journey. This explains the gap in running time of two orders of magnitude. As the maximum values are significantly higher than the 95%-quantile, we can conclude that the graphs are in most cases of manageable size with a few outlines that distort the average values. Upon closer manual inspection, we discover that most outliers with large decision graphs connect remote rural areas, where even no “good” delay-free journey exists. We can therefore not expect to find any form of robust travel plan. In Figure \[fig:display\] we evaluate the value of the display window such that the extracted graphs have less than 25 arcs in the compact representation. Recall that this modifies what is displayed to the user. It is still guaranteed that backups exist. As the 1.0-bounded graphs are smaller than 2.0-bounded graphs we can display more, explaining the larger display window. The difference between 2.0-bounded graphs and unbounded graphs is small. A greater relaxation parameter also reduces the graph size and thus allows for slightly larger display windows. If there is no “good” way to travel the decision graphs degenerate to single-path-journeys. #### Section Conclusions. We described the Minimum Expected Arrival Time (MEAT) problem and described an efficient CSA-based algorithm to solve it. This demonstrates that the CSA-framework is very flexible and can adapt to complex problem settings. The achieved query running times of 100ms on average are fast enough for interactive systems. This is further demonstrated by our proof of concept implementation accessible at <http://meatdemo.iti.kit.edu>. However, the fast query running times were bought by removing most buses from the instance. For the full Germany instance the running times are unfortunately prohibitively large. Fortunately, decision graphs make most sense in long-distance travel where most high-frequency local bus lines do not play a role. The size of the computed decision graphs can become large. Fortunately, using careful engineering it is possible to sufficiently reduce their size to a manageable size. Overall, we believe that the MEAT problem and our CSA-based algorithm are a promising basis on which an innovative timetable information system can be built. Conclusion ========== We described the Connection Scan family of algorithms (CSA). The algorithms are a simple solution to various routing problems in timetable-based networks. We presented profile and non-profile variants of the algorithm. CSA optimizes the arrival time and optionally the number of transfers in the Pareto sense. CSA can adjust to a new timetable in mere seconds enabling the computation of journeys with respect to the current delay situation. We combined CSA with multilevel overlay techniques yielding Connection Scan Accelerated (CSAccel). 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[^5]: $d$ is global since we lack per-train data. Our approach can be adjusted, if such data became available.

--- abstract: | A scheme for conditional generating a Hermite polynomial excited squeezed vacuum states (HESVS) is proposed. Injecting a two-mode squeezed vacuum state (TMSVS) into a beam splitter (BS) and counting the photons in one of the output channels, the conditional state in the other output channel is just a HESVS. To exhibit a number of nonclassical effects and non-Guassianity, we mainly investigate the photon number distribution, sub-Poissonian distribution, quadrature component distribution, and quasi-probability distribution of the HPESVS. We find that its nonclassicality closely relates to the control parameter of the BS, the squeezed parameter of the TMSVS, and the photon number of conditional measurement. These further demonstrate that performing the conditional measurement on a BS is an effective approach to generate non-Guassian state. **ocis:** (270.5570) Quantum detectors; (270.4180) Multiphoton processes; (270.5290) Photon statistics **Keywords:**Conditional measurement; beam splitter; Wigner function; Nonclassicality author: - 'Xue-xiang Xu$^{1,\dag }$, Hong-chun Yuan$^{2}$ and Hong-yi Fan$^{3}$' title: Generating Hermite polynomial excited squeezed states by means of conditional measurements on a beam splitter --- Introduction ============ Quantum state engineering has been a subject of increasing interest to construct various novel nonclassical states in quantum optics and quantum information processing[@a1; @a2]. From a theoretical point of view, the simplest way of generating nonclassical field states is to apply the photon creation operation to classical states such as the thermal and coherent states[@a3; @a4; @a5]. These nonclassical states, such as the single-photon added coherent state[@a6] and single-photon-added thermal state[@a7], have been realized experimentally. Subsequently, it has been demonstrated that subtracting photons from traditional quantum states exhibit an abundance of nonclassical properties[@a8; @a9; @a10; @a11]. Photon subtraction or addition can improve entanglement between Guassian states[@a12], loophole-free tests of Bell’s inequality[@a13], and quantum computing[@a14]. To meet the requirement of the development of quantum optics and quantum information tasks, some nonclassical states are explored by performing the different combination of photon subtraction and photon addition[a15,a16,a17,a18,a19,a20]{}, which have different properties. Kim et al[a21]{} discussed single photon adding then subtracting (or single photon subtracting then adding) coherent state (or thermal state) to probe quantum commutation rules $\left[ a,a^{\dag}\right] =1$. Lee et al[@a17] investigated the nonclassicality of field states when photon subtraction-then-addition operation or the photon addition-then-subtraction operation is applied to the coherent state (or thermal state), respectively. Yang and Li[@a18] analyzed multiphoton addition followed by multiphoton subtraction ($a^{l}a^{\dag k}$) and its inverse ($a^{\dag l}a^{k}$) on an arbitrary state. Recently, Lee and Nha[@a23] proposed a coherent superposition of photon addition and subtraction, $ta+ra^{\dag}$ ($% \left \vert t\right \vert ^{2}+\left \vert r\right \vert ^{2}=1$) acting on a coherent state and a thermal state. More recently, we investigated the nonclassical properties of optical fields generated by Hermite-excited coherent state[@a24] and Hermite-excited squeezed thermal states[a25]{}. On the other hand, another promising method for generating highly nonclassical states of optical fields is known to be conditional measurement[@a25a; @a26; @a27; @a28; @a28a]. Namely, when a system is prepared in an entangled state of two subsystems and a measurement is performed on one subsystem, then the quantum state of the other subsystem can be reduced to a new state. In particular, it turned out that conditional measurement on a beam splitter may be advantageously used for generating new classes of quantum states[@a28; @a28a]. Dakna’s group used conditional measurement on the BS to generate cat-like state[@a29]. Podoshvedov et al[@a27] proposed optical scheme for generating both a displaced photon and a displaced qubit via conditional measurement. In Ref.[@a30], they proposed to create arbitrary Fock states via conditional measurement on the BS. In addition, conditional output measurement on the BS may be used to produce photon-added states for a large class of signal-mode quantum states, such as thermal state, coherent state, and squeezed states[@a31]. Similarly, photon-subtracted states can be produced by means of conditional measurement on the BS[@a32]. Therefore, based on conditional measurement on the BS, it is possible to generate and manipulate various nonclassical optical fields in a real laboratory. In this paper, we study the Hermite polynomial excited squeezed vacuum state (HESVS), a kind non-Gaussian quantum state, generated by conditional output measurement on a BS. The calculations show that when a two-mode squeezed vacuum state (TMSVS) is injected in the input channels and the photon number of the mode in one of the output channels is measured, then the mode in the other output channel is prepared in a conditional state that has the typical features of a Hermite polynomial excited squeezed state. To exhibit the nonclassical properties of this conditional state, we mainly analyze the states in terms of the photon number distribution, sub-Poissonian distribution, quadrature component distribution, and Quasi-probability distribution including the Wigner function(WF) and Husimi function(HF). The paper is organized as follows. Section 2 presents the basic scheme for generation of the HESVS and its normalization related to Legendre polynomial. The nonclassical properties of the HESVS are analytically and numerically studied in Section 3-4. The results indicate that the conditional HESVS is strongly noncassical and non-Gaussian due to the presence of the partial negative WF. Finally, a summary and concluding remarks are given in Section 5. Generation of Hermite polynomial excited squeezed state ======================================================= It is well known that the input-output relations at a lossless beam splitter can be characterized by the SU(2) Lie algebra. In the Schrödinger picture, the role played by the beam splitter (BS) upon the input state $% \rho _{in}$ results in the output state $$\rho _{out}=\hat{B}\rho _{in}\hat{B}^{\dag }, \label{1.1}$$where $\hat{B}=\exp \left[ \theta \left( a^{\dag }b-ab^{\dag }\right) \right] $ corresponds to the unitary operator in terms of the creation (annihilation) operator $a^{\dag }$($a$) and $b^{\dag }$($b$) for mode $a$ and $b$, whose transformations satisfy[@a30] $$\begin{aligned} \hat{B}a\hat{B}^{\dag }& =a\cos \theta -b\sin \theta , \notag \\ \hat{B}b\hat{B}^{\dag }& =a\sin \theta +b\cos \theta . \label{1.1a}\end{aligned}$$Moreover, $\cos \theta $ and $\sin \theta $ are the transmittance and reflectance of the beam splitter, respectively. Note that the globe phase factor of BS may be omitted without loss of generality. For the sake of simplicity, we also assume that $\theta $ is tunable in the range of $\left[ 0,\pi /2\right] $. Under special circumstances, when $\theta =0$ or $\theta =\pi /2$, the BS corresponds to the cases of total transmission and total reflection, respectively. For $\theta =\pi /4$, the BS is just the symmetrical, i.e. 50/50 BS. Hermite polynomial excited squeezed state ----------------------------------------- A two-mode squeezed vacuum state (TMSVS) is the correlated state of two field modes $a$ and $b$ (signal and idle) that can be generated by a nonlinear medium. Theoretically, the TMSVS is obtained by applying the unitary operator $S_{2}\left( r\right) $ on the two-mode vacuum,$$\left\vert \Psi \right\rangle _{ab}=S_{2}\left( r\right) \left\vert 0,0\right\rangle =\cosh ^{-1}re^{a^{\dag }b^{\dag }\tanh r}\left\vert 0,0\right\rangle , \label{1.2}$$where $S_{2}\left( r\right) =\exp \left[ r\left( a^{\dag }b^{\dag }-ab\right) \right] $ is the two-mode squeezed operator and the values of $r$ determines the degree of squeezing. The larger $r$, the more the state is squeezed. Especially, when $r=0$, $\left\vert \Psi \right\rangle _{ab}$ reduces to two-mode vacuum state $\left\vert 0,0\right\rangle $. \[Fig1-1\] ![[Preparation scheme of HPESVS. When a TMSVS is mixed by a beam splitter and the number of photons ]{}$\left\vert m\right\rangle $[is measured in one of the output channels, then the conditional quantum state in the other output channel is generated.]{}](BS.eps "fig:"){width="0.9\columnwidth"} The conceptual scheme of the experimental setup is depicted in Fig.1. The two input modes prepared in the two-mode squeezed state ($\rho _{in}=\left\vert \Psi \right\rangle _{ab}\left\langle \Psi \right\vert $) is mixed at BS, so the output-state density operator can be given by $\rho _{out}=\hat{B}\left\vert \Psi \right\rangle _{ab}\left\langle \Psi \right\vert \hat{B}^{\dag }.$ In fact the output modes in $\rho _{out}$ are generally highly correlated. When the photon number of the mode in the second output channel is measured and $m$ photons are detected, then the mode in the first output channel is prepared in a quantum state, whose density operator $\rho _{out}^{a}$ reads as$$\rho _{out}^{a}=N_{m}^{-1}\left. _{b}\left\langle m\right\vert \hat{B}% \left\vert \Psi \right\rangle _{ab}\left\langle \Psi \right\vert \hat{B}% ^{\dag }\left\vert m\right\rangle _{b}\right. =\left\vert \Psi _{m}\right\rangle \left\langle \Psi _{m}\right\vert , \label{1.3}$$where $\left\vert \Psi _{m}\right\rangle $ is the normalized output conditional state (a pure state) and $N_{m}$ is the normalization factor determined by $\mathrm{Tr}\left( \rho _{out}^{a}\right) =1$. Next, using the integration of the TMSVS[@a33], $$\left\vert \Psi \right\rangle _{ab}=\frac{1}{\sinh r}\int \frac{d^{2}\alpha }{\pi }e^{-\left\vert \alpha \right\vert ^{2}/\tanh r+\alpha a^{\dag }+\alpha ^{\ast }b^{\dag }}\left\vert 0_{a},0_{b}\right\rangle , \label{1.3a}$$where $\left\vert 0_{a},0_{b}\right\rangle =\left\vert 0\right\rangle _{a}\otimes \left\vert 0\right\rangle _{b}$ is two-mode vacuum state, and the transformation relation in Eq.(\[1.1a\]), after some algebra we derive that$$\begin{aligned} & _{b}\left\langle m\right\vert \hat{B}\left\vert \Psi \right\rangle _{ab} \notag \\ & =\frac{1}{\sinh r}\int \frac{d^{2}\alpha }{\pi }e^{-\left\vert \alpha \right\vert ^{2}/\tanh r}e^{\left( \alpha \cos \theta +\alpha ^{\ast }\sin \theta \right) a^{\dag }}\left\vert 0\right\rangle _{a} \notag \\ & \times \frac{1}{\sqrt{m!}}\frac{\partial ^{m}}{\partial \tau ^{m}}e^{\tau \left( \alpha ^{\ast }\cos \theta -\alpha \sin \theta \right) }|_{\tau =0} \notag \\ & =\frac{1}{\cosh r\sqrt{m!}}\frac{\partial ^{m}}{\partial \tau ^{m}}e^{% \frac{\mu }{2}a^{\dag 2}-\frac{\mu }{2}\tau ^{2}+\tau \nu a^{\dag }}\left\vert 0\right\rangle _{a}|_{\tau =0}, \label{1.3b}\end{aligned}$$where $\mu =\sin 2\theta \tanh r$ and $\nu =\cos 2\theta \tanh r$. Hence the output conditional state $\left\vert \Psi _{m}\right\rangle $ is explicitly expressed as $$\left\vert \Psi _{m}\right\rangle =\Omega _{m}^{1/2}H_{m}\left( \frac{\nu a^{\dag }}{\sqrt{2\mu }}\right) S_{1}\left( \lambda \right) \left\vert 0\right\rangle \label{1.4}$$with $\Omega _{m}=\mu ^{m}\cosh \lambda /(2^{m}m!N_{m}\cosh ^{2}r)$, where we have used the generating function of the single-variable $m$-order Hermite polynomial $H_{m}\left( x\right) =\partial _{\tau }^{m}e^{2x\tau -\tau ^{2}}|_{\tau =0}\ $and the expression of single-mode squeezed vacuum $% S_{1}\left( \lambda \right) \left\vert 0\right\rangle =\cosh ^{-1/2}\lambda e^{\left( \tanh \lambda /2\right) a^{\dag 2}}\left\vert 0\right\rangle $ with the single-mode squeezed operator $S_{1}\left( \lambda \right) =\exp % \left[ \lambda \left( a^{\dag 2}-a^{2}\right) /2\right] $. Eq.(\[1.4\]) indicates that the conditional state $\left\vert \Psi _{m}\right\rangle $ is actually a single-mode $m$-order Hermite polynomial excited squeezed vacuum state. It is worth noticing that the degree of squeezing $\lambda $ of the conditional state is not the same squeezing parameter $r$ of the TMSVS but related to the parameters of the TMSVS and the BS satisfying $\tanh \lambda =\sin 2\theta \tanh r$. For the symmetrical case, i.e., $\theta =\pi /4$, $% \hat{B}\left\vert \Psi \right\rangle _{ab}|_{\theta =\pi /4}=S_{1a}\left( r\right) \left\vert 0\right\rangle \otimes S_{1b}\left( -r\right) \left\vert 0\right\rangle $ is just the product state of two separate single-mode squeezed vacuum state. In this case, when we detect $m$ photons in the second output channel, the conditional state is always $S_{1}\left( r\right) \left\vert 0\right\rangle $ with the same squeezed parameter $r$. This is that the effect of the symmetrical BS splits the entangled TMSVS into two independent single-mode squeezed vacuum state. Note that when no photons are detected, $m=0$, then $\left\vert \Psi _{m}\right\rangle $ also reduces to $% S_{1}\left( \lambda \right) \left\vert 0\right\rangle $. Normalization via probability of such event ------------------------------------------- In addition, the normalization factor $N_{m}$ is determined by the probability $p\left( m\right) $ of such an event given by$$\begin{aligned} N_{m} &=&p\left( m\right) \notag \\ &=&\mathrm{Tr}\left( \left. _{b}\left\langle m\right\vert \hat{B}\left\vert \Psi \right\rangle _{ab}\left\langle \Psi \right\vert \hat{B}^{\dag }\left\vert m\right\rangle _{b}\right. \right) \notag \\ &=&\frac{1}{m!\sqrt{A}\cosh ^{2}r}\allowbreak \frac{\partial ^{2m}}{\partial s^{m}\partial \tau ^{m}}e^{-B_{1}s^{2}/2-B_{1}\tau ^{2}/2+\allowbreak B_{2}s\tau }\allowbreak |_{s=\tau =0} \notag \\ &=&\frac{\left( -\sqrt{B_{3}}\right) ^{m}}{\cosh ^{2}r\sqrt{A}}P_{m}\left( \sqrt{B_{4}}\right) \label{1.9}\end{aligned}$$ where we have set $A=1-\mu ^{2}$, $B_{1}=\mu /\left( A\cosh ^{2}r\right) $, $% B_{2}=\nu ^{2}/A$, $B_{3}=\left( \tanh ^{4}r-\mu ^{2}\right) /A$, $B_{4}=\nu ^{4}/\left( A^{2}B_{3}\right) $, and in the last step we have used the formula of $m$-order Legendre polynomial P$_{m}\left( x\right) $, i.e.,$$\frac{\partial ^{2m}}{\partial t^{m}\partial \tau ^{m}}e^{-t^{2}-\tau ^{2}+% \frac{2x}{\sqrt{x^{2}-1}}\tau t}|_{t,\tau =0}=\frac{2^{m}m!}{\left( x^{2}-1\right) ^{m/2}}\text{P}_{m}\left( x\right) . \label{1.10}$$Especially, when $r=0$, $\left\vert \Psi \right\rangle \rightarrow \left\vert 0,0\right\rangle $, there is no photons in the output channels. In this case, there is no necessary to making conditional measurement. So the event is happen only for $r\neq 0$. When $\theta =0$ or $\theta =\pi /2$, leading to $\mu =0$ and $\nu =\tanh r$ or $\nu =-\tanh r$ then $A=1$, $% B_{1}=0$, $B_{2}=\tanh ^{2}r$, $B_{3}=\tanh ^{4}r$, and $B_{4}=1$, $% N_{m}|_{\theta =0or\theta =\pi /2}=\tanh ^{2m}r/\cosh ^{2}r$ and $\left\vert \Psi _{m}\right\rangle $ is just the Fock $\left\vert m\right\rangle $, which is rational because of the inherent properties of the TMSVS. If $% \theta =\pi /4$, the BS is just the symmetrical, i.e., 50/50 BS, leading to $% \mu =\tanh r,$ $\nu =0$ then $A=1-\tanh ^{2}r$, $B_{1}=-\frac{\tanh r}{2}$, $% B_{2}=0$, then the output states is the the product of two independent single-mode SVS. According to Eq.(\[1.9\]), we discuss the probability of observing such a conditional $m$-order HPESVS. In Fig.2 the probability $p\left( m\right) $is plotted for two parameter values of the BS. For a given transmittance of BS, $p\left( m\right) $ as a function of the input squeezing parameter $r$ can attain a maximum and the maximum is shifted towards larger values of $r$ when $m$ is increased (see Fig.2a). In this figure, for each $r$, we should use the transmittance of the BS in a way to optimize the success probability. The ideal procedure would use the appropriate transmittance for each value of $r$ and $m$. By tuning the parameters of the interaction, namely, the control parameter of the BS, the squeezed parameter $r$ of the TMSVS, and the photon number of conditional measurement $m$, the HPESVS may be modulated, generating a wide range of nonclassical phenomena, as described below. \[Fig2-1\] ![[The probability of producing ]{}$\left\vert \Psi _{m}\right\rangle $ [is shown as a function of the parameter ]{}$r$ [of the TMSVS for two parameter values of the BS \[(a) ]{}$% \protect\theta =\protect\pi /7$[; (b) ]{}$\protect\theta =2% \protect\pi /7$[\] and various values of ]{}$m$[, where ]{}$m=1,2,3,4$[correspond to the solid, dashed, dotted and dotdashed lines, respectively.]{}](PM.eps "fig:"){width="0.9\columnwidth"} Observable nonclassical effects of the conditional HPESVS ========================================================= To study the nonclassical properties of the conditional states in more detail, we shall calculate the photon number distribution, sub-Poissonian distribution, and quadrature component distribution. Photon number distribution -------------------------- The photon number distribution (PND), the probability of finding $n$ photons, is a key characteristic of every quantum state. Recalling Eq.([1.4]{}), the PND of the conditional HPESVS reads as $$P\left( n|m\right) =\left\vert \left\langle n\right. \left\vert \Psi _{m}\right\rangle \right\vert ^{2}. \label{2.1}$$Using the unnormalized coherent $\left\vert z\right\rangle =\exp [za^{\dag }]\left\vert 0\right\rangle $, leading to $\left\vert n\right\rangle =\frac{1% }{\sqrt{n!}}\frac{\partial ^{n}}{\partial z^{n}}\left\vert z\right\rangle |_{z=0}$, and combining with Eq.(\[1.4\]), we finally obtain $$\begin{aligned} & P\left( n|m\right) \notag \\ & =\frac{1}{m!n!N_{m}\cosh ^{2}r}\allowbreak \left\vert \frac{\partial ^{2m}% }{\partial \tau ^{m}\partial s^{n}}e^{\frac{\mu }{2}s^{2}-\frac{\mu }{2}\tau ^{2}+\nu s\tau }|_{s=\tau =0}\right\vert ^{2} \notag \\ & =\frac{m!n!}{N_{m}\cosh ^{2}r}\allowbreak \left\vert \sum_{g=0}^{\min [m,n]}\frac{\left( -1\right) ^{\frac{m-g}{2}}\left( \allowbreak \frac{\mu }{2% }\right) ^{\frac{m+n-2g}{2}}\nu ^{g}}{\left( \frac{n-g}{2}\right) !\left( \frac{m-g}{2}\right) !g!}\right\vert ^{2}, \label{2.2}\end{aligned}$$where we have used $\frac{\partial ^{m}}{\partial x^{m}}x^{n}|_{x=0}=m!% \delta _{mn}$ ($\delta _{mn}$ is Krocher function) and $N_{m}$ is given in Eq.(\[1.9\]). In the summation of Eq.(\[2.2\]), the value of $g$ must make $\frac{n-g}{2}$ and $\frac{m-g}{2}$ be integer. To see clearly the variation of the PND, in Fig.3 we plot the bar graph of the PND for the conditional HPESVS with different values of parameters $m$, $\theta $, and $r $. From Fig.3 we easily see that when the number $m$ of the conditional measurement is odd (even), then the photon-number distribution is nonzero only for odd (even) photon numbers. The probability $P\left( n|m\right) $with different parity between $m$ and $n$ is zero. For given $m$ and $\theta $, the bigger the squeezing parameter $r$, the wider the distribution (see Figs.3(a) and 3(b)). \[Fig3-0\] ![[Photon-number distribution of the conditional HPESVS for (a) ]{}$m=1,\protect\theta =2\protect\pi /7,r=0.5$[; (b) ]{}$m=1,\protect\theta =2\protect\pi /7,r=1.0$[; (c) ]{}$m=4,% \protect\theta =2\protect\pi /7,r=1.0$[; (d) ]{}$m=4,\protect% \theta =3\protect\pi /7,r=1.0$[, respectively.]{}](PND.eps "fig:"){width="0.9\columnwidth"} Sub-Poissonian distribution --------------------------- In order to study the photon-number statistics of this conditional state, we first calculate $\left\langle a^{k}a^{\dag l}\right\rangle =\left\langle \Psi _{m}\right\vert a^{k}a^{\dag l}\left\vert \Psi _{m}\right\rangle $. Using the completeness of coherent state $\int \frac{d^{2}\alpha }{\pi }% \left\vert \alpha \right\rangle \left\langle \alpha \right\vert =1$ as well as $y^{k}=\frac{\partial ^{k}}{\partial t^{k}}e^{ty}|_{t=0}$ yields$$\begin{aligned} & \left\langle a^{k}a^{\dag l}\right\rangle \notag \\ & =\frac{1}{N_{m}m!\cosh ^{2}r\sqrt{A}}\frac{\partial ^{2m}}{\partial s^{m}\partial \tau ^{m}}e^{-B_{1}s^{2}/2-B_{1}\tau ^{2}/2+B_{2}\tau s} \notag \\ & \times \frac{\partial ^{k+l}}{\partial x^{k}\partial y^{l}}e^{\left[ \mu \nu \left( xs+y\tau \right) +\nu \left( ys+x\tau \right) +\mu \left( x^{2}+y^{2}\right) /2+xy\right] /A}|_{s=\tau =x=y=0}. \label{2.3}\end{aligned}$$Thus, the mean photon number $$\left\langle n\right\rangle =\left\langle a^{\dag }a\right\rangle =\left\langle aa^{\dag }\right\rangle -1, \label{2.4}$$can be determined by Eq.(\[2.3\]) with $k=l=1$. Examples are shown in Figs.4(a) and 4(c). We see that when $\theta =\pi /5$, the number of photons that can be found in $\left\vert \Psi _{m}\right\rangle $ increases with $m$ for the given larger $r$, and $\left\langle n\right\rangle $ as a function of $\theta $ is symmetric distribution for $\theta =\pi /4$. This is simply a consequence of the BS transformation. In particular, when no photons are detected, $m=0$, then $\left\langle n\right\rangle =\sinh ^{2}r$ reduces to the mean photon number of single-mode squeezed vacuum state. A measure of the deviation of the photon number distribution from a Poissonian is the Mandel $Q$ factor defined by[@a34]$$\begin{aligned} Q & =\frac{\left \langle n^{2}\right \rangle -\left \langle n\right \rangle ^{2}}{\left \langle n\right \rangle }-1 \notag \\ & =\frac{\left \langle a^{2}a^{\dagger2}\right \rangle -\left \langle aa^{\dagger}\right \rangle ^{2}-2\left \langle aa^{\dagger}\right \rangle +1% }{\left \langle aa^{\dagger}\right \rangle -1}, \label{2.5}\end{aligned}$$ It holds that $Q\geqslant0$ and the equality is achieved for the Fock state. The light is sub-Poissonian when the photon-number variance $% \left \langle n^{2}\right \rangle -\left \langle n\right \rangle ^{2}$ is less than $\left \langle n\right \rangle $. This is indicated by a negative value of $Q$. The statistics are Poissonian when $Q=0$, and super- (sub-) Poissonian if $Q>0$ ($Q<0$). \[Fig3-1\] ![[(a) Mean photon number ]{}$\left \langle n\right \rangle $ [and (b) Mandel ]{}${\protect\small Q}$ [factor versus ]{}$r$ [for different ]{}$m=1$[(solid line), ]{}$m=2$[(dashed line), ]{}$m=3$[(dotted line), and ]{}$m=4$[(dotdashed line) with the same ]{}$\protect\theta=\protect\pi /5$[. (c) Mean photon number ]{}$\left \langle n\right \rangle $ [and (d) Mandel Q factor versus ]{}$\protect\theta$ [for different ]{}$m=1$[(solid line), ]{}$m=2$[(dashed line), ]{}$m=3$[(dotted line), and ]{}$m=4$[(dotdashed line) with the same ]{}$r=0.5$[.]{}](NQ.eps "fig:"){width="1.0\columnwidth"} \[Fig4-1\] ![[The sub-Poissonian properties of the conditional state with ]{}$Q=0$[(solid line), ]{}$Q=-0.2$[(dashed line), ]{}$Q=-0.5$[(dotted line), and ]{}$Q=-0.8$[(dotdashed line) in the plane space of two parameters (]{}$% \protect\theta$[and ]{}$r$[) with different number of conditional measurement: (a) ]{}$m=1$[; (b) ]{}$m=2$[; (c) ]{}$m=3$[; (d) ]{}$m=4$[, respectively.]{}](MQ.eps "fig:"){width="1.0\columnwidth"} According to Eqs.(\[2.3\]) and (\[2.5\]), we plot the variation of $Q$ for HPESVS versus $r$ or $\theta $ for different $m=1,2,3,4$ in Fig.4(b) and 4(d). It is clearly seen that $Q$ as a function of $\theta $ is also symmetric distribution for $\theta =\pi /4$ and the HPESVS has sub-Poissonian statistics behavior due to the emergence of the negativity of $Q$. With the increasing values of $m$, the increasing intrend of $Q$ is accelerated for larger $r.$ To further exhibit the high nonclassicality, Fig.5 shows the dependence of this conditional state on $\theta $ and $r$ for four different $Q$ factors. Especially, we consider first the boundary case of the Poissonian distribution, $Q=0$ (see the solid line in Fig.5). Quadrature component distribution --------------------------------- Next we pay attention to the conditional quadrature component distribution (QCD)[@a28a]$$P\left( x,\varphi |m\right) =\left\vert \left\langle x,\varphi \right. \left\vert \Psi _{m}\right\rangle \right\vert ^{2}, \label{3.1}$$which can be measured in balanced homodyne detection. Here $\left\vert x,\varphi \right\rangle $ is the eigenstate of the quadrature component $% X\left( \varphi \right) =\left( ae^{-i\varphi }+a^{\dag }e^{i\varphi }\right) $, expressed as in the Fock basis$$\left\vert x,\varphi \right\rangle =\pi ^{-1/4}e^{-\frac{x^{2}}{2}+\sqrt{2}% xa^{\dag }e^{i\varphi }-\frac{a^{\dag 2}e^{2i\varphi }}{2}}\left\vert 0\right\rangle . \label{3.2}$$Using Eqs.(\[1.4\]) and (\[3.2\]) and inserting the completeness of coherent state $\int \frac{d^{2}\alpha }{\pi }\left\vert \alpha \right\rangle \left\langle \alpha \right\vert =1$, after integration$% \allowbreak $, the wave function $\left\langle x,\varphi \right. \left\vert \Psi _{m}\right\rangle $ reads $$\left\langle x,\varphi \right. \left\vert \Psi _{m}\right\rangle =\frac{\pi ^{-1/4}\left( \sqrt{\Gamma /2}\right) ^{m}e^{-\frac{\Pi }{2}x^{2}}}{\sqrt{% N_{m}m!\left( 1+\mu e^{-2i\varphi }\right) }\cosh r}H_{m}\left( \frac{\Delta }{\sqrt{\Gamma }}x\right) , \label{3.3}$$where we have set $\Theta =1+\mu e^{-2i\varphi }$, $\Pi =\left( 1-\mu e^{-2i\varphi }\right) /\Theta $, $\Gamma =\left( \mu +e^{-2i\varphi }\tanh ^{2}r\right) /\Theta $, and $\Delta =e^{-i\varphi }\nu /\Theta $. As a result of Eq.(\[3.3\]), we easily obtain the conditional QCD defined by Eq.(\[3.1\]) and plot the variation of $P\left( x,\varphi |m\right) $ for the HPESVS as a function of $x$ or $\varphi $ for different $m=1,2,3,4$ in Fig.6. Ones see that for $\varphi $ near $\pi /2$ the QCD $P\left( x,\varphi |m\right) $ with $m>0$ exhibits two separated peaks, wheresas for $% \varphi $ close to $0$ and $\pi $ an interference pattern is observed. \[Fig6-1\] ![[Quadrature-component distribution ]{}$P(x,\protect% \varphi |m)$[of the conditional state ]{}$\left\vert \Psi _{m}\right\rangle $[for ]{}$\protect\theta =\protect\pi /7,$$% r=0.5$[and various numbers ]{}${\protect\small m}$[of measured photons with (a) ]{}$m=1$[, (b) ]{}$% m=2$[, (c) ]{}$m=3$[, and (d) ]{}$m=4$[, respectively.]{}](QCD.eps "fig:"){width="1.0\columnwidth"} Quasi-probability distribution of the conditional HPESVS ======================================================== Quasi-probability distribution function in the phase space is a very useful tool for a comprehensive description of the nonclassical state. Thus, in this section, we shall analytically discuss several quasi-probability distributions, including Wigner function and Husimi function to characterize the nonclassicality of the conditional HPESVS. Wigner function --------------- The WF was first introduced by Wigner in 1932 to calculate quantum correction to a classical distribution function of a quantum-mechanical system. The presence of negativity of the WF is a signature of its nonclassicality[@a36; @a37]. For a single-mode density operator $\rho$, the WF in the coherent state representation $\left \vert z\right \rangle $ can be expressed as $$W(\alpha )=\frac{2e^{2\left\vert \alpha \right\vert ^{2}}}{\pi }\int \frac{% d^{2}z}{\pi }\left\langle -z\right\vert \rho \left\vert z\right\rangle e^{-2\left( z\alpha ^{\ast }-z^{\ast }\alpha \right) }, \label{4.1}$$ where $\alpha =\left( x+ip\right) /\sqrt{2}$. The Wigner function of the conditional state $\rho _{out}^{a}=\left\vert \Psi _{m}\right\rangle \left\langle \Psi _{m}\right\vert $, can be calculated in a straightforward way.$$\begin{aligned} & W(x,p|m) \notag \\ & =\frac{2}{\pi N_{m}m!\cosh ^{2}r\sqrt{A}}e^{-2\Xi \left\vert \alpha \right\vert ^{2}+\frac{2\mu }{A}\alpha ^{2}+\frac{2\mu }{A}\alpha ^{\ast 2}} \notag \\ & \times \frac{\partial ^{2m}}{\partial \tau ^{m}\partial s^{m}}% e^{Rs+R^{\ast }\tau -B_{2}\tau s-\frac{B_{1}}{2}s^{2}-\frac{B_{1}}{2}\tau ^{2}}|_{s=\tau =0} \notag \\ & =\frac{2m!}{\pi N_{m}\cosh ^{2}r\sqrt{A}}e^{-2\Xi \left\vert \alpha \right\vert ^{2}+\frac{2\mu }{A}\alpha ^{2}+\frac{2\mu }{A}\alpha ^{\ast 2}} \notag \\ & \times \sum_{l=0}^{m}\frac{\left( -B_{2}\right) ^{l}\left( -B_{1}/2\right) ^{m-l}}{l!\left[ \left( m-l\right) !\right] ^{2}}\left\vert H_{m-l}\left( -% \frac{R}{\sqrt{2B_{1}}}\right) \right\vert ^{2}, \label{4.2}\end{aligned}$$where we have set $\Xi =\left( 1+\mu ^{2}\right) /\left( 1-\mu ^{2}\right) $ and $R=2\nu \left( \alpha -\mu \alpha ^{\ast }\right) /A$. Especially, when no photons are detected, $m=0$, then $W(x,p|0)$ $\rightarrow \exp \left( -p^{2}e^{-2\lambda }-x^{2}e^{2\lambda }\right) /\pi $ is a Gaussian form in phase space, which is just WF of single-mode SVS, as expected. \[Fig7-1\] ![[Wigner functions ]{}$W(x,p|m)$[of the conditional state ]{}$\left\vert \Psi _{m}\right\rangle $[for ]{}$% \protect\theta =\protect\pi /7,$$r=0.5$[and various numbers m of measured photons with (a) ]{}$m=1$[, (b) ]{}$m=2$[, (c) ]{}$m=3$[, (d) ]{}$m=4$[, respectively.]{}](WF.eps "fig:"){width="1.0\columnwidth"} The WFs of the conditional HPESVS $\left \vert \Psi_{m}\right \rangle $ in Fig.7 are plotted for the same parameters as in Fig.6. The figures indicate that the conditional HPESVS is a noncassical non-Gaussian state, since the partial negative regions in phase space are observed in Fig.7. This further demonstrates that performing the conditional output measurement on a BS is an effective approach to generate non-Guassian state. In addition, it is seen from Fig.7 that for odd $m$ there exists a negative valley in the center region, whereas for even $m$ there exists a main peak. In fact, for the center region $W(0,0|m)=\frac{2}{\pi}\left( -1\right) ^{m}$, as expected. Husimi function --------------- The Husimi function $Q(x,p|m)$ of the state $\left \vert \Psi_{m}\right \rangle $ is defined by[@a28] $$Q\left( x,p|m\right) =\frac{1}{\pi}\left \vert \left \langle \beta \right. \left \vert \Psi_{m}\right \rangle \right \vert ^{2}, \label{4.3}$$ where $\left \vert \beta \right \rangle $ is a coherent state and $% \beta=\left( x+ip\right) /\sqrt{2}$. Using Eq.(\[1.4\]), the scalar product $\left \langle \beta \right. \left \vert \Psi_{m}\right \rangle $ can be easily calculated as follow $$\left \langle \beta \right. \left \vert \Psi_{m}\right \rangle =\frac{% \allowbreak e^{-\frac{\left \vert \beta \right \vert ^{2}}{2}+\allowbreak \frac{\mu}{2}\beta^{\ast2}}}{\sqrt{m!N_{m}}\cosh r}\frac{\partial^{m}}{% \partial \tau^{m}}e^{-\tau^{2}\frac{\mu}{2}+\tau \nu \beta^{\ast}}|_{\tau=0}, \label{4.4}$$ So we find that $Q\left( x,p|m\right) $ can be written as$$Q\left( x,p|m\right) =\frac{\left( \mu/2\right) ^{m}e^{-\left \vert \beta \right \vert ^{2}+\allowbreak \frac{\mu}{2}\left( \beta^{2}+\beta^{\ast 2}\right) }}{\pi N_{m}m!\cosh^{2}r}\left \vert \allowbreak H_{m}\left( \frac{% \nu}{\sqrt{2\mu}}\beta^{\ast}\right) \right \vert ^{2}. \label{4.5}$$ \[Fig8-1\] ![[Husimi functions ]{}${\protect\small Q(x,p|m)}$ [of the conditional state ]{}$\left \vert \Psi_{m}\right \rangle $ [for ]{}$\protect\theta% {\protect\small =}\protect\pi{\protect\small /7}$[, ]{}$% {\protect\small r=0.5}$[and various numbers ]{}${\protect\small % m}$ [of measured photons with (a)]{}${\protect\small m=1}$[, (b) ]{}${\protect\small m=2}$[, (c) ]{}$% {\protect\small m=3}$[, (d) ]{}${\protect\small m=4}$[, respectively.]{} ](QF.eps "fig:"){width="1.0\columnwidth"} As expected, for $m=0$ the Husimi function is Gaussian, while for odd $m$ a two-peak structure and for even m a single peak are observed in Fig.8, whose parameters are the same as WFs in Fig.7. Note that the Husimi function is a phase-space function that can be measured in multiport balanced homodyning. Since the Husimi function can be regarded as a smoothed Wigner function, it is always non-negative and the oscillating behavior, typical of WF (see Fig.7), cannot be observed. Conclusions and Discussions =========================== In summary, we have shown that Hermite polynomial excited squeezed states can be generated by conditional measurements using a simple beam splitter scheme. When a two-mode squeezed vacuum state is mixed by a beam splitter and the number of photons is measured in one of the output channels, then the conditional quantum state in the other output channel reveals all properties of a Hermite polynomial excited squeezed state. Then, we also have numerically analyzed the conditional HPESVS in terms of the photon-number statistics, quadrature-component distribution and quasi-probability distribution such as the Wigner and Husimi functions. 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--- abstract: 'The CELESTE atmospheric Cherenkov detector ran until June 2004. It has observed the blazars Mrk 421, 1ES 1426+428 and Mrk 501. We significantly improved our understanding of the atmosphere using a LIDAR, and of the optical throughput of the detector using stellar photometry. The new data analysis provides better background rejection. We present our light curve for Mrk 421 for the 2002-2004 season and a comparison with X-ray data and the 2004 observation of 1ES 1426+428. The new analysis will allow a more sensitive search for a signal from Mrk 501.' address: 'CENBG, Domaine du Haut-Vigneau, BP 120, 33175 Gradignan Cedex, France' author: - 'Brion, E.' - and the CELESTE collaboration title: 'Blazar observations above 60 GeV: the Influence of CELESTE’s Energy Scale on the Study of Flares and Spectra' --- CELESTE,Cherenkov,Mrk 421,1ES 1426+428,Mrk 501 95.85.Pw ,98.54.Cm Introduction ============ CELESTE was a Cherenkov experiment using 53 heliostats of the former Électricité de France solar plant in the French Pyrenees at the Thémis site. It detected Cherenkov light from electromagnetic showers produced in the atmosphere by the $\gamma$-rays coming from high energy astrophysical sources. The light is reflected to secondary optics and photomultipliers installed at the top of the tower. Finally it is sampled to be analysed (Paré 2002). To constrain the energy scale of the experiment we improved the optics simulation, now in good agreement with the data. The data analysis has also been improved so that we have better background rejection. We present the light curve for Mrk 421. Constraining the energy scale {#sec:simulation} ============================= The simulation has been reexamined to reduce the uncertainties on the energy scale of the experiment (Brion 2003). The LIDAR operating on the site for atmospheric monitoring provided a better determination of the atmospheric extinction (Bussóns Gordo 2004). A stellar photometry study, focussing on the comparison between simulations and data on bright stars’ currents, has been done on the star 51 UMa (M$_\mathrm{B}=6.16$) which is in the field of view (FOV) of Mrk 421. This showed that the old simulation was too optimistic. All mirror reflectivities were decreased subsequent to new measurements. The nominal focussing of the heliostats was degraded after a study of star image sizes. We also verified the photomultiplier gains. The results for the old data set (40 heliostats, new is 53 helisotats, see § \[sec:analysis\]) are presented in figure \[fig:simulation\]: the effect of these changes is smaller for $\gamma$-ray showers (an extended light source) than for stars (point sources). ![ON$-$OFF illumination from the star 51 UMa (M$_\mathrm{B}=6.16$) in the FOV of Mrk 421 as a function of hour angle for pointing at 11 km: the new simulation with our corrections (red stars) fits the data well (black squares) whereas the old simulation (blue circles) was 50 % too high.[]{data-label="fig:simulation"}](brion_fig1.ps){width="7.0cm"} This study helped us to perform our selection criteria for the data: all data that are too low in currents on the star have also low trigger rates. In order to trigger CELESTE, the heliostats, except the *veto* (see § \[sec:veto\]), are split into 6 groups. For each of them, the analog sum of signals gives the first level trigger. Then, a logical pattern is defined on the majority of the triggering groups. Thus, in the case of the source Mrk 421, we reject all data with trigger rates under 20 Hz for the old data set and under 16 Hz for the new data set. We also looked at the proton rates as a standard candle for the detector which should be stable for good quality nights. These rates are determined with high offline threshold cuts to avoid trigger bias, and are therefore low (typical trigger rates $\sim 22$ Hz). We’ve shown that they are correlated with the currents on star 51 UMa for selected data (new data set, figure \[fig:ProtonRate\] (a)). The data with low rates are now still rejected but perhaps some of them could be corrected. Indeed, three doubtful zones, A, B and C on the figure, can be distinguished. The zone A can be interpreted as bad nights with thick cloud cover (weak star, little Cherenkov light), the zone B as nights with aerosols and above average extinction of Cherenkov and starlight, and the zone C as nights with high clouds that stop starlight but not Cherenkov light. The data in this last zone may therefore be used. Figure \[fig:ProtonRate\] (b) shows the correlation between the proton rate and the trigger rate for the same data set. Defining a selection criteria for each type of acquisition, and based only on these rates, would be very interesting for sources that don’t have any star nearby for a photometry study. ![(a) Proton rate as a function of ON$-$OFF current for heliostat E03 that sees 51 UMa when pointing Mrk 421. For a typical photomultiplier tube gain of $5.6\times10^{4}$, $2.5$ A corresponds to 0.28 p.e./ns, as seen in figure \[fig:simulation\]. (b) Proton rate as a function of trigger rate for data on Mrk 421.[]{data-label="fig:ProtonRate"}](brion_fig2.eps "fig:"){width="6.0cm"} ![(a) Proton rate as a function of ON$-$OFF current for heliostat E03 that sees 51 UMa when pointing Mrk 421. For a typical photomultiplier tube gain of $5.6\times10^{4}$, $2.5$ A corresponds to 0.28 p.e./ns, as seen in figure \[fig:simulation\]. (b) Proton rate as a function of trigger rate for data on Mrk 421.[]{data-label="fig:ProtonRate"}](brion_fig3.eps "fig:"){width="6.0cm"} Analysis improvement {#sec:analysis} ==================== Since the 2000 status of the experiment (de Naurois 2002) three main changes were made for the analysis. First, the selection criteria of the data are stricter for the current and trigger rate stability, and as we’ve shown for the trigger and proton rate value (work in progress). The experiment has been upgraded from 40 to 53 heliostats. We use part of them to broaden our narrow FOV. Finally, we have found a new method to exploit the FADC information to reject the hadronic background (Manseri 2004 a, b). The second and third point are developed hereafter. The *veto* configuration {#sec:veto} ------------------------ Hadronic showers have a more chaotic and extended development than electromagnetic showers. To measure the extent of the shower, we artificially broaden the FOV: as before, all heliostats aim at 11 km above the ground in the direction of the source, where the maximum of the shower is supposed to occur in our energy range. But 12 heliostats, distributed around the edge of the field, sample a ring of 150 m around that point (figure \[fig:PrincipeVeto\]). Because of the compactness of electromagnetic showers, the light does not illuminate these 12 heliostats, named [*veto*]{}, contrary to hadronic showers (figure \[fig:NbVetoMC\]). So we require that no *veto* be illuminated. ![Distribution of the number of illuminated *veto* heliostats (simulations).[]{data-label="fig:NbVetoMC"}](brion_fig4.eps){width="\linewidth"} ![Distribution of the number of illuminated *veto* heliostats (simulations).[]{data-label="fig:NbVetoMC"}](brion_fig5.eps){width="\linewidth"} FADC information ---------------- To detect low energy $\gamma$-rays, we use the sum of the individual digitized signals to increase the signal-to-background ratio. The summation includes a correction for the sphericity of the wavefront, assumed to be centered in the 11 km plane. Assuming a wrong position for this center (impact parameter) broadens the sum: the height-over-width ratio, $(H/W)$, decreases. We compute $(H/W)$ for different assumed positions. The impact parameter is the position for which the $(H/W)$ ratio is maximum, denoted by $(H/W)_{max}$. This is valid for $\gamma$-rays (figure \[fig:Timing\] (a)) but not for protons for which the wavefront is not spherical (figure \[fig:Timing\] (b)). A measurement of the flatness of these 2D-distributions is given by the following estimator: $$\xi = \mathrm{average}\,\left(\frac{\displaystyle(H/W)_{200m}}{\displaystyle(H/W)_{max}}\right)_{\mbox{\small over 24 positions}},$$ where $(H/W)_{200m}$ is the average of $(H/W)$ over 24 positions along a ring 200 m from the maximum position. For $\gamma$-rays there is a clear maximum and this estimator takes low values. For hadrons, it is usually larger (figure \[fig:Timing\] (c)). We have applied $\xi<0.35$. ![Left: the shape of the FADC sum depends on the position of the centre of the wavefront we assume. Right: views of the height-over-width ratio (H/W) computed for discrete positions in the 11 km plane (a) for a 100 GeV $\gamma$ ray and (b) for a 500 GeV proton. (c) Distribution of the $\xi$ parameter for $\gamma$-rays and OFF data.[]{data-label="fig:Timing"}](brion_fig6.eps){width="\textwidth"} A cut on this estimator is the biggest contribution to an improvement in Crab sensitivity from 2.2 $\sigma/\sqrt{\mathrm{h}}$ to 5.8 $\sigma/\sqrt{\mathrm{h}}$. The Crab detection is also stable (figure \[fig:Crab\]). ![Stability of the Crab detection after hour angle efficiency corrections. Line is $6.5\pm0.5\ \gamma/$min.[]{data-label="fig:Crab"}](brion_fig7.ps){width="7.0cm"} Blazar observations {#sec:blazars} =================== This stable analysis and good sensitivity provide a good detection of the Mrk 421 flares. A 19 $\sigma$ detection during 10 h since October 2002 gives a mean of 5.6 $\gamma/$min. The light curve is presented in figure \[fig:Mrk421\]. With the new acceptances we will deduce a spectral measurement for Mrk 421 with smaller uncertainties. ![Light curve of Mrk 421. Left: since 2002 seen by CELESTE (red points in $\gamma$/min corrected for hour angle efficiency, 1 point is 20 min data, Crab transit rate shown for reference) and RXTE/ASM (blue squares in mCrab, 6 day bins). Right: light curve of Mrk 421 zoomed on the $17^{\mathrm{th}}$ of March 2004 flare seen by CELESTE.[]{data-label="fig:Mrk421"}](brion_fig8.ps "fig:"){width="6.0cm"} ![Light curve of Mrk 421. Left: since 2002 seen by CELESTE (red points in $\gamma$/min corrected for hour angle efficiency, 1 point is 20 min data, Crab transit rate shown for reference) and RXTE/ASM (blue squares in mCrab, 6 day bins). Right: light curve of Mrk 421 zoomed on the $17^{\mathrm{th}}$ of March 2004 flare seen by CELESTE.[]{data-label="fig:Mrk421"}](brion_fig9.ps "fig:"){width="6.0cm"} The source 1ES 1426+428 is still not detected with our new analysis during the March 2004 observation for 4.4 h data. Since the data have been taken in the same conditions as for the Crab, we can use the sensitivity of 5.8 $\sigma/\sqrt{\mathrm{h}}$ to remark that for a 3 $\sigma$ observation for these 4.4 h data, the flux $f_\mathrm{1ES\,1426}$ would have to be $f_\mathrm{1ES\,1426} = \frac{3\sigma}{5.8\sigma\sqrt{4.4}}f_\mathrm{Crab} \approx \frac{f_\mathrm{Crab}}{4}$. A better upper limit determination is in preparation. Finally, the blazar Mrk 501 has been observed during 14.5 h with the previous experiment at 40 heliostats. The old analysis gave a 2.5 $\sigma$ significance (with 850 photons which would give 1 $\gamma/$min). The new analysis will allow an improved investigation of Mrk 501’s behaviour in this unexplored energy range. Brion, E., Smith, D. and The CELESTE Collaboration, Testing the Energy Scale of the CELESTE experiment, SF2A-2003: Semaine de l’Astrophysique Francaise, pp. 463-+, 2003 (astro-ph/0310148). Bussóns Gordo, J., Debiais, G., Espigat, P. , LIDAR atmospheric monitoring for the CELESTE gamma-ray experiment, submitted to Astroparticle Physics, 2004. Manseri, H., Astronomie gamma au-dessus de 30 GeV. Une nouvelles méthode d’identification des rayons $\gamma$ cosmiques à partir du sol avec le détecteur CELESTE, Ph.D. thesis of the École Polytechnique, Paris, France *(in french)*, March 2004. Manseri, H. and The CELESTE Collaboration, Gamma-ray astronomy above 30 GeV with the CELESTE experiment (1996-2004), International Symposium on High Energy Gamma-Ray Astronomy, 2004. de Naurois, M., Holder, J., Bazer-Bachi, R. , Measurement of the Crab Flux above 60 GeV with the CELESTE Cerenkov Telescope, ApJ, 566:343-357, February 2002. Paré, E., Balauge, B., Bazer-Bachi, R. , CELESTE: an atmospheric Cherenkov telescope for high energy gamma astrophysics, NIM, 490:71-89, September 2002.

--- author: - | [Austin Hounsel]{}\ Princeton University - | [Prateek Mittal]{}\ Princeton University - | [Nick Feamster]{}\ Princeton University bibliography: - '../common/bibliography.bib' title: | **Automatically Generating a Large,\ Culture-Specific Blocklist for China** ---

--- abstract: 'Hysteresis and commonly observed p-doping of graphene based field effect transistors (FET) was already discussed in reports over last few years. However, the interpretation of experimental works differs; and the mechanism behind the appearance of the hysteresis and the role of charge transfer between graphene and its environment are not clarified yet. We analyze the relation between electrochemical and electronic properties of graphene FET in moist environment extracted from the standard back gate dependence of the graphene resistance. We argue that graphene based FET on a regular SiO$_2$ substrate exhibits behavior that corresponds to electrochemically induced hysteresis in ambient conditions, and can be caused by charge trapping mechanism associated with sensitivity of graphene to the local pH.' author: - Alina Veligura - 'Paul J. Zomer' - 'Ivan J. Vera-Marun' - Csaba Józsa - 'Pavlo I. Gordiichuk' - 'Bart J. van Wees' title: Relating Hysteresis and Electrochemistry in Graphene Field Effect Transistors --- Introduction ============ Graphene, as a single atom thick layer of carbon atoms, has already showed potential for application in electronics and biosensing[@Ratinac]. However graphene as a truly 2D system is ultrasensitive [@Schedin2007] to the underlying substrate and surface chemistry, which alters the charge transport properties of pristine graphene. One of the main issues in graphene devices is a hysteretic behavior of its resistance observed in ambient conditions, when a gate voltage is swept back and forth. The presence of hysteresis and commonly observed p-doping of graphene based field effect transistors (FET) was already discussed in recent reports[@Lafkioti; @Wang2010; @Sabri2009; @Levesque]. The interpretation of experimental works differs; and the mechanism behind the appearance of hysteresis and the role of charge transfer between graphene and its environment are not clarified yet. In an ideal case of grounded graphene its charge neutrality point (CNP) is located at zero back gate voltage. However, in ambient conditions most of the graphene based FETs show initial p-doping (CNP is positioned at positive Vg) and hysteresis. We point out that these two effects can be related but do not necessarily have the same nature. The doping of graphene can be caused either by the adsorbates on top or underneath the graphene surface[@Schedin2007; @Lafkioti; @Wang2010] or by the electrochemical processes involving graphene[@Sabri2009; @Levesque; @Sidorov2011]. Depending on the nature of the dopant or the electrochemical environment, the initial doping can be either p or n, which introduces a shift of the graphene CNP to positive or negative gate voltages respectively. One should keep in mind that even in the absence of a net doping the dynamic response of the graphene resistance, namely hysteresis, can be different. There are two types of directions defined for hysteresis; positive and negative [@Wang2010]. The positive direction of hysteresis corresponds to the CNP shifting towards negative voltages while the gate voltage is swept further into the negative regime. In case of negative hysteresis the shift of the resistance with respect to the gate voltage is in the opposite direction: the CNP shifts toward more positive values while sweeping the gate into the negative regime. Wang et al.[@Wang2010] proposed that negative and positive hysteresis directions can be attributed to two competing mechanisms: capacitive coupling and charge trapping from/to graphene, respectively. Capacitive coupling enhances the local electrical field near graphene, inducing more charge carriers and causing a negative direction of hysteresis. An example of a mechanism for capacitive coupling is a dipole layer placed in between graphene and the back gate. In moist air and without additional treatment of the silicon oxide substrate (a common insulator for a GFET) this dipole layer exists as adsorbed water molecules at room temperature [@Moser; @Lafkioti] or ordered ice at low temperature[@Wang2010; @Wehling]. The capacitive coupling mechanism is also dominant in electrolyte-gating devices, via ions in the electrical double layer [@Wang2010]. The positive direction of hysteresis is caused by a charge trapping mechanism. Accumulated charge in trap centers will start screening the electric field of the back gate. One of the examples of trap centers are surface states in between SiO$_2$ and graphene [@Wang2010; @Romero2008; @Liao; @Shin]. In case of graphene based FET traps in bulk SiO$_2$ or SiO$_2$/Si interface were excluded in a recent report by Lee et al.[@Lee2011], who measured time scales which were too fast for these types of trapped centers. A separate charge transfer mechanism which was observed for the hydrogenated surface of diamond [@Chakrapani], carbon nanotubes [@Aguirre] and graphene based FETs [@Sabri2009; @Levesque; @Sidorov2011], is the dissociation of adsorbed water and oxygen on the carbon surface. Since water in equilibrium with air is slightly acidic (pH=6), the electrochemical potential of the carbon surface is higher than that of the solution, resulting in electron transfer from graphene. Therefore, a graphene FET possesses a net p-doping in moist air. The electron transfer is mediated by oxygen solvated in water and can occur in opposite direction with increasing pH. This redox can therefore influence the dynamic response of graphene devices under an applied back gate and cause a positive hysteresis. A recent report by Fu et al.[@Fu] opened the discussion whether graphene pH sensitivity is caused by charge transfer directly between graphene and the solution [@Ang2008; @Ohno; @Heller] or if the sensitivity is mediated by a layer on top or next to graphene (either oxide or polymer residue). This layer can provide terminal hydroxyl groups which can be protonized or deprotonized depending on the proton concentration in the solution (pH), yielding a bound surface charge layer, which can electrostatically induce carriers in graphene. Recently it was reported that application of a gate potential can lead to a local change of pH in a thin water film next to an oxide substrate [@Veenhuis]. We argue that a combination of these two effects can result in a positive hysteresis in graphene, where the residues act as mediators for charge trapping actuated by pH changes induced via gate electrical field. We emphasize that both cases, independent whether the charge trapping is direct or mediated by residues, would lead to the same direction in hysteresis and will be undistinguishable in transport experiments. Though replacement of the silicon oxide with either a hydrophobic [@Lafkioti; @Shin] or an oxygen free [@Sabri2009] substrate did show suppression of both initial p-doping and hysteretic behavior, none of the reports link the chemical redox to the direction of hysteresis. In this work we analyze the relation between electrochemical and electronic properties of graphene FET in moist environment. We argue that graphene based FET on a regular SiO$_2$ substrate exhibits behavior that corresponds to electrochemically induced hysteresis in ambient conditions, caused by charge trapping mechanisms associated with the sensitivity of graphene to the local pH. Methods ======= Samples were obtained by mechanical exfoliation of graphite (Highly Ordered Pyrolytic Graphite or Kish) on an oxidized n$^+$-doped silicon substrate (300 or 500 nm thick oxide layer), which functions as a back gate. The SiO$_2$ wafers are commercially available from Silicon Quest International, where the oxide is prepared by dry oxidation. Single layer graphene flakes were chosen based on their optical contrast and thickness measured by atomic force microscopy. A small number of samples were inspected with Raman spectroscopy to verify the number of layers. Ti/Au (5/40 nm thick) electrodes were prepared using standard electron beam lithography and lift off techniques. For electrical measurements samples are placed in a vacuum can with base pressure of $~5\cdot 10^{-6}$ mbar, using a standard low frequency AC lock-in technique with an excitation current of 100 nA. The carrier density in graphene is varied by applying DC voltage (Vg) between the back gate electrode and the graphene flake, as depicted in Fig. \[fig:Fig1\](a). The charge carrier mobilities ($\mu$) ranged from 2.500 up to 5.000 cm$^2$/Vs at a charge carrier density of $n=2\cdot 10^{11}cm^{-2}$. The sensor properties of the devices were studied in the following way. First, we pumped down the sample can (95 cm$^3$ in volume) to the base pressure. Then a valve connecting the can to a volume, containing liquid water and filled with saturated vapor (H$_2$O or D$_2$O at  32 mbar saturation pressure) at 25 C$^o$, was kept open for 1 s (short exposure to the vapor). After measurement, the valve to the sample was fully opened, connecting the sample volume to the water container (flooding with water vapor). In case of ethanol vapor exposure the procedure was kept the same, but the partial pressure of ethanol in the liquid cavity was 78 mbar. The purity of heavy water and ethanol was 99.9%. A graphene based FET on a hydrophobic substrate was also prepared by exposure of SiO$_2$ to hexamethyldisilazane (HMDS) vapor prior to graphene deposition. HMDS forms a self assembled monolayer which protects graphene from the influence of dangling bonds in silicon dioxide and prevents adsorbtion of water molecules in the vicinity of graphene. Results and discussion ====================== In ambient conditions the devices appear to be p-doped, with a pronounced positive hysteresis in the dependence of resistivity versus gate voltage (not shown). To remove adsorbates from the graphene surface we perform global annealing of the device in vacuum at 130$^o$C for 1,5 hrs. After annealing, the gate dependence does not show hysteresis and becomes symmetric around the CNP (Fig. \[fig:Fig1\](c)), which is located at a negative gate voltage (-11 V), indicating electron doping. Similar shifts towards negative gate voltages were observed by Romero et al.[@Romero2008] and associated with SiO$_2$ surface states. We will call this position of the charge neutrality point the initial position (after annealing). Short exposure to water does not cause hysteresis, but reduces $\mu$ by 25 % compared to the initial state and can be attributed to the increase of a number of the scatter centers for charge carriers[@Schedin2007] (Fig. \[fig:Fig1\](d)). Since graphene is hydrophobic, we assume that during the short exposure adsorbates only occasionally agglomerate on the graphene surface in the vicinity of polymer leftovers which are unavoidably present after the lithography step (Fig. \[fig:Fig1\](a)).  a) Scheme of a graphene based device with a discontinuous layer of adsorbed water in case of a short exposure to H$_2$O vapor. Dangling bonds in SiO$_2$, lithographic polymer remains (in red) on the graphene surface and electric field lines between graphene and the back gate are schematically drawn; b) A continuous thin layer of water on the graphene surface in case of flooding the sample with water vapor; c) Graphene resistance versus gate voltage after annealing (initial state); d) After a short exposure to water vapor; e) Positive hysteresis developed after further flooding with water vapor.](Fig1_Device.eps){width="8.5cm"} Flooding the sample chamber with H$_2$O vapor assures full coverage of the previously annealed SiO$_2$ and graphene surface with a thin film of water ( 3 nm thick [@Kim]), similar to ambient conditions. After flooding we observe both electron-hole asymmetry and a highly hysteretic behavior of the graphene device, where the CNP for trace and retrace are situated at Vg of opposite signs (Fig. \[fig:Fig1\](e)). Moreover, a decrease of the scanning rate in gate voltage sweeps (V/s) leads to more pronounced hysteresis with the spacing between trace and retrace maxima increasing from 6.5 V at 1 V/s up to 23,5 V for 0,1 V/s. The cycle of annealing and water exposure was repeated a few times showing reproducible results. The positive direction of hysteresis indicates charge trapping mechanism, while electron -hole asymmetry can be explained in two ways: real asymmetry due to doping of graphene under the contacts[@Huard] or an artifact of charging and discharging graphene due to the hysteresis. Since we do not observe asymmetry in the initial curve, the latter situation will be assumed in further discussions. Next, we present a novel analysis of hysteretic back gate voltage sweeps from the point of view of time dependent shifts in CNP. These shifts represent a change in carrier density within a certain time, equivalent to a current. We estimate this current corresponding either to the charge flow in or out of graphene, or induced charge, in the following way. Charge current is extracted by comparing the non hysteretic Dirac curve of graphene, which is shortly exposed to water vapor, to the curves after the sample is flooded, measured at different scan rates: 0,5; 0,25 and 0,1 V/s. The exact procedure is shown in Fig. \[fig:Hysteresis\]a), b). For each scan rate the gate voltage axis was divided into fixed regions $\Delta V_{fixed}$. A change in voltage $\Delta V_{fixed}$ induces a change in the carrier density and resistance $\Delta R$ accordingly. Due to the charge trapping mechanism induced by water, the same $\Delta R$ will require a different value of gate voltage $\Delta V_i$ in case of the non-hysteretic curve. The difference between $\Delta V_{fixed}$ and $\Delta V_i$ will be proportional to the amount of additionally induced or transferred charge in graphene. The charge current (A/$\mu$m$^2$) in graphene can then be calculated as: ![\[fig:Hysteresis\] (Color online) Calculation of the charge current in graphene. a) Gate voltage dependence of graphene resistance “flooded” with water vapor and measured at a rate of 0,1 V/s. The curve is divided into parts with a fixed step in gate voltage $\Delta V_{fixed}$, corresponding to the change in resistance $\Delta R$; b) Gate dependence of graphene resistance shortly exposed to H$_2$O vapor. Due to the charge transfer now the same change $\Delta$R requires different value of applied voltage $\Delta V_i$ ; c) Calculated charge current versus gate voltage for three different scan rates: 0,5; 0,25 and 0,1 V/s; d) Linear scaling of the peak, at positive gate voltage shown in c), with the scan rate. ](Fig2_Electrochem.eps){width="8.5cm"} $$I_{i} = \frac{e\alpha(\Delta V_i-\Delta V_{fixed})} {\Delta V_{fixed}/\beta} \,$$ where $e$ is the elementary charge, $\alpha=2\cdot 10^{14}m^{-2}V^{-2}$ with $e\cdot\alpha$ the charge capacitance per unit area for 500 nm SiO$_2$, and $\beta$ is the scan rate of the gate sweep (V/s). The calculated charge current curves (Fig. \[fig:Hysteresis\](c)) resemble the electrovoltaic characteristics of graphene based electrochemical cells with controlled pH [@Fu]. A graphene based device on a SiO$_2$ substrate can act as a working electrode in the thin layer of water covering the hydrophilic oxide surface. Thus we can consider graphene based devices as electrochemical cells. Moreover, the height of the observed peaks scales linearly with the scan rate of the applied gate voltage (Fig. \[fig:Hysteresis\](d)) which, for an electrochemical cell, suggests that these peaks originate from a non-Faradaic or non-diffusion limited process involving the adsorbed ions on the graphene surface[@Ang2008]. We performed the same sequence of experiments with graphene devices on HMDS primed SiO$_2$. In contrast to graphene on hydrophilic SiO$_2$ we observe neither hysteresis nor any changes in the graphene resistance under water vapor exposure.\ From the fact that the initial curve (after annealing) has no hysteresis we can exclude charge trapping in the surface states of SiO$_2$. Comparing to a local current annealing procedure [@Wang2010], here we globally annealed the sample which assures desorption of H$_2$O molecules from the whole SiO$_2$ surface and prohibits their diffusion back to the graphene surface. The hysteresis appears only when the amount of water in the system is high enough to form a continuous layer. The linear scaling of extracted height of current peaks with scan rate indicates the reversible charging of an ionic layer at the graphene surface (electrode) by an applied gate voltage. The absence of hysteresis of the graphene resistance when HMDS is used supports the idea that the trapping mechanism happens by the presence of a water layer on the SiO$_2$ surface. The dielectric constant of water is $\varepsilon_{H2O}=80$ much higher than $\varepsilon_{oxide}=3.9$. Therefore the electrical field lines in the device deviate from plane capacitor and can be present in the water layer (Fig. \[fig:Fig1\](b)). The strong electrical field across the water layer can either cause dissociation of water molecules [@Teoh] or proton release/uptake by terminal OH$^-$ groups at the oxide surface, as previously described[@Fu; @Veenhuis]. Both these mechanisms lead to a local pH change in the graphene vicinity. Depending on the pH, the dangling bonds of the oxide or polymer remains on graphene will change their charge state, inducing an opposite charge in graphene[@Fu; @Teoh]. At the present state we can not pinpoint the exact identity of the ionic species causing the change of environment around the graphene. A possible electrochemical reaction on the unprotected Au electrodes is not relevant as this was ruled out by Wang at al[@Wang2010]. where both samples with protected and unprotected gold contacts showed the same type of hysteresis. Since the dipole nature of water molecules is often discussed in relation to the hysteresis observed in graphene devices [@Wang2010; @Lafkioti; @Wehling], we decided to study the response of graphene resistance to ethanol vapors. A pure neutral ethanol solution has at least 100 times less concentration of H$^+$ and OH$^-$ ions than pure water[@Hansen]. However the dipole moment of an ethanol molecule $\vec{p}_e=1.68 D$ is comparable to that of water $\vec{p}_w=1.85 D$ [@Hansen], which makes it possible to separate the electrochemical from electrostatic influences on the charge carrier density in graphene. In Fig. \[fig:Ethanol\](a,b) the changes in graphene resistivity under ethanol vapor exposure are presented. Except for the reduction of charge carrier mobility by  25 % (comparable to water exposure) neither considerable hysteresis nor doping were observed. ![\[fig:Ethanol\] Changes in graphene resistance versus gate voltage under exposure to ethanol and D$_2$O vapors. a) The initial state; b) After further flooding with ethanol vapor; c) The initial state (another sample); d) Previously mentioned sample after further flooding with D$_2$O vapor. ](Fig3_Ethanol.eps){width="8.5cm"} We also performed similar experiments using D$_2$O vapor with another set of samples. Chemically, D$_2$O molecules behave similar to H$_2$O. However, D$^+$ ions are two times heavier than H$^+$, whereas the relative increase in mass of OD$^-$ ions compared to OH$^-$ is negligible. If the electrochemical process on graphene surface is proton diffusion limited, one expects to observe a different behavior of the hysteresis at various scan rates. Experimentally we do not observe any significant difference in graphene’s response between H$_2$O and D$_2$O. Heavy water exposure causes doping and direction of the hysteresis comparable to normal water values (Fig. \[fig:Ethanol\](c,d)). Our experiment with ethanol vapor supports the idea that the polarity of molecules adsorbed in the graphene vicinity does not influence the dynamic response of graphene resistance to a gate voltage. We suggest that the main reason of the observed hysteresis in ambient conditions is the electrochemical activity of water molecules in the graphene environment. Conclusions =========== In conclusion, we have shown that the commonly observed positive hysteresis in graphene FETs can be derived from the electrochemical activity of water adsorbates on the SiO$_2$ substrate. In a moist environment a standard graphene FET can act as an effective electrochemical-cell with graphene being an electrode in the thin layer of water. Therefore the application of the back gate voltage may lead to local changes of pH which in turn affect the carrier density in graphene. From this point of view we suggest that, next to contact doping effect, the observed electron-hole asymmetry in graphene resistance appears as an artifact of the hysteresis caused by charge trapping. Conducted experiments with ethanol vapor and heavy water did not show a relation between the hysteresis and neither dipole moment nor mass of adsorbed molecules, supporting the idea of electrochemical activity of water as a key element in the dynamic response to gate voltage sweeping. These findings give a further insight to graphene-related electrochemistry outside an ideal electrochemical cell and open perspectives for the application of a graphene FET as a memory element. We would like to thank Bernard Wolfs, Siemon Bakker, and Johan G. Holstein for technical assistance and Daniele Fausti for measuring Raman spectra. This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM) and is supported by NanoNed, NWO, and the Zernike Institute for Advanced Materials. 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--- abstract: 'We investigate the space of abelian relations of planar webs admitting infinitesimal automorphisms. As an application we construct $4k - 14$ new algebraic families of global exceptional $k$-webs on the projective plane, for each $k\ge 5$.' address: - | Departament de Matem[à]{}tiques\ Universitat Aut[ò]{}noma de Barcelona\ E-08193 Bellaterra (Barcelona)\ Spain - | Instituto de Matem[á]{}tica Pura e Aplicada\ Est. D. Castorina, 110\ 22460-320, Rio de Janeiro, RJ, Brasil - | IRMAR\ Campus de Beaulieu\ 35042 Rennes Cedex, France author: - 'D. Mar[í]{}n' - 'J. V. Pereira' - 'L. Pirio' title: On Planar Webs with Infinitesimal Automorphisms --- [^1] Introduction and statement of the results ========================================= Planar Webs ----------- A germ of regular $k$-web $\mathcal W=\mathcal F_1 \boxtimes \cdots \boxtimes \mathcal F_k$ on $(\mathbb C^2,0)$ is a collection of $k$ germs of smooth foliations $\mathcal F_i$ subjected to the condition that any two of these foliations have distinct tangent spaces at the origin. One of the most intriguing invariants of a web is its [*space of abelian relations*]{} $\mathcal A(\mathcal W)$. If the foliations $\mathcal F_i$ are induced by $1$-forms $\omega_i$ then by definition $$\begin{aligned} \mathcal A(\mathcal W) = \Big\lbrace {\big(\eta_i\big)}_{i=1}^k \in (\Omega^1(\mathbb C^2,0))^k \, \, \Big| \, \,\forall i \, \, d\eta_i =0 \,, \; \eta_i\wedge \omega_i=0 \, \, \text{ and } \sum_{i=1}^k\eta_i = 0 \Big\rbrace \, .\end{aligned}$$ The dimension of $\mathcal A(\mathcal W)$ is commonly called the [*rank*]{} of $\mathcal W$ and noted by $\mathrm{rk}(\mathcal W)$. It is a theorem of Bol that $\mathcal A(\mathcal W)$ is a finite-dimensional $\mathbb C$-vector space and moreover $$\begin{aligned} \label{bolbound} \mathrm{rk}( \mathcal W) \le \frac{1}{2}\, (k-1)(k-2)\, .\end{aligned}$$ An interesting chapter of the theory of webs concerns the characterization of webs of [*maximal rank*]{}, [*i.e*]{} webs for which (\[bolbound\]) is in fact an equality. It follows from Abel’s Addition Theorem that all the webs $\mathcal W_C$ obtained from reduced plane curves $C$ by projective duality are of maximal rank ([*cf.*]{} §\[S:action\] for details). The webs analytically equivalent to some $\mathcal W_C$ are the so called [*algebrizable webs*]{}. It can be traced back to Lie a remarkable result that says that all $4$-webs of maximal rank are in fact algebrizable. In the early 1930’s Blaschke claimed to have extended Lie’s result to $5$-webs of maximal rank. Not much latter Bol came up with a counter-example: a $5$-web of maximal rank that is not algebrizable. The non-algebrizable webs of maximal rank are nowadays called [*exceptional webs*]{}. For a long time Bol’s web remained as the only example of exceptional planar web in the literature. The following quote illustrates quite well this fact. > [*(…) we cannot refrain from mentioning what we consider to be the fundamental problem on the subject, which is to determine the maximum rank non-linearizable webs. The strong conditions must imply that there are not many. It may not be unreasonable to compare the situation with the exceptional simple Lie groups.*]{} > > Chern and Griffiths in [@Jbr]. A comprehensive account of the current state of the art concerning the exceptional webs is available at [@theseluc Introduction §3.2.1], [@Robert] and [@PT §1.4]. Here we will just mention that before this work no exceptional $k$-web with $k\ge 10$ appeared in the literature. At first glance, the list of known exceptional webs up today does not reveal common features among them. Although at a second look one sees that many of them (but not all, not even the majority) have one property in common: infinitesimal automorphisms. Infinitesimal Automorphisms --------------------------- In [@cartan], [É]{}. Cartan proves that [*a 3-web which admits an 2-dimensional continuous group of transformations is hexagonal*]{}. It is then an exercise to deduce that a $k$-web ($k> 3$) which admits $2$ linearly independent infinitesimal automorphisms is parallelizable and in particular algebrizable. Cartan’s result naturally leads to the following question: > [*What can be said about webs which admit one infinitesimal automorphism?*]{} In fact, Cartan answers this question for 3-webs. In [*loc. cit.*]{} he establishes that such a web is equivalent to those induced by the $1$-forms $dx,dy, dy -u(x+y)dx$, where $u$ is a germ of holomorphic function. It is very surprising that this story stops here… To our knowledge, there is no other study concerning webs with infinitesimal automorphisms, although they are particularly interesting. Indeed, on the one hand their study is considerably simplified by the presence of an infinitesimal automorphism, but on the other hand, these webs can be very interesting from a geometrical point of view: we will show they are connected to the theory of exceptional webs. Variation of the Rank --------------------- Let ${\mathcal W}$ be a regular web in $({\mathbb C}^2,0)$ which admits an infinitesimal automorphism $X$, [*i.e.*]{} $X$ is a germ of vector field whose local flow preserves the foliations of $\mathcal W$. As we will see in §\[S:geral\] the Lie derivative $L_X=i_Xd + di_X$ with respect to $X$ induces a linear operator on $\mathcal A(\mathcal W)$. Most of our results will follow from an analysis of such operator. In §\[S:liouville\] we use this operator to give a simple description of the abelian relations of $\mathcal W$ and from this we will deduce in §\[S:rank\] what we consider our main result: \[T:1\] Let ${\mathcal W}$ be a $k$–web which admits a transverse infinitesimal automorphism $X$. Then $$\mathrm{rk}(\mathcal W \boxtimes \mathcal F_X) =\mathrm{rk}(\mathcal W) + (k -1)\, .$$ In particular, ${\mathcal W}$ is of maximal rank if and only if $\mathcal W \boxtimes \mathcal F_{X}$ is of maximal rank. We will derive from Theorem \[T:1\] the existence of new families of exceptional webs. New Families of Exceptional Webs -------------------------------- If we start with a reduced plane curve $C$ invariant under an algebraic ${\mathbb C}^*$-action on ${\mathbb P}^2$ then we obtain a dual algebraic ${\mathbb C}^*$-action on $\check{\mathbb P}^2$, letting invariant the algebraic web ${\mathcal W}_C$ ([*cf.*]{} §\[S:action\] for details). Combining this construction with Theorem \[T:1\] we deduce our second main result. \[T:2\] For every $k \geq 5$ there exist a family of dimension at least $\lfloor k/2 \rfloor -1$ of pairwise non-equivalent exceptional global $k$-webs on ${\mathbb P}^2$. In fact, for each $k\ge 5$, we obtain $4k - 15$ other families of smaller dimension. We also give a complete classification of all the exceptional 5-webs of the type $ {\mathcal W}\boxtimes {\mathcal F}_X$ where $X$ is an infinitesimal automorphism of $ {\mathcal W}$ ([*cf.*]{} Corollary \[C:classi\]). Generalities on webs with infinitesimal automorphisms {#S:geral} ===================================================== Let $\mathcal F$ be a regular foliation on $(\mathbb C^2,0)$ induced by a (germ of) $1$-form $\omega$. We say that a (germ of) vector field $X$ is an infinitesimal automorphism of $\mathcal F$ if the foliation $\mathcal F$ is preserved by the local flow of $X$. In algebraic terms: $ L_X \omega \wedge \omega = 0 \, .$ When the infinitesimal automorphism $X$ is transverse to $\mathcal F$, [*i.e*]{} when $\omega(X)\neq0$, then a simple computation ([*cf*]{}. [@Percy Corollary 2]) shows that the $1$-form $$\eta = \frac{\omega}{i_X \omega}$$ is closed and satisfies $L_X \eta =0$. By definition, the integral $$u(z) = \int^z_0 \eta$$ is the [*canonical first integral*]{} of ${\mathcal F}$ (with respect to $X$). Clearly, we have $u(0)=0$ and $L_X(u)= 1$. Now let $\mathcal W$ be a germ of regular $k$-web on $(\mathbb C^2,0)$ induced by the (germs of) $1$-forms $\omega_1, \ldots, \omega_k$ and let $X$ be an infinitesimal automorphism of $\mathcal W$. Here, of course, we mean that $X$ is an infinitesimal automorphism for all the foliations in $\mathcal W$. By hypothesis, we have $L_X\, \omega_i \wedge \omega_i=0$ for $i=1,\ldots,k$. Then because the Lie derivative $L_X$ is linear and commutes with $d$, it induces a linear map $$\begin{aligned} \label{themap!} L_X : \mathcal A(\mathcal W) & \longrightarrow & \mathcal A(\mathcal W) \\ (\eta_1,\ldots ,\eta_k) & \longmapsto& (L_X\eta_1 ,\ldots, L_X \eta_k) \, . \nonumber\end{aligned}$$ This map is central in this paper: all our results come from an analysis of the $L_X$-invariant subspaces of $ {\mathcal A}(\mathcal W)$. Abelian relations of webs with infinitesimal automorphisms ========================================================== Description of ${\mathcal A}({\mathcal W})$ in presence of an infinitesimal automorphism {#S:liouville} ---------------------------------------------------------------------------------------- In this section, ${\mathcal W}=\mathcal F_1 \boxtimes \cdots \boxtimes \mathcal F_k $ denotes a $k$-web in $({\mathbb C}^2,0)$ which admits an infinitesimal automorphism $X$, regular and transverse to the foliations ${\mathcal F}_i$ in a neighborhood of the origin. Let $i\in \{1,\ldots,k\}$ be fixed. We note ${\mathcal A}^i({\mathcal W})$ the vector subspace of $\Omega^1(\mathbb C^2,0)$ spanned by the $i$-th components $\alpha_i$ of abelian relations $(\alpha_1,\ldots,\alpha_k)\in {\mathcal A}({\mathcal W})$. If $u_i=\int \eta_i$ denotes the canonical first integral of ${\mathcal F}_i$ with respect to $X$, then for $\alpha_i \in {\mathcal A}^i({\mathcal W})$, there exists a holomorphic germ $f_i \in \mathbb C\{t\}$ such that $\alpha_i=f_i(u_i)\,du_i$. Assume now that ${\mathcal A}^i({\mathcal W})$ is not trivial and let $\big\{\alpha_i^\nu=f_\nu(u_i)\,du_i \, | \, \nu=1,\ldots,n_i\}$ be a basis. Since $L_X : \mathcal A^i(\mathcal W) \to \mathcal A^i(\mathcal W)$ is a linear map, there exist complex constants $c_{\nu \mu}$ such that, for $\nu=1,\ldots,n_i$ we have $$\begin{aligned} \label{baz} L_X(\alpha_i^\nu)=\sum_{\mu=1}^{n_i} c_{\nu \mu}\, \alpha_i^\mu \; .\end{aligned}$$ But $ L_X(\alpha_i^\nu)=L_X\big(f_\nu(u_i)\,du_i\big)=X\big( f_\nu(u_i)\big)du_i+f_\nu(u_i)\,L_X\big(du_i\big)=f_\nu'(u_i)\,{du_i} $ for any $\nu$, so relations (\[baz\]) are equivalent to the scalar ones $$\begin{aligned} \label{diffeq} \quad \quad f_\nu' =\sum_{\mu=1}^{n_i} c_{\nu \mu}\, {f_\mu} \, , \quad \quad \, \nu=1,\ldots,n_i \, .\end{aligned}$$ Now let $\lambda_1,\ldots,\lambda_\tau \in {\mathbb C}$ be the eigenvalues of the map $L_X$ acting on ${\mathcal A}({\mathcal W})$ corresponding to maximal eigenspaces with corresponding dimensions $\sigma_1,\ldots,\sigma_\tau $. The differential equations (\[diffeq\]) give us the following description of $ {\mathcal A}({\mathcal W})$: \[description\] The abelian relations of $ {\mathcal W}$ are of the form $$P_1(u_1)\,e^{\lambda_i\,u_1}\,du_1+\cdots+ P_k(u_k)\,e^{\lambda_i\,u_k}\,du_{k}=0$$ where $P_1, \ldots, P_k$ are polynomials of degree less or equal to $ \sigma_i$. A non-zero eigenvalue $\lambda$ of the map (\[themap!\]) corresponds to a functional equation of the form $ c_1\,e^{\lambda\,u_1}+\cdots+ c_k\,e^{\lambda\,u_k}={cst.}$ where the $c_i$’s are complex constants. Using Abel’s method to solve functional equations (see [@theseluc Chapter 2]), such $\lambda$ can be computed effectively from the $u_i$’s, which can be effectively computed from the $\omega_i$’s. In a few words: Proposition \[description\] gives an effective tool to compute the abelian relations of $\mathcal W$. Proof of Theorem \[T:1\] {#S:rank} ------------------------ With Proposition \[description\] at hand we are able to prove our main result. Let $\mathcal W= \mathcal F_1 \boxtimes \cdots \boxtimes \mathcal F_k$ and for $i= 1 \ldots k$, set $\eta_i = du_i$ as the differential of the canonical first integral of $\mathcal F_i$ relatively to $X$. We note $x$ a first integral of the foliation ${\mathcal F}_X$, normalized such that $x(0)=0$. When $j$ varies from $2$ to $k$, we have $$i_X(\eta_1 - \eta_j) =0 \qquad\quad \text{ and } \qquad\quad L_X(\eta_1 - \eta_j) = 0 \, .$$ Consequently there exists $g_j \in \mathbb C\{x\}$ such that $$\label{E:xxx} du_1 - du_j - g_j(x)\,dx = 0 \, .$$ Clearly these are abelian relations for the web $\mathcal W \boxtimes \mathcal F_X$. They span a $(k-1)$-dimensional vector subspace $\mathcal V$ of the maximal eigenspace of $L_X$ associated to the eigenvalue zero, noted $\mathcal A_0(\mathcal W\boxtimes \mathcal F_X)$. Observe that $\mathcal V$ fits in the following exact sequence ($i$ is the natural inclusion): $$\begin{aligned} \label{exactseq} 0 \to {\mathcal V} \stackrel{i}{\longrightarrow} {\mathcal A_0(\mathcal W \boxtimes \mathcal F_X)}\stackrel{L_X}{\longrightarrow} {\mathcal A_0(\mathcal W)} \,.\end{aligned}$$ Indeed, the kernel $ K:= \ker \{ L_X : \mathcal A_0(\mathcal W \boxtimes \mathcal F_X) \to {\mathcal A_0(\mathcal W)} \}$ is generated by abelian relations of the form $\sum_{i=1}^k c_i du_i + g(x)\,dx = 0$, where $c_i \in \mathbb C$ and $g \in \mathbb C\{x\}$. Since $i_X du_i=1$ for each $i$, it follows that the constants $c_i$ satisfy $\sum_{i=1}^k c_i =0$. It implies that the abelian relations in the kernel of $L_X$ can be written as linear combinations of abelian relations of the form (\[E:xxx\]). Therefore $$\label{E:yyy} K = \mathcal V$$ and consequently $\ker L_X \subset \mathrm{Im } \, \, i$. The exactness of (\[exactseq\]) follows easily. From general principles we deduce that the sequence $$0 \to \frac{\mathcal V}{\mathcal A_0(\mathcal W)\cap \mathcal V} \stackrel{i}{\longrightarrow} \frac{\mathcal A_0(\mathcal W \boxtimes \mathcal F_X)}{\mathcal A_0(\mathcal W)} \stackrel{L_X}{\longrightarrow} \frac{\mathcal A_0(\mathcal W)}{L_X \mathcal A_0(\mathcal W)} \, ,$$ is also exact. Thus to prove the Theorem it suffices to verify the following assertions: 1. $\mathcal V$ is isomorphic to $$\frac{\mathcal V}{\mathcal A_0(\mathcal W)\cap \mathcal V} \oplus \frac{\mathcal A_0(\mathcal W)}{L_X \mathcal A_0(\mathcal W)} \, ;$$ 2. the morphism $L_X:\mathcal A_0(\mathcal W \boxtimes \mathcal F_X) \to \mathcal A_0(\mathcal W)$ is surjective; 3. the vector spaces $$\frac{\mathcal A_0(\mathcal W \boxtimes \mathcal F_X)}{\mathcal A_0(\mathcal W)} \, \, \text{ and } \, \, \frac{\mathcal A(\mathcal W \boxtimes \mathcal F_X)}{\mathcal A(\mathcal W)}$$ are isomorphic. To verify assertion (a), notice that the nilpotence of $L_X$ on $\mathcal A_0(\mathcal W)$ implies that $\frac{\mathcal A_0(\mathcal W)}{L_X \mathcal A_0(\mathcal W)}$ is isomorphic to $\mathcal A_0(\mathcal W) \cap K$. Combined with (\[E:yyy\]), it implies assertion (a). To prove assertion (b), it suffices to construct a map $\Phi:\mathcal A_0(\mathcal W) \to \mathcal A_0(\mathcal W \boxtimes \mathcal F_X)$ such that $ L_X \circ \Phi = \mathrm{Id}$. Proposition \[description\] implies that $\mathcal A_0(\mathcal W)$ is spanned by abelian relations of the form $ \sum_{i=1}^k c_i u_i^r du_i =0 ,$ where $c_i$ are complex numbers and $r$ is a non-negative integer. For such an abelian relation, since $$\sum_{i=1}^k c_i u_i^r du_i = \frac{1}{r+1}\,L_X \Big( \sum_{i=1}^k c_i u_i^{r+1} du_i \Big) =0 \, ,$$ there exists an unique $g\in \mathbb C \{ x\}$ satisfying $ \sum_{i=1}^k c_i u_i^{r+1} du_i + g(x)\, dx =0 \, . $ If we set $$\Phi\Big(\sum_{i=1}^k c_i u_i^r du_i \Big) = \frac{1}{r+1}\, \Big( \sum_{i=1}^k c_i u_i^{r+1} du_i + g(x)\, dx \Big) \,$$ then $L_X \circ \Phi= \mathrm{Id}$ on $\mathcal A_0(\mathcal W)$ and assertion (b) follows. To prove assertion (c) we first notice that $$\mathcal A(\mathcal W \boxtimes \mathcal F_X)=\mathcal A_0(\mathcal W \boxtimes \mathcal F_X) \oplus \mathcal A_*(\mathcal W \boxtimes \mathcal F_X) \,$$ where $\mathcal A_*(\mathcal W \boxtimes \mathcal F_X)$ denotes the sum of eigenspaces corresponding to non-zero eigenvalues. Of course $\mathcal A_*(\mathcal W \boxtimes \mathcal F_X)$ is invariant and moreover we have the equality $$L_X\big(\mathcal A_*(\mathcal W \boxtimes \mathcal F_X)\big) = \mathcal A_*(\mathcal W \boxtimes \mathcal F_X) \, .$$ But $L_X$ [*kills*]{} the $\mathcal F_X$-components of abelian relations. In particular, it implies $$L_X\big(\mathcal A_*(\mathcal W \boxtimes \mathcal F_X)\big) \subset \mathcal A_*(\mathcal W ).$$ This is sufficient to show that $\mathcal A_*(\mathcal W \boxtimes \mathcal F_X) = \mathcal A_*(\mathcal W )$ and deduce assertion (c) and, consequently that $$\mathrm{rk}(\mathcal W \boxtimes \mathcal F_X) =\mathrm{rk}(\mathcal W) + (k -1)\, .$$ Because $k(k-1)/2=(k-1)(k-2)/2+(k-1)$, the above inequality implies immediately the last assertion of Theorem 1. New Families of exceptional webs ================================ Algebrizable Webs with Infinitesimal Automorphisms {#S:action} -------------------------------------------------- Let $C \subset \mathbb P^2$ be a degree $k$ reduced curve. If $U\subset \check{\mathbb P}^2$ is a simply-connected open set not intersecting $\check C$ and if $\gamma_1, \ldots, \gamma_k: U \to C$ are the holomorphic maps defined by the intersections of lines in $U$ with $C$ then Abel’s Theorem implies that $$\mathrm{Tr}(\omega)= \sum_{i=1}^k \gamma_i^* \omega = 0 \,$$ for every $\omega \in H^0(C,\omega_C)$, where $\omega_C$ denotes the dualizing sheaf of $C$. The maps $\gamma_i$ define the $k$-web $\mathcal W_C$ on $U$ and the trace formula above associates an abelian relation of $\mathcal W_C$ to each $\omega \in H^0(C,\omega_C)$. Since $h^0(C,\omega_C)=(k-1)(k-2)/2$, the web $\mathcal W_C$ is of maximal rank. Suppose now that $C$ is invariant by a $\mathbb C^*$-action $\varphi: \mathbb C^* \times \mathbb P^2 \to \mathbb P^2.$ Notice that $\varphi$ induces a dual action $\check{\varphi}:\mathbb C^* \times \check{\mathbb P^2} \to \check{\mathbb P^2}$ satisfying $ \varphi_t \circ \gamma_i = \gamma_i \circ \check{\varphi_t} $ for $i=1,\ldots,k$. Consequently the web $\mathcal W_C$ admits an infinitesimal automorphism. In a suitable projective coordinate system $[x:y:z]$, a plane curve $C$ invariant by a ${\mathbb C}^*$-action is cut out by an equation of the form $$\label{E:aluffi} x^{\epsilon_1}\cdot y^{\epsilon_2}\cdot z^{\epsilon_3} \cdot \prod_{i=1}^k ( x^a + \lambda_i y^b z^{a-b} ) \,$$ where $\epsilon_1, \epsilon_2, \epsilon_3 \in \{ 0,1 \}$, $k,a,b \in \mathbb N$ are such that $k\ge1$, $a\ge 2$, $1\le b\le a/2$, $\gcd(a,b)=1$ and the $\lambda_i$ are distinct non zero complex numbers ([*cf.*]{} [@aluffi §1] for instance). Notice that here the $\mathbb C^*$-action in question is $$\label{E:zzz} \begin{array}{l c l c l} \varphi &:& \mathbb C^* \times \mathbb P^2 &\to& \mathbb P^2 \\ & & ( t, [x:y:z] ) &\mapsto& [t^{b(a-b)}x:t^{a(a-b)} y: t^{ab}z ] \, . \end{array}$$ Moreover once we fix $\epsilon_1,\epsilon_2,\epsilon_3,k,a,b$ we can always choose $\lambda_1=1$ and in this case the set of $k-1$ complex numbers $\{\lambda_2,\ldots, \lambda_k\}$ projectively characterizes the curve $C$. In particular one promptly sees that there exists a $(d-1)$-dimensional family of degree $2d$ (or $2d +1$) reduced plane curves all projectively distinct and invariant by the same $\mathbb C^*$-action: for a given $2d+ \delta$ with $\delta\in \{0,1\}$ set $a=2$, $b=1$, $\epsilon_1=\delta$ and $\epsilon_2=\epsilon_3=0$. A moment of reflection shows that the number of discrete parameters giving distinct families of degree $d$ curves of the form (\[E:aluffi\]) is $$\underbrace{\Big\lfloor \frac{d}{2} \Big\rfloor}_{\epsilon_1=\epsilon_2=\epsilon_3=0} + \underbrace{ 3 \Big\lfloor \frac{d-1}{2} \Big\rfloor}_{\epsilon_i=\epsilon_j=0, \, \epsilon_k=1} + \underbrace{3\Big\lfloor \frac{d-2}{2} \Big\rfloor}_{\epsilon_i=\epsilon_j=1, \, \epsilon_k=0} + \underbrace{\Big\lfloor \frac{d-3}{2} \Big\rfloor}_{\epsilon_1=\epsilon_2=\epsilon_3=1} -\; 2 = 4d - 10 \, .$$ Notice that the $-2$ appears on left hand side because the curves $\{y=0\}$ and $\{z=0\}$ are indistinguishable when $a=2$. Proof of Theorem \[T:2\] ------------------------ If $C$ is a reduced curve of the form (\[E:aluffi\]) then $\mathcal W_C$ is invariant by an algebraic $\mathbb C^*$-action $\check \varphi$. We will note by $X$ the infinitesimal generator of $\check \varphi$ and by $\mathcal F_X$ the corresponding foliation. From the discussion on the last paragraph, Theorem \[T:2\] follows at once from the stronger: \[T:geral\] If $\deg C\ge 4$ then $\mathcal W_C \boxtimes \mathcal F_X$ is exceptional. Moreover if $C'$ is another curve invariant by $\varphi$ then $\mathcal W_C \boxtimes \mathcal F_X$ is analytically equivalent to $\mathcal W_{C'} \boxtimes \mathcal F_X$ if and only if the curve $C$ is projectively equivalent to $C'$. Since $\mathcal W_C$ has maximal rank it follows from Theorem \[T:1\] that $\mathcal W_C \boxtimes \mathcal F_X$ is also of maximal rank. Suppose that its localization at a point $p\in \mathbb P^2$ is algebrizable and let $\psi:(\mathbb P^2,p) \to (\mathbb C^2,0)$ be a holomorphic algebrization. Since both $\mathcal W_C$ and $\psi_*(\mathcal W_C)$ are linear webs of maximal rank it follows from a result of Nakai [@nakai] that $\psi$ is the localization of an automorphism of $\mathbb P^2$. But the generic leaf of $\mathcal F_X$ is not contained in any line of $\mathbb P^2$ and consequently $\psi_*(\mathcal W \boxtimes \mathcal F_X)$ is not linear. This concludes the proof of the theorem. We do not know if the families above are [*irreducible*]{} in the sense that the generic element does not admit a deformation as an exceptional web that is not contained in the family. Due to the presence of automorphism one could imagine that they are indeed degenerations of some other exceptional webs. A characterization result ------------------------- Combining Theorem \[T:1\] with Lie’s Theorem we can easily prove the \[C:classi\] Let $\mathcal W$ be a 4-web that admits a transverse infinitesimal automorphism $Y$. If $\mathcal W\boxtimes \mathcal F_Y$ is exceptional then it is analytically equivalent to an exceptional 5-web $\mathcal W_C \boxtimes \mathcal F_X$ described in Theorem \[T:geral\]. It follows from Theorem \[T:1\] that $\mathcal W$ is of maximal rank. Lie’s Theorem implies that $\mathcal W$ is analytically equivalent to $\mathcal W_C$ for some reduced plane quartic $C$. Since the local flow of $Y$ preserves $\mathcal W$ there exists a (germ) of vector field $X$ whose local flow preserves $\mathcal W_C$. Using again Nakai’s result we deduce that the germs of automorphisms on the local flow of $X$ are indeed projective automorphisms. This is sufficient to conclude that $X$ is a global vector field preserving $\mathcal W_C$. The example below shows that Theorem \[T:geral\] does not give all the exceptional webs admitting an infinitesimal automorphism. In [@crasluc], it is proved that the web ${\mathcal W}$ induced by the functions $x,y,x+y,x-y,x^2+y^2$ is exceptional. Moreover it admits the radial vector field $R=x\,\partial/\partial_x+y\,\partial/\partial_y$ as a transverse infinitesimal automorphism. Theorem \[T:1\] implies that the $6$-web $\mathcal W \boxtimes \mathcal F_R$ is also exceptional. This result was previously obtained by determining an explicit basis of the space of abelian relations, see [@theseluc p. 253]. Problems ======== A conjecture about the nature of the abelian relations ------------------------------------------------------ It is clear from Proposition \[description\] that for webs $\mathcal W$ admitting infinitesimal automorphisms there exists a Liouvillian extension of the field of definition of $\mathcal W$ containing all its abelian relations. We believe that a similar statement should hold for arbitrary webs $\mathcal W$. The abelian relations of a web are Liouvillian. Our belief is supported by the recent works of H[é]{}naut [@Henaut] and Ripoll [@Ripoll] on abelian relations and of Casale [@Guy] on non-linear differential Galois Theory. When $\mathcal W$ is of maximal rank the main result of [@Henaut] shows that there exists a Picard-Vessiot extension of the field of definition of $\mathcal W$ containing all the abelian relations. In the general case, one should be able to deduce a similar result from the above mentioned work of Ripoll. On the other hand, and at least over polydiscs, [@Guy Theorem 6.4] implies that the foliations with first integrals on Picard-Vessiot extension are transversely projective. Since the first integrals in question are components of abelian relations they are of finite determinacy and hopefully this should imply that they are indeed Liouvillian. Restricted Chern’s Problem -------------------------- With the techniques now available, the classification of all exceptional 5-webs (“[*Chern’s problem*]{}”) seems completely out of reach. So we propose the Classify exceptional 5-webs admitting infinitesimal automorphisms. Notice that this restricted version is not completely hopeless. The linear map $L_X$ can be “integrated” giving birth to a holomorphic action on $\mathbb P(\mathcal A(\mathcal W))$. The Poincar[é]{}-Blaschke curves will be orbits of this action and the dual action will induce an automorphism of the associated Blaschke surface. This seems valuable extra data that may lead to a solution of the restricted Chern’s problem. For a definition of the above mentioned concepts see [@theseluc Chapter 8]. [9]{} P. Aluffi and C. Faber, , Internat. J. Math. **11** (2000), pp. 591–608. . Cartan, , uvres compl[è]{}tes, Vol. 3, pp. 78–83. G. Casale, *Le groupo[ï]{}de de Galois d’un germe de feuilletage de codimension un*, to appear in Ann. Inst. Fourier. S.-S. Chern and P.A. Griffiths, *Corrections and addenda to our paper: “Abel’s theorem and webs”.* Jahresber. Deutsch. Math.-Verein. **83** (1981), pp. 78–83. A. H[é]{}naut, . Ann. of Math. (2) **159** (2004), pp. 425–445. I. Nakai, , [Topology]{} **26** ([1987]{}), [pp. 475–504]{}. J.V. Pereira, and P.F. S[á]{}nchez, *Transformation groups of holomorphic foliations*, Comm. Anal. Geom. **10** (2002), pp. 1115–1123. L. Pirio, , [C. R. Math. Acad. Sci. Paris]{} **339** ([2004]{}), pp. [131–136]{}. L. Pirio, , Th[è]{}se de l’Universit[é]{} Paris VI, soutenue le 15 d[é]{}cembre 2004. L. Pirio and J.-M. Tr[é]{}preau, , Ann. Inst. Fourier **55** (2005), pp. 2209–2237. O. Ripoll, , C.R. Acad. Sci. Paris, Ser. I **341** (2005), pp. 247–252. G. Robert, , Pr[é]{}publication **146**, Universit[é]{} Bordeaux 1 (2002). [^1]: The second author is supported by Cnpq and Instituto Unibanco. The third author was partially supported by the International Cooperation Agreement Brazil-France.

--- abstract: 'Region-based methods have proven necessary for improving segmentation accuracy of neuronal structures in electron microscopy (EM) images. Most region-based segmentation methods use a scoring function to determine region merging. Such functions are usually learned with supervised algorithms that demand considerable ground truth data, which are costly to collect. We propose a semi-supervised approach that reduces this demand. Based on a merge tree structure, we develop a differentiable unsupervised loss term that enforces consistent predictions from the learned function. We then propose a Bayesian model that combines the supervised and the unsupervised information for probabilistic learning. The experimental results on three EM data sets demonstrate that by using a subset of only $3\%$ to $7\%$ of the entire ground truth data, our approach consistently performs close to the state-of-the-art supervised method with the full labeled data set, and significantly outperforms the supervised method with the same labeled subset.' author: - Ting Liu - Miaomiao Zhang - Mehran Javanmardi - Nisha Ramesh - Tolga Tasdizen - Ting Liu - Miaomiao Zhang - Mehran Javanmardi - Nisha Ramesh - Tolga Tasdizen bibliography: - 'refs-arxiv.bib' subtitle: Supplementary Materials title: - 'SSHMT: Semi-supervised Hierarchical Merge Tree for Electron Microscopy Image Segmentation' - 'SSHMT: Semi-supervised Hierarchical Merge Tree for Electron Microscopy Image Segmentation' --- Introduction ============ Connectomics researchers study structures of nervous systems to understand their function [@sporns2005human]. Electron microscopy (EM) is the only modality capable of imaging substantial tissue volumes at sufficient resolution and has been used for the reconstruction of neural circuitry [@famiglietti1991synaptic; @briggman2011wiring; @helmstaedter2013cellular]. The high resolution leads to image data sets at enormous scale, for which manual analysis is extremely laborious and can take decades to complete [@briggman2006towards]. Therefore, reliable automatic connectome reconstruction from EM images, and as the first step, automatic segmentation of neuronal structures is crucial. However, due to the anisotropic nature, deformation, complex cellular structures and semantic ambiguity of the image data, automatic segmentation still remains challenging after years of active research. Similar to the boundary detection/region segmentation pipeline for natural image segmentation [@arbelaez2011contour; @ren2013image; @arbelaez2014multiscale; @liu2016image], most recent EM image segmentation methods use a membrane detection/cell segmentation pipeline. First, a membrane detector generates pixel-wise confidence maps of membrane predictions using local image cues [@sommer2011ilastik; @ciresan2012deep; @seyedhosseini2013image]. Next, region-based methods are applied to transforming the membrane confidence maps into cell segments. It has been shown that region-based methods are necessary for improving the segmentation accuracy from membrane detections for EM images [@arganda2015crowdsourcing]. A common approach to region-based segmentation is to transform a membrane confidence map into over-segmenting superpixels and use them as “building blocks” for final segmentation. To correctly combine superpixels, greedy region agglomeration based on certain boundary saliency has been shown to work [@nunez2013machine]. Meanwhile, structures, such as loopy graphs [@kaynig2015large; @krasowski2015improving] or trees [@liu2014modular; @funke2015learning; @uzunbas2016efficient], are more often imposed to represent the region merging hierarchy and help transform the superpixel combination search into graph labeling problems. To this end, local [@liu2014modular; @krasowski2015improving] or structured [@funke2015learning; @uzunbas2016efficient] learning based methods are developed. Most current region-based segmentation methods use a scoring function to determine how likely two adjacent regions should be combined. Such scoring functions are usually learned in a supervised manner that demands considerable amount of high-quality ground truth data. Obtaining such ground truth data, however, involves manual labeling of image pixels and is very labor intensive, especially given the large scale and complex structures of EM images. To alleviate this demand, Parag et al. recently propose an active learning framework [@parag2014small; @parag2015efficient] that starts with small sets of labeled samples and constantly measures the disagreement between a supervised classifier and a semi-supervised label propagation algorithm on unlabeled samples. Only the most disagreed samples are pushed to users for interactive labeling. The authors demonstrate that by using $15\%$ to $20\%$ of all labeled samples, the method can perform similar to the underlying fully supervised method with full training set. One disadvantage of this framework is that it does not directly explore the unsupervised information while searching for the optimal classification function. Also, retraining is required for the supervised algorithm at each iteration, which can be time consuming especially when more iterations with fewer samples per iteration are used to maximize the utilization of supervised information and minimize human effort. Moreover, repeated human interactions may lead to extra cost overhead in practice. In this paper, we propose a semi-supervised learning framework for region-based neuron segmentation that seeks to reduce the demand for labeled data by exploiting the underlying correlation between unsupervised data samples. Based on the merge tree structure [@liu2014modular; @funke2015learning; @uzunbas2016efficient], we redefine the labeling constraint and formulate it into a differentiable loss function that can be effectively used to guide the unsupervised search in the function hypothesis space. We then develop a Bayesian model that incorporates both unsupervised and supervised information for probabilistic learning. The parameters that are essential to balancing the learning can be estimated from the data automatically. Our method works with very small amount of supervised data and requires no further human interaction. We show that by using only $3\%$ to $7\%$ of the labeled data, our method performs stably close to the state-of-the-art fully supervised algorithm with the entire supervised data set (Section \[sec:res\]). Also, our method can be conveniently adopted to replace the supervised algorithm in the active learning framework [@parag2014small; @parag2015efficient] and further improve the overall segmentation performance. Hierarchical Merge Tree {#sec:hmt} ======================= Starting with an initial superpixel segmentation $S_o$ of an image, a merge tree $T=(\mathcal{V},\mathcal{E})$ is a graphical representation of superpixel merging order. Each node $v_i\in\mathcal{V}$ corresponds to an image region $s_i$. Each leaf node aligns with an initial superpixel in $S_o$. A non-leaf node corresponds to an image region combined by multiple superpixels, and the root node represents the whole image as a single region. An edge $e_{i,c}\in\mathcal{E}$ between $v_i$ and one of its child $v_c$ indicates $s_c\subset s_i$. Assuming only two regions are merged each time, we have $T$ as a full binary tree. A clique $p_i=(\{v_i,v_{c_1},v_{c_2}\},\{e_{i,c_1},e_{i,c_2}\})$ represents $s_i=s_{c_1}\cup s_{c_2}$. In this paper, we call clique $p_i$ is at node $v_i$. We call the cliques $p_{c_1}$ and $p_{c_2}$ at $v_{c_1}$ and $v_{c_2}$ the child cliques of $p_i$, and $p_i$ the parent clique of $p_{c_1}$ and $p_{c_2}$. If $v_i$ is a leaf node, $p_i=(\{v_i\},\varnothing)$ is called a leaf clique. We call $p_i$ a non-leaf/root/non-root clique if $v_i$ is a non-leaf/root/non-root node. An example merge tree, as shown in Fig. \[fig:sub:toy\_tree\], represents the merging of superpixels in Fig. \[fig:sub:toy\_segi\]. The red box in Fig. \[fig:sub:toy\_tree\] shows a non-leaf clique $p_7=(\{v_7,v_1,v_2\},\{e_{7,1},e_{7,2}\})$ as the child clique of $p_9=(\{v_9,v_7,v_3\},\{e_{9,7},e_{9,3}\})$. A common approach to building a merge tree is to greedily merge regions based on certain boundary saliency measurement in an iterative fashion [@liu2014modular; @funke2015learning; @uzunbas2016efficient]. Given the merge tree, the problem of finding a final segmentation is equivalent to finding a complete label assignment $\mathbf{z}=\{z_i\}_{i=1}^{|\mathcal{V}|}$ for every node being a final segment ($z=1$) or not ($z=0$). Let $\rho(i)$ be a query function that returns the index of the parent node of $v_i$. The $k$-th ($k=1,\ldots d_i$) ancestor of $v_i$ is denoted as $\rho^k(i)$ with $d_i$ being the depth of $v_i$ in the tree, and $\rho^0(i)=i$. For every leaf-to-root path, we enforce the *region consistency constraint* that requires $\sum_{k=0}^{d_i}z_{\rho^k(i)}=1$ for any leaf node $v_i$. As an example shown in Fig. \[fig:sub:toy\_tree\], the red nodes ($v_6$, $v_8$, and $v_9$) are labeled $z=1$ and correspond to the final segmentation in Fig. \[fig:sub:toy\_seg\]. The rest black nodes are labeled $z=0$. Supervised algorithms are proposed to learn scoring functions in a local [@liu2014modular; @liu2016image] or a structured [@funke2015learning; @uzunbas2016efficient] fashion, followed by greedy [@liu2014modular] or global [@funke2015learning; @liu2016image; @uzunbas2016efficient] inference techniques for finding the optimal label assignment under the constraint. We refer to the local learning and greedy search inference framework in [@liu2014modular] as the hierarchical merge tree (HMT) method and follow its settings in the rest of this paper, as it has been shown to achieve state-of-the-art results in the public challenges [@arganda2015crowdsourcing; @isbichallenge2013]. A binary label $y_i$ is used to denote whether the region merging at clique $p_i$ occurs (“merge”, $y_i=1$) or not (“split”, $y_i=0$). For a leaf clique, $y=1$. At training time, $\mathbf{y}=\{y_i\}_{i=1}^{|\mathcal{V}|}$ is generated by comparing both the “merge” and “split” cases for non-leaf cliques against the ground truth segmentation under certain error metric (e.g. adapted Rand error [@arganda2015crowdsourcing]). The one that causes the lower error is adopted. A binary classification function called the boundary classifier is trained with $(\mathbf{X},\mathbf{y})$, where $\mathbf{X}=\{\mathbf{x}_i\}_{i=1}^{|\mathcal{V}|}$ is a collection of feature vectors. Shape and image appearance features are commonly used. At testing time, each non-leaf clique $p_i$ is assigned a likelihood score $P(y_i|\mathbf{x}_i)$ by the classifier. A potential for each node $v_i$ is defined as $$u_i=P(y_i=1|\mathbf{x}_i)\cdot P(y_{\rho(i)}=0|\mathbf{x}_{\rho(i)}).\label{eq:node_potential}$$ The greedy inference algorithm iteratively assigns $z=1$ to an unlabeled node with the highest potential and $z=0$ to its ancestor and descendant nodes until every node in the merge tree receives a label. The nodes with $z=1$ forms a final segmentation. Note that HMT is not limited to segmenting images of any specific dimensionality. In practice, it has been successfully applied to both 2D [@liu2014modular; @arganda2015crowdsourcing] and 3D segmentation [@isbichallenge2013] of EM images. SSHMT: Semi-supervised Hierarchical Merge Tree {#sec:sshmt} ============================================== The performance of HMT largely depends on accurate boundary predictions given fixed initial superpixels and tree structures. In this section, we propose a semi-supervised learning based HMT framework, named SSHMT, to learn accurate boundary classifiers with limited supervised data. Merge consistency constraint ---------------------------- Following the HMT notation (Section \[sec:hmt\]), we first define the *merge consistency constraint* for non-root cliques: $$y_i\geq y_{\rho(i)},\forall i.\label{eq:mcc}$$ Clearly, a set of consistent node labeling $\mathbf{z}$ can be transformed to a consistent $\mathbf{y}$ by assigning $y=1$ to the cliques at the nodes with $z=1$ and their descendant cliques and $y=0$ to the rest. A consistent $\mathbf{y}$ can be transformed to $\mathbf{z}$ by assigning $z=1$ to the nodes in $\{v_i\in\mathcal{V}|\forall i,\textrm{s.t.\ }y_i=1\wedge(v_i\textrm{ is the root}\vee y_{\rho(i)}=0)\}$ and $z=0$ to the rest, vice versa. Define a clique path of length $L$ that starts at $p_i$ as an ordered set $\boldsymbol{\pi}^L_i=\{p_{\rho^l(i)}\}^{L-1}_{l=0}$. We then have \[theorem:cp\_mono\] Any consistent label sequence $\mathbf{y}^L_i=\{y_{\rho^l(i)}\}_{l=0}^{L-1}$ for $\boldsymbol{\pi}^L_i$ under the merge consistency constraint is monotonically non-increasing. Assume there exists a label sequence $\mathbf{y}^L_i$ subject to the merge consistency constraint that is not monotonically non-increasing. By definition, there must exist $k\geq0$, s.t. $y_{\rho^k(i)}<y_{\rho^{k+1}(i)}$. Let $j=\rho^k(i)$, then $\rho^{k+1}(i)=\rho(j)$, and thus $y_j<y_{\rho(j)}$. This violates the merge consistency constraint , which contradicts the initial assumption that $\mathbf{y}^L_i$ is subject to the merge consistency constraint. Therefore, the initial assumption must be false, and all label sequences that are subject to the merge consistency constraint must be monotonically non-increasing. Intuitively, Theorem \[theorem:cp\_mono\] states that while moving up in a merge tree, once a split occurs, no merge shall occur again among the ancestor cliques in that path. As an example, a consistent label sequence for the clique path $\{p_7,p_9,p_{11}\}$ in Fig. \[fig:sub:toy\_tree\] can only be $\{y_7,y_9,y_{11}\}=\{0,0,0\}$, $\{1,0,0\}$, $\{1,1,0\}$, or $\{1,1,1\}$. Any other label sequence, such as $\{1,0,1\}$, is not consistent. In contrast to the region consistency constraint, the merge consistency constraint is a local constraint that holds for the entire leaf-to-root clique paths as well as any of their subparts. This allows certain computations to be decomposed as shown later in Section \[sec:res\]. Let $f_i$ be a predicate that denotes whether $y_i=1$. We can express the non-increasing monotonicity of any consistent label sequence for $\boldsymbol{\pi}^L_i$ in disjunctive normal form (DNF) as $$F^L_i=\bigvee_{j=0}^{L}\left(\bigwedge_{k=0}^{j-1}f_{\rho^k(i)}\wedge\bigwedge_{k=j}^{L-1}\neg f_{\rho^k(i)}\right),\label{eq:dnf}$$ which always holds $true$ by Theorem \[theorem:cp\_mono\]. We approximate $F^L_i$ with real-valued variables and operators by replacing $true$ with $1$, $false$ with $0$, and $f$ with real-valued $\tilde{f}$. A negation $\neg f$ is replaced by $1-\tilde{f}$; conjunctions are replaced by multiplications; disjunctions are transformed into negations of conjunctions using De Morgan’s laws and then replaced. The real-valued DNF approximation is $$\tilde{F}^L_i=1-\prod_{j=0}^L\left(1-\prod_{k=0}^{j-1}\tilde{f}_{\rho^k(i)}\cdot\prod_{k=j}^{L-1}\left(1-\tilde{f}_{\rho^k(i)}\right)\right),\label{eq:rdnf}$$ which is valued $1$ for any consistent label assignments. Observing $\tilde{f}$ is exactly a binary boundary classifier in HMT, we further relax it to be a classification function that predicts $P(y=1|\mathbf{x})\in[0,1]$. The choice of $\tilde{f}$ can be arbitrary as long as it is (piecewise) differentiable (Section \[sec:sub:sslearn\]). In this paper, we use a logistic sigmoid function with a linear discriminant $$\tilde{f}(\mathbf{x};\boldsymbol{w})=\frac{1}{1+\exp(-\boldsymbol{w}^{\top}\mathbf{x})},\label{eq:logsig}$$ which is parameterized by $\boldsymbol{w}$. We would like to find an $\tilde{f}$ so that its predictions satisfy the DNF  for any path in a merge tree. We will introduce the learning of such $\tilde{f}$ in a semi-supervised manner in Section \[sec:sub:sslearn\]. Bayesian semi-supervised learning {#sec:sub:sslearn} --------------------------------- To learn the boundary classification function $\tilde{f}$, we use both supervised and unsupervised data. Supervised data are the clique samples with labels that are generated from ground truth segmentations. Unsupervised samples are those we do not have labels for. They can be from the images that we do not have the ground truth for or wish to segment. We use $\mathbf{X}_s$ to denote the collection of supervised sample feature vectors and $\mathbf{y}_s$ for their true labels. $\mathbf{X}$ is the collection of all supervised and unsupervised samples. Let $\boldsymbol{\tilde{f}}_{\boldsymbol{w}}=[\tilde{f}_{j_1},\ldots,\tilde{f}_{j_{N_s}}]^{\top}$ be the predictions about the supervised samples in $\mathbf{X}_s$, and $\boldsymbol{\tilde{F}}_{\boldsymbol{w}}=[\tilde{F}^L_{i_1},\ldots,\tilde{F}^L_{i_{N_u}}]^{\top}$ be the DNF values  for all paths from $\mathbf{X}$. We are now ready to build a probabilistic model that includes a regularization prior, an unsupervised likelihood, and a supervised likelihood. The prior is an i.i.d. Gaussian $\mathcal{N}(0,1)$ that regularizes $\boldsymbol{w}$ to prevent overfitting. The unsupervised likelihood is an i.i.d. Gaussian $\mathcal{N}(0,\sigma_u)$ on the differences between each element of $\boldsymbol{\tilde{F}}_{\boldsymbol{w}}$ and $1$. It requires the predictions of $\tilde{f}$ to conform the merge consistency constraint for every path. Maximizing the unsupervised likelihood allows us to narrow down the potential solutions to a subset in the classifier hypothesis space without label information by exploring the sample feature representation commonality. The supervised likelihood is an i.i.d. Gaussian $\mathcal{N}(0,\sigma_s)$ on the prediction errors for supervised samples to enforce accurate predictions. It helps avoid consistent but trivial solutions of $\tilde{f}$, such as the ones that always predict $y=1$ or $y=0$, and guides the search towards the correct solution. The standard deviation parameters $\sigma_u$ and $\sigma_s$ control the contributions of the three terms. They can be preset to reflect our prior knowledge about the model distributions, tuned using a holdout set, or estimated from data. By applying Bayes’ rule, we have the posterior distribution of $\boldsymbol{w}$ as $$\begin{split} P(\boldsymbol{w}\,|\,\mathbf{X},\mathbf{X}_s,\mathbf{y}_s,\sigma_u,\sigma_s)\propto & \,P(\boldsymbol{w})\cdot P(\mathbf{1}\,|\,\mathbf{X},\boldsymbol{w},\sigma_u)\cdot P(\mathbf{y}_s\,|\,\mathbf{X}_s,\boldsymbol{w},\sigma_s)\\ \propto & \,\exp\left(-\frac{\|\boldsymbol{w}\|_2^2}{2}\right)\\ & \cdot\frac{1}{\left(\sqrt{2\pi}\sigma_u\right)^{N_u}}\exp\left(-\frac{\|\mathbf{1}-\boldsymbol{\tilde{F}}_{\boldsymbol{w}}\|_2^2}{2\sigma_u^2}\right)\\ & \cdot\frac{1}{\left(\sqrt{2\pi}\sigma_s\right)^{N_s}}\exp\left(-\frac{\|\mathbf{y}_s-\boldsymbol{\tilde{f}}_{\boldsymbol{w}}\|_2^2}{2\sigma_s^2}\right),\label{eq:post} \end{split}$$ where $N_u$ and $N_s$ are the number of elements in $\boldsymbol{\tilde{F}}_{\boldsymbol{w}}$ and $\boldsymbol{\tilde{f}}_{\boldsymbol{w}}$, respectively; $\mathbf{1}$ is a $N_u$-dimensional vector of ones. ### Inference We infer the model parameters $\boldsymbol{w}$, $\sigma_u$, and $\sigma_s$ using maximum a posteriori estimation. We effectively minimize the negative logarithm of the posterior $$\begin{split} J(\boldsymbol{w},\sigma_u,\sigma_s)= & \frac{1}{2}\|\boldsymbol{w}\|_2^2+\frac{1}{2\sigma_u^2}\|\mathbf{1}-\boldsymbol{\tilde{F}}_{\boldsymbol{w}}\|_2^2+N_u\log\sigma_u\\ & +\frac{1}{2\sigma_s^2}\|\mathbf{y}_s-\boldsymbol{\tilde{f}}_{\boldsymbol{w}}\|_2^2+N_s\log\sigma_s.\label{eq:energy} \end{split}$$ Observe that the DNF formula in  is differentiable. With any (piecewise) differentiable choice of $\tilde{f}_{\boldsymbol{w}}$, we can minimize  using (sub-) gradient descent. The gradient of  with respect to the classifier parameter $\boldsymbol{w}$ is $$\nabla_{\boldsymbol{w}}J=\boldsymbol{w}^{\top}-\frac{1}{\sigma_u^2}\left(\mathbf{1}-\boldsymbol{\tilde{F}}_{\boldsymbol{w}}\right)^{\top}\nabla_{\boldsymbol{w}}\boldsymbol{\tilde{F}}_{\boldsymbol{w}}-\frac{1}{\sigma_s^2}\left(\mathbf{y}_s-\boldsymbol{\tilde{f}}_{\boldsymbol{w}}\right)^{\top}\nabla_{\boldsymbol{w}}\boldsymbol{\tilde{f}}_{\boldsymbol{w}},$$ Since we choose $\tilde{f}$ to be a logistic sigmoid function with a linear discriminant , the $j$-th ($j=1,\ldots,N_s$) row of $\nabla_{\boldsymbol{w}}\boldsymbol{\tilde{f}}_{\boldsymbol{w}}$ is $$\nabla_{\boldsymbol{w}}\tilde{f}_j=\tilde{f}_j(1-\tilde{f}_j)\cdot\mathbf{x}_j^{\top}.\label{eq:dlogsig}$$ where $\mathbf{x}_j$ is the $j$-th element in $\mathbf{X}_s$. Define $g_j=\prod_{k=0}^{j-1}\tilde{f}_{\rho^k(i)}\cdot\prod_{k=j}^{L-1}(1-\tilde{f}_{\rho^k(i)})$, $j=0,\ldots,L$, we write  as $\tilde{F}^L_i=1-\prod_{j=0}^L(1-g_j)$ as the $i$-th ($i=1,\ldots,N_u$) element of $\boldsymbol{\tilde{F}}_{\boldsymbol{w}}$. Then the $i$-th row of $\nabla_{\boldsymbol{w}}\boldsymbol{\tilde{F}}_{\boldsymbol{w}}$ is $$\nabla_{\boldsymbol{w}}\tilde{F}^L_i=\sum_{j=0}^L\left(g_j\prod_{\substack{k=0\\k\neq j}}^L\left(1-g_k\right)\right)\left(\sum_{k=0}^{j-1}\frac{\nabla_{\boldsymbol{w}}\tilde{f}_{\rho^k(i)}}{\tilde{f}_{\rho^k(i)}}-\sum_{k=j}^{L-1}\frac{\nabla_{\boldsymbol{w}}\tilde{f}_{\rho^k(i)}}{1-\tilde{f}_{\rho^k(i)}}\right),\label{eq:ddnf}$$ where $\nabla_{\boldsymbol{w}}\tilde{f}_{\rho^k(i)}$ can be computed using . We also alternately estimate $\sigma_u$ and $\sigma_s$ along with $\boldsymbol{w}$. Setting $\nabla_{\sigma_u}J=0$ and $\nabla_{\sigma_s}J=0$, we update $\sigma_u$ and $\sigma_s$ using the closed-form solutions $$\begin{aligned} \sigma_u= & \frac{\|\mathbf{1}-\boldsymbol{\tilde{F}}_{\boldsymbol{w}}\|_2}{\sqrt{N_u}}\\ \sigma_s= & \frac{\|\mathbf{y}_s-\boldsymbol{\tilde{f}}_{\boldsymbol{w}}\|_2}{\sqrt{N_s}}.\end{aligned}$$ At testing time, we apply the learned $\tilde{f}$ to testing samples to predict their merging likelihood. Eventually, we compute the node potentials with  and apply the greedy inference algorithm to acquire the final node label assignment (Section \[sec:hmt\]). Results {#sec:res} ======= We validate the proposed algorithm for 2D and 3D segmentation of neurons in three EM image data sets. For each data set, we apply SSHMT to the same segmentation tasks using different amounts of randomly selected subsets of ground truth data as the supervised sets. Data sets --------- ### Mouse neuropil data set [@deerinck2010enhancing] consists of $70$ 2D SBFSEM images of size $700\times700\times700$ at $10\times10\times50$ nm/pixel resolution. A random selection of $14$ images are considered as the whole supervised set, and the rest $56$ images are used for testing. We test our algorithm using $14$ ($100\%$), $7$ ($50\%$), $3$ ($21.42\%$), $2$ ($14.29\%$), $1$ ($7.143\%$), and half ($3.571\%$) ground truth image(s) as the supervised data. We use all the $70$ images as the unsupervised data for training. We target at 2D segmentation for this data set. ### Mouse cortex data set [@isbichallenge2013] is the original training set for the ISBI SNEMI3D Challenge [@isbichallenge2013]. It is a $1024\times1024\times100$ SSSEM image stack at $6\times6\times30$ nm/pixel resolution. We use the first $1024\times1024\times50$ substack as the supervised set and the second $1024\times1024\times50$ substack for testing. There are $327$ ground truth neuron segments that are larger than $1000$ pixels in the supervised substack, which we consider as all the available supervised data. We test the performance of our algorithm by using $327$ ($100\%$), $163$ ($49.85\%$), $81$ ($24.77\%$), $40$ ($12.23\%$), $20$ ($6.116\%$), $10$ ($3.058\%$), and $5$ ($1.529\%$) true segments. Both the supervised and the testing substack are used for the unsupervised term. Due to the unavailability of the ground truth data, we did not experiment with the original testing image stack from the challenge. We target at 3D segmentation for this data set. ### *Drosophila melanogaster* larval neuropil data set [@knott2008serial] is a $500\times500\times500$ FIBSEM image volume at $10\times10\times10$ nm/pixel resolution. We divide the whole volume evenly into eight $250\times250\times250$ subvolumes and do eight-fold cross validation using one subvolume each time as the supervised set and the whole volume as the testing data. Each subvolume has from $204$ to $260$ ground truth neuron segments that are larger than $100$ pixels. Following the setting in the mouse cortex data set experiment, we use subsets of $100\%$, $50\%$, $25\%$, $12.5\%$, $6.25\%$, and $3.125\%$ of all true neuron segments from the respective supervised subvolume in each fold of the cross validation as the supervised data to generate boundary classification labels. We use the entire volume to generate unsupervised samples. We target at 3D segmentation for this data set. Experiments ----------- We use fully trained Cascaded Hierarchical Models [@seyedhosseini2013image] to generate membrane detection confidence maps and keep them fixed for the HMT and SSHMT experiments on each data set, respectively. To generate initial superpixels, we use the watershed algorithm [@beucher1992morphological] over the membrane confidence maps. For the boundary classification, we use features including shape information (region size, perimeter, bounding box, boundary length, etc.) and image intensity statistics (mean, standard deviation, minimum, maximum, etc.) of region interior and boundary pixels from both the original EM images and membrane detection confidence maps. We use the adapted Rand error metric [@arganda2015crowdsourcing] to generate boundary classification labels using whole ground truth images (Section \[sec:hmt\]) for the 2D mouse neuropil data set. For the 3D mouse cortex and *Drosophila melanogaster* larval neuropil data sets, we determine the labels using individual ground truth segments instead. We use this setting in order to match the actual process of analyzing EM images by neuroscientists. Details about label generation using individual ground truth segments are provided in Appendix \[app\]. We can see in  and  that computing $\tilde{F}^L_i$ and its gradient involves multiplications of $L$ floating point numbers, which can cause underflow problems for leaf-to-root clique paths in a merge tree of even moderate height. To avoid this problem, we exploit the local property of the merge consistency constraint and compute $\tilde{F}^L_i$ for every path subpart of small length $L$. In this paper, we use $L=3$ for all experiments. For inference, we initialize $\boldsymbol{w}$ by running gradient descent on  with only the supervised term and the regularizer before adding the unsupervised term for the whole optimization. We update $\sigma_u$ and $\sigma_s$ in between every $100$ gradient descent steps on $\boldsymbol{w}$. We compare SSHMT with the fully supervised HMT [@liu2014modular] as the baseline method. To make the comparison fair, we use the same logistic sigmoid function as the boundary classifier for both HMT and SSHMT. The fully supervised training uses the same Bayesian framework only without the unsupervised term in  and alternately estimates $\sigma_s$ to balance the regularization term and the supervised term. All the hyperparameters are kept identical for HMT and SSHMT and fixed for all experiments. We use the adapted Rand error [@arganda2015crowdsourcing] following the public EM image segmentation challenges [@arganda2015crowdsourcing; @isbichallenge2013]. Due to the randomness in the selection of supervised data, we repeat each experiment $50$ times, except in the cases that there are fewer possible combinations. We report the mean and standard deviation of errors for each set of repeats on the three data sets in Table \[tab:res\]. For the 2D mouse neuropil data set, we also threshold the membrane detection confidence maps at the optimal level, and the adapted Rand error is $0.2023$. Since the membrane detection confidence maps are generated in 2D, we do not measure the thresholding errors of the other 3D data sets. In addition, we report the results from using the globally optimal tree inference [@liu2016image] in the supplementary materials for comparison. \ \ Examples of 2D segmentation testing results from the mouse neuropil data set using fully supervised HMT and SSHMT with $1$ ($7.143\%$) ground truth image as supervised data are shown in Fig. \[fig:viz:ncmir\]. Examples of 3D individual neuron segmentation testing results from the *Drosophila melanogaster* larval neuropil data set using fully supervised HMT and SSHMT with $12$ ($6.25\%$) true neuron segments as supervised data are shown in Fig. \[fig:viz:fibsem\]. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-209_gray.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-209_1-sssegi.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-209_1-ssseg.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-209_truth.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-212_gray.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-212_1-sssegi.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-212_1-ssseg.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-212_truth.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-234_gray.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-234_1-sssegi.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-234_1-ssseg.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-234_truth.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-241_gray.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-241_1-sssegi.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-241_1-ssseg.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-241_truth.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-259_gray.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-259_1-sssegi.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-259_1-ssseg.png "fig:"){width="23.00000%"} ![Examples of the 2D segmentation testing results for the mouse neuropil data set, including (a) original EM images, (b) HMT and (c) SSHMT results using $1$ ground truth image as supervised data, and (d) the corresponding ground truth images. Different colors indicate different individual segments.[]{data-label="fig:viz:ncmir"}](fig_ncmir-259_truth.png "fig:"){width="23.00000%"} \(a) Original \(b) HMT \(c) SSHMT \(d) Ground truth ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_sssegi_1d.png "fig:"){width="27.00000%"} ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_ssseg_1d.png "fig:"){width="27.00000%"} ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_truth_1d.png "fig:"){width="27.00000%"} ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_sssegi_2d.png "fig:"){width="27.00000%"} ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_ssseg_2d.png "fig:"){width="27.00000%"} ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_truth_2d.png "fig:"){width="27.00000%"} ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_sssegi_4d.png "fig:"){width="27.00000%"} ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_ssseg_4d.png "fig:"){width="27.00000%"} ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_truth_4d.png "fig:"){width="27.00000%"} ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_sssegi_5d.png "fig:"){width="27.00000%"} ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_ssseg_5d.png "fig:"){width="27.00000%"} ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_truth_5d.png "fig:"){width="27.00000%"} ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_sssegi_14d.png "fig:"){width="27.00000%"} ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_ssseg_14d.png "fig:"){width="27.00000%"} ![Examples of individual neurons from the 3D segmentation testing results for the *Drosophila melanogaster* larval neuropil data set, including (a) HMT and (b) SSHMT results using $12$ ($6.25\%$) 3D ground truth segments as supervised data, and (c) the corresponding ground truth segments. Different colors indicate different individual segments. The 3D visualizations are generated using Fiji [@schindelin2012fiji].[]{data-label="fig:viz:fibsem"}](fig_fibsem_truth_14d.png "fig:"){width="27.00000%"} \(a) HMT \(b) SSHMT \(c) Ground truth ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- From Table \[tab:res\], we can see that with abundant supervised data, the performance of SSHMT is similar to HMT in terms of segmentation accuracy, and both of them significantly improve from optimally thresholding (Table \[tab:sub:ncmir\]). When the amount of supervised data becomes smaller, SSHMT significantly outperforms the fully supervised method with the accuracy close to the HMT results using the full supervised sets. Moreover, the introduction of the unsupervised term stabilizes the learning of the classification function and results in much more consistent segmentation performance, even when only very limited ($3\%$ to $7\%$) label data are available. Increases in errors and large variations are observed in the SSHMT results when the supervised data become too scarce. This is because the few supervised samples are incapable of providing sufficient guidance to balance the unsupervised term, and the boundary classifiers are biased to give trivial predictions. Fig. \[fig:viz:ncmir\] shows that SSHMT is capable of fixing both over- and under-segmentation errors that occur in the HMT results. Fig. \[fig:viz:fibsem\] also shows that SSHMT can fix over-segmentation errors and generate highly accurate neuron segmentations. Note that in our experiments, we always randomly select the supervised data subsets. For realistic uses, we expect supervised samples of better representativeness to be provided with expertise and the performance of SSHMT to be further improved. We also conducted an experiment with the mouse neuropil data set in which we use only $1$ ground truth image to train the membrane detector, HMT, and SSHMT to test a fully semi-supervised EM segmentation pipeline. We repeat $14$ times for every ground truth image in the supervised set. The optimal thresholding gives adapted Rand error $0.3603\pm 0.06827$. The error of the HMT results is $0.2904\pm 0.09303$, and the error of the SSHMT results is $0.2373\pm 0.06827$. Despite the increase of error, which is mainly due to the fully supervised nature of the membrane detection algorithm, SSHMT again improves the region accuracy from optimal thresholding and has a clear advantage over HMT. We have open-sourced our code at <https://github.com/tingliu/glia>. It takes approximately 80 seconds for our SSHMT implementation to train and test on the whole mouse neuropil data set using $50$ $2.5$ GHz Intel Xeon CPUs and about $150$ MB memory. Conclusion ========== In this paper, we proposed a semi-supervised method that can consistently learn boundary classifiers with very limited amount of supervised data for region-based image segmentation. This dramatically reduces the high demands for ground truth data by fully supervised algorithms. We applied our method to neuron segmentation in EM images from three data sets and demonstrated that by using only a small amount of ground truth data, our method performed close to the state-of-the-art fully supervised method with full labeled data sets. In our future work, we will explore the integration of the proposed constraint based unsupervised loss in structural learning settings to further exploit the structured information for learning the boundary classification function. Also, we may replace the current logistic sigmoid function with more complex classifiers and combine our method with active learning frameworks to improve segmentation accuracy. ### Acknowledgment This work was supported by NSF IIS-1149299 and NIH 1R01NS075314-01. We thank the National Center for Microscopy and Imaging Research at the University of California, San Diego, for providing the mouse neuropil data set. We also thank Mehdi Sajjadi at the University of Utah for the constructive discussions. Appendix: Generating Boundary Classification Labels Using Individual Ground Truth Segments {#app} ========================================================================================== Assume we only have individual annotated image segments instead of entire image volumes as ground truth. Given a merge tree, we generate the best-effort ground truth classification labels for a subset of cliques as follows: 1. For every region represented by a tree node, compute the Jaccard indices of this region against all the annotated ground truth segments. Use the highest Jaccard index of each node as its eligible score. 2. Mark every node in the tree as “eligible” if its eligible score is above certain threshold ($0.75$ in practice) or “ineligible” otherwise. 3. Iteratively select a currently “eligible” node with the highest eligible score; mark it and its ancestors and descendants as “ineligible”, until every node is “ineligible”. This procedure generates a set of selected nodes. 4. For every selected node, label the cliques at itself and its descendants as $y=1$ (“merge”) and the cliques at its ancestors as $y=0$ (“split”). Eventually, the clique samples that receive merge/split labels are considered as the supervised data. Under the same experiment setting as in Section 4.2, we report the results from using \[9\] in Table S1, which considers the merge tree structure as a constrained conditional model (CCM) and computes globally optimal solutions based on supervised learning for inference. Comparing Table S1 with Table 1 in Section 4.2, we can see that even though the supervised CCM improves from HMT, our SSHMT still consistently outperforms it with a clear margin. Also, the globally optimal inference algorithm in CCM can be used in combination with the proposed semi-supervised learning framework conveniently. We experienced a data loss of the mouse cortex dataset due to power outage, so we did not experiment on this dataset, but we expect similar results.

--- abstract: 'We study almost periodic orbits of quantum systems and prove that for periodic time-dependent Hamiltonians an orbit is almost periodic if, and only if, it is precompact. In the case of quasiperiodic time-dependence we present an example of a precompact orbit that is not almost periodic. Finally we discuss some simple conditions assuring dynamical stability for nonautonomous quantum system.' address: - 'Departamento de Matemática – UFSCar, São Carlos, SP, 13560-970 Brazil' - 'Departamento de Matemática – UFSCar, São Carlos, SP, 13560-970 Brazil\' author: - 'César R. de Oliveira' - 'Mariza S. Simsen' title: 'Almost Periodic Orbits and Stability for Quantum Time-Dependent Hamiltonians' --- [^1] [^2] [Keywords]{}: almost periodicity; quantum stability; time-dependent systems; precompact orbits. Introduction {#IntroductionSection} ============ The time evolution of a quantum mechanical system with time-dependent Hamiltonians $H(t)$ is determined by the Schrödinger equation $$i\frac{d\psi(t)}{dt}=H(t)\psi(t),$$ where $H(t)$ is a family of self-adjoint operators in the Hilbert space $\mathcal{H}$ and $\psi(t)\in\mathcal{H}$ for all $t\in{\ensuremath{{\mathrm{I\!R}}}}$. The initial value problem $\psi(0)=\psi$ has a unique solution $$\psi(t)\doteq U(t,0)\psi,$$ under suitable conditions on $H(t)$ (see [@RS; @K; @K1; @I]) and the propagators, or time evolution operators $U(t,s)$, form a strongly continuous family of unitary operators acting on $\mathcal{H}$, such that $$U(t,r)U(r,s)=U(t,s),\qquad \forall r,s,t,\in{\ensuremath{{\mathrm{I\!R}}}}$$ $$U(t,t)= {{\mathrm{I_d}}},\qquad \forall t.$$ ${{\mathrm{I_d}}}$ denotes the identity operator. If the Hamiltonian is time-periodic with period $T$, then $U(t+T,r+T)=U(t,r)$ and the Floquet operator at $s$ is defined by $U_F(s)\doteq U(s+T,s)$; $U_F(0)$ is simply called Floquet operator and denoted by $U_F$, and $U_F(s)$ is unitarily equivalent to $U_F(r)$, $\forall r,s$. Let $$\mathcal{O}(\psi)\doteq\{ U(t,0)\psi:t\in{\ensuremath{{\mathrm{I\!R}}}}\}$$ be the orbit of a vector $\psi\in\mathcal{H}$. If $H(t)=H$ is independent of $t$ the time evolution operators are $U(t,s)=e^{-iH(t-s)}$. In this case, it is a well-known fact that if $\psi$ is in the point subspace of $H$ then the quantum time evolution of the state $\psi$, $\psi(t)$, is almost periodic, since it can be expanded in terms of the eigenfunctions $\varphi_n$ of $H$, with eigenvalues $E_n$, $$\psi(t)=\sum_nc_ne^{-iE_nt}\varphi_n.$$ Reciprocally, if $\psi(t)$ is almost periodic then using the results in [@Kat] (Chapter VI) it holds true that $\mathcal{O}(\psi)$ is precompact and then $\psi$ is in the point subspace of $H$ (see Theorem \[teo.31\] ahead). In this work, we prove that this fact remains true in the periodic case, that is, $\psi$ is in the point subspace of $U_F$ if, and only if, $\psi(t)$ is almost periodic (see Theorem \[teo.31a\]). In the studies of time-dependent systems it is common to consider the quasienergy operator, i.e., a self-adjoint operator formally given by $$K=-i\frac{d}{dt}+H(t)$$ acting in some enlarged Hilbert space. The quasienergy operator $K$ was previously defined for periodic Hamiltonians [@Y; @H] and then generalized for general time dependence in [@H1]. In the periodic case it was proved that $$e^{-iKT}\simeq {{\mathrm{I_d}}}\otimes U_F,$$ where $\simeq$ means unitary equivalence. A natural framework for considering general time-dependent perturbations, which includes both periodic and the random potentials as special cases, is to write $H(t)$ in the form $$H(t)=H(g_t(\theta))=H_0+V(g_t(\theta)),$$ where $g_t:\Omega\rightarrow\Omega$ is an invertible flow on a compact manifold $\Omega$ with a probability ergodic measure $\mu$ and $H_0$ is the Hamiltonian of the isolated system (see [@JL; @BJL]). Again, under suitable conditions on $V$ there exists a unitary time evolution operator $U_{\theta}(t,s)$ and the generalized quasienergy operator is defined [@JL] on $L^2(\Omega,\mathcal{H},d\mu)$ by $$(e^{-i\tilde{K}t}f)_{\theta}=\mathcal{F}_{-t}U_{\theta}(t,0)f_{\theta}= U_{\theta}(0,-t)\mathcal{F}_{-t}f_{\theta},$$ where $\mathcal{F}_{-t}f_{\theta}=f_{g_{-t}(\theta)};$ we refer to this construction as [*Jauslin-Lebowitz formulation.*]{} The operator $\tilde{K}$ acts as $$(\tilde{K}f)_{\theta}=i\frac{d}{dt}f_{g_{-t}(\theta)}\Big|_{t=0}+H_{\theta}f_{\theta}.$$ In the case of a periodic potential one has $\Omega=S^1\equiv[0,2\pi)$, $g_t(\theta)=\theta+\omega t$ and $d\mu=\frac{d\theta}{2\pi}$. For quasiperiodic potentials with two incommensurate frequencies $\omega_1/\omega_2\notin{\ensuremath{{\mathrm{Q\hspace{-2.1mm}\rule{0.3mm}{2.6mm}\;}}}}$ the manifold $\Omega$ is $S^1\times S^1$, $g_t(\theta_1,\theta_2)=(\theta_1+\omega_1t,\theta_2+\omega_2t)$ and $d\mu=\frac{d\theta_1}{2\pi}\frac{d\theta_2}{2\pi}$. We denote the two periods by $T_j=\frac{2\pi}{\omega_j}$. In this case the generalized Floquet operator acting on $\mathcal{K}_1\doteq L^2(S^1,\mathcal{H},\frac{d\theta_1}{2\pi})$ is defined by $$\label{FloquetGenEq} U_{\mathrm{F}}=\mathcal{T}_{-T_2}u_1,$$ where $u_1(\theta_1)=U_{(\theta_1,0)}(T_2,0)$ ($\doteq$ monodromy operator) and $(\mathcal{T}_{-T_2}\phi)(\theta_1)=\phi(\theta_1-\omega_1T_2)$. Let $A:{{\mathrm{dom}~}}A\subset\mathcal{H}\rightarrow\mathcal{H}$ be an unbounded positive self-adjoint operator with discrete spectrum which we call a [*probe operator*]{}. Assuming that if $\psi\in{{\mathrm{dom}~}}A$, then $U(t,0)\psi\in{{\mathrm{dom}~}}A$ for all $t\geq0$, a very interesting question is about the behavior of the expectation value of $A$, that is, $$E_{\psi}^{A}(t)\equiv\langle U(t,0)\psi,AU(t,0)\psi\rangle.$$ We say the system is $A$-dynamically stable if $E_{\psi}^{A}(t)$ is a bounded function of time, and $A$-dynamically unstable otherwise. A particular case is when the Hamiltonian has the form $H(t)=H_0+V(t)$ and $A=H_0$. In this work we discuss some simple conditions assuring dynamical stability, mainly when either the Floquet or quasienergy operator has purely point spectrum; recall that in the periodic case it is known that continuous spectrum of the Floquet operator implies dynamical instability (see Section \[PreliminarSection\]). Usually it is not a simple task to get results on dynamical (in)stability in the original Hilbert space $H$ through properties of $K$ or $\tilde{K}$ acting in the corresponding enlarged space. We present some theoretical results about this point in Section \[BoundedSection\]. An important result in the periodic case was proved in [@DSSV], i.e., that the applicability of the KAM method for the quasienergy operator $K$, which is a technique to find out a unitary operator $U$ such that $UKU^{-1}=D,$ where $D$ is pure point, gives a uniform bound at the expectation value of the energy for a class of time-periodic Hamiltonians of the form $H(t)=H_0+V(t)$ considered in [@DLSV]. The study of precompacity (and related properties) of orbits of a time-dependent quantum system and their connection with spectral type and stability was carried out, e.g., in [@EV; @dOT; @dO; @BJLPN; @JL; @BJL]. In this work we prove that in the periodic case (including the autonomous case) the orbit $\mathcal{O}(\psi)$ is precompact if, and only if, $\psi(t)$ is an almost periodic function. Moreover, already in the quasiperiodic case we present an example with precompact orbits which are not almost periodic. This paper is organized as follows. In Section \[PreliminarSection\] we recall some subspaces of $\mathcal{H}$ that were studied in the literature and the results that connect this subspaces with dynamical (in)stability and spectral properties of the Floquet or quasienergy operators. In Section \[PeneperiodicSection\] we present ours results about almost periodic orbits. In Section \[BoundedSection\] we discuss some simple conditions assuring dynamical stability; we pay special attention to connection between enlarged spaces and the original quantum Hilbert space. A number of known results are recalled in the text in order to make it as readable as possible. Preliminaries {#PreliminarSection} ============= In this section we present a short account of suitable subspaces and relations among them, in order to put our results in context. Consider a time-dependent Hamiltonian $H(t)$ acting in a separable Hilbert space $\mathcal{H}$, which may be nonperiodic, and let $U(t,0)$ the corresponding propagators. Denote by $A:{{\mathrm{dom}~}}A\subset\mathcal{H}\rightarrow\mathcal{H}$ a probe operator, such that ${{\mathrm{dom}~}}A$ is invariant under time evolution $U(t,0)$. Let $F(A>E)$ be the spectral projection onto the closed space spanned by the eigenvectors of $A$ corresponding to the eigenvalues larger than $E\in{\ensuremath{{\mathrm{I\!R}}}}$. The relevant definitions are as follows [@EV; @dOT; @dO; @BJLPN]. \[def.26\] (i) $\mathcal{H}_{pc}\doteq\{\xi\in\mathcal{H}: \mathcal{O}(\xi)\;\textrm{is precompact in}\;\mathcal{H}\}$. (ii) $\mathcal{H}_{\mathrm{f}}\doteq\big\{\xi\in\mathcal{H}: \lim_{\tau\rightarrow\infty}\frac{1}{\tau} \int_0^{\tau}\|CU(t,0)\xi\|dt=0\;\textrm{for any compact}$$ \textrm{operator}\; C\big\}$. (iii) $\mathcal{H}_{\mathrm{be}}\doteq\{0\neq\xi\in\mathcal{H}: \lim_{E\rightarrow\infty}\sup_{t\in{\ensuremath{{\mathrm{I\!R}}}}} \|F(A>E)U(t,0)\frac{\xi}{\|\xi\|}\|=0\}\cup\{0\}$. (iv) $\mathcal{H}_{\mathrm{ue}}\doteq\{0\neq\xi\in\mathcal{H}: \lim_{E\rightarrow\infty}\sup_{t\in{\ensuremath{{\mathrm{I\!R}}}}} \|F(A>E)U(t,0)\frac{\xi}{\|\xi\|}\|=1\}\cup\{0\}$. (v) $\mathcal{S}^{\mathrm{bd}}(A)\doteq\{\xi\in{{\mathrm{dom}~}}A:\text{the function}\;t\mapsto E_{\xi}^A(t)\;\textrm{is bounded}\}$. (vi) $\mathcal{S}^{\mathrm{un}}(A)\doteq\{\xi\in{{\mathrm{dom}~}}A:\text{the function}\;t\mapsto E_{\xi}^A(t)\;\textrm{is unbounded}\}$. Important compact operators are the projections onto finite subspaces of $\mathcal{H}$, so that the elements of $\mathcal{H}_{\mathrm{f}}$ are interpreted as the vectors that under time evolution leave, on average, any finite-dimensional subspace of $\mathcal{H}$. Some basic properties of the sets that appeared in the above definition are summarized ahead. For proofs we refer the reader to [@dO; @dOT; @EV; @BJLPN]. \[GeneralTheo\] Let $H(t)$ be a time-dependent Hamiltonian and $A$ as above; then: (a) $\mathcal{H}_{\mathrm{f}}$ and $\mathcal{H}_{\mathrm{pc}}$ are closed subspaces of $\mathcal{H}$. (b) $\mathcal{H}_{\mathrm{pc}}\perp\mathcal{H}_{\mathrm{f}}$. (c) $\mathcal{H}_{\mathrm{be}}=\mathcal{H}_{\mathrm{pc}}$ and $\mathcal{H}_{\mathrm{f}}\subset\mathcal{H}_{\mathrm{ue}}$. (d) If $\xi\in{{\mathrm{dom}~}}A$ and $\xi\notin\mathcal{H}_{\mathrm{pc}}$ then $\xi\in\mathcal{S}^{\mathrm{un}}(A)$, that is, $\mathcal{S}^{\mathrm{bd}}(A)\subset\mathcal{H}_{\mathrm{pc}}$. In particular, $({{\mathrm{dom}~}}A\cap\mathcal{H}_{\mathrm{f}})\setminus\{0\} \subset\mathcal{S}^{\mathrm{un}}(A)$. Note that if the Hamiltonian $H(t)$ has the form $H(t)=H_0+V(t)$ with $H_0$ an unbounded, positive, self-adjoint operator with discrete spectrum, then Theorem \[GeneralTheo\](d) holds true for $A=H_0$. Periodic Case {#PeriodicSection} ------------- If $H(t)$ is periodic of period $T$ and $U_F=U(T,0)$ is the corresponding Floquet operator, we denote by $\mathcal{H}_{\mathrm{p}}$ the point spectral subspace and by $\mathcal{H}_{\mathrm{c}}$ the continuous subspace of the Floquet operator $U_{\mathrm{F}}$. Recall the important \[teo.31R\] Let $C:\mathcal{H}\rightarrow\mathcal{H}$ be a compact operator and $\xi\in\mathcal{H}_{\mathrm{c}}$, then $$\lim_{\tau\rightarrow\infty}\frac{1}{\tau}\int_0^{\tau}\|CU(t,0)\xi\|dt=0.$$ A detailed proof of Theorem \[teo.31R\] can be found in [@EV]; this result was firstly proved for the autonomous case (see, e.g., [@AG]). As a consequence of this theorem it follows that if $\xi\in\mathcal{H}_{\mathrm{c}}$ then $\xi\in\mathcal{H}_{\mathrm{f}}$, so by Theorem \[GeneralTheo\](d) it follows that $\langle U(t,0)\xi,AU(t,0)\xi\rangle$ is unbounded. Thus, as it is well known, the presence of continuous spectrum for the Floquet operator is a signature of quantum instability. In principle, one would expect that a Floquet operator with purely point spectrum would imply quantum stability, however there are examples with purely point spectrum and dynamically unstable; see [@dRJLS; @JSS; @dOP] for examples in the autonomous case and [@dOS] for the time-periodic case. Using the above theorem and a series of technical lemmas in [@dOT], one gets \[teo.31\] If the Hamiltonian operator is periodic in time, then (a) $\mathcal{H}_{\mathrm{p}}=\mathcal{H}_{\mathrm{be}}= \mathcal{H}_{\mathrm{pc}}$; (b) $\mathcal{H}_{\mathrm{c}}=\mathcal{H}_{\mathrm{ue}}=\mathcal{H}_{\mathrm{f}}$. We observe that Theorem \[teo.31\] also holds in the autonomous case $H(t)=H$ and with $\mathcal{H}_{\mathrm{p}}$ and $\mathcal{H}_{\mathrm{c}}$ denoting, respectively, the point and continuous subspace of the Hamiltonian $H$. According to the above-quoted results, for periodic systems we have $$\label{eq.22} \mathcal{H}=\mathcal{H}_{\mathrm{pc}}\oplus\mathcal{H}_{\mathrm{f}}.$$ In [@dO] was presented an example for which relation (\[eq.22\]) does not hold for nonperiodic time dependence. It was defined the “unusual" subspace $\mathcal{H}_{\mathrm{a}}$ by the relation $$\mathcal{H}=\mathcal{H}_{\mathrm{pc}}\oplus\mathcal{H}_{\mathrm{f}} \oplus\mathcal{H}_{\mathrm{a}},$$ and constructed a nonperiodic Hamiltonian such that $\mathcal{H}=\mathcal{H}_{\mathrm{a}}$. The example is given by the Floquet operator generated by the kicked Hamiltonian $$H(t)=p^2+x\sum_{n=1}^{\infty}\epsilon_n\delta(t-n),\qquad x\in[0,2\pi),$$ acting on $\mathcal{H}=L^2(\mathbb{T})$ and $\epsilon_n\in\{-1,0,1\}$ adequately chosen. This example illustrates some possible unusual properties of nonstationary quantum systems. Quasiperiodic Case {#QuasiperiodicSection} ------------------ In this case we have the generalized Floquet operator $U_{\mathrm{F}}$ as defined in (\[FloquetGenEq\]), acting on the enlarged space $\mathcal{K}_1=L^2(S^1,\mathcal{H},\frac{d\theta_1}{2\pi})$, and the generalized quasienergy operator $\tilde{K}$ acting in $L^2(S^1\times S^1,\mathcal{H},\frac{d\theta_1}{2\pi}\frac{d\theta_2}{2\pi})$. We denote, respectively, by $\mathcal{K}_{1,p}$ and $\mathcal{K}_{1,\mathrm{c}}$ the point and continuous subspace of the generalized Floquet operator $U_{\mathrm{F}}$. For each fixed $t$ let the unitary operator $U(t):\mathcal{K}_1\rightarrow\mathcal{K}_1$ be given by $(U(t)\psi)(\theta_1)=U_{(\theta_1,0)}(t,0)\psi(\theta_1)$, that is, $$U(t)=\int_{S^1}^{\oplus}U_{(\theta_1,0)}(t,0)\frac{d\theta_1}{2\pi},$$ and given $\psi\in\mathcal{K}_1$ let $\tilde{\mathcal{O}}(\psi)=\{U(t)\psi:t\in{\ensuremath{{\mathrm{I\!R}}}}\}$ be the orbit of $\psi$ in the enlarged space $\mathcal{K}_1$. Let $A:{{\mathrm{dom}~}}A\subset\mathcal{K}_1\rightarrow\mathcal{K}_1$ be a probe operator with $U(t){{\mathrm{dom}~}}A\subset{{\mathrm{dom}~}}A$ and $F(A>E)$ as before. The relevant definitions are as follows [@JL; @BJL; @dOT]: \[def.33\] (a) $\mathcal{K}_{1,\mathrm{f}}\doteq\big\{\psi\in \mathcal{K}_1:\lim_{\tau\rightarrow\infty}\frac{1}{\tau}\int_0^{\tau} \|CU(t)\psi\|_{\mathcal{K}_1}dt=0\;\textrm{for any}$ $\textrm{compact operator}\; C\;\textrm{in}\;\mathcal{K}_1\big\}$. (b) $\mathcal{K}_{1,\mathrm{pc}}=\{\psi\in\mathcal{K}_1:\tilde{\mathcal{O}}(\psi)\; \textrm{is precompact in}\;\mathcal{K}_1\}$. (c) $\mathcal{K}_{1,\mathrm{be}}\doteq\{0\neq\psi\in\mathcal{K}_1: \lim_{E\rightarrow\infty}\sup_{t\in{\ensuremath{{\mathrm{I\!R}}}}} \|F(A>E)U(t)\frac{\psi}{\|\psi\|}\|=0\}\cup\{0\}$. (d) $\mathcal{K}_{1,\mathrm{ue}}\doteq\{0\neq\psi\in\mathcal{K}_1: \lim_{E\rightarrow\infty}\sup_{t\in{\ensuremath{{\mathrm{I\!R}}}}} \|F(A>E)U(t)\frac{\psi}{\|\psi\|}\|=1\}\cup\{0\}$. In [@JL] it was proved the analog of the RAGE Theorem for the quasiperiodic case. The proof is an adaptation of the similar statement in the periodic case discussed in [@EV]. As in the periodic case one has: \[teo.35\] If the Hamiltonian operator is quasiperiodic in time, then (a) $\mathcal{K}_{1,p}=\mathcal{K}_{1,\mathrm{pc}}=\mathcal{K}_{1,\mathrm{be}}$; (b) $\mathcal{K}_{1,\mathrm{c}}=\mathcal{K}_{1,\mathrm{ue}}=\mathcal{K}_{1,\mathrm{f}}$. It is worth mentioning that the relation between the energy growth and the characterizations in Definition \[def.33\] is not as direct as in the case of periodic and autonomous potentials. The above theorem holds on the enlarged space $\mathcal{K}_1$ so that a generalized operator with continuous spectrum does not ensure unbounded energy growth in the original Hilbert space $\mathcal{H}$, although it does in $\mathcal{K}_1$. See [@JL; @BJL] for interesting examples on systems with time-quasiperiodic dependence. Almost Periodic Orbits {#PeneperiodicSection} ====================== Let $\mathcal{B}$ be a Banach space. A continuous function $f:{\ensuremath{{\mathrm{I\!R}}}}\rightarrow\mathcal{B}$ is called *almost periodic* if for any number $\epsilon>0$, one can find a number $l(\epsilon)>0$ such that any interval of the real line of length $l(\epsilon)$ contains at least one point $\tau$ with the property that $$\|f(t+\tau)-f(t)\|<\epsilon, \qquad\qquad \forall t\in{\ensuremath{{\mathrm{I\!R}}}}.$$ For properties of almost periodic functions we refer the reader to [@COR; @Kat]. Now we introduce the following subset of $\mathcal{H}$: $$\mathcal{H}_{\mathrm{ap}}\doteq\{\xi\in\mathcal{H}: \textrm{the function}\;{\ensuremath{{\mathrm{I\!R}}}}\ni t\mapsto\xi(t)=U(t,0)\xi\;\textrm{is almost periodic}\}.$$ By abuse of language sometimes we say that the orbit $\mathcal{O}(\xi)$ is almost periodic. For general time dependence one has \[prop.27a\] $\mathcal{H}_{\mathrm{ap}}$ is a closed subspace of $\mathcal{H}$ and $\mathcal{H}_{\mathrm{ap}}\subset\mathcal{H}_{\mathrm{pc}}$. Clearly $0\in\mathcal{H}_{\mathrm{ap}}$. If $\xi,\psi\in\mathcal{H}_{\mathrm{ap}}$ then $\xi(t)=U(t,0)\xi$ and $\psi(t)=U(t,0)\psi$ are almost periodic functions. Since the sum of two almost periodic functions with values in $\mathcal{H}$ is an almost periodic function, it follows that $\xi(t)+\psi(t)=U(t,0)\xi+U(t,0)\psi=U(t,0)(\xi+\psi)=(\xi+\psi)(t)$ is an almost periodic function. So $\xi+\psi\in\mathcal{H}_{\mathrm{ap}}$. Now, let $\xi\in\mathcal{H}_{\mathrm{ap}}$ and $\lambda$ a complex number, then $\xi(t)=U(t,0)\xi$ is an almost periodic function. Since $\lambda\xi(t)=\lambda U(t,0)\xi=U(t,0)(\lambda\xi)$ is an almost periodic function, it follows that $\lambda\xi\in\mathcal{H}_{\mathrm{ap}}$. So $\mathcal{H}_{\mathrm{ap}}$ is a vector subspace of $\mathcal{H}$. Suppose that $\{\xi_j\}\subset\mathcal{H}_{\mathrm{ap}}$ and $\lim_{j\rightarrow\infty}\xi_j=\xi$. Given $\epsilon>0$ there exists $N\in{\ensuremath{{\mathrm{I\!N}}}}$ such that $\|\xi_j-\xi\|<\epsilon$ for all $j\geq N$; thus, there exists $N$ as above such that $j\geq N$ implies that $\forall\;t\in{\ensuremath{{\mathrm{I\!R}}}}$ $$\|\xi(t)-\xi_j(t)\|=\|U(t,0)\xi-U(t,0)\xi_j\|\leq\|\xi-\xi_j\|<\epsilon.$$ So $\xi_j(t)\rightarrow\xi(t)$ uniformly in ${\ensuremath{{\mathrm{I\!R}}}}$ in the sense of convergence in the norm. Since each $\xi_j(t)$ is an almost periodic function, it follows that $\xi(t)$ is an almost periodic function (Theorem 6.4 in [@COR]) and $\xi\in\mathcal{H}_{\mathrm{ap}}$, which shows that $\mathcal{H}_{\mathrm{ap}}$ is a closed vector subspace of $\mathcal{H}$. Since the set of values of an almost periodic function with values in $\mathcal{H}$ is precompact in $\mathcal{H}$ (Theorem 6.5 in [@COR]), it follows that $\mathcal{H}_{\mathrm{ap}}\subset\mathcal{H}_{\mathrm{pc}}$. Periodic Systems ---------------- If the Hamiltonian time dependence is periodic (or autonomous) more can be said. \[EigenAlmostProp\] If the Hamiltonian operator is periodic in time and $\xi\in\mathcal{H}_{\mathrm{p}}$ is an eigenvector of $U_{\mathrm{F}}$, that is, $U_{\mathrm{F}}\xi=e^{-i\alpha}\xi$, $\alpha\in{\ensuremath{{\mathrm{I\!R}}}}$, then $\xi\in\mathcal{H}_{\mathrm{ap}}\subset\mathcal{H}_{\mathrm{pc}}$. Since $U(t,0)$ is strongly continuous the map $t\mapsto\xi(t)$ is continuous. Any $t\in{\ensuremath{{\mathrm{I\!R}}}}$ can be written in the form $t=nT+s$, with $n\in{\ensuremath{{\mathsf{Z\!\!Z}}}}$ and $0\leq s<T$. We have $U_{\mathrm{F}}\xi=e^{-i\alpha}\xi$ and $U_{\mathrm{F}}^{-1}\xi=e^{i\alpha}\xi$. Since for $t\geq0$ ($n\geq0$) $$\begin{aligned} U(t,0)\xi &=& U(s+nT,nT)U(nT,(n-1)T)\ldots U(T,0)\xi\\ &=& U(s,0)\underbrace{U(T,0)\ldots U(T,0)}_{n\;\textrm{factors}}\xi=U(s,0)e^{-in\alpha}\xi,\end{aligned}$$ and for $t<0$ ($n<0$) $$\begin{aligned} U(t,0)\xi &=& U(s+nT,nT)U(nT,(n+1)T)\ldots U(-T,0)\xi\\ &=& U(s,0)\underbrace{U(T,0)^{-1}\ldots U(T,0)^{-1}}_{n\;\textrm{factors}}\xi=U(s,0)e^{-in\alpha}\xi,\end{aligned}$$ it follows that $$U(t,0)\xi=U(s,0)e^{-in\alpha}\xi,$$ for $t=nT+s\in{\ensuremath{{\mathrm{I\!R}}}}$, $n\in{\ensuremath{{\mathsf{Z\!\!Z}}}}$ and $0\leq s<T$. So for each $t=nT+s\in{\ensuremath{{\mathrm{I\!R}}}}$ $$\begin{aligned} \xi(t+T)&=&U(t+T,0)\xi=U(s,0)e^{-i(n+1)\alpha}\xi\\&=& e^{-i\alpha}U(s,0)e^{-in\alpha}\xi=e^{-i\alpha}\xi U(t,0)\xi=e^{-i\alpha}\xi(t),\end{aligned}$$ so $t\rightarrow\xi(t)$ is an almost periodic function and the result is proved. Summing up, we conclude: \[teo.31a\] If the Hamiltonian operator is periodic in time, then (a) $\mathcal{H}_{\mathrm{p}}=\mathcal{H}_{\mathrm{be}}= \mathcal{H}_{\mathrm{pc}}=\mathcal{H}_{\mathrm{ap}}$; (b) $\mathcal{H}_{\mathrm{c}}=\mathcal{H}_{\mathrm{ue}}= \mathcal{H}_{\mathrm{f}}$. It is enough to prove that $\mathcal{H}_{\mathrm{pc}}=\mathcal{H}_{\mathrm{ap}}$. The inclusion $\mathcal{H}_{\mathrm{ap}}\subset\mathcal{H}_{\mathrm{pc}}$ was proved in Proposition \[prop.27a\]. On the other hand, it is a consequence of Propositions  \[prop.27a\] and \[EigenAlmostProp\] that $\mathcal{H}_{\mathrm{p}}\subset\mathcal{H}_{\mathrm{ap}}$. Since $\mathcal{H}_{\mathrm{p}}=\mathcal{H}_{\mathrm{pc}}$, it follows that $\mathcal{H}_{\mathrm{pc}}\subset\mathcal{H}_{\mathrm{ap}}$. Theorem \[teo.31a\] holds also for autonomous Hamiltonians. Quasiperiodic Systems --------------------- In the above theorem we proved that for time-periodic Hamiltonians an orbit $\mathcal{O}(\xi)$ is precompact if, and only if, $t\mapsto\xi(t)$ is almost periodic. Now we construct an example showing that already in the case of time-quasiperiodic Hamiltonians there are precompact orbits that are not almost periodic. [**Example**]{} Given the matrix $$u_1(\theta_1)= \left(\begin{array}{cc} e^{i\theta_1} & 0\\ 0 & e^{-i\theta_1}\end{array}\right),$$ it is known (see Lemma 5.1 in [@BJL]) that there exists a quasiperiodic Hamiltonian $H_{\theta}(t)$, $\theta=(\theta_1,\theta_2)$, acting on $\mathcal{H}={\ensuremath{{\mathrm{C\hspace{-1.7mm}\rule{0.3mm}{2.6mm}\;}}}}^2$, of the form $$\label{eq.25} H_{\theta}(t)=h_0(t){{\mathrm{I_d}}}+\sum_{j=1}^3h_j(t)\sigma_j,$$ where $\sigma_j$ are the Pauli matrices, and $h_j(t)$ are real quasiperiodic functions, i.e., $h_j(t)= \bar{h}_j(\omega_1t+\theta_1,\omega_2t+\theta_2)$, where $\bar{h}_j(\theta_1,\theta_2)$ are continuous and $2\pi$-periodic in the two arguments $\theta_1,\;\theta_2\in S^1,$ and $\omega_1,\;\omega_2$ are positive real numbers so that $u_1(\theta_1)=U_{(\theta_1,0)}(T_2,0)$ is the corresponding monodromy operator. Moreover, the corresponding generalized Floquet operator $U_F=\mathcal{T}_{-T_2}u_1$ has absolutely continuous spectrum for any irrational $\alpha\doteq\frac{\omega_1}{\omega_2}$. By the construction in the proof of Lemma 5.1 in [@BJL], it is found that for $k\in{\ensuremath{{\mathsf{Z\!\!Z}}}}$, $k>0$, $$\begin{aligned} U_{(\theta_1,0)}(kT_2,0) &=& u_1(\theta_1+(k-1)2\pi\alpha)\ldots u_1(\theta_1+2\pi\alpha)u_1(\theta_1)\\ &=& \left(\begin{array}{cc} e^{i(\theta_1+(k-1)2\pi\alpha)}& 0 \\ 0 & e^{-i(\theta_1+(k-1)2\pi\alpha)}\\ \end{array}\right)\ldots\left(\begin{array}{cc} e^{i\theta_1}& 0 \\ 0 & e^{-i\theta_1}\\ \end{array}\right)\\ &=& \left(\begin{array}{cc} e^{i(\theta_1+(k-1)2\pi\alpha)}\ldots e^{i\theta_1} & 0 \\ 0 & e^{-i(\theta_1+(k-1)2\pi\alpha)}\ldots e^{-i\theta_1}\\ \end{array}\right)\\ &=& \left(\begin{array}{cc} e^{i(k\theta_1+(1+2+\ldots(k-1))2\pi\alpha)} & 0 \\ 0 & e^{-i(k\theta_1+(1+2+\ldots(k-1))2\pi\alpha)} \\ \end{array}\right)\\ &=& \left(\begin{array}{cc} e^{ik(\theta_1+(k-1)\pi\alpha)} & 0 \\ 0 & e^{-ik(\theta_1+(k-1)\pi\alpha)} \\ \end{array}\right);\end{aligned}$$ for $k<0$ the same expression is found. Therefore, for all $k\in{\ensuremath{{\mathsf{Z\!\!Z}}}}$ $$U_{(\theta_1,0)}(kT_2,0)=\left(\begin{array}{cc} e^{ik(\theta_1+(k-1)\pi\alpha)} & 0 \\ 0 & e^{-ik(\theta_1+(k-1)\pi\alpha)} \\ \end{array}\right).$$ Moreover, for $\theta_1\in S^1$, $0\leq t\leq T_2$, define $$v(t;\theta_1)=\left(\begin{array}{cc} e^{i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)} & 0 \\ 0 & e^{-i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)} \\ \end{array}\right),$$ which is differentiable with respect to $t$ and satisfies $$v(0;\theta_1)=\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array}\right)={{\mathrm{I_d}}},\qquad v(T_2;\theta_1)=\left(\begin{array}{cc} e^{i\theta_1} & 0 \\ 0 & e^{-i\theta_1} \\ \end{array}\right)=u_1(\theta_1).$$ So for $t\in{\ensuremath{{\mathrm{I\!R}}}}$, $t=kT_2+\delta_t$, $0\leq\delta_t\leq T_2,$ one has $$U_{(\theta_1,0)}(t,0)=v(\delta_t;\theta_1+k2\pi\alpha)U_{(\theta_1,0)}(kT_2,0).$$ Therefore, for $\xi\in\mathcal{H}={\ensuremath{{\mathrm{C\hspace{-1.7mm}\rule{0.3mm}{2.6mm}\;}}}}^2$, $\xi=\left(\begin{array}{cc} \xi_1 \\ \xi_2 \\ \end{array}\right),$ we have $$U_{(\theta_1,0)}(t,0)\xi=\left(\begin{array}{cc} e^{i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}\xi_1 \\ e^{-i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}\xi_2 \\ \end{array}\right).$$ Since the map, for $0\neq a\in{\ensuremath{{\mathrm{I\!R}}}}$, $t\mapsto\sin at^2$ is not almost periodic, because it is not uniformly continuous, we conclude that the map $t\mapsto e^{iat^2}$ is not almost periodic. Thus, $$t\mapsto g(t)=e^{i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}=e^{i\frac{t}{T_2}\theta_1} e^{it^2\frac{\omega_1\omega_2}{4\pi}}e^{-it\frac{\omega_1}{2}}$$ is not almost periodic, because on the contrary the map $$e^{-i\frac{t}{T_2}\theta_1}g(t)e^{it\frac{\omega_1}{2}}= e^{it^2\frac{\omega_1\omega_2}{4\pi}}$$ would be almost periodic. Therefore, if $\xi\neq0$ then the map $t\mapsto U_{(\theta_1,0)}(t,0)\xi$ is not almost periodic for all $\theta_1\in S^1$. Hence we have got an example of precompact orbit (a closed and bounded set on ${\ensuremath{{\mathrm{C\hspace{-1.7mm}\rule{0.3mm}{2.6mm}\;}}}}^2$ is compact) which is not almost periodic. This finishes the example. The above example can be extend to the infinite dimensional Hilbert space $\mathcal{H}=\bigoplus_{n\in{\ensuremath{{\mathrm{I\!N}}}}}{\ensuremath{{\mathrm{C\hspace{-1.7mm}\rule{0.3mm}{2.6mm}\;}}}}^2$ of the elements $\xi=(\xi_n)_{n\in{\ensuremath{{\mathrm{I\!N}}}}}$ with $\xi_n\in{\ensuremath{{\mathrm{C\hspace{-1.7mm}\rule{0.3mm}{2.6mm}\;}}}}^2$ and $\sum_n|\xi_n|^2<\infty$. Denote $$\tilde{u_1}(\theta_1)=\left(\begin{array}{cc} e^{i\theta_1} & 0\\ 0 & e^{-i\theta_1}\\ \end{array}\right);$$ we know that there exists a quasiperiodic $\tilde{H}_{\theta}(t)$ such that $\tilde{u_1}(\theta_1)$ is the corresponding monodromy operator. Moreover, $\sigma(\tilde{U_{\mathrm{F}}})$ is absolutely continuous for all irrational $\alpha$. Let $$u_1(\theta_1)=\left(\begin{array}{cccc} \left(\begin{array}{cc} e^{i\theta_1} & 0\\ 0 & e^{-i\theta_1}\\ \end{array}\right) & & & \\ & \left(\begin{array}{cc} e^{i\theta_1} & 0\\ 0 & e^{-i\theta_1}\\ \end{array}\right) & & \\ & & \left(\begin{array}{cc} e^{i\theta_1} & 0\\ 0 & e^{-i\theta_1}\\ \end{array}\right) & \\ & & & \ddots \\ \end{array}\right)$$ or, writing in the another way, $u_1(\theta_1)=\bigoplus\tilde{u_1}(\theta_1)$. For $\xi\in\mathcal{H}$ one has $u_1(\theta_1)\xi=\bigoplus\tilde{u_1}(\theta_1)\xi_n$. The Floquet operator corresponding to $u_1(\theta_1)$, $U_{\mathrm{F}}=\mathcal{T}_{-T_2}u_1:L^2(S^1,\mathcal{H}, \frac{d\theta_1}{2\pi})\rightarrow L^2(S^1,\mathcal{H},\frac{d\theta_1}{2\pi})$ has absolutely continuous spectrum for all irrational $\alpha$. If $H_{\theta}(t)=\bigoplus_{n\in{\ensuremath{{\mathrm{I\!N}}}}}\tilde{H}_{\theta}(t)$ then the propagator of $H_{\theta}(t)$ is $U_{\theta}(t,0)=\break\bigoplus\tilde{U}_{\theta}(t,0)$. Thus, $H_{\theta}(t)$ has $u_1(\theta_1)$ as the corresponding monodromy operator, and given $0\neq\xi\in\mathcal{H}$ and $\theta=(\theta_1,0)\in S^1\times S^1$ one has $$\begin{aligned} U_{(\theta_1,0)}(t,0)\xi &=& \bigoplus_n\left(\begin{array}{cc} e^{i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)} & 0\\ 0 & e^{-i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}\\ \end{array}\right) \xi_n \\ &=& \bigoplus_n\left(\begin{array}{c} e^{i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}\xi_n^1\\ e^{-i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}\xi_n^2\\ \end{array}\right).\end{aligned}$$ So $t\mapsto U_{(\theta_1,0)}(t,0)\xi$ is not almost periodic. If $\xi$ satisfies $\xi=\oplus\xi_n$ with $\xi_n\neq0$ if, and only if, $n=l$, then $$U_{(\theta_1,0)}(t,0)\xi=\left(\begin{array}{c} e^{i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}\xi_l^1\\ e^{-i\frac{t}{T_2}(\theta_1+(\frac{t}{T_2}-1)\pi\alpha)}\xi_l^2\\ \end{array}\right),$$ and the orbit is precompact since it lives in a finite dimension subspace. In the same way, if $\xi$ is of the form $\xi=\oplus\xi_n$ with $\xi_n\neq0$ only for finitely many indices $n$, we have an example of a theoretical quantum model with precompact orbits which are not almost periodic. Quasienergy Operator and Almost Periodic Orbits ----------------------------------------------- Let $H(t)$ be a general time-dependent Hamiltonian in a Hilbert space $\mathcal{H}$ such that the propagator $U(t,s)$ is well defined. In this case we have defined the quasienergy operator $K=-i\frac{d}{dt}+H(t)$ acting in the extended Hilbert space $\mathcal{K}=L^2({\ensuremath{{\mathrm{I\!R}}}},\mathcal{H},dt)$. It is known [@H; @H1] that the quasienergy operator and the propagator are connected by the relation $$\label{RelEq} (e^{-iK\sigma}f)(t)=U(t,t-\sigma)f(t-\sigma).$$ Let $\mathcal{K}_{\mathrm{p}}(K)$ and $\mathcal{K}_{\mathrm{c}}(K)$ denote, respectively, the point and continuous subspaces of $K$. We get the following result: \[prop.31\] Let $\xi\in\mathcal{H}$ be such that $1\otimes\xi\in\mathcal{K}_{\mathrm{p}}(K)$. Then: - The map $t\mapsto U(t,0)^{-1}\xi$ is almost periodic. - If the eigenvectors of $K$ have the form $\psi_m=1\otimes\xi_m$, with $\xi_m\in\mathcal{H}$, then $\xi\in\mathcal{H}_{\mathrm{ap}}$. If $1\otimes\xi\in\mathcal{K}_{\mathrm{p}}(K)$ then $1\otimes\xi=\sum_mc_m\psi_m,$ with $K\psi_m=\lambda_m\psi_m$. So $$e^{iK\sigma}(1\otimes\xi)=\sum_mc_me^{i\lambda_m\sigma}\psi_m,$$ therefore by (\[RelEq\]) for each $t\in{\ensuremath{{\mathrm{I\!R}}}},$ $$U(t,t+\sigma)\xi=(e^{iK\sigma}(1\otimes\xi))(t)= \sum_mc_me^{-i\lambda_m\sigma}\psi_m(t)$$ and we conclude that, for each fixed $t$, the map $\sigma\mapsto U(t,t+\sigma)\xi$ is almost periodic. In particular taking $t=0$ we obtain that $\sigma\mapsto U(0,\sigma)\xi$ is almost periodic and i) is proved. Now, if the eigenvectors of $K$ have the form $\psi_m=1\otimes\xi_m$, then $$\begin{aligned} \xi(t)&=&U(t,0)\xi=(e^{-iKt}(1\otimes\xi))(t)\\&=& \sum_mc_me^{-i\lambda_mt}\psi_m(t)\\&=& \sum_mc_me^{-i\lambda_mt}\xi_m.\end{aligned}$$ If the sum is finite the map $t\mapsto\xi(t)$ is almost periodic since it is a trigonometric polynomial. If the sum is infinite then $\sum_{m=1}^kc_me^{-i\lambda_mt}\xi_m\rightarrow\sum_{m=1}^{\infty}c_me^{-i\lambda_mt}\xi_m$ uniformly as $k\rightarrow\infty$ and so the map $t\mapsto\xi(t)$ is almost periodic, that is, $\xi\in\mathcal{H}_{\mathrm{ap}}$, which is ii). Bounded Energy {#BoundedSection} ============== In this section we consider time-dependent Hamiltonians $H(t)=H_0+V(t)$ for which $H_0$ is a probe operator. If $\psi_0\in{{\mathrm{dom}~}}H_0$ and $\psi(t)=U(t,0)\psi_0$ is the solution of the Schrödinger equation, under which conditions $$E_{\psi_0}^0(t)=\langle\psi(t),H_0\psi(t)\rangle$$ is a bounded function on $t$? Also, when $$E_{\psi_0}(t)=\langle\psi(t),H(t)\psi(t)\rangle$$ is a bounded function? Next we present a set of simple general conditions related to the boundedness of such energy functions. General Systems --------------- \[prop.38\] If $V(t)$ is an uniformly bounded family of operators, that is, $\sup_t\|V(t)\|<\infty$, then $E_{\psi_0}^0(t)$ is bounded if, and only if, $E_{\psi_0}(t)$ is bounded. It is sufficient to note that $$E_{\psi_0}(t) = \langle\psi(t),H(t)\psi(t)\rangle = E_{\psi_0}^0(t)+\langle\psi(t),V(t)\psi(t)\rangle$$ and $$\sup_t|\langle\psi(t),V(t)\psi(t)\rangle|\leq\sup_t\|\psi(t)\|^2\|V(t)\|=\sup_t \|\psi_0\|^2\|V(t)\|<\infty.$$ \[prop.39\] If $\psi(t)\in C^1({\ensuremath{{\mathrm{I\!R}}}};\mathcal{H})$ is almost periodic and $\psi'(t)$ is uniformly continuous, then $E_{\psi_0}(t)$ is bounded. For each $n\in{\ensuremath{{\mathrm{I\!N}}}}^*$ define $$f_n(t)=n\left[\psi\left(t+\frac{1}{n}\right)-\psi(t)\right]= n\int_{t}^{t+\frac{1}{n}}\psi'(s)ds.$$ Since $\psi$ is almost periodic it follows that $f_n$ is almost periodic for each $n$. As $\psi'(t)$ is uniformly continuous, for each $\epsilon>0$ there exists $\delta>0$ such that $|s-t|<\delta$ implies $\|\psi'(t)-\psi'(s)\|<\epsilon$. Given $\epsilon>0$ let $N(\epsilon)$ the smallest integer larger or equal to $\frac{1}{\delta}$; then for all $n>N(\epsilon)$ and $t\in{\ensuremath{{\mathrm{I\!R}}}}$ $$\begin{aligned} \left\|f_n(t)-\psi'(t)\right\| &=& \left\|n\int_{t}^{t+\frac{1}{n}}(\psi'(s)-\psi'(t))ds\right\|\\ &\leq& n\int_{t}^{t+\frac{1}{n}}\left\|\psi'(s)-\psi'(t)\right\|ds<\epsilon.\end{aligned}$$ So $f_n\rightarrow\psi'$ uniformly and therefore $\psi'(t)$ is almost periodic. Hence $i\psi'(t)$ and $\psi(t)$ are bounded maps. Since $$E_{\psi_0}(t)=\langle\psi(t),H(t)\psi(t)\rangle= \langle\psi(t),i\frac{d\psi}{dt}(t)\rangle$$ the result follows. Note that the boundedness of energy follows if $t\mapsto\psi(t)$ and $t\mapsto\psi'(t)$ are bounded maps. Though well known, it is worth mentioning Proposition \[prop.40\] in this set of conditions. \[prop.40\] If $t\mapsto V(t)$ is strongly $C^1$ and $\psi'(t)\in{{\mathrm{dom}~}}H(t)$ for all $t$, then: - The map $t\mapsto E_{\psi}(t)$ is differentiable and $$\frac{d}{dt}E_{\psi}(t)=\langle\psi(t),V'(t)\psi(t)\rangle.$$ - $\left|E_{\psi}(t)-E_{\psi}(0)\right|\leq t\times\sup_s\|V'(s)\|$. - If there are $C>0, a>1$ so that $\|V'(t)\|\leq\frac{C}{(1+|t|)^a}$, then $E_{\psi}(t)$ and $E_{\psi}^0(t)$ are bounded functions. \(a) $E_{\psi}(t)=\langle\psi(t),(H_0+V(t))\psi(t)\rangle$ and so $$\begin{aligned} \frac{d}{dt}E_{\psi}(t) &=& \langle\psi'(t),H(t)\psi(t)\rangle + \langle\psi(t),H(t)\psi'(t)\rangle + \langle\psi(t),V'(t)\psi(t)\rangle\\ &=& \langle\psi'(t),i\psi'(t)\rangle + \langle i \psi'(t),\psi'(t)\rangle + \langle\psi(t),V'(t)\psi(t)\rangle \\ &=& \langle\psi(t),V'(t)\psi(t)\rangle.\end{aligned}$$ (b) Since $$E_{\psi}(t)-E_{\psi}(0)=\int_0^t\frac{d}{ds}E_{\psi}(s)ds= \int_0^t\langle\psi(s),V'(s)\psi(s)\rangle ds$$ the result follows. (c) Similar to (b). A possibility for the proposition above is $V(t)=B_1\sin t+\frac{B_2}{(1+|t|)^2}$ with $B_1,\;B_2\in B(\mathcal{H})$ and self-adjoint. From this we see that certainly the choices of $\psi$ depend on $B_1,B_2$, since $B_1\psi$ and $B_2\psi$ must be kept in suitable domains so that $E_{\psi}(t)$ is meaningful. Purely Point Systems -------------------- The next result is restricted to periodic time dependence and Floquet operators with nonempty point spectrum (see [@DSSV]). \[prop.41\] Let $V$ be periodic with period $T$. If the subset $\{\xi_1,\ldots,\xi_n\}$ of eigenvectors of $U_{\mathrm{F}}$ is in ${{\mathrm{dom}~}}H_0$ and $t\mapsto\xi_j(t)$ are $C^1$ maps, then for $\psi=\sum_{j=1}^na_j\xi_j$, where $a_j\in{\ensuremath{{\mathrm{C\hspace{-1.7mm}\rule{0.3mm}{2.6mm}\;}}}},\;j=1,\cdots,n$, the map $E_{\psi}(t)$ is bounded. If, moreover, $V(t)$ are bounded operators and $\sup_t\|V(t)\|<\infty$, then $E_{\psi}^0(t)$ is also bounded. Suppose $U_{\mathrm{F}}\xi_j=e^{i\lambda_j}\xi_j$ with $\lambda_j\in{\ensuremath{{\mathrm{I\!R}}}}$, $1\leq j\leq n$. We have $$E_{\xi_j,\xi_k}(t)\doteq\langle\xi_j(t),H(t)\xi_k(t)\rangle= \langle\xi_j(t),i\frac{d}{dt}\xi_k(t)\rangle$$ and so $t\mapsto E_{\xi_j,\xi_k}(t)$ is continuous. Now $$\begin{aligned} E_{\xi_j,\xi_k}(t+T) &=& \langle U(t+T,0)\xi_j,H(t+T)U(t+T,0)\xi_k\rangle\\ &=& \langle U(t+T,T)U_{\mathrm{F}}\xi_j,H(t)U(t+T,T)U_{\mathrm{F}}\xi_k\rangle\\ &=& e^{-i\lambda_j} e^{i\lambda_k}\langle U(t,0)\xi_j,H(t)U(t,0)\xi_k\rangle\\ &=& e^{i(\lambda_k-\lambda_j)}E_{\xi_j,\xi_k}(t)\end{aligned}$$ and then $t\mapsto E_{\xi_j,\xi_k}(t)$ is an almost periodic function. Since for $\psi=\sum_{j=1}^na_j\xi_j$ we have $E_{\psi}(t)=\sum_{j,k=1}^n\bar{a_j}a_kE_{\xi_j,\xi_k}(t)$ it follows that $E_{\psi}(t)$ is almost periodic and so bounded. The second statement follows by Proposition \[prop.38\]. According to Proposition \[prop.41\], in order to get dynamical stability in the periodic case we need conditions assuring the eigenvectors of $U_{\mathrm{F}}$ are in ${{\mathrm{dom}~}}H_0$ and $t\mapsto\xi_j(t)$ to be $C^1$ functions. We present some sufficient conditions in terms of the quasienergy operator $K$. \[lema.42\] Let $\xi\in\mathcal{H}$ be such that $H(t)U(t,s)\xi$ is well defined. Then the map $t\mapsto H(t)U(t,s)\xi$ is a $C^r$ function if, and only if, $t\mapsto e^{i\lambda(t-s)}U(t,s)\xi$ is a $C^{r+1}$ function for fixed $\lambda,\;s\in{\ensuremath{{\mathrm{I\!R}}}}$. Note that $$\frac{d}{dt}(e^{i\lambda(t-s)}U(t,s)\xi)=i\lambda e^{i\lambda(t-s)}U(t,s)\xi- ie^{i\lambda(t-s)}H(t)U(t,s)\xi.$$ Thus, if $t\mapsto H(t)U(t,s)\xi$ is $C^r$ then $t\mapsto e^{i\lambda(t-s)}U(t,s)\xi$ is $C^{r+1}$ and reciprocally if $t\mapsto e^{i\lambda(t-s)}U(t,s)\xi$ is $C^{r+1}$ then $t\mapsto H(t)U(t,s)\xi$ is $C^r$. \[cor.43\] If $f^{(\lambda)}$ is an eigenvector of $K$, $Kf^{(\lambda)}=\lambda f^{(\lambda)}$, then the map $t\mapsto f^{(\lambda)}(t)$ is $C^r$ if, and only if, there exists $s\in{\ensuremath{{\mathrm{I\!R}}}}$ so that $t\mapsto H(t)U(t,s)f^{(\lambda)}(s)$ is $C^{r-1}$. If $Kf^{(\lambda)}=\lambda f^{(\lambda)}$, then by relation (\[RelEq\]), $$e^{-i\lambda\sigma}f^{(\lambda)}(t)=U(t,t-\sigma)f^{(\lambda)}(t-\sigma);$$ so $f^{(\lambda)}(t)=e^{i\lambda\sigma}U(t,t-\sigma)f^{(\lambda)}(t-\sigma)$ for all $\sigma\in{\ensuremath{{\mathrm{I\!R}}}}$. Denoting $t-\sigma=s$ it follows that $f^{(\lambda)}(t)=e^{i\lambda(t-s)}U(t,s)f^{(\lambda)}(s)$ and the result follows by Lemma \[lema.42\]. By using relation (\[RelEq\]) one can easily show \[Lemma12\] For periodic systems with period $T$, one has: - If $Kf=\lambda f$ then $U_{\mathrm{F}}(s)f(s)=e^{-i\lambda T}f(s)$, $\forall\;s\in{\ensuremath{{\mathrm{I\!R}}}}$. - If $U_{\mathrm{F}}(s)\xi_s=e^{-i\lambda T}\xi_s$, $\xi_s\in\mathcal{H}$, $\forall\;s$, then $$f_{\xi}(t)\doteq \break e^{i\lambda(t-s)}U(t,s)\xi_s\in{{\mathrm{dom}~}}K$$ and $Kf_{\xi}=\lambda f_{\xi}$. \[cor.44\] (a) If $H(t+T)=H(t)$, and $\xi^{(\lambda)}$ is an eigenvector of $U_{\mathrm{F}}(s)$, $U_{\mathrm{F}}(s)\xi^{(\lambda)}=e^{-i\lambda T}\xi^{(\lambda)}$, then $\xi^{(\lambda)}\in{{\mathrm{dom}~}}H(s)$ if, and only if, there exists an eigenvector $f_{\xi^{(\lambda)}}$ of $K$, $Kf_{\xi^{(\lambda)}}=\lambda f_{\xi^{(\lambda)}}$, with $t\mapsto f_{\xi^{(\lambda)}}(t)$ continuous and differentiable. (b) In particular, $U_{\mathrm{F}}(s)$ has a basis of eigenvectors in ${{\mathrm{dom}~}}H(s)$ if, and only if, $K$ has a basis of eigenvectors $\{f_j\}$ such that $t\mapsto f_j(t)$ is continuous and differentiable for each $j$. \(a) Suppose that $\xi^{(\lambda)}\in{{\mathrm{dom}~}}H(s)$. By Lemma \[Lemma12\] $f_{\xi^{\lambda}}(t)=\break e^{i\lambda(t-s)}U(t,s)\xi^{\lambda}\in{{\mathrm{dom}~}}K$ and $Kf_{\xi^{\lambda}}=\lambda f_{\xi^{\lambda}}$. Since $\xi^{(\lambda)}\in{{\mathrm{dom}~}}H(s)$ it follows that $U(t,s)\xi^{\lambda}\in{{\mathrm{dom}~}}H(t)$ and $i\partial_tU(t,s)\xi^{\lambda}=H(t)U(t,s)\xi^{\lambda}$. Thus, $t\mapsto f_{\xi^{(\lambda)}}(t)$ is continuous and differentiable. Reciprocally, it there exists an eigenvector $f_{\xi^{(\lambda)}}$ of $K$ with $t\mapsto f_{\xi^{(\lambda)}}(t)$ continuous and differentiable, then $f_{\xi^{\lambda}}(t)=e^{i\lambda(t-s)}U(t,s)\xi^{\lambda}$ and $Kf_{\xi^{(\lambda)}}=\lambda f_{\xi^{(\lambda)}}$ implies $-i\partial_tf_{\xi^{\lambda}}(t)+H(t)f_{\xi^{\lambda}}(t)=\lambda f_{\xi^{\lambda}}(t)$; therefore, $\xi^{\lambda}\in{{\mathrm{dom}~}}H(s)$. \(b) It is a directly consequence of (a). Jauslin-Lebowitz Formulation ---------------------------- We want to study an analogue of the expectation value of probe operators $A:{{\mathrm{dom}~}}A\subset\mathcal{H}\rightarrow\mathcal{H}$ on the formulation presented by Jauslin and Lebowitz [@JL; @BJL] briefly recalled in the Introduction. If the generalized quasienergy operator $\tilde{K}$ has pure point spectrum, there exists an orthonormal basis $B\doteq\{f_n\}_{n=1}^{\infty}$ of $\tilde{\mathcal{K}}$ with $\tilde{K}f_n=\lambda_nf_n$. By Theorem 4.2 in [@JL], if $f=1\otimes\xi$ is in the point subspace of $\tilde{K}$ the function $t\mapsto U_{\theta}(t,0)\xi$ is almost periodic a.e. $\theta$ with respect to the ergodic measure $\mu$ on the compact manifold $\Omega$ (see Section \[IntroductionSection\]). Denote $$B_{n,m}(A)\doteq\int_{\Omega}\left\langle f_n(\theta),Af_m(\theta)\right\rangle_{\mathcal{H}}d\mu(\theta)=\left\langle f_n,(1\otimes A)f_m\right\rangle_{\tilde{\mathcal{K}}}.$$ If $f\in\tilde{\mathcal{K}}$ then $f=\sum_na_nf_n$, with $\sum_n|a_n|^2=\|f\|^2_{\tilde{\mathcal{K}}}$. For each time $t$, consider the average over $\Omega$ of the expectation value of $A$, that is, $$\begin{aligned} A_f(t) &\doteq& \int_{\Omega}\left\langle U_{\theta}(t,0)f(\theta),AU_{\theta}(t,0)f(\theta)\right\rangle_{\mathcal{H}}d\mu(\theta)\\ &=& \int_{\Omega}\langle (\mathcal{F}_te^{-i\tilde{K}t}f)(\theta),A(\mathcal{F}_te^{-i\tilde{K}t}f)(\theta) \rangle_{\mathcal{H}}d\mu(\theta)\end{aligned}$$ $$\begin{aligned} &=& \left\langle\mathcal{F}_te^{-i\tilde{K}t}f,(1\otimes A)\mathcal{F}_te^{-i\tilde{K}t}f\right\rangle_{\tilde{\mathcal{K}}}\\ &=& \left\langle e^{-i\tilde{K}t}f,(1\otimes A)e^{-i\tilde{K}t}f\right\rangle_{\tilde{\mathcal{K}}}\\ &=& \sum_{n,m}\overline{a_n}a_me^{-it(\lambda_m-\lambda_n)}\left\langle f_n,(1\otimes A)f_m\right\rangle_{\tilde{\mathcal{K}}}\\ &=& \sum_{n,m}\overline{a_n}a_me^{-it(\lambda_m-\lambda_n)}B_{n,m}(A).\end{aligned}$$ Note that if this sum is absolutely convergent then $A_f(t)$ is a bounded and almost periodic function of $t$, and $$t\mapsto\left\langle U_{\theta}(t,0)f(\theta),AU_{\theta}(t,0)f(\theta)\right\rangle_{\mathcal{H}}$$ is bounded a.e. $\theta$. We conclude \[cor.45\] If $f=\sum_{j=1}^{m}a_jf_j$, where $f_j$ are eigenvectors of $\tilde{K}$ and $f_j(\theta)\in{{\mathrm{dom}~}}A$, for all $\theta$, then $t\mapsto A_f(t)$ is a bounded and almost periodic function. Moreover, $$t\mapsto\left\langle U_{\theta}(t,0)f(\theta),AU_{\theta}(t,0)f(\theta)\right\rangle_{\mathcal{H}}$$ is bounded for almost every $\theta$. More generally we obtain the following result: \[teo.47\] Suppose that $\Omega$ is a compact manifold, $g_t:\Omega\rightarrow\Omega$ a $C^1$ flow with $\sup_{t,\theta}\|\partial_tg_t(\theta)\|<\infty$, and $\tilde{K}f^{(\lambda)}=\lambda f^{(\lambda)}$ with $\theta\mapsto f^{(\lambda)}(\theta)$ a $C^1$ map. Then for $\mu$ almost every $\theta$ one has $U_{\theta}(t,0)f^{(\lambda)}(\theta)\in{{\mathrm{dom}~}}H_{\theta}(t)$ and $$\left\langle U_{\theta}(t,0)f^{(\lambda)}(\theta),H_{\theta}(t) U_{\theta}(t,0)f^{(\lambda)}(\theta)\right\rangle$$ is a bounded function of $t$. Moreover, if $H_{\theta}(t)=H_0+V(g_t(\theta))$ with $V(g_t(\theta))$ bounded and $\sup_{t,\theta}\|V(g_t(\theta))\|<\infty$, then the energy expectation $$\left\langle U_{\theta}(t,0)f^{(\lambda)}(\theta),H_0 U_{\theta}(t,0)f^{(\lambda)}(\theta)\right\rangle$$ is also bounded. Since $\tilde{K}f^{(\lambda)}=\lambda f^{(\lambda)}$ then $f^{(\lambda)}(\theta)\in{{\mathrm{dom}~}}H_{\theta}(0)$ a.e. $\theta$ and therefore $U_{\theta}(t,0)f^{(\lambda)}(\theta)\in{{\mathrm{dom}~}}H_{\theta}(t)$ a.e. $\theta$. On the other hand $$U_{\theta}(t,0)f^{(\lambda)}(\theta)= \mathcal{F}_te^{-i\tilde{K}t}f^{(\lambda)}(\theta)= \mathcal{F}_te^{-i\lambda t}f^{(\lambda)}(\theta)=e^{-i\lambda t}f^{(\lambda)}(g_t(\theta))$$ and from the differentiability hypothesis it follows that $$i\frac{\partial}{\partial t}U_{\theta}(t,0)f^{(\lambda)}(\theta)=\lambda e^{-i\lambda t}f^{(\lambda)}(g_t(\theta))+ie^{-i\lambda t}\frac{d}{d\theta}f^{(\lambda)}(g_t(\theta))\frac{d}{dt}g_t(\theta),$$ which implies that $$i\frac{\partial}{\partial t}U_{\theta}(t,0)f^{(\lambda)}(\theta)=H_{\theta}(t) U_{\theta}(t,0)f^{(\lambda)}(\theta)$$ is bounded and the first part of the result is proved. The second one follows as in Proposition \[prop.38\]. \[cor.48\] Suppose the hypotheses of the above theorem hold and that for each eigenvector $f^{(\lambda_n)}\in\tilde{\mathcal{K}}$ the function $\theta\mapsto f^{(\lambda_n)}(\theta)$ is $C^1$. 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Ishii: Linear evolution equations $du/dt+A(t)u=0$: a case where $A(t)$ is strongly uniform-measurable. [*J. Math. Soc. Japan*]{} [**34**]{}, 413 (1982). H. R. Jauslin and Lebowitz: Spectral and stability aspects of quantum chaos. [*Chaos*]{} [**1**]{}, 114 (1991). S. Jitomirskaya, H. Schulz-Baldes and G. Stolz: Delocalization in Random Polymer Models. [*Commun. Math. Phys.*]{} [**233**]{}, 27 (2003). T. Kato: Linear evolution equations of “hyperbolic" type. [*J. Fac. Sci. Univ. Tokyo*]{} Sect. I, A Math. [**17**]{}, 241 (1970). T. Kato: Linear evolution equations of “hyperbolic" type II. [*J. Math. Soc. Japan*]{} [**25**]{}, 648 (1973). Y. Katznelson Y.: [*An Introduction to Harmonic Analysis.*]{} John Wiley, New York 1968. M. Reed and B. Simon: [*Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness.*]{} Academic Press, New York 1975. K. Yajima: Scattering theory for Schrödinger equations with potential periodic in time. [*J. Math. Soc. 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--- abstract: 'Differential resistance measurements are conducted for point contacts (PCs) between tungsten tip approaching along the $c$ axis direction and the $ab$ plane of Sr$_{2}$RuO$_{4}$ single crystal. Three key features are found. Firstly, within 0.2 mV there is a dome like conductance enhancement due to Andreev reflection at the normal-superconducting interface. By pushing the W tip further, the conductance enhancement increases from 3% to more than 20%, much larger than that was previously reported, probably due to the pressure exerted by the tip. Secondly, there are also superconducting like features at bias higher than 0.2 mV which persists up to 6.2 K, resembling the enhanced superconductivity under uniaxial pressure for bulk Sr$_{2}$RuO$_{4}$ crystals but more pronounced here. Third, the logarithmic background can be fitted with the Altshuler-Aronov theory of tunneling into quasi two dimensional electron system, consistent with the highly anisotropic electronic system in Sr$_{2}$RuO$_{4}$.' author: - 'He Wang (王贺)' - 'Weijian Lou (娄伟坚)' - 'Jiawei Luo (骆佳伟)' - 'Jian Wei (危健)' - 'Y. Liu' - 'J.E. Ortmann' - 'Z.Q. Mao' title: 'Enhanced superconductivity at the interface of W/Sr$_{2}$RuO$_{4}$ point contact' --- [UTF8]{}[gbsn]{} The layered perovskite ruthenate Sr$_{2}$RuO$_{4}$ (SRO) has shown evidence for spin-triplet, odd-parity superconductivity (SC) which may be useful for topological quantum computation. [@Maeno1994nature; @Mackenzie2003rmp; @Maeno2012jpsj] The possible chiral orbital order parameter for the two-dimensional SC is $p_{x}\pm ip_{y}$ as suggested by the time-reversal symmetry breaking experiments. [@Luke1998nature; @Xia2006prl] Such chiral order is expected to generate edge currents, but the expected magnetic field due to edge currents has not been directly observed with local field imaging, [@Kirtley2007prb; @Hicks2010prb; @Curran2014prb] though there is indirect evidence of edge currents revealed by in-plane tunneling spectroscopy [@Kashiwaya2011prl; @Kashiwaya2014pe] and point contact spectroscopy (PCS), [@Laube2000prl]both with assumptions to fit the conductance spectra. The surface properties of SRO is very critical for field imaging with scanning quantum interference devices, as well as for the tunneling and point contact spectroscopy. It is known that the SRO surface can undergo reconstruction and the intrinsic SC may not be probed, [@Upward2002prb; @Firmo2013prb] and it may even show ferromagnetism (FM) due to lattice distortion. [@Matzdorf2000science] Very careful *in situ* preparation of devices is required for making good tunnel junctions using microfabrication techniques. [@Kashiwaya2011prl] Recently there is also theory proposal that surface disorder indeed can destroy the spontaneous currents. [@Lederer2014prb] One way to overcome the surface problem is to use a hard tip for the point contact (PC) measurement. If the tip is hard enough, it may pierce through the surface dead layer and probe the SC underneath. [@Gonnelli2002jpcs] In fact, for this reason tungsten tip has been used for PCS of heavy fermion superconductors. [@Gloos1996jltp_scaling] A consequence of using a hard tip is that the tip will exert some pressure on the surface which may affect the SC, [@Daghero2010sst] possibly due to local distortion of lattice. [@Gloos1995pb; @Miyoshi2005prb] It is known that for SRO a very low uniaxial pressure of 0.2 GPa along the $c$ axis can enhance the superconducting transition temperature ($T_c$) of pure SRO from 1.5 K up to 3.2 K, [@Kittaka2010prb; @Kittaka2009jpsj_b] and recently in-plane strain (0.23%) along $\langle 100 \rangle$ direction is also shown to enhance $T_c$ from 1.3 K up to 1.9 K. [@Hicks2014science] The pressure in abovementioned measurements were applied to bulk samples, while for PCS the pressure is exerted locally. In the latter case it may be less affected by the inhomogenity of the applied pressure and the sample is less tend to developing cracks, thus locally higher pressure may be reached though absolute pressure is not known. Here we report greatly enhanced SC observed at the interface of the point contact junction between a tungsten tip approaching along the $c$ axis direction and the $ab$ plane surface of a SRO single crystal. SRO single crystals are grown by floating zone methods and are from two different batches, details of sample preparation can be found in previous reports. [@Mao2000mrb] Sample S1 is from the first batch and is easier to cleave and shows no Ru inclusions. Sample S2 is from the second batch, too hard to cleave, and contains a lot of Ru inclusions (for optical images see Appendix \[appendix\_Ru\_inclusions\]). Only on the cleaved surface of S1 do we observe SC feature. Tungsten wire of 0.25 mm diameter is etched to form the tip, and then fixed pointing to the $ab$ plane of the SRO sample. A Si chip with the sample and thermometer glued on top is mounted on an attoCube nanopositioner stack. Since the tip and sample are both fixed to the copper housing, relative displacement between the tip and sample is suppressed, which ensures a stable contact and reproducible PCS. The housing is suspended with springs at the bottom of a insertable probe for a Leiden dilution fridge. With such customization the sample position is not at the field center of the magnet, and the field value is estimated with the tabled values from the magnet manufacturer. Differential resistance ($dV/dI$) is measured with standard lock-in technique. ![(Color online) Bias dependence of $dV/dI$ (a, c, e) and magnetoresistance (b, d, f) of three different point contact (PC) resistance at the same location between the W tip and SRO single crystal S1 at 0.35 K. The resistance at zero bias and zero field is 9.3, 4.3, 3.2 $\Omega$ respectively. For clarity, in (a) and (c) the $dV/dI$ curves at 625 Oe (Green) are shifted up by 0.2 $\Omega$. Arrows in (b), (d), (f) show the sweeping direction of the magnetic field. The reproducibility of the measurements is demonstrated by the overlapping of $dV/dI$ curves in (a), (c), (e) with bias ramping in both directions. The discontinuity around $\pm$625 Oe is related to the ramping speed of the field, and can be smaller when the field ramping speed is reduced, while the hysteresis is almost the same.[]{data-label="fig_dVdI_pressure"}](Fig1){width="9cm"} At the same location, by pushing the tungsten tip towards the SRO surface (more precisely it is the SRO moving towards the tip), the PC resistance is reduced and the pressure is increased. The zero bias and zero field resistance ($R_0$) is: 9.3, 4.3, 3.2 $\Omega$ respectively (see Appendix \[appendix\_R\_pc\] for a discussion of PC resistance). The bias dependence of $dV/dI$ is shown in Figs. \[fig\_dVdI\_pressure\]a, \[fig\_dVdI\_pressure\]c, and \[fig\_dVdI\_pressure\]e, at nominal temperature 0.35 K. SC is clearly shown by the resistance dip within $\pm$0.2 mV without any applied field. With a 625 Oe magnetic field applied along the $c$ axis (H$_{\perp}$), SC is almost fully suppressed for the 9.3 $\Omega$ PC as shown by the recover of the resistance peak at zero bias. However, for the 4.3 $\Omega$ PC there is still a small dip, suggesting that SC is not fully suppressed, *i.e.*, SC is enhanced with increased pressure. Enhancement of SC is further confirmed by the temperature dependence of $dI/dV$ at zero field as shown in Fig. \[fig\_dVdI\_9ohm\]b and Fig. \[fig\_dVdI\_4ohm\]b, where $T_c$ is increased from the bulk value of 1.5 k to about 2 K and 2.5 K for the 9.3 $\Omega$ and 4.3 $\Omega$ PC respectively. This enhanced T$_{c}$ is consistent with previous susceptibility measurements on bulk SRO sample under uniaxial pressure, where the mechanism of $T_c$ enhancement was ascribed to anisotropic lattice distortion, [@Kittaka2009jpsj_b; @Kittaka2010prb; @Taniguchi2012jpcs] similar to that found in the eutectic 3K phase. [@Ying2009prl; @Ying2013ncomms] In Figs. \[fig\_dVdI\_9ohm\] and  \[fig\_dVdI\_4ohm\], for easy comparison with theoretical description, $dV/dI$ is converted to $dI/dV$. The magnetoresistance (MR) is shown in Figs. \[fig\_dVdI\_pressure\]b, \[fig\_dVdI\_pressure\]d, and \[fig\_dVdI\_pressure\]f for the three PCs. The resistance starts to increase quickly at around 400 Oe, and there is clearly a hysteresis with steps which gets sharper and more pronounced for higher PC pressure. MR hysteresis is usually observed for ferromagnetic samples, and the observation of both SC and MR hysteresis was linked to the coexistence of SC and ferromagnetism (FM) for SC at the oxides interface. [@Dikin2011prl] If indeed a FM-like internal field exists, could it be related to the long sought-after time-reversal symmetry-breaking fields? [@Kirtley2007prb; @Hicks2010prb; @Curran2014prb] ![(Color online) (a) Bias dependence of $dI/dV$ for the 9.3 $\Omega$ point contact at 0.35 K and with increasing $H_{\bot}$ and (b) zero field $dI/dV$ with increasing T. Curves are shifted for clarity except for the zero field 0.35 K curve. (c) Fitting with EEI theory in the 2D limit for curves in (a) with fitting temperature $T_{fit}$=1.0 K, and (d) fitting for curves in (b) with $T=$ 0.35 (1.0), 0.6 (1.1), 0.8 (1.24), 1.5 (1.65), 1.6 (1.85), 1.8 (1.95), and 2.0 (2.2) K from top to bottom ($T_{fit}$ is indicated in the parentheses). After normalized by the EEI fits with corresponding $T_{fit}$, the data curves are shown in (e) for different $H_{\bot}$ and (f) for different T. Curves are shifted for clarity. []{data-label="fig_dVdI_9ohm"}](Fig2){width="9cm"} First the possibility of conventional vortex pinning needs be considered. The field value above which $dV/dI$ starts to increase quickly is around 400 Oe, in the same order of magnitude with the upper critical field $H_{c2}||c$ about 710 Oe for pure SRO crystal, but much larger than the critical field $H_{c1}||c$ about 70 Oe (by specific heat measurements). [@Deguchi2004jpsj] Sharp increase of resistance may indicate that vortices enter the PC interface and SC is suppressed. However, it is not clear whether such strong pinning could be reduced by PC. The average distance between vortices is $\sim\sqrt{\Phi_{0}/H}$, about 0.3 $\mu$m for 400 Oe, so the diameter of the PC should be much larger to include multiple vortices, which is inconsistent with conventional understanding of the PC. Moreover, it is difficult to explain why the step-like features become sharper with higher pressure. Besides the external pinning due to defects, the intrinsic pinning due to chiral domain wall [@Dumont2002prb] seems also unlikely to reach 400 Oe. One variation of vortex pinning is chiral domain wall motion, where with ramping field the DW wall moves and the edge current can affect the transport of the PC, [@Kambara2008prl] which seems reasonable. Surface FM also needs to be considered since among other layered perovskite ruthenates in the series $A_{n+1}Ru_{n}O_{3n+1}$, SrRuO$_{3}$ is a ferromagnetic metal with T$_{c}$=160 K, and Sr$_{3}$Ru$_{2}$O$_{7}$ is at the boarder of FM and shows pressure-induced FM. [@Ikeda2000prb] Thus it is natural to expect that FM could be induced for SRO, or there might be some eutectic phase [@Mao2000mrb] on the surface which leads to FM. Previously, experimental attempts to measure the bulk magnetic susceptibility of SRO with uniaxial pressure were not successful, since above 0.4 Gpa SRO sample tends to crush, [@Ikeda2004jmmm] while no drastic change of the temperature dependence of susceptibility was observed. On the other hand, doping the Sr with Ca does show a ground state of *static* magnetic order due to rotation of $RuO_{6}$ octahedra. [@Carlo2012nmat; @Ortmann2013sr] Thus it is possible that the pressure under the tip may be higher than 0.4 GPa [@Gonnelli2002jpcs] and its influence is comparable with that by doping. However, this is inconsistent with the fact that the hysteresis diminishes together with SC at higher temperatures, which also indicates that the hysteresis is not due to eutectic phase impurities. Both field and temperature dependences of $dI/dV$ resemble those found for in-plane Au/SRO tunneling junctions in Ref. \[\], as shown by detailed field and temperature dependences in Fig. \[fig\_dVdI\_9ohm\] and Fig. \[fig\_dVdI\_4ohm\], for the 9.3 and 4.3 $\Omega$ PC respectively. However, in Ref. \[\] the gap is about 0.7 mV instead of 0.2 mV, and the conductance enhancement of the dome like feature is less than 1% (see see Appendix \[appendix\_reproducibility\] for similar PC spectra with a Au tip). The dome like feature may be fitted considering chiral *p*-wave symmetry [@Kashiwaya2011prl], but here we focus on experimental findings and methodology while leave the fittings in the future. ![(Color online) (a) Bias dependence of $dI/dV$ for the 4.3 $\Omega$ point contact at 0.35 K and with increasing $H_{\bot}$ and (b) zero field $dI/dV$ with increasing T. Curves are shifted for clarity except for the zero field 0.35 K curve. (c) Fitting with EEI theory in the 2D limit for $H_{\bot}$ = 0 (blue) and 625 Oe (yellow) curves in (a) with $T_{fit}$=0.8 K, and (d) fitting for different T curves in (b) with $T$ = 0.35 (0.8), 1.5 (1.5) K ($T_{fit}$ is indicated in the parentheses). After normalized by the EEI fits with corresponding $T_{fit}$, the data curves are shown in (e) for different $H_{\bot}$ and (f) for different T. Curves are shifted for clarity.[]{data-label="fig_dVdI_4ohm"}](Fig3){width="9cm"} The broad background resistance hump as shown by $dV/dI$ at 625 Oe in Fig. \[fig\_dVdI\_difference\](a) (same as in Fig. \[fig\_dVdI\_pressure\]a) is generally called zero-bias anomaly (ZBA), which is frequently observed in tunnel junctions [@Kashiwaya2011prl] as well as PCs. [@Gloos2009ltp; @Gloos2012jpcs] The possible origins for ZBA in PCs include “extrinsic” magnetic impurities, two-level systems, Kondo scattering due to spontaneous electron spin polarization etc, as well as “intrinsic” density of states (DOS) effect, as shown for chromium where DOS is reduced due to the spin density wave gap, [@Meekes1988prb] and more recently for iron pnictides where DOS is enhanced due to strong electron correlations. [@Arham2012prb] Here ZBA apparently coexists with SC in SRO, which is very sensitive to impurities, thus the origin of ZBA is more likely due to some “intrinsic” origin. The background ZBA can be normalized when the bias dependence is replotted using $\ln{(eV/k_{B}T)}$. In Figs. \[fig\_dVdI\_9ohm\](c) and  \[fig\_dVdI\_9ohm\](d), the normalized change of conductance shows a linear dependence for $eV \gg k_{B}T$, similar to what was observed in tunneling measurements for disordered metal films, [@Gershenzon1986jetp] and also for layered cuprates and manganites. [@Abrikosov2000prb; @Mazur2007prb] In the tunneling case, the reduction of DOS is due to electron-electron interaction (EEI). As proposed by Altshuler and Aronov, [@Altshuler1979; @*Altshuler1980; @*Altshuler1985] for low dimensional systems the exchange interaction between electrons can cause quantum corrections to the conductivity as well as DOS, which depends on the dimensionality of the systems. For $eV \gg k_{B}T$, the DOS correction $\sim\ln{(eV/k_{B}T)}$ in 2D (see Appendix \[appendix\_fit\_eei\] for details). When the full formula is used, we get good fits in the full bias range as shown by the dashed lines in Figs. \[fig\_dVdI\_9ohm\](c) and  \[fig\_dVdI\_9ohm\](d) (also in Figs. \[fig\_dVdI\_pressure\]a and b). We note that in order for all normalized $dI/dV$ curves to collapse onto a single curve, enhanced temperature ($T_{fit}$) needs to be assumed for $dI/dV$ measured at lower temperatures. This may indicate there is local heating in the small PC region, possibly due to inadequate filtering of the external microwave noise. [@Liu2014prb] ![(Color online) (a) $dV/dI$ curves for the 9.3 $\Omega$ PC at 0 (blue), 625 Oe (black), and EEI fits (red dashed lines). (b) $dV/dI$ curves for the 4.3 $\Omega$ PC at 0 (blue), 521 (yellow), 625 Oe (black), and the EEI fit. (c) Zoom-in of the zero bias resistance dip regime with curves shifted for clarity except for the 3.2 $\Omega$ curve. The curves are reproducible for both ramping directions of the bias. (d) Zero field conductance enhancement after normalized with the fitted EEI background. All at 0.35 K.[]{data-label="fig_dVdI_difference"}](Fig4){width="9cm"} The fitted ZBA can be considered as the normal state background and divided from the normalized conductance, [@Kashiwaya2011prl] the resulted curves are shown in Figs. \[fig\_dVdI\_9ohm\](e) and  \[fig\_dVdI\_9ohm\](f), with the dome like conductance enhancement well demonstrated. Another feature of the PC spectra is a small periodic “wiggling” outside $\pm$0.2 mV, which also diminishes with increasing field and temperature, suggesting that it is probably due to interference of quasiparticles at the NS interface. Similar feature was also observed for multiple band superconductor MgB$_2$ [@Gonnelli2002jpcs; @Daghero2010sst] but detailed analyses are lacking. The ZBA background becomes less pronounced when the PC resistance is reduced from 9.3 $\Omega$ to 4.3 $\Omega$, as shown in Figs. \[fig\_dVdI\_4ohm\]c and d, while the conductance enhancement gets larger. This is better illustrated by the normalized enhancement (Fig. \[fig\_dVdI\_difference\]d), and by direct comparison of the zero field $dV/dI$ (Fig. \[fig\_dVdI\_difference\]c). What parameters may change when the PC resistance is reduced from 9.3 $\Omega$ to 4.3 $\Omega$? In the standard theory for PCS (see Appendix \[appendix\_R\_pc\] for details), the PC resistance $$R_{PC}=R_{Sh}+R_{Max}, \label{Eq_R_pc}$$ where $R_{Sh}$ is the Sharvin resistance corresponding to the ballistic limit, and $R_{Max}$ is the Maxwell resistance corresponding to the diffusive limit and related to the resistivity. For the simplest metallic PC, $R_{Sh}$ is considered to be energy independent as the energy dependence of velocity cancels that of the DOS. This can be changed when complicated Fremi surface is involved and the effective DOS may be probed by $R_{Sh}$. [@Meekes1988prb; @Arham2012prb] Here for single crystal SRO the mean free path is large, and if the interface is clean and barrier-free, the PC should be close to the ballistic limit. As $R_{Sh} \propto (1/d)^2$ and $R_{Max} \propto (1/d)$, where $d$ is the diameter of PC, and if the anisotropic electronic state in SRO is not considered, the reduction of resistance from 9.3 $\Omega$ to 4.3 $\Omega$ would lead to an increase of $d$ by roughly $\sqrt{9.3/4.3}$=1.47 times in the ballistic limit (twice increase of the area); or by 2 times in the thermal limit(quadruple increase of the area). ![(Color online) (a) For $R_0$=3.2 $\Omega$ three SC transitions are shown by $dV/dI$ curves of at different temperatures. Red curves are guide to the eye. (b) The position of the $dV/dI$ peaks vs temperature. (c) Zero bias $dV/dI$ vs temperature. Both (b) and (c) are derived from (a).[]{data-label="fig_dVdI_3ohm"}](Fig5){width="9cm"} With the increase of contact area, the PC may show a larger critical current ($I_C$) if the critical current density is constant and $I_C$ is *only* determined by the PC itself. As the additional $dI/dV$ dips shown in Fig. \[fig\_dVdI\_4ohm\] is ascribed to the critical current effect [@Sheet2004prb], $I_C$ can be estimate from the dip position. At 1.6 K the dip position is about 1.2 and 2.3 mV for the 4.3 $\Omega$ and 9.3 $\Omega$ PCs, so the calculated $I_C$ is around 0.28 and 0.25 mA respectively, inconsistent with the expected 2-4 times increase of $I_C$ if $I_C$ is proportional to the contact area. This may suggest that $I_C$ is determined by a fixed region, e.g., chiral domains under the PC, instead of by the area of the PC itself. Thus, with increasing bias the region of SRO under the PC reaches its $I_C$, and $R_{PC}$ shows a finite increase due to $R_{Max}$, as described in Eq.(\[Eq\_R\_pc\]). When the PC resistance is reduced further to 3.2 $\Omega$, even larger conductance enhancement is observed as shown in Fig. \[fig\_dVdI\_pressure\]e and Fig. \[fig\_dVdI\_difference\]. After normalization by the background, the conductance enhancement at zero bias is about 3% for the 9.3 $\Omega$ PC, 14% for the 4.3 $\Omega$ PC, and 22% for the 3.2 $\Omega$ PC (Fig. \[fig\_dVdI\_difference\]d). The original $dV/dI$ curves without normalization and the EEI fits are also shown in Figs. \[fig\_dVdI\_difference\]a and b, and the fits can very well reproduce the $dV/dI$ curves when an effective temperature $T_{fit}$ is taken into account. Zoom-in the zero bias regime, the absolute amplitude of the $dV/dI$ dip and of the “wiggling” part outside of the dip are clearly shown in Fig. \[fig\_dVdI\_difference\]c. For all three PCs, the $dV/dI$ dip evolves to ZBA at around 0.2 mV (Fig. \[fig\_dVdI\_difference\]c), which is consistent with the gap value of SRO from the weak-coupling theory ($2\Delta/e=3.5k_{B}T_{c}$) with $T_{c}\sim 1.5$ K. This value is much smaller compared with previous PCS and $ab$ plane tunneling results where 0.7-0.9 mV were obtained, [@Laube2000prl; @Kashiwaya2011prl] and also smaller compared with scanning tunnel spectroscopy (STS) measurements where 0.3-0.5 mV were reported. [@Upward2002prb; @Firmo2013prb] For the 3.2 $\Omega$ PC, the surprising feature that the critical current effect persists to much higher temperatures is better illustrated in Fig. \[fig\_dVdI\_3ohm\]. In Fig. \[fig\_dVdI\_3ohm\]a, besides the first $dV/dI$ peak at around 0.2 mV, there are two additional $dV/dI$ peaks, one persists up to about 5 K, while the other persists up to about 6.2 K. These two $dV/dI$ peaks are likely to be SC features as the measured MR up to around 625 Oe also shows hysteresis, which decreases with increasing temperature and diminishes along with the resistance dip near zero bias. At 8 K, the $dV/dI$ within $\pm$1.5 mV, and the MR of the zero bias $dV/dI$ within $\pm$625 Oe, becomes practically flat and changes less than 0.03 $\Omega$. The temperature dependence of the position of $dV/dI$ peaks is plotted in Fig. \[fig\_dVdI\_3ohm\]b, and the zero bias $dV/dI$ from the spectra in Fig. \[fig\_dVdI\_3ohm\]a is plotted in Fig. \[fig\_dVdI\_3ohm\]c. There are clearly two resistance drops at around 4 and 6 K, and we note a similar but smaller drop around 4 k was also observed in Ref. \[\]. Since the bulk $T_{c}\sim$ 1.5 K for S1, and even for the 3-K phase $T_{c}\sim$ 3 K, thus the greatly enhanced $T_{c}$ could be only due to the W/SRO PC. In summary, an ultralow temperature point contact setup using nanopositioners was used to measure differential resistance of W/SRO point contact junctions. We find: 1) a superconducting gap around 0.2 mV and a dome like shape of conductance enhancement, consistent with chiral *p*-wave symmetry; 2) SC-like features persisting up to 6.2 K, much higher than the bulk T$_c$ of SRO, presumably due to the pressure exerted by the W tip and a mechanism similar to that of the 3K-phase; 3) a broad resistance hump coexisting with superconductivity, which is ascribed to density of states effect due to 2D electron-electron interaction, consistent with the highly anisotropic electronic system of SRO. We believe PCS may provide useful information beyond the surface problem for SRO. We thank Hu JIN for contribution in the earlier stage of this project, Liang LIU for various help with experiments and data analysis, Xin LU for helpful discussions on point contact measurements, and Fa WANG for discussions on correlated systems respectively. Work at Peking University is supported by National Basic Research Program of China (973 Program) through Grant No. 2011CBA00106 and No. 2012CB927400. The work at Tulane is supported by the DOE under grant DE-SC0012432. Basics of point contact resistance {#appendix_R_pc} ================================== There are many reviews on point contact spectroscopy [@Duif1989jpcm] and particular on unconventional heavy fermion systems. [@Naidyuk1998jpcm; @Park2009jpcm] Here we introduce the basics of the PC resistance following Ref. . In the simple theoretical model, PC is formed with an orifice with diameter $d$ between two bulk metallic electrodes. Depending on the relative ratio between $d$ and different mean free path $l$, PC can be categorized into three regimes: ballistic ($d<l_{elastic}$), diffusive ($l_{elastic}<d<l_{inelastic}$), and thermal ($d>l_{inelastic}$). In the ballistic regime, the Fermi surface in the two electrodes has a difference of $eV$, similar to the tunneling junction case; while in the thermal regime, the Fermi surface evolves smoothly within the PC and there is a well defined equilibrium temperature profile. [@Pothier1997] The current density in the orifice along its normal direction ($z$-axis) is $$j_{z}=2e \sum_{k}(v_{k})_{z}f_{k}(E),$$ where $v_{k}$ is the electron velocity, and $f_{k}(E)$ is the Fermi-Dirac distribution function. For a voltage biased ballistic PC, considering the energy difference $eV$, $$j_{z}=e \int_{E_{F}-eV/2}^{E_{F}+eV/2}dE \int \frac{d\Omega}{4\pi} v_{z}(E)f(E)N(E),$$ where $N(E)$ is the electronic DOS. In the simplified case, $v_{z}(E)$ is inversely proportional to $N(E)$, thus there is no non-linearity caused by energy dependence of DOS. The resulted Ohmic resistance is $$R_{Sh}=\frac{16R_{q}}{(k_{F}d)^{2}}=\frac{16\rho l}{3\pi d^{2}},$$ where $\rho$ is the bulk resistivity, $l$ the elastic mean free path, $R_{q}=h/2e^{2}=12.9$ k$\Omega$ the quantum resistance. With the assumption that the Drude picture holds, $\rho l = p_{F}/ne^{2}$ is a constant for a particular metal (Note that the quantities $p_{F}$ and $n$ were used in the original derivation). Thus, in the ballistic regime the diameter of the orifice $d$ can be estimated using the zero bias resistance $R_{0}$. To get a rough number, in the case of copper and other simple metals, $d\sim 30/\sqrt{R_{0}(\Omega)}$ nm. At finite bias, the electron can also be backscattered by phonons, magnons etc, at characteristic bias energy. So $I$-$V$ curve of the ballistic PC can be nonlinear and second derivative is often used to identify phonon and magnon spectra. More generally, for correlated materials with complex Fermi surface, $v_{z}(E)$ is no longer inversely proportional to $N(E)$, $I$-$V$ curve is nonlinear and $R_{Sh}(E)$ may reflect the change of DOS. [@Meekes1988prb; @Arham2012prb] For PC in the diffusive or thermal regime, electrons in the PC are scattered by impurities or defects, whose contribution to $R_{PC}$ can be estimated from the bulk resistivity, and the orifice just provides a geometric limitation. In the limit $d\gg l_{inelastic}$, the Maxwell resistance is $$R_{Max}=\frac{\rho}{d}.$$ As it depends on $d^{-1}$ instead of $d^{-2}$, it dominates over $R_{Sh}$ when $d$ is large. And when inelastic scattering happens inside the PC, the equilibrium temperature in the PC can be elevated following $$T_{PC}^{2}=T_{bath}^{2}+\frac{V^{2}}{4L}$$ where $L$ is the Lorentz number. For a rough estimation, when $T_{bath}\ll T_{PC}$, assume a standard $L=2.45\times 10^{-8}$ V$^{2}$K$^{-2}$, then $eV\sim 3.63k_{B}T_{PC}$, or $T_{PC}$ (K)$\simeq$ 3.2$V$ (mV). That explains for a thermal PC similar feature can be found in $dV/dI(V)$ and in $dV/dI(T)$. For the gap energy around 0.2 mV in this work, in the thermal limit a rough estimation of $T_{PC}$ at 0.2 mV is 0.64 K, which is below the $T_C$ of SRO, so the bias will not drive the PC out of the SC state even in the thermal limit. In the intermediate regime, Wexler derived an interpolation formula $$R_{PC}(T)\simeq\frac{16\rho l}{3\pi d^{2}}+\frac{\rho(T)}{d}.$$ For a heterocontact between two different electrodes (1 and 2), the resistance has contribution from both sides. For geometrically symmetric PC with almost equal $p_{F}$, $$R_{PC}(T)\simeq\frac{16\rho l}{3\pi d^{2}}+\frac{\rho_{1}(T)+\rho_{2}(T)}{2d}.$$ Since the resistivity of simple metal tip like tungsten is usually much smaller than that of the correlated electron systems (in normal state), we may just keep the resistivity term of the correlated systems being probed. The assumption of equal $p_{F}$ is very rough, the difference between $k_F$ of tungsten and SRO is shown in Table \[table\_k\_F\]. Here $k_F$ of tungsten is roughly estimated by assuming two valence electrons and simple spherical Fermi surface. Fermi sheet $\alpha$ $\beta$ $\gamma$ Tungsten ------------------------ -------------------- -------------------- --------------------- --------------------- $k_{F}$ (${\AA}^{-1}$) 0.304 0.622 0.753 1.55 $v_{F}$ ($ms^{-1}$) $1.0\times 10^{5}$ $1.0\times 10^{5}$ $5.5\times 10^{4} $ $1.8\times 10^{6} $ $m^{*}$ ($m_{e}$) 3.3 7.0 16 1 : Summary of quasiparticle parameters of Sr$_2$RuO$_4$ ($\alpha$,$\beta$,$\gamma$)[@Mackenzie2003rmp] and Tungsten. \[table\_k\_F\] For a heterocontact between a normal metal and a superconductor, Blonder-Tinkham-Klapwijk (BTK) model [@Blonder1982prb] is widely used to explain the conductance enhancement within the gap energy and a tunnel barrier $Z$ parameter is used to characterize the interface. Whether the Fermi velocity mismatch can be represented with an effective $Z$ parameter is not yet clear. [@Park2009jpcm] Note that in BTK model the scattering in the metals and the interface is not considered, even for finite $Z$. So its transparent interface limit ($Z=0$) corresponds to the ballistic limit of the PC model, *i.e.*, the point contact Andreev reflection spectroscopy can only be applied to *ballistic* contacts. Since the BTK model can be used for various interface transparencies, it has wider application than the simple $Z=0$ point contact model. To take into account additional scattering at or near the interface, *i.e.*, $R_{Max}$, a normal resistor in series [@Sheet2004prb] or a normal current in parallel [@Miyoshi2005prb; @Peng2013prb] can be added. Thus even in the so-called thermal regime, the gap value can be roughly estimated with consideration of a combination of the BTK model and PC model. [@Sheet2004prb] In some cases it is believed that although the footprint of the PC can be tens of microns, much larger than $l$, but still ballistic limit can be applied because there are multiple smaller PC junctions randomly distributed across the contact area, [@Bugoslavsky2005prb; @Peng2013prb] and the BTK model can be used directly. Although conceptually this is different from the picture that there is an interface barrier which contributes to the PC resistance like a real tunneling junction, but in both cases ballistic limit can be applied as $R_{Max}$ is smaller than $R_{Sh}$. When the SC has unconventional pairing symmetries, generalized BTK model is developed to fit the data by taking into account various parameters including order parameter symmetry, incidence angle, Fermi surface mismatch, life time broadening due to inter or intra band scattering etc. PCS for unconventional SC has been reviewed in Ref. . It is still not clear whether the order parameter symmetry can be verified strictly from the shape of the point contact Andreev reflection spectra. [@Gloos1996jltp_scaling; @Park2009jpcm] In this work we mainly report the temperature and field dependence of the PC spectra rather than quantitatively fit the data with the generalized BTK model. [@Kashiwaya2011prl; @Kashiwaya2014pe] Fitting with electron-electron interaction {#appendix_fit_eei} ========================================== The difference between PCS and planar tunneling is whether the in-plane momentum is conserved. Since there is no well-known theory for incorporation of quantum correction of DOS into PCS, here we use the theory for the planar tunneling junctions. Correction to tunneling conductance by electron-electron interaction (EEI) is quantitatively described by the Altshuler-Aronov (AA) theory, [@Altshuler1985; @Gershenzon1986jetp] in the 2D limit, $$\frac{G(V,T)-G(0,T)}{G(0,T)}=\frac{e^{2}R_{sq}}{8\pi^{2}\hbar}\ln{\frac{4\pi\delta}{\mathcal{D}R_{sq}}}[\Phi_{2}(\frac{eV}{k_{B}T})-\Phi_{2}(0)], \label{Eq_AA_2d}$$ where $R_{sq}$ is the resistance per square of the metal film, $\delta$ the thickness of the insulating barrier, $\mathcal{D}$ the diffusion constant, and $\Phi_{2}$ a integral for 2D as defined in Ref. \[\]. The integral is $$\begin{aligned} & \Phi_{d}(A)= \int_{-\infty}^{\infty}dx{\frac{\cosh(x+A)-1}{{\cosh(x/2)}^{2}}} \\ & \times\int_{0}^{\infty}dx\frac{\sinh{y}dy}{[\cosh{y}+\cosh(x+A)](1+\cosh{y}){y^{2-d/2}}}, \end{aligned} \label{Eq_AA_Phi}$$ where $x=\epsilon/kT$ and $A=eV/kT$. The prefactor before the bracket in Eq. (\[Eq\_AA\_2d\]) can be lumped into one parameter $S$ and it is the only fitting parameter. When $eV\gg k_{B}T$ but still within the 2D limit, Eq. (\[Eq\_AA\_2d\]) approaches $S\ln{\frac{eV}{k_{B}T}}$ and $S$ is just the slope shown in Fig. 2. Since $R_{sq}=\rho/a$, $a$ the thickness of the metal film, the resistivity $\rho=(e^{2}\nu \mathcal{D})^{-1}$, the slope $S \propto R_{sq}\ln(c\nu)$, where $c$ is a constant. For the 3D limit, $$\frac{G(V,T)-G(0,T)}{G(0,T)}=\frac{e^{2}\rho}{8\sqrt{2}\pi^{2}\hbar}(\frac{k_{B}T}{\hbar\mathcal{D}})^{1/2}[\Phi_{3}(\frac{eV}{k_{B}T})-\Phi_{3}(0)], \label{Eq_AA_3d}$$ which shows a linear dependence on $\sqrt{eV/k_{B}T}$ when $eV\gg k_{B}T$. Optical images of the SRO surface {#appendix_Ru_inclusions} ================================= Optical images for SRO samples S1 and S2 are shown in Fig. \[fig\_micro\_images\] for comparison. Dense Ru inclusions of width about 1 $\mu$m and length a few $\mu$m are clearly seen in the micro image for S2, which is also harder to cleave than S1. This is consistent to the observation of Lichtenberg in Ref.  that SRO with Ru vacancies is much easier to cleave and the surface dead layer probably is also easier to pierce through. We note that although here the surface was polished by sandpaper to improve image quality, the Ru inclusions can easily be observed on the surface of S2 without any treatment. ![(Color online) Comparison of polarized optical microscope images of two samples: S1 (left), S2 (right). Ru inclusions are clearly seen in the right image for S2. []{data-label="fig_micro_images"}](Fig6){width="8cm"} Reproducibility {#appendix_reproducibility} =============== PC spectra for more than 10 locations were measured in several runs. In each run a few locations are tried to search for SC-like features. With increasing force the tip eventually became blunted and bent, and small cracks can also develop on the surface of the SRO. A set of PC spectra similar to that in Fig. \[fig\_dVdI\_4ohm\] is shown in Fig. \[fig\_other\_PCS\]a, for a W/SRO PC on S1 but obtained in another run. Besides W tip, Au tip (0.5 mm dia.) was also tried on S1 and the PC spectra are shown in Fig. \[fig\_other\_PCS\]b. For the Au/SRO PC, gap value around 0.5 mV is observed, the conductance enhancement is only about 1%, and instead of the dome like conductance peak, a split peak is observed, similar to that was reported in Ref. \[\]. ![(Color online) (a) Temperature dependence of the PC spectra for a W/SRO PC on S1 but obtained in another run, showing similar gap features as in the main text. The resistance is only about 1 $\Omega$, but the gap around 0.15 mV, and the critical current effect are clearly demonstrated. (b) Temperature dependence of the PC spectra for a Au/SRO PC on S1, a split peak within $\pm$0.5 mV is observed, similar to that in Ref. \[\].[]{data-label="fig_other_PCS"}](Fig7){width="9cm"} ZBA is less obvious for Au/SRO PCs. For W/SRO PCs, ZBA is frequently observed, which could be due to a thin oxide layer or a defective layer on the surface as observed in other PC measurements. [@Sasaki2011prl; @Peng2013prb] For those PC spectra showing clear ZBA, there are two typical types as shown in Fig. \[fig\_two\_kinds\_ZBA\]. One type is similar to that in Fig. \[fig\_dVdI\_9ohm\] with a logarithmic dependence consistent with 2D EEI, and SC feature sometimes coexists with ZBA; the other type has a $\sqrt{V}$ dependence which is consistent with 3D EEI, no SC feature is observed with this type of ZBA. For the 2D EEI type, e.g., for a 35 $\Omega$ PC on S2 as shown in Fig. \[fig\_two\_kinds\_ZBA\], the slope 0.07 is close to the slope 0.11 for S1 in Fig. \[fig\_dVdI\_9ohm\], and 0.08 in Fig. \[fig\_dVdI\_4ohm\], indicating similar 2D EEI is probed, though here $T_{fit}$ = 2 K is higher than the bath temperature about 0.52 K, which is probably the reason that SC feature is not observed. ![(Color online) (a) Bias dependence of the normalized conductance for point contact on S2 with resistance 35 (blue) and 13 $\Omega$ (green) at T=0. 52 K, and (b) the conductance after normalized by the background EEI fits (the green symbols are shifted up for clarity). (c) EEI 2D fitting (black dash line) for the 13 $\Omega$ PC with $T_{fit}$ = 2 K, slope 0.07. (d) EEI 3D fitting for the 35 $\Omega$ PC with $T_{fit}$ = 0.52 K, slope (3D) = 0.015. []{data-label="fig_two_kinds_ZBA"}](Fig8){width="9cm"}

--- abstract: 'We have used the GALEX ultraviolet telescope to study stellar populations and star formation morphology in a well-defined sample of more than three dozen nearby optically-selected pre-merger interacting galaxy pairs. We have combined the GALEX NUV and FUV images with broadband optical maps from the Sloan Digitized Sky Survey to investigate the ages and extinctions of the tidal features and the disks. We have identified a few new candidate tidal dwarf galaxies in this sample, as well as other interesting morphologies such as accretion tails, ‘beads on a string’, and ‘hinge clumps’. In only a few cases are strong tidal features seen in HI maps but not in GALEX.' author: - 'Beverly J. Smith$^1$, Mark L. Giroux$^1$, Curtis Struck$^2$, Mark Hancock$^{3}$, and Sabrina Hurlock$^1$' title: 'Tidal Dwarf Galaxies, Accretion Tails, and ‘Beads on a String’ in the ‘Spirals, Bridges, and Tails’ Interacting Galaxy Survey' --- Introduction ============ Tidal disturbances have played an important role in reshaping galaxies and triggering star formation over cosmic time. This is confirmed by H$\alpha$, far-infrared, and mid-infrared studies showing that the mass-normalized star formation rates of pre-merger optically-selected interacting galaxies are enhanced by a factor of two on average compared to normal spirals [@bushouse87; @kennicutt87; @bushouse88; @smith07]. With the advent of the Galaxy Evolution Explorer (GALEX), a new window on star formation in galaxies is now available. The addition of UV helps to break the age$-$extinction degeneracy in population synthesis modeling (e.g., [@smith08]). Furthermore, since the UV traces somewhat older and lower mass stars ($\le$400 Myrs; O to early-B stars) than H$\alpha$ ($\le$10 Myrs; early- to mid-O stars), it provides a measure of star formation over a longer timescale than H$\alpha$ studies. GALEX imaging has shown that some tidal features in interacting galaxies are quite bright in the UV (e.g., [@neff05]). In some cases, tidal features previously thought to be purely gaseous have been detected by GALEX (e.g., [@hancock07]). In other systems, GALEX images have been used to identify new tidal features (e.g., [@boselli05]). To address these issues, we have used the GALEX telescope to image more than three dozen strongly interacting galaxies in the UV (the ‘Spirals, Bridges, and Tails’ (SB&T) sample). These galaxies were selected from the Arp (1966) Atlas using the following criteria: 1) They are relatively isolated binary systems; we eliminated merger remnants, close triples, and multiple systems in which the galaxies have similar optical brightnesses. 2) They are tidally disturbed. 3) They have radial velocities less than 10,350 km/s. 4) Their total angular size is $>$ 3$'$, to allow for good resolution with GALEX. Each galaxy was imaged for $\ge$1500 seconds in the FUV and NUV broadband filters of GALEX, which have effective bandpasses of 1350 $-$ 1705Å  and 1750 $-$ 2800Å, respectively. Some of the galaxies that fit our selection criteria were previously observed by guaranteed time projects. For these galaxies, we used the archival GALEX images. The circular GALEX field of view has a diameter of 1.2 degrees. The pixel size is 15, and the spatial resolution $\sim$ 5$''$. About 2/3rds of our galaxies have broadband optical images available from the Sloan Digitized Sky Survey (SDSS), while 3/4 have broadband Spitzer infrared images available [@smith07]. About half have published 21 cm HI maps. Morphologies ============ The SB&T galaxies have a large range of collisional morphologies, including M51-like systems, wide pairs with long tails and/or bridges, wide pairs with short tails, close pairs with long tails, and close pairs with short tails. In the current paper, we discuss unusual tidal morphologies in a subset of the galaxies. In @giroux10, we present an Atlas of UV images of additional SB&T galaxies. The full survey is described in detail in @smith10. For four of the galaxies in the SB&T sample, we have already published the GALEX images as part of detailed studies of the individual galaxies, and compared with numerical simulations of the interaction [@hancock07; @hancock09; @hancock10; @smith08; @peterson09; @peterson10]. There is a large variety of star formation morphologies within the tidal features in this sample. In many cases, the tidal features are quite bright in the UV. This is illustrated by Arp 72 (Figure 1a), whose eastern tail is very prominent in the GALEX images, and has very blue UV/optical colors. Arp 72 is also a good example of the so-called ‘beads on a string’ morphology, in which regularly-spaced clumps of star formation are seen along spiral arms and tidal features. These clumps are generally spaced about 1 kpc apart, the characteristic scale for gravitational collapse of molecular clouds [@elmegreen96]. Such beads are seen in many other systems in our sample, including the northern tail of the western galaxy in Arp 65 (Figure 1b), Arp 82 [@hancock07], and Arp 285 [@smith08]. In a few systems, we see luminous star forming regions at the base of a tidal feature. We call these features ‘hinge clumps’ [@hancock09]. These lie near the intersection of the spiral density wave in the inner disk and the material wave in the tail. These may form when dense material in the inner disk gets pulled out into a tail. This lowers the shear, which may allow more massive clouds to gravitationally collapse. Hinge clumps are visible at the eastern end of the Arp 72 bridge (Figure 1a) and the base of the northern tail of Arp 65 (Figure 1b). Hinge clumps are also seen in Arp 82 [@hancock07] and Arp 305 [@hancock09]. Our sample also includes some candidate ‘tidal dwarf galaxies’ (TDGs), massive concentrations of young stars near the tips of tidal features. The prototypical TDGs in Arp 244 and Arp 245 [@mirabel92; @duc00] are included in the SB&T sample, along with the bridge TDG in Arp 305 [@hancock09; @hancock10]. Another possible TDG is seen in Arp 202 (Figure 2), an interaction between an edge-on disk galaxy and a smaller irregularly-shaped galaxy to the south. A long clumpy tail is visible to the west of the southern galaxy. The tip of this tail is particularly prominent in the GALEX images, and has very blue UV/optical colors. Our optical spectrum shows that this clump is at the same redshift as Arp 202. This source was not detected in our Spitzer 8 $\mu$m map [@smith07] or in our SARA H$\alpha$ map, suggesting that it is in a post-starburst stage. Another SB&T system that may have TDGs is Arp 181 (Figure 3). A clump is visible near the end of the western tail in the GALEX and SDSS images, with very blue optical/UV colors. However, no optical spectrum is available, thus it is unclear whether it is at the same redshift as Arp 181. Further west, another galaxy is visible, without any obvious link to the tail. Our optical spectrum shows that it is at the same redshift as Arp 181. In the SDSS image it looks like a spiral galaxy or a disturbed disk with short tidal tails. It is extremely blue in NUV $-$ g, and is detected at 8 $\mu$m [@smith07]. This may be either a pre-existing dwarf galaxy or a recently detached TDG. The SB&T sample also contains numerous examples of accretion from one galaxy to another. One of the best-studied examples is the northern tail of Arp 285, which was likely produced from material accreted from the southern galaxy [@toomre72; @smith08]. According to our numerical simulations, the material in this tail fell into the gravitational potential of the northern galaxy, overshot that potential, and is now gravitationally collapsing and forming stars [@smith08]. We call such features ‘accretion tails’, to distinguish them from classical tidal features. The inner western tail of Arp 284 was likely produced by the same process [@struck03]. Another system which may have an accretion tail is Arp 105 (Figure 4). The spiral in this system has a long tail extending to the north, previously classified as a TDG [@duc97]. The spiral is connected by a bridge to an elliptical galaxy to the south. South of the elliptical is a bright star formation knot [@stockton72]. Both the northern TDG candidate and the southern knot of star formation are luminous in HI maps [@duc97]. In the GALEX images, the spiral and the northern TDG are quite bright in the UV, but the highest UV surface brightness is found in the knot of star formation south of the elliptical. We suggest, based on analogy to Arp 285 [@smith08] and proximity to the elliptical, that the southern star forming region in Arp 105 is an accretion tail, rather than simply a classical tidal tail coincidently seen in projection behind the elliptical. Another system with possible mass transfer is Arp 269 (Figure 5a), an unequal-mass pair of galaxies with a bridge. In both the GALEX and SDSS images, an off-center group of blue star forming regions is visible in the smaller galaxy NGC 4485, as well as along the bridge. As noted by @elmegreen98, several of these knots of star formation lie in a tail-like structure southwest of NGC 4485. @clemens00 suggest that NGC 4485 passed through the disk of the larger galaxy NGC 4490, and ram pressure caused an offset in the location of the interstellar gas in NGC 4485, and thus the observed offset in star formation. We suggest an alternative possibility, that the star formation was triggered by gas flow from NGC 4490 along the bridge. Thus this may be an example of accretion from one galaxy to another. Our sample contains only a few tidal features with high HI column densities that are not detected in our GALEX maps or published optical maps. One of these systems is Arp 269. In the @clemens98a HI maps, two large plume-like features extend 10$'$ ($\sim$20 kpc) to the north and south of the pair. Neither of these plumes is strongly detected in the GALEX or SDSS images, although smoothed SDSS images show a possible hint of the southern plume. This is surprising in light of the relatively high N$_{HI}$ in the inner 5$'$ (10 kpc) sections of these plumes of 4 $\times$ 10$^{20}$ cm$^{-2}$. It was suggested by @clemens98a that these HI features were produced by SN-driven outflow from the main galaxy NGC 4490. Instead, we suggest that the HI plumes are simply gas-rich tidal features. Deeper optical and UV imaging is needed to search for a stellar component to these features. Another system with a gas-rich tidal feature without a GALEX counterpart is Arp 84 (‘The Heron’), a pair of unequal mass spiral galaxies connected by a bridge (Figure 5b). In HI maps, a large gaseous plume extends to the south of the main galaxy [@kaufman99]. In spite of its high N$_{HI}$ $\sim$ 4 $\times$ 10$^{20}$ cm$^{-2}$ and narrow HI line width of $\sim$100 km s$^{-1}$, no diffuse UV emission is seen in this feature, although a few UV-bright clumps are present. No redshifts are available at present for these clumps. Arp 84 also exhibits ‘beads on a string’ along the inner edge of the northern tail of the smaller galaxy. The most northern knot along this ‘string’ is bright in the GALEX image, as well as at 8 $\mu$m [@smith07] and H$\alpha$ [@kaufman99]. Summary ======= In this paper, we present GALEX UV images of a subset of the SB&T interacting galaxy pairs. We identify examples of ‘beads on a string’, accretion from one galaxy to another, and candidate tidal dwarf galaxies. There are only a few tidal features that are bright in HI maps that are not detected in GALEX. This research was supported by GALEX grant\ GALEXGI04-0000-0026, NASA LTSA grant NAG5-13079, and Spitzer grant RSA 1353814. We thank the Lick Observatory staff for their support of our optical spectroscopy campaign. Arp, H. 1966, Atlas of Peculiar Galaxies (Pasadena: Caltech) Boselli, A., et al. 2005, ApJ, 623, L13 Bushouse, H. A. 1987, ApJ, 320, 49 Bushouse, H. A., Lamb, S. A., & Werner, M. W. 1988, ApJ, 335, 74 Clemens, M. 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--- abstract: 'The fermions of the Standard Model are integrated out to obtain the effective Lagrangian in the sector violating $P$ and $CP$ at zero temperature. We confirm that no contributions arise for operators of dimension six or less and show that the leading operators are of dimension eight. To assert this we explicitly compute one such non-vanishing contribution, namely, that with three $Z^0$, two $W^+$ and two $W^-$. Terms involving just gluons and $W$’s are also considered, however, they turn out to vanish in the $P$-odd sector to eighth order. The analogous gluonic term in the $CP$-odd and $P$-even ($C$-odd) sector is non-vanishing and it is also computed. The expressions derived apply directly to Dirac massive neutrinos. All $CP$-violating results display the infrared enhancement already found at dimension six.' author: - 'L. L. Salcedo' title: 'Leading order one-loop $CP$ and $P$ violating effective action in the Standard Model' --- Introduction {#sec:1} ============ A full understanding of $CP$-violation remains a challenge and for this reason it is a fruitful field of research both in relation with the Standard Model of particle physics and in extensions thereof [@Xing:2003ez; @Buras:1997fb; @Neubert:1996qg; @Grossman:1997pa; @Winstein:1992sx; @Paschos:1989ur; @Wolfenstein:1987pe; @Donoghue:1987wu; @Charles:2004jd]. $CP$-violation enters in very different phenomena, like non vanishing of the electric dipole momentum of elementary particles, baryogenesis [@Sakharov:1967dj], or, assuming $CPT$ invariance, the puzzling $T$-violation. Yet, in the Standard Model, $CP$-violation is rather elusive. There is no trace of it in the QCD sector, while in the electroweak sector it enters through a small parameter in the CKM matrix for quarks [@Kobayashi:1973fv], and possibly also for leptons, for massive neutrinos [@Maki:1962mu]. Even in the electroweak sector manifestation of $CP$ breaking requires a subtle combination, the Jarlskog determinant $\Delta$, which requires order twelve in the quark (or leptons) masses and would vanish if two up-like or two down-like quarks were degenerated in mass [@Jarlskog:1985ht]. In any case only through fermions $CP$ can be broken in the Standard Model. The structure of the Standard Model action implies that integration of the fermions results in an effective Lagrangian of the form (we assume the unitary gauge throughout) $$\mathcal{L}^\text{eff} {} (x) = \sum_\alpha g_\alpha \left(\frac{v}{\phi(x)}\right)^{d_\alpha-4}\mathcal{O}_\alpha(x) ,$$ where $\mathcal{O}_\alpha(x)$ represents any possible operator, of mass dimension $d_\alpha$, constructed as a Lorentz and gauge invariant product of the gauge fields, their derivatives and derivatives of the Higgs field. $g_\alpha$ is the operator coupling constant, with mass dimension $4-d_\alpha$. $\phi(x)$ denotes the Higgs field and $v$ its vacuum expectation value. The coupling constant (which may vanish for some operators) has two additive contributions, one from the quark loop and another from the lepton loop. In the $CP$-odd sector, $g_\alpha$ must contain the Jarlskog determinant. In terms of the Yukawa coupling this yields a tiny dimensionless number, $\Delta/v^{12}$, of the order of $10^{-24}$. This fact has occasionally been presented as an indication of an intrinsic limitation of the Standard Model to produce enough $CP$-breaking to account for observations, including the baryon asymmetry. While this might be true, qualitative arguments should eventually be supported by a detailed computation. Smit argued in [@Smit:2004kh] that the coupling $g_\alpha$ is just a homogeneous function of the quarks (or leptons) masses of the appropriate degree. This implies that $g_\alpha \sim \Delta \times I_\alpha$, where $I_\alpha$ has a large negative degree to compensate that of $\Delta$. Both $\Delta$ and $I_\alpha$ depend only on the fermion masses and do not involve $v$. On the other hand, the various quark masses are very different and widely different result can be obtained by combining them at random. Actual calculations have been carried out in [@Hernandez:2008db; @GarciaRecio:2009zp] for operators of dimension six, which is the first possible $CP$-violating contribution at one-loop. They show that $g_\alpha \sim J\kappa/m_c^2$ where is $m_c$ the charm quark mass, $J=2.9(2)\times 10^{-5}$ is the Jarlskog invariant [@Nakamura:2010zzi] and $\kappa$ is a dimensionless coefficient of the order of unity. Implications for cold electroweak baryogenesis have been considered in [@Tranberg:2009de; @Tranberg:2010af]. Unfortunately, these two references differ in that [@Hernandez:2008db] finds such a dimension six contribution in the $P$-odd sector whereas [@GarciaRecio:2009zp] finds a contribution in the $C$-odd sector but none in the $P$-odd one. The purpose of this note is manyfold. First, to reduce to the simplest and more transparent terms the calculation of these couplings constants. Second, to confirm that, although dimension six $CP$-odd and $P$-odd operators do exist, their coupling vanish in the Standard Model. Third, to verify that the order six cancellation is accidental, and non vanishing contributions in the $CP$-odd and $P$-odd sector appear for the first time at dimension eight. The purely gluonic leading (eighth) order term is also computed since it is particularly simple. As it turns out, this term breaks $C$ but not $P$. Lastly, to verify that the enhancement (as compared to the naive estimate) found at order six is displayed also at higher orders. The method {#sec:2} ========== We will integrate out the fermions in the Standard Model to extract the $CP$ violating contribution of the resulting effective action. This is the one-loop approximation to the effective action with full one-particle irreducible bosonic lines and vertices. We work at zero temperature. Quarks will be explicitly considered. Leptons would not contribute to the $CP$-odd sector if neutrinos are assumed to be exactly massless. For massive Dirac neutrinos the contribution of the leptons will be completely analogous to the one obtained for quarks. The quark-sector Lagrangian of the Standard Model, in its Euclidean version and in the unitary gauge, can be written as [@Huang:1992bk]: $${\cal L}(x) = \bar{q}(x){{\bf D}}q(x) = (\bar{q}_L,\bar{q}_R) \left(\begin{matrix} m & {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L \\ {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R & m \end{matrix} \right) \left(\begin{matrix} q_R \\ q_L \end{matrix} \right).$$ Here $q_{L,R}$ carry Dirac, generation (family), $ud$ and color indices ($ud$ space distinguishes the up-like from down-like quarks in each generation). Expanding further the matrices in $ud$ space: $$\begin{aligned} m &=& \left(\begin{matrix} \frac{\phi}{v} m_u &0 \\ 0 & \frac{\phi}{v} m_d \end{matrix} \right) , \quad {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L = \left(\begin{matrix} {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_u + {\hbox{$ \mathrel{\mathop{Z\!\!\!\!/}}$}} + {\hbox{$ \mathrel{\mathop{G\!\!\!\!/}}$}} & {\mathrel{\mathop{W\!\!\!\!\!/}}}{}^+ C \\ {\mathrel{\mathop{W\!\!\!\!\!/}}}{}^-C^{-1} & {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_d - {\hbox{$ \mathrel{\mathop{Z\!\!\!\!/}}$}} + {\hbox{$ \mathrel{\mathop{G\!\!\!\!/}}$}} \end{matrix} \right) , \quad {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R = \left(\begin{matrix} {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_u + {\hbox{$ \mathrel{\mathop{G\!\!\!\!/}}$}} & 0 \\ 0 & {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_d + {\hbox{$ \mathrel{\mathop{G\!\!\!\!/}}$}} \end{matrix} \right) . \label{eq:2.2}\end{aligned}$$ Here $m_{u,d}$ are the diagonal matrices (in generation space) with the up-like and down-like quarks masses, respectively. $G_\mu$ the gluon field, $Z_\mu$ the $Z^0$ field, $W_\mu^\pm$ the $W$ boson fields, $C$ is the CKM matrix, finally $(D_\mu)_{u,d}= \partial_\mu+q_{u,d}B_\mu$ where $q_u=2/3$, $q_d=-1/3$, and $B_\mu$ is the weak hypercharge gauge connection. For convenience, in all cases the coupling constant has been included in the corresponding gauge connection. Further details can be found in [@GarciaRecio:2009zp]. After integration of the quark loop, the corresponding Euclidean effective action is just $$\Gamma = -{{\rm Tr}}\log{{\bf D}}.$$ $\Gamma$ is the sum of the all Feynman graphs with one quark loop and any number of bosonic legs, gauge fields and Higgs. This sum is written as a functional which will be expressed within a covariant derivative expansion of these bosonic fields. Certainly, the effective action can be computed following the efficient method outlined in [@GarciaRecio:2009zp] and based on [@Salcedo:2008tc], applied there to sixth order in the derivative expansion. However, one of our goals here is to present a derivation as transparent as possible, and closer to the method introduced in [@Salcedo:2000hx] on which the calculation of [@Hernandez:2008db] is based. To this end, we will use the relation $$\delta\Gamma = -{{\rm Tr}}(\delta{{\bf D}}\, {{\bf D}}^{-1}) = -\int d^4x \,{{\rm tr \,}}\!\!\left[\delta{{\bf D}}\langle x| {{\bf D}}^{-1}|x\rangle\right] . \label{eq:2.4}$$ In the second equality it has been used that the variation $\delta{{\bf D}}$, induced by the variation in the gauge and Higgs fields in ${{\bf D}}$, contains no derivatives. The method is then to choose a suitable variation of these fields, compute $\delta\Gamma$ in the desired sector, and subsequently seek a functional fulfilling such a variation. The virtues of this approach are i) the current $\langle x| {{\bf D}}^{-1}|x\rangle$ is easier to obtain than the ${{\rm Tr}}\log{{\bf D}}$ itself, ii) the condition on $\delta\Gamma$ of being a consistent variation provides a nontrivial check of the calculation, and iii) even if one where to compute $\Gamma$ directly, the simplest way to avoid integration by parts identities (i.e., redundant operators in the final expression) is to obtain its functional derivative, $\delta\Gamma/\delta{{\bf D}}=-\langle x| {{\bf D}}^{-1}|x\rangle$. This quantity is local and so free from $x$-integration by parts identities. This issue becomes increasingly important as the number of derivatives increases. A convenient field to use as variation in [eq. (\[eq:2.4\])]{} is $Z$ which appears just in $D^L_\mu$. We adopt such a choice, namely, $$\delta{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L=\left(\begin{matrix} \delta {\hbox{$ \mathrel{\mathop{Z\!\!\!\!/}}$}} & 0 \\ 0 & -\delta {\hbox{$ \mathrel{\mathop{Z\!\!\!\!/}}$}} \end{matrix} \right) :=\delta {\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}}, \qquad \delta{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R = \delta m = 0 . \label{eq:2.5}$$ The Dirac operator can be written as ${{\bf D}}= P_L {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R P_R + P_R {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L P_L + P_R m P_R + P_L m P_L $ where $P_{R,L}=\frac{1}{2}(1\pm\gamma_5)$ project on the $R$ or $L$ spaces, respectively. ${{\bf D}}$ can be explicitly inverted in chiral blocks. In particular, for the $RL$ block, $P_R {{\bf D}}^{-1} P_L = P_R ({\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L - m {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R\!\!^{-1} m)^{-1} P_L$. Hence, from [eq. (\[eq:2.4\])]{} and (\[eq:2.5\]), $$\delta\Gamma = -{{\rm Tr}}(P_R \,\delta{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}} \, ({\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L - m {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R\!\!^{-1} m)^{-1} ) .$$ Therefore, if only the $P$-odd sector (i.e., that with a $\gamma_5$) is retained, we will have $$\begin{aligned} \delta\Gamma^- &=& -\frac{1}{2}{{\rm Tr}}\!\left[\gamma_5 \,\delta {\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}} \, ({\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L- m {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R\!\!^{-1} m)^{-1} \right] \nonumber \\ &=& -\frac{1}{2}{{\rm Tr}}\!\left[\gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}} \, ( {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L\!\!^{-1} + {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L\!\!^{-1} m {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R\!\!^{-1} m {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L\!\!^{-1} + \cdots )\right] .\end{aligned}$$ It can be noted that each term of the expansion between parenthesis starts and ends with a label $L$. Moreover, as the chiral label propagates through the term it flips (from $L$ to $R$ and vice versa) at $m$ but not at $D_R$ or $D_L$. Keeping these rules in mind we can simply write $$\begin{aligned} \delta\Gamma^- &=& -\frac{1}{2}{{\rm Tr}}\!\left[\gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}}\, ( {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}\,{}^{-1} + {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}\,{}^{-1} m {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}\,{}^{-1} m {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}\,{}^{-1} + \cdots )\right] \nonumber \\ &=& -\frac{1}{2}{{\rm Tr}}\!\left[\gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}} \, ({\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}+m)^{-1} )\right] .\end{aligned}$$ The second equality follows from the fact that terms with an odd number of $m$’s are automatically discarded since they cannot start and end with a label $L$. A simple and convenient technique to compute $\langle x| ({\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}+m)^{-1} |x\rangle$ is the method of symbols [@Nepomechie:1984wt; @Salcedo:1996qy; @GarciaRecio:2009zp]. This method is suitable for computing one-loop Feynman graphs with external legs at zero momentum or more generally, for expansions around zero momentum. In the present case it takes the form $$\begin{aligned} \delta\Gamma^- &=& -\frac{1}{2}\int \frac{d^4x d^4p}{(2\pi)^4} \, {{\rm tr \,}}\!\!\left[ \gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}}\, ( {\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}+i\!{\mathrel{\mathop{p\!\!\!/}}}+m)^{-1} \right] . \label{eq:2.9}\end{aligned}$$ Two remarks apply here: i) the momentum variable $p_\mu$, like $D^{R,L}_\mu$, does not introduce a flip in the chiral label, and ii) after momentum integration all $D_\mu$ appear only in the form $[D_\mu,~]$ and so there are no longer differential (or pseudo differential) operators acting; only an ordinary function of $x$ survives. At the same time gauge invariance is ensured. The covariant derivative expansion is just an expansion in powers of $D_\mu$. Only even orders contribute (the space-time dimension being even). Besides, in the $P$-odd sector the effective action starts at fourth order in four dimensions, since a Levi-Civita pseudo tensor must be present. Hence: $$\begin{aligned} \delta\Gamma^- &=& \frac{1}{2}\int \frac{d^4x d^4p}{(2\pi)^4} \, {{\rm tr \,}}\!\!\left[ \gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}}\, \sum_{n=0}^\infty \left( \frac{i\!{\mathrel{\mathop{p\!\!\!/}}}-m}{p^2+m^2}{\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}} \right)^{2n+3} \frac{i\!{\mathrel{\mathop{p\!\!\!/}}}-m}{p^2+m^2} \right] . \label{eq:2.11}\end{aligned}$$ Sixth order $P$-odd terms {#sec:3} ========================= Taking the appropriate values of $n$ in [eq. (\[eq:2.11\])]{} and the appropriate contributions in $D^{R,L}$ and $m$, one can select the desired terms of the effective action. At least two $W^+W^-$ pairs must be present in the $CP$-odd sector terms, since the quark loop must visit the three generations [@Kobayashi:1973fv].[^1] A term with just $(W^+W^-)^2$ would count as fourth order, however no $CP$-odd term can be constructed without introducing further fields or derivatives. To sixth order such a $P$-odd and $CP$-odd term can be written using $(W^+W^-)^2ZD$. ($D$ here refers to either $D_u$ or $D_d$.) The question is whether this operator appears with a non vanishing coefficient in the Standard Model or not. Ref. [@Hernandez:2008db] claims that it does whereas the calculation in [@GarciaRecio:2009zp] concludes that it does not. Therefore we will start by reconsidering such a term within our present approach. Under a variation of $Z$, the contribution of the candidate term to be found in [eq. (\[eq:2.11\])]{} is of the form $\delta Z(W^+W^-)^2D$. We can set $\phi=v$, since we are not interested in contributions from Higgs, and likewise we can set $Z_\mu$ and $G_\mu$ to zero in $D^{R,L}_\mu$. Moreover, we can even set $D_\mu^u=D_\mu^d=\partial_\mu$ in $D^{R,L}_\mu$. The ordinary derivatives can be unambiguously replaced by covariant ones at the end without loss of information since no $F_{\mu\nu}$ tensor can be present in the term considered. The computation is tedious but straightforward. Let us spell out the main steps in the calculation. We select terms with $n=1$ in [eq. (\[eq:2.11\])]{} and restore the $L,R$ labels. We keep only terms starting and ending with the label $L$ and only $m$ introduces a chiral label flip $L\leftrightarrow R$. No flip is introduced by $m^2$, $p$ or $D$. This yields terms of the type $$\delta\Gamma^- = \int \frac{d^4x d^4p}{(2\pi)^4} \, {{\rm tr \,}}\!\!\left[ \frac{1}{2} \gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}}\, N \!{\mathrel{\mathop{p\!\!\!/}}} \, {\mathrel{\mathop{\hat W\!\!\!\!\!/}}} N m {\mathrel{\mathop{\partial\!\!\!/}}} N m {\mathrel{\mathop{\hat W\!\!\!\!\!/}}} N \!{\mathrel{\mathop{p\!\!\!/}}} \, {\mathrel{\mathop{\hat W\!\!\!\!\!/}}} N \! {\mathrel{\mathop{p\!\!\!/}}} \, {\mathrel{\mathop{\hat W\!\!\!\!\!/}}} N \! {\mathrel{\mathop{p\!\!\!/}}} +\cdots \right] + \text{o.t.} \label{eq:2.12}$$ Here we have set ${\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_R={{\mathrel{\mathop{\partial\!\!\!/}}}}$ and ${\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L={\mathrel{\mathop{\partial\!\!\!/}}}+{\mathrel{\mathop{\hat W\!\!\!\!\!/}}}$ and ${\mathrel{\mathop{\hat W\!\!\!\!\!/}}}$ represents the off diagonal (charged) part of ${\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L$ as a matrix in $ud$ space. We have kept terms with four $\hat{W}$’s and one derivative. Also we have introduced the quantity $N= (p^2+m^2)^{-1}$. The dots in [eq. (\[eq:2.12\])]{} refer to further terms of the same type, while “o.t.” refers to other terms which cannot have a contribution to the pattern $\delta Z(W^+W^-)^2D$. Next, we expand the $ud$ labels using [eq. (\[eq:2.2\])]{} for $m$ and ${\mathrel{\mathop{\hat W\!\!\!\!\!/}}}$, and [eq. (\[eq:2.5\])]{} for $\delta\hat{Z}$. This produces $$\delta\Gamma^- = \int \frac{d^4x d^4p}{(2\pi)^4} \, {{\rm tr \,}}\!\!\left[ \frac{1}{2} \gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{Z\!\!\!\!/}}$}}\, N_u {\mathrel{\mathop{p\!\!\!/}}} \, {\mathrel{\mathop{W\!\!\!\!\!/}}}{}^+ C N_d m_d {\mathrel{\mathop{\partial\!\!\!/}}} N_d m_d {\mathrel{\mathop{W\!\!\!\!\!/}}}{}^- C^{-1} N_u {\mathrel{\mathop{p\!\!\!/}}} \, {\mathrel{\mathop{W\!\!\!\!\!/}}}{}^+ C N_d {\mathrel{\mathop{p\!\!\!/}}} \, {\mathrel{\mathop{W\!\!\!\!\!/}}}{}^- C^{-1} N_u \! {\mathrel{\mathop{p\!\!\!/}}} \, + \cdots \right] + \text{o.t.} ,$$ where $ N_{u,d}= (p^2+m_{u,d}^2)^{-1} $. At this point we can already factorize the trace between quantities which act only in generation space, namely, $N_{u,d}$, $m_{u,d}$ and $C$, and all the other quantities, which do not act on that space: $$\delta\Gamma^- = \int \frac{d^4x d^4p}{(2\pi)^4} \, \!\Big( \frac{1}{2} {{\rm tr \,}}[N_u C N^2_d m_d^2 C^{-1} N_u C N_d C^{-1} N_u] {{\rm tr \,}}\!\!\left[ \gamma_5 \,\delta{\hbox{$ \mathrel{\mathop{Z\!\!\!\!/}}$}}\, {\mathrel{\mathop{p\!\!\!/}}} \, {\mathrel{\mathop{W\!\!\!\!\!/}}}{}^+ \! {\mathrel{\mathop{\partial\!\!\!/}}} \, {\mathrel{\mathop{W\!\!\!\!\!/}}}{}^- \! {\mathrel{\mathop{p\!\!\!/}}} \, {\mathrel{\mathop{W\!\!\!\!\!/}}}{}^+ \! {\mathrel{\mathop{p\!\!\!/}}} \, {\mathrel{\mathop{W\!\!\!\!\!/}}}{}^- \! {\mathrel{\mathop{p\!\!\!/}}} \, \right] + \cdots \Big) + \text{o.t.} \label{eq:2.15}$$ It is a general rule that $m_u$ or $m_d$ can only appear raised to even powers and can be eliminated in favor of $N_u$ and $N_d$.[^2] Eventually, all required momentum integrals and traces on $3\times 3$ matrices in generation space can be cast in the form [@GarciaRecio:2009zp] $$I^k_{a,b,c,d} = \int \frac{d^4p}{(2\pi)^4} (p^2)^k \,{{\rm tr \,}}\!\! \left[ N_u^a C N_d^b C^{-1} N_u^c C N_d^d C^{-1} \right] , \label{eq:2.16}$$ where the exponents $k,a,b,c,d$ are non negative integers. On the other hand, only the $CP$-odd contribution is of interest to us. This is the component antisymmetric under the exchange $C\to C^*$, $$\hat{I}^k_{a,b,c,d} = i\,{{\rm Im \,}}I^k_{a,b,c,d} .$$ Due to cyclic and hermiticity properties of the trace and matrices involved, these integral satisfy $$\hat{I}^k_{a,b,c,d} = -\hat{I}^k_{c,b,a,d} = -\hat{I}^k_{a,d,c,b} \,. \label{eq:2.18}$$ Such antisymmetry under exchange of the labels $a$ and $c$, or $b$ and $d$ implies that many terms in [eq. (\[eq:2.15\])]{} do not have a contribution to the $CP$-odd sector and this greatly alleviates the amount of subsequent computation. Performing an angular average over the momentum and taking the color and Dirac traces in [eq. (\[eq:2.15\])]{} yields then, for $CP$-odd terms ($N_c=3$ is the number of colors) $$\delta\Gamma^- = N_c\int d^4x \, \!\Big( \frac{1}{3} \hat{I}^3_{1,1,2,2} \, \epsilon_{\mu\nu\alpha\beta} \,\delta Z_\mu W^-_\nu \partial_\alpha W^+_\beta W^-_\lambda W^+_\lambda + \cdots \Big) + \text{o.t.}$$ It only remains to apply the derivative on all fields at its right. It can be checked that after this operation is carried out all terms so obtained cancel. So there is no term of the type $(W^+W^-)^2ZD$ in the $CP$-odd, $P$-odd sector of the Standard Model, in agreement with the alternative and more systematic calculation in [@GarciaRecio:2009zp]. Dimension eight operators {#sec:4} ========================= In this section we show that at eighth order in the derivative expansion there are non vanishing contributions in the $CP$-odd and $P$-odd sector of the Standard Model. Concretely we consider terms of the form $(W^+W^-)^2 Z^3D$, with no gluons nor derivatives of the Higgs field, and apply the technique just described. In this case we select terms with $n=2$ in [eq. (\[eq:2.11\])]{} and seek terms of the type $\delta Z (W^+W^-)^2 Z^2D$. The calculation is analogous to the one shown previously, except that now $Z$ is not set to zero in ${\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L$, instead we use ${\hbox{$ \mathrel{\mathop{D\!\!\!\!/}}$}}_L={\mathrel{\mathop{\partial\!\!\!/}}}+{\mathrel{\mathop{\hat W\!\!\!\!\!/}}}+{\hbox{$ \mathrel{\mathop{\hat Z\!\!\!\!/}}$}}$. Then we keep terms with precisely four $\hat{W}$’s, two $\hat{Z}$’s and one derivative. After restoring the $L,R$ labels and $u,d$ labels, and carrying out the momentum integration, the trace in color, Dirac and generation space, and applying the derivative to the right, one obtains: $$\delta\Gamma^- = N_c\int d^4x \, \!\Big( 2 \hat{I}^4_{1,1,2,4} \, \epsilon_{\mu\nu\alpha\beta} \,\delta Z_\mu Z_\lambda Z_{\nu\alpha} W^+_\beta W^+_\lambda W^-_\sigma W^-_\sigma + \cdots \Big) + \text{o.t.} \label{eq:2.20}$$ Here $Z_{\nu\alpha}$ stands for the $\nu$ derivative of $Z_\alpha$. We have eliminated the $\hat{I}^3_{a,b,c,d}$ in favor of $\hat{I}^4_{a,b,c,d}$ and have used identities involving $\delta_{\mu\nu}$ and $\epsilon_{\mu\nu\alpha\beta}$ to bring the expression to a canonical form. The expression contains 62 operators, each one weighted with various integrals $\hat{I}^4_{a,b,c,d}$. It remains to find out the effective action from which the variation in [eq. (\[eq:2.20\])]{} derives. This serves also as a non trivial check of the computation. The method is just to propose all allowed independent terms of the form $(W^+W^-)^2Z^3D$ with arbitrary coefficients, and take a first order variation with respect to $Z$ to fix those coefficients. The Minkowski space result (see [@GarciaRecio:2009zp] for further details in the conventions) is $$\begin{aligned} \mathcal{L}^{\rm eff}(x) &=& \frac{N_c}{15}\frac{v^4}{\phi^4} \epsilon_{\mu\nu\alpha\beta}\Big[ (12 \hat{I}_1 -16 \hat{I}_2) Z_\mu Z_\lambda^2 W^+_\nu W^+_\sigma W^-_\alpha W^-_{\beta \sigma} \nonumber\\&& + (4 \hat{I}_1 +23 \hat{I}_2) Z_\mu Z_\lambda^2 W^+_\sigma W^+_{\nu \alpha} W^-_\beta W^-_\sigma + (6 \hat{I}_1 -23 \hat{I}_2) Z_\mu Z_\lambda^2 W^+_\nu W^+_{\alpha \beta} W^-_\sigma{}^2 \nonumber\\&& +(32 \hat{I}_1 +4 \hat{I}_2) Z_\mu Z_\lambda Z_\sigma W^+_\nu W^+_{\alpha \lambda } W^-_\beta W^-_\sigma +(16 \hat{I}_1 -38 \hat{I}_2) Z_\mu Z_\lambda Z_\sigma W^+_\lambda W^+_{\nu \alpha} W^-_\beta W^-_\sigma \nonumber\\&& +(16 \hat{I}_1 +22 \hat{I}_2) Z_\mu Z_\lambda Z_\sigma W^+_\lambda W^+_\sigma W^-_\nu W^-_{\alpha \beta} +(-20 \hat{I}_1 -15 \hat{I}_2) Z_\lambda^2 Z_\sigma W^+_\mu W^+_\sigma W^-_\nu W^-_{\alpha \beta} \nonumber\\&& +10 \hat{I}_1 Z_\mu Z_\lambda Z_{\nu \alpha} W^+_\sigma{}^2 W^-_\beta W^-_\lambda -20 \hat{I}_2 Z_\mu Z_\lambda Z_{\nu \sigma} W^+_\alpha W^+_\lambda W^-_\beta W^-_\sigma +\text{c.c} \Big] \label{eq:4.2}\end{aligned}$$ We have defined $\hat{I}_1= \hat{I}^4_{1,1,2,4}-\hat{I}^4_{1,1,4,2}$ and $\hat{I}_2= \hat{I}^4_{1,2,2,3}-\hat{I}^4_{2,1,3,2}$, and “c.c” refers to complex conjugate; $Z_\mu$ is real, $(W^{\pm}_\mu)^*=W^{\mp}_\mu$ and $\hat{I}_{1,2}$ are imaginary. The (underivated) Higgs field has been restored using that it scales as the mass dimension of $\hat{I}_{1,2}$. Also the derivative includes the field $B_\mu$ when it acts on the $W$’s [@GarciaRecio:2009zp]. Numerically,[^3] $$\hat{I}_{1,2} = \frac{iJ}{(4\pi)^2}\frac{\kappa_{1,2}}{m_s^2 m_c^2}, \quad \kappa_1 = 0.226,\quad \kappa_2 = 0.456. \label{eq:2.23}$$ We can see that the values of these coefficients are considerably larger than simple estimates based on the Jarlskog determinant divided by the appropriate power of $v$. At sixth order the enhancement is driven by the small mass of the light quarks and so this can be considered as a kind of chiral enhancement. The possibility of such an effect was first pointed out in [@Smit:2004kh] and confirmed in [@Hernandez:2008db; @GarciaRecio:2009zp]. The momentum integrals $\hat{I}^k_{a,b,c,d}$ are completely explicit but rather complicated homogeneous functions of the fermion masses [@GarciaRecio:2009zp]. At sixth order these integrals are not continuous at $\bar{m}_u,\bar{m}_d,m_s=0$, yet one can take the limit $\bar{m}_u,\bar{m}_d\to 0$ and subsequently $m_s\to 0$ and this approximation gives a value fairly close to the exact one [@GarciaRecio:2009zp]. To discuss the situation at eighth order we will consider the simpler case of $m_b,m_t\to\infty$ which is a quite good approximation for $\kappa_{1,2}$. At eighth order the momentum integrals are more ultraviolet convergent and also more infrared divergent than at sith order. Specifically, $\hat{I}_2$ diverges as $1/m_s^2$ when $\bar{m}_u=\bar{m}_d=0$, with $\kappa_2=1/2$. The other integral, $\hat{I}_1$, is more infrared divergent: in the same limit $\kappa_1$ depends on the ratio $\bar{m}_u/\bar{m}_d$, varying continuously between $-1/6$ for $\bar{m}_d\ll\bar{m}_u$ to $3/2$ for $\bar{m}_u\ll\bar{m}_d$ . In [eq. (\[eq:2.23\])]{} we have used $\bar{m}_u=2.55\,{\rm MeV}$ and $\bar{m}_d=5.04\,{\rm MeV}$. We have also considered $CP$-violating terms containing only $W$’s and gluons. Such terms appear for the first time at eighth order since (at one loop) at least four $W$’s are needed to violate $CP$ and two $G_{\mu\nu}$ are required to make a color singlet. In the $P$-odd sector one can write three independent operators, however we find that they have zero coupling in the Standard Model. On the other hand, in the $P$-even ($C$-violating) sector, there are also three operators of which one has zero coupling while the other two terms result in the following effective Lagrangian (in Minkowski space)[^4] $$\mathcal{L}^{\rm eff}(x) = -\frac{4}{3} \frac{v^4}{\phi^4}\hat{I}^2_{1,1,2,2} \left( W^+_\lambda{}^2 W^-_\mu W^-_\nu G^a_{\mu\alpha} G^a_{\nu\alpha} - \text{c.c.} \right) . \label{eq:4.4}$$ Numerically, $\hat{I}^2_{1,1,2,2}=iJ \kappa_3/(4\pi)^2m_s^2 m_c^2$, with $\kappa_3=3.76$. This coefficient diverges logarithmically as $\bar{m}_u,\bar{m}_d\to 0$. Note that, at the order considered, the dimension four gluon condensate[^5] does not induce a $CP$-violating interaction between the four $W$’s. Such term vanishes identically, as it should be, since no $CP$-odd term can be written using just $W$’s without derivatives or other fields. Conclusions {#sec:6} =========== We have shown that, to one-loop and at zero temperature, the leading $P$-violating $CP$-odd operators in the effective action of the Standard Model are of dimension eight. We have computed explictly the couplings for the operators of the form $Z^3(W^+W^-)^2$ plus one covariant derivative, [eq. (\[eq:4.2\])]{}. These operators come from Feynman graphs with one quark-loop, four $W$ legs, three $Z$ legs and possibly one $B\sim Z+\gamma$ leg. In principle, dimension six operators could develop beyond one-loop or at finite temperature due to the breaking of Lorentz invariance. Purely gluonic operators of dimension eight have also been computed, [eq. (\[eq:4.4\])]{}, and they are $C$-odd and $P$-even. Remarkably the corresponding coupling constants we find are not vanishingly small, rather they have a natural scale related to intermediate mass quarks times the Jarlskog invariant. All formulas derived for quarks extend directly to massive Dirac leptons. This implies that even if the neutrino masses are small their contribution to the $CP$-violating couplings needs not be small, due to infrared sensitivity in the momentum integrals on which the couplings depend. As a consequence, such couplings will be strongly dependent on the mass ratios between neutrinos of the different generations. I am grateful to C. Garcia-Recio for discussions. Research supported by DGI under contract FIS2008-01143, Junta de Andalucía grant FQM-225, the Spanish Consolider-Ingenio 2010 Programme CPAN contract CSD2007-00042, and the European Community-Research Infrastructure Integrating Activity [*Study of Strongly Interacting Matter*]{} (HadronPhysics2, Grant Agreement no. 227431) under the 7th Framework Programme of EU. [10]{} Z.-z. Xing, Int. J. Mod. Phys. [**A19**]{}, 1 (2004), \[hep-ph/0307359\]. A. J. Buras and R. Fleischer, Adv. Ser. Direct. High Energy Phys. [**15**]{}, 65 (1998), \[hep-ph/9704376\]. M. Neubert, Int. J. Mod. Phys. [**A11**]{}, 4173 (1996), \[hep-ph/9604412\]. Y. Grossman, Y. Nir and R. Rattazzi, Adv. Ser. Direct. High Energy Phys. [**15**]{}, 755 (1998), \[hep-ph/9701231\]. B. Winstein and L. Wolfenstein, Rev. Mod. Phys. [**65**]{}, 1113 (1993). E. A. Paschos and U. Turke, Phys. Rept. [**178**]{}, 145 (1989). L. Wolfenstein, Ann. Rev. Nucl. Part. Sci. [**36**]{}, 137 (1986). J. F. Donoghue, B. R. 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A. Tranberg, A. Hernandez, T. Konstandin and M. G. Schmidt, Phys. Lett. [**B690**]{}, 207 (2010), \[0909.4199\]. A. Tranberg, JHEP (2010), \[1009.2358\]. K. Huang, (World Scientific, Singapore, 1992). L. L. Salcedo, Eur. Phys. J. [**C58**]{}, 423 (2008), \[0807.1696\]. L. L. Salcedo, Eur. Phys. J. [**C20**]{}, 161 (2001), \[hep-th/0012174\]. R. I. Nepomechie, Phys. Rev. [**D31**]{}, 3291 (1985). L. L. Salcedo and E. Ruiz Arriola, Ann. Phys. [**250**]{}, 1 (1996), \[hep-th/9412140\]. [^1]: Of course, terms with a single $W^+W^-$ pair are allowed beyond one-loop. [^2]: The label $u$ or $d$ does not change between two consecutive $W$’s. This implies two consecutive $D_L$ and so an even number of $m$’s. That the presence of $m_{u,d}$ can be obviated follows also from eq. (7.1) of [@GarciaRecio:2009zp]. [^3]: $\bar{m}_u,\bar{m}_d,m_c,m_s,m_t,m_b$ denote the quark masses. [^4]: The gluon field strength tensor has been normalized according to $[D_\mu,D_\nu]=i(\lambda_a/2) G^a_{\mu\nu}$. [^5]: In the presence of the gluon condensate $G^a_{\mu\nu} G^a_{\alpha\beta} = \frac{1}{12} (\delta_{\mu\alpha} \delta_{\nu\beta} - \delta_{\mu\beta} \delta_{\nu\alpha})\langle (G^a_{\lambda\sigma})^2\rangle + \text{fluctuations} $.

--- abstract: 'Let $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}\subseteq{\operatorname{GL}}_r$ be a Levi subgroup of ${\operatorname{GL}}_r$, where $r=r_1+\cdots+r_k$, and ${\widetilde{M}}$ its metaplectic preimage in the $n$-fold metaplectic cover ${\widetilde{\operatorname{GL}}}_r$ of ${\operatorname{GL}}_r$. For automorphic representations $\pi_1,\dots,\pi_k$ of ${\widetilde{\operatorname{GL}}}_{r_1}({\mathbb{A}}),\dots,{\widetilde{\operatorname{GL}}}_{r_k}({\mathbb{A}})$, we construct (under a certain technical assumption, which is always satisfied when $n=2$) an automorphic representation $\pi$ of ${\widetilde{M}}({\mathbb{A}})$ which can be considered as the “tensor product” of the representations $\pi_1,\dots,\pi_k$. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place $v$, $\pi_v$ is equivalent to the local metaplectic tensor product of $\pi_{1,v},\dots,\pi_{k,v}$ defined by Mezo. Then we show that if all of $\pi_i$ are cuspidal (resp. square-integrable modulo center), then the metaplectic tensor product is cuspidal (resp. square-integrable modulo center). We also show that (both locally and globally) the metaplectic tensor product behaves in the expected way under the action of a Weyl group element, and show the compatibility with parabolic inductions.' address: 'Shuichiro Takeda: Mathematics Department, University of Missouri, Columbia, 202 Math Sciences Building, Columbia, MO, 65211' author: - Shuichiro Takeda title: 'Metaplectic tensor products for automorphic representations of ${\widetilde{\operatorname{GL}}}(r)$' --- **Introduction** ================ Let $F$ be either a local field of characteristic 0 or a number field, and $R$ be $F$ if $F$ is local and the ring of adeles ${\mathbb{A}}$ if $F$ is global. Consider the group ${\operatorname{GL}}_r(R)$. For a partition $r=r_1+\cdots+r_k$ of $r$, one has the Levi subgroup $$M(R):={\operatorname{GL}}_{r_1}(R)\times\cdots\times{\operatorname{GL}}_{r_k}(R)\subseteq{\operatorname{GL}}_r(R).$$ Let $\pi_1,\dots,\pi_k$ be irreducible admissible (resp. automorphic) representations of ${\operatorname{GL}}_{r_1}(R),\dots,{\operatorname{GL}}_{r_k}(R)$ where $F$ is local (resp. $F$ is global). Then it is a trivial construction to obtain the representation $\pi_1\otimes\cdots\otimes\pi_k$, which is an irreducible admissible (resp. automorphic) representation of the Levi $M(R)$. Though highly trivial, this construction is of great importance in the representation theory of ${\operatorname{GL}}_r(R)$. Now if one considers the metaplectic $n$-fold cover ${\widetilde{\operatorname{GL}}}_r(R)$ constructed by Kazhdan and Patterson in [@KP], the analogous construction turns out to be far from trivial. Namely for the metaplectic preimage ${\widetilde{M}}(R)$ of $M(R)$ in ${\operatorname{GL}}_r(R)$ and representations $\pi_1,\dots,\pi_k$ of the metaplectic $n$-fold covers ${\widetilde{\operatorname{GL}}}_{r_1}(R),\dots,{\widetilde{\operatorname{GL}}}_{r_k}(R)$, one cannot construct a representation of ${\widetilde{M}}(R)$ simply by taking the tensor product $\pi_1\otimes\cdots\otimes\pi_k$. This is simply because ${\widetilde{M}}(R)$ is not the direct product of ${\widetilde{\operatorname{GL}}}_{r_1}(R),\dots,{\widetilde{\operatorname{GL}}}_{r_k}(R)$, namely $${\widetilde{M}}(R)\ncong{\widetilde{\operatorname{GL}}}_{r_1}(R)\times\dots\times{\widetilde{\operatorname{GL}}}_{r_k}(R),$$ and even worse there is no natural map between them. When $F$ is a local field, for irreducible admissible representations $\pi_1,\dots,\pi_k$ of ${\widetilde{\operatorname{GL}}}_{r_1}(F),\dots,{\widetilde{\operatorname{GL}}}_{r_k}(F)$, P. Mezo ([@Mezo]), whose work, we believe, is based on the work by Kable [@Kable2], constructed an irreducible admissible representation of the Levi ${\widetilde{M}}(F)$, which can be called the “metaplectic tensor product” of $\pi_1,\dots,\pi_k$, and characterized it uniquely up to certain character twists. (His construction will be reviewed and expanded further in Section \[S:Mezo\].) The theme of the paper is to carry out a construction analogous to Mezo’s when $F$ is a number field, and our main theorem is Let $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$ be a Levi subgroup of ${\operatorname{GL}}_r$, and let $\pi_1,\dots,\pi_k$ be unitary automorphic subrepresentations of ${\widetilde{\operatorname{GL}}}_{r_1}({\mathbb{A}}),\dots,{\widetilde{\operatorname{GL}}}_{r_k}({\mathbb{A}})$. Assume that $M$ and $n$ are such that Hypothesis ($\ast$) is satisfied, which is always the case if $n=2$. Then there exists an automorphic representation $\pi$ of ${\widetilde{M}}({\mathbb{A}})$ such that $$\pi\cong{\widetilde{\otimes}}'_v\pi_v,$$ where each $\pi_v$ is the local metaplectic tensor product of Mezo. Moreover, if $\pi_1,\dots,\pi_k$ are cuspidal (resp. square-integrable modulo center), then $\pi$ is cuspidal (resp. square-integrable modulo center). In the above theorem, ${\widetilde{\otimes}}_v'$ indicates the metaplectic restricted tensor product, the meaning of which will be explained later in the paper. The existence and the local-global compatibility in the main theorem are proven in Theorem \[T:main\], and the cuspidality and square-integrability are proven in Theorem \[T:cuspidal\] and Theorem \[T:square\_integrable\], respectively. Let us note that by unitary, we mean that $\pi_i$ is equipped with a Hermitian structure invariant under the action of the group. Also we require $\pi_i$ be an automorphic subrepresentation, so that it is realized in a subspace of automorphic forms and hence each element in $\pi_i$ is indeed an automorphic form. (Note that usually an automorphic representation is a subquotient.) We need those two conditions for technical reasons, and they are satisfied if $\pi_i$ is in the discrete spectrum, namely either cuspidal or residual. Also we should emphasize that if $n>2$, we do not know if our construction works unless we impose a technical assumption as in Hypothesis ($\ast$). We will show in Appendix \[A:topology\] that this assumption is always satisfied if $n=2$, and if $n>2$ it is satisfied, for example, if $\gcd(n, r-1+2cr)=1$, where $c$ is the parameter to be explained. We hope that even for $n>2$ it is always satisfied, though at this moment we do not know how to prove it. As we will see, strictly speaking the metaplectic tensor product of $\pi_1,\dots,\pi_k$ might not be unique even up to equivalence but is dependent on a character $\omega$ on the center $Z_{{\widetilde{\operatorname{GL}}}_r}$ of ${\widetilde{\operatorname{GL}}}_r$. Hence we write $$\pi_\omega:=(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_\omega$$ for the metaplectic tensor product to emphasize the dependence on $\omega$.\ Also we will establish a couple of important properties of the metaplectic tensor product both locally and globally. The first one is that the metaplectic tensor product behaves in the expected way under the action of the Weyl group. Namely \ [**Theorem \[T:Weyl\_group\_local\] and \[T:Weyl\_group\_global\].**]{} [*Let $w\in W_M$ be a Weyl group element of ${\operatorname{GL}}_r$ that only permutes the ${\operatorname{GL}}_{r_i}$-factors of $M$. Namely for each $(g_1,\dots,g_k)\in{\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$, we have $w (g_1,\dots,g_k)w^{-1}=(g_{\sigma(1)},\dots,g_{\sigma(k)})$ for a permutation $\sigma\in S_k$ of $k$ letters. Then both locally and globally, we have $$^{w}(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_\omega \cong(\pi_{\sigma(1)}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_{\sigma(k)})_\omega,$$ where the left hand side is the twist of $(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_\omega$ by $w$.* ]{}\ The second important property we establish is the compatibility of the metaplectic tensor product with parabolic inductions. Namely \ [**Theorem \[T:induction\_local\] and \[T:induction\_global\].**]{} [ *Both locally and globally, let $P=MN\subseteq{\operatorname{GL}}_r$ be the standard parabolic subgroup whose Levi part is $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$. Further for each $i=1,\dots,k$ let $P_i=M_iN_i\subseteq{\operatorname{GL}}_{r_i}$ be the standard parabolic of ${\operatorname{GL}}_{r_i}$ whose Levi part is $M_i={\operatorname{GL}}_{r_{i,1}}\times\cdots\times{\operatorname{GL}}_{r_{i, l_i}}$. For each $i$, we are given a representation $$\sigma_i:=(\tau_{i,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{i,l_i})_{\omega_i}$$ of ${\widetilde{M}}_i$, which is given as the metaplectic tensor product of the representations $\tau_{i,1},\dots,\tau_{i,l_i}$ of ${\widetilde{\operatorname{GL}}}_{r_{i,1}},\dots,{\widetilde{\operatorname{GL}}}_{r_{i, l_i}}$. Assume that $\pi_i$ is an irreducible constituent of the induced representation ${\operatorname{Ind}}_{{\widetilde{P}}_i}^{{\widetilde{\operatorname{GL}}}_{r_i}}\sigma_i$. Then the metaplectic tensor product $$\pi_\omega:=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$$ is an irreducible constituent of the induced representation $${\operatorname{Ind}}_{{\widetilde{Q}}}^{{\widetilde{M}}}(\tau_{1,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{1, l_1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{k,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{k,l_k})_\omega,$$ where $Q$ is the standard parabolic subgroup of $M$ whose Levi part is $M_1\times\cdots\times M_k$.* ]{}\ In the above two theorems, it is implicitly assumed that if $n>2$ and $F$ is global, the metaplectic tensor products in the theorems exist in the sense that Hypothesis ($\ast$) is satisfied for the relevant Levi subgroups. Finally at the end, we will discuss the behavior of the global metaplectic tensor product when restricted to a smaller Levi. Namely for each automorphic form $\varphi\in(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_\omega$ in the metaplectic tensor product, we would like to know which space the restriction $\varphi|_{{\widetilde{M}}_2}$ belongs to, where $M_2=\{I_{r_1}\}\times{\operatorname{GL}}_{r_2}\times\cdots\times{\operatorname{GL}}_{r_k}\subset M$, viewed as a subgroup of $M$, is the Levi for the smaller group ${\operatorname{GL}}_{r-r_1}$. Somehow similarly to the non-metaplectic case, the restriction $\varphi|_{{\widetilde{M}}_2}$ belongs to the metaplectic tensor product of $\pi_2, \dots,\pi_k$. But the precise statement is a bit more subtle. Indeed, we will prove \ [**Theorem \[T:restriction\].**]{} [*Assume Hypothesis ($\ast\ast$) is satisfied, which is always the case if $n=2$ or $\gcd(n,r-1+2cr)=\gcd(n,r-r_1-1+2c(r-r_1))=1$. Then there exists a realization of the metaplectic tensor product $\pi_\omega=(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega}$ such that, if we let $$\pi_\omega\|_{{\widetilde{M}}_2({\mathbb{A}})}=\{{\widetilde{\varphi}}|_{{\widetilde{M}}_2({\mathbb{A}})}:{\widetilde{\varphi}}\in \pi_\omega\},$$ then $$\pi_\omega\|_{{\widetilde{M}}_2({\mathbb{A}})}\subseteq \bigoplus_\delta m_\delta(\pi_2{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega_\delta},$$ as a representation of ${\widetilde{M}}_2({\mathbb{A}})$, where $(\pi_2{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega_\delta}$ is the metaplectic tensor product of $\pi_2,\dots,\pi_k$, $\omega_\delta$ is a certain character twisted by $\delta$ which runs through a finite subset of ${\operatorname{GL}}_{r_1}(F)$ and $m_\delta\in{\mathbb{Z}}^{\geq 0}$ is a multiplicity.* ]{}\ The precise meanings of the notations will be explained in Section \[S:restriction\]. Even though the theory of metaplectic groups is an important subject in representation theory and automorphic forms and used in various important literatures such as [@Banks2; @F; @BBL; @BFH; @BH; @Suzuki] to name a few, and most importantly for the purpose of this paper, [@BG] which concerns the symmetric square $L$-function on ${\operatorname{GL}}(r)$ , it has an unfortunate history of numerous technical errors and as a result published literatures in this area are often marred by those errors which compromise their reliability. As is pointed out in [@BLS], this is probably due to the deep and subtle nature of the subject. At any rate, this has made people who work in the area particularly wary of inaccuracies in new works. For this reason, especially considering the foundational nature of this paper, we tried to provide detailed proofs for most of our assertions at the expense of the length of the paper. Furthermore, for large part, we rely only on the two fundamental works, namely the work on the metaplectic cocycle by Banks, Levy and Sepanski ([@BLS]) and the local metaplectic tensor product by Mezo ([@Mezo]), both of which are written carefully enough to be reliable. Finally, let us mention that the result of this paper will be used in our forthcoming [@Takeda2], which will improve the main result of [@Takeda1]. [**Notations**]{} Throughout the paper, $F$ is a local field of characteristic zero or a number field. If $F$ is a number field, we denote the ring of adeles by ${\mathbb{A}}$. As we did in the introduction we often use the notation $$R=\begin{cases} F\quad\text{if $F$ is local}\\ {\mathbb{A}}\quad\text{if $F$ is global}. \end{cases}$$ The symbol $R^\times$ has the usual meaning and we set $$R^{\times n}=\{a^n: a\in R^\times\}.$$ Both locally and globally, we denote by ${\mathcal{O}_F}$ the ring of integers of $F$. For each algebraic group $G$ over a global $F$, and $g\in G({\mathbb{A}})$, by $g_v$ we mean the $v^{{{\text{th}}}}$ component of $g$, and so $g_v\in G(F_v)$. For a positive integer $r$, we denote by $I_r$ the $r\times r$ identity matrix. Throughout we fix an integer $n\geq 2$, and we let $\mu_n$ be the group of $n^{\text{th}}$ roots of unity in the algebraic closure of the prime field. We always assume that $\mu_n\subseteq F$, where $F$ is either local or global. So in particular if $n\geq 3$, for archimedean $F$, we have $F={\mathbb C}$, and for global $F$, $F$ is totally complex. The symbol $(-,-)_F$ denotes the $n^{\text{th}}$ order Hilbert symbol of $F$ if $F$ is local, which is a bilinear map $$(-,-)_F:F^\times\times F^\times\rightarrow\mu_n.$$ If $F$ is global, we let $(-,-)_{\mathbb{A}}:=\prod_v(-,-)_{F_v}$, where the product is finite. We sometimes write simply $(-,-)$ for $(-,-)_R$ when there is no danger of confusion. Let us recall that both locally and globally the Hilbert symbol has the following properties: $$\begin{gathered} (a,b)^{-1}=(b,a)\\ (a^n,b)=(a, b^n)=1\\ (a, -a)=1\end{gathered}$$ for $a, b\in R^\times$. Also for the global Hilbert symbol, we have the product formula $(a,b)_{\mathbb{A}}=1$ for all $a, b\in F^\times$. We fix a partition $r_1+\cdots+r_k=r$ of $r$, and we let $$M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}\subseteq{\operatorname{GL}}_r$$ and assume it is embedded diagonally as usual. We often denote each element $m\in M$ by $$m=\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}, \quad\text{or}\quad m={\operatorname{diag}}(g_1,\dots,g_k)$$ or sometimes simply $m=(g_1,\dots,g_k)$, where $g_i\in{\operatorname{GL}}_{r_i}$. For ${\operatorname{GL}}_r$, we let $B=TN_B$ be the Borel subgroup with the unipotent radical $N_B$ and the maximal torus $T$. If $\pi$ is a representation of a group $G$, we denote the space of $\pi$ by $V_{\pi}$, though we often confuse $\pi$ with $V_\pi$ when there is no danger of confusion. We say $\pi$ is unitary if $V_\pi$ is equipped with a Hermitian structure invariant under the action of $G$, but we do not necessarily assume that the space $V_\pi$ is complete. Now assume that the space $V_\pi$ is a space of functions or maps on the group $G$ and $\pi$ is the representation of $G$ on $V_\pi$ defined by right translation. (This is the case, for example, if $\pi$ is an automorphic subrepresentation.) Let $H\subseteq G$ be a subgroup. Then we define $\pi\|_H$ to be the representation of $H$ realized in the space $$V_{\pi\|_H}:=\{f|_H: f\in V_\pi\}$$ of restrictions of $f\in V_\pi$ to $H$, on which $H$ acts by right translation. Namely $\pi\|_H$ is the representation obtained by restricting the functions in $V_\pi$. Occasionally, we confuse $\pi\|_H$ with its space when there is no danger of confusion. Note that there is an $H$-intertwining surjection $\pi|_H\rightarrow\pi\|_H$, where $\pi|_H$ is the (usual) restriction of $\pi$ to $H$. For any group $G$ and elements $g, h\in G$, we define $^gh=ghg^{-1}$. For a subgroup $H\subseteq G$ and a representation $\pi$ of $H$, we define $^g\pi$ to be the representation of $gHg^{-1}$ defined by $^g\pi(h')=\pi(g^{-1}h'g)$ for $h'\in gHg^{-1}$. We let $W$ be the set of all $r\times r$ permutation matrices, so for each element $w\in W$ each row and each column has exactly one 1 and all the other entries are 0. The Weyl group of ${\operatorname{GL}}_r$ is identified with $W$. Also for our Levi $M$, we let $W_M$ be the subset of $W$ that only permutes the ${\operatorname{GL}}_{r_i}$-blocks of $M$. Namely $W_M$ is the collection of block matrices $$W_M:=\{(\delta_{\sigma(i),j}I_{r_j})\in W : \sigma\in S_k\},$$ where $S_k$ is the permutation group of $k$ letters. Though $W_M$ is not a group in general, it is in bijection with $S_k$. Note that if $w\in W_M$ corresponds to $\sigma\in S_k$, we have $$^w{\operatorname{diag}}(g_1,\dots,g_k)=w{\operatorname{diag}}(g_1,\dots,g_k)w^{-1} ={\operatorname{diag}}(g_{\sigma^{-1}(1)},\dots, g_{\sigma^{-1}(k)}).$$ In addition to $W$, in order to use various results from [@BLS], which gives a detailed description of the 2-cocycle $\sigma_r$ defining our metaplectic group ${\widetilde{\operatorname{GL}}}_r$, one sometimes needs to use another set of representatives of the Weyl group elements, which we as well as [@BLS] denote by ${\mathfrak{M}}$. The set ${\mathfrak{M}}$ is chosen to be such that for each element $\eta\in\mathfrak{M}$ we have $\det(\eta)=1$. To be more precise, each $\eta$ with length $l$ is written as $$\eta=w_{\alpha_1}\cdots w_{\alpha_l}$$ where $w_{\alpha_i}$ is a simple root refection corresponding to a simple root $\alpha_i$ and is the matrix of the form $$w_{\alpha_i}=\begin{pmatrix} \ddots&&&\\ &&-1&\\ &1&&\\ &&&\ddots \end{pmatrix}.$$ Though the set $\mathfrak{M}$ is not a group, it has the advantage that we can compute the cocycle $\sigma_r$ in a systematic way as one can see in [@BLS]. For each $w\in W$, we denote by $\eta_w$ the corresponding element in ${\mathfrak{M}}$. If $w\in W_M$, one can see that $\eta_w$ is of the form $(\varepsilon_j\delta_{\sigma(i),j}I_{r_j})$ for $\varepsilon_j\in\{\pm 1\}$. Namely $\eta_w$ is a $k\times k$ block matrix in which the non-zero entries are either $I_{r_j}$ or $-I_{r_j}$. [**Acknowledgements**]{} The author would like to thanks Paul Mezo for reading an early draf and giving him helpful comments, and Jeff Adams for sending him the preprint [@Adams] and explaining the construction of the metaplectic tensor product for the real case. The author is partially supported by NSF grant DMS-1215419. Also part of this research was done when he was visiting the I.H.E.S. in the summer of 2012 and he would like to thank their hospitality. **The metaplectic cover ${\widetilde{\operatorname{GL}}}_r$ of ${\operatorname{GL}}_r$** {#S:metaplectic_cover} ======================================================================================== In this section, we review the theory of the metaplectic $n$-fold cover ${\widetilde{\operatorname{GL}}}_r$ of ${\operatorname{GL}}_r$ for both local and global cases, which was originally constructed by Kazhdan and Patterson in [@KP]. **The local metaplectic cover ${\widetilde{\operatorname{GL}}}_r(F)$** ---------------------------------------------------------------------- Let $F$ be a (not necessarily non-archimedean) local field of characteristic $0$ which contains all the $n^\text{th}$ roots of unity. In this paper, by the metaplectic $n$-fold cover ${\widetilde{\operatorname{GL}}}_r(F)$ of ${\operatorname{GL}}_r(F)$ with a fixed parameter $c\in\{0,\dots,n-1\}$, we mean the central extension of ${\operatorname{GL}}_r(F)$ by $\mu_n$ as constructed by Kazhdan and Patterson in [@KP]. To be more specific, let us first recall that the $n$-fold cover ${\widetilde{\operatorname{SL}}}_{r+1}(F)$ of ${\operatorname{SL}}_{r+1}(F)$ was constructed by Matsumoto in [@Matsumoto], and there is an embedding $$\label{E:embedding0} l_0:{\operatorname{GL}}_r(F)\rightarrow{\operatorname{SL}}_{r+1}(F),\quad g\mapsto\begin{pmatrix}\det(g)^{-1}&\\ &g\end{pmatrix}.$$ Our metaplectic $n$-fold cover ${\widetilde{\operatorname{GL}}}_r(F)$ with $c=0$ is the preimage of $l_0({\operatorname{GL}}_r(F))$ via the canonical projection ${\widetilde{\operatorname{SL}}}_{r+1}(F)\rightarrow{\operatorname{SL}}_{r+1}(F)$. Then ${\widetilde{\operatorname{GL}}}_r(F)$ is defined by a 2-cocycle $$\sigma_r:{\operatorname{GL}}_r(F)\times{\operatorname{GL}}_r(F)\rightarrow\mu_n.$$ For arbitrary parameter $c\in\{0,\dots, n-1\}$, we define the twisted cocycle $\sigma_r^{(c)}$ by $$\sigma_r^{(c)}(g, g')=\sigma_r(g, g')(\det(g),\det(g'))^c_F$$ for $g, g'\in{\operatorname{GL}}_r(F)$, where recall from the notation section that $(-,-)_F$ is the $n^{\text{th}}$ order Hilbert symbol for $F$. The metaplectic cover with a parameter $c$ is defined by this cocycle. In [@KP], the metaplectic cover with parameter $c$ is denoted by ${\widetilde{\operatorname{GL}}}_r^{(c)}(F)$ but we avoid this notation. This is because later we will introduce the notation ${\widetilde{\operatorname{GL}}^{(n)}}_r(F)$, which has a completely different meaning. Also we suppress the superscript $(c)$ from the notation of the cocycle and always agree that the parameter $c$ is fixed throughout the paper. By carefully studying Matsumoto’s construction, Banks, Levy, and Sepanski ([@BLS]) gave an explicit description of the 2-cocycle $\sigma_r$ and shows that their 2-cocycle is “block-compatible” in the following sense: For the standard $(r_1,\dots,r_k)$-parabolic of ${\operatorname{GL}}_r$, so that its Levi $M$ is of the form ${\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$ which is embedded diagonally into ${\operatorname{GL}}_r$, we have $$\begin{aligned} \label{E:compatibility} &\sigma_r(\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}, \begin{pmatrix}g'_1&&\\ &\ddots&\\ &&g'_k\end{pmatrix})\\ =&\prod_{i=1}^k\sigma_{r_i}(g_i,g_i') \prod_{1\leq i<j\leq k}(\det(g_i), \det(g_j'))_F \prod_{i\neq j}(\det(g_i), \det(g_j'))_F^c,\notag \end{aligned}$$ for all $g_i, g_i'\in{\operatorname{GL}}_{r_i}(F)$. (See [@BLS Theorem 11, §3]. Strictly speaking in [@BLS] only the case $c=0$ is considered but one can derive the above formula using the bilinearity of the Hilbert symbol.) This 2-cocycle generalizes the well-known cocycle given by Kubota [@Kubota] for the case $r=2$. Also we should note that if $r=1$, this cocycle is trivial. Note that ${\widetilde{\operatorname{GL}}}_r(F)$ is not the $F$-rational points of an algebraic group, but this notation seems to be standard. Let us list some other important properties of the cocycle $\sigma_r$, which we will use in this paper. \[P:BLS\] Let $B=TN_B$ be the Borel subgroup of ${\operatorname{GL}}_r$ where $T$ is the maximal torus and $N_B$ the unipotent radical. The cocycle $\sigma_r$ satisfies the following properties: (1) $\sigma_r(g,g')\sigma_r(gg', g'')=\sigma_r(g, g'g'')\sigma_r(g',g'')$ for $g,g',g''\in{\operatorname{GL}}_r$. (2) $\sigma_r(ng, g'n')=\sigma_r(g, g')$ for $g, g'\in{\operatorname{GL}}_r$ and $n, n'\in N_B$, and so in particular $\sigma_r(ng, n')=\sigma_r(n, g'n')=1$. (3) $\sigma_r(gn, g')=\sigma_r(g, ng')$ for $g, g'\in{\operatorname{GL}}_r$ and $n\in N_B$. (4) $\sigma(\eta, t)=\underset{\substack{\alpha=(i,j)\in\Phi^+\\ \eta\alpha<0}}{\prod}(-t_j, t_i)$ for $\eta\in{\mathfrak{M}}$ and $t={\operatorname{diag}}(t_1,\dots,t_r)\in T$, where $\Phi^+$ is the set of positive roots and each root $\alpha\in\Phi^+$ is identified with a pair of integers $(i,j)$ with $1\leq i<j\leq r$ as usual. (5) $\sigma_r(t,t')=\underset{i<j}{\prod}(t_i, t'_j)(\det(t),\det(t'))^c$ for $t={\operatorname{diag}}(t_1,\dots,t_r), t'={\operatorname{diag}}(t'_1,\dots,t'_r)\in T$. (6) $\sigma_r(t,\eta)=1$ for $t\in T$ and $\eta\in{\mathfrak{M}}$. The first one is simply the definition of 2-cocycle and all the others are some of the properties of $\sigma_r$ listed in [@BLS Theorem 7, p.153]. We need to recall how this cocycle is constructed. As mentioned earlier, Matsumoto constructed ${\widetilde{\operatorname{SL}}}_{r+1}(F)$. It is shown in [@BLS] that ${\widetilde{\operatorname{SL}}}_{r+1}(F)$ is defined by a cocycle $\sigma_{{\operatorname{SL}}_{r+1}}$ which satisfies the block-compatibility in a much stronger sense as in [@BLS Theorem 7, §2, p145]. (Note that our ${\operatorname{SL}}_{r+1}$ corresponds to ${\mathbb{G}}^\flat$ of [@BLS].) Then the cocycle $\sigma_r$ is defined by $$\sigma_r(g, g')=\sigma_{{\operatorname{SL}}_{r+1}}(l(g), l(g'))(\det(g), \det(g'))_F (\det(g), \det(g'))^c_F,$$ where $l$ is the embedding defined by $$\label{E:embedding} l:{\operatorname{GL}}_r(F)\rightarrow{\operatorname{SL}}_{r+1}(F),\quad g\mapsto\begin{pmatrix}g&\\ &\det(g)^{-1}\end{pmatrix}.$$ See [@BLS p.146]. (Note the difference between this embedding and the one in (\[E:embedding0\]). This is the reason we have the extra Hilbert symbol in the definition of $\sigma_r$.) Since we would like to emphasize the cocycle being used, we denote ${\widetilde{\operatorname{GL}}}_r(F)$ by ${{^{\sigma}\widetilde{\operatorname{GL}}}}_r(F)$ when the cocycle $\sigma$ is used. Namely ${{^{\sigma}\widetilde{\operatorname{GL}}}}_r(F)$ is the group whose underlying set is $${{^{\sigma}\widetilde{\operatorname{GL}}}}_r(F) ={\operatorname{GL}}_r(F)\times\mu_n=\{(g,\xi):g\in{\operatorname{GL}}_r(F), \xi\in\mu_n\},$$ and the group law is defined by $$(g,\xi)\cdot (g',\xi')=(gg',\sigma_r(g, g')\xi\xi').$$ To use the block-compatible 2-cocycle of [@BLS] has obvious advantages. In particular, it has been explicitly computed and, of course, it is block-compatible. Indeed, when we consider purely local problems, we always assume that the cocycle $\sigma_r$ is used. However it does not allow us to construct the global metaplectic cover ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$. Namely one cannot define the adelic block-combatible 2-cocycle simply by taking the product of the local block-combatible 2-cocycles over all the places. Namely for $g, g'\in{\operatorname{GL}}_r({\mathbb{A}})$, the product $$\prod_v\sigma_{r,v}(g_v,g_v')$$ is not necessarily finite. This can be already observed for the case $r=2$. (See [@F p.125].) For this reason, we will use a different 2-cocycle $\tau_r$ which works nicely with the global metaplectic cover ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$. To construct such $\tau_r$, first assume $F$ is non-archimedean. It is known that an open compact subgroup $K$ splits in ${\widetilde{\operatorname{GL}}}_r(F)$, and moreover if $|n|_F=1$, we have $K={\operatorname{GL}}_r({\mathcal{O}_F})$. (See [@KP Proposition 0.1.2].) Also for $k,k'\in K$, a property of the Hilbert symbol gives $(\det(k),\det(k'))_F=1$. Hence one has a continuous map $s_r:{\operatorname{GL}}_r(F)\rightarrow\mu_n$ such that $\sigma_r(k,k')s_r(k)s_r(k')=s_r(kk')$ for all $k,k'\in K$. Then define our 2-cocycle $\tau_r$ by $$\label{E:tau_sigma} \tau_r(g, g'):=\sigma_r(g, g')\cdot\frac{s_r(g)s_r(g')}{s_r(gg')}$$ for $g, g'\in{\operatorname{GL}}_r(F)$. If $F$ is archimedean, we set $\tau_r=\sigma_r$. The choice of $s_r$ and hence $\tau_r$ is not unique. However when $|n|_F=1$, there is a canonical choice with respect to the splitting of $K$ in the following sense: Assume that $F$ is such that $|n|_F=1$. Then the Hilbert symbol $(-,-)_F$ is trivial on ${\mathcal{O}_F}^\times\times{\mathcal{O}_F}^\times$, and hence, when restricted to ${\operatorname{GL}}_r({\mathcal{O}_F})\times{\operatorname{GL}}_r({\mathcal{O}_F})$, the cocycle $\sigma_r$ is the restriction of $\sigma_{{\operatorname{SL}}_{r+1}}$ to the image of the embedding $l$. Now it is known that the compact group ${\operatorname{SL}}_{r+1}({\mathcal{O}_F})$ also splits in ${\widetilde{\operatorname{SL}}}_{r+1}(F)$, and hence there is a map ${\mathfrak{s}}_r:{\operatorname{SL}}_{r+1}(F)\rightarrow\mu_n$ such that the section ${\operatorname{SL}}_{r+1}(F)\rightarrow{\widetilde{\operatorname{SL}}}_{r+1}(F)$ given by $(g,{\mathfrak{s}}_r(g))$ is a homomorphism on ${\operatorname{SL}}_{r+1}({\mathcal{O}_F})$. (Here we are assuming ${\widetilde{\operatorname{SL}}}_{r+1}(F)$ is realized as ${\operatorname{SL}}_{r+1}(F)\times\mu_n$ as a set and the group structure is defined by the cocycle $\sigma_{{\operatorname{SL}}_{r+1}}$.) Moreover ${\mathfrak{s}}_r|_{{\operatorname{SL}}_{r+1}({\mathcal{O}_F})}$ is determined up to twists by the elements in $H^1({\operatorname{SL}}_{r+1}({\mathcal{O}_F}), \mu_n)={\operatorname{Hom}}({\operatorname{SL}}_{r+1}({\mathcal{O}_F}), \mu_n)$. But ${\operatorname{Hom}}({\operatorname{SL}}_{r+1}({\mathcal{O}_F}), \mu_n)=1$ because ${\operatorname{SL}}_{r+1}({\mathcal{O}_F})$ is a perfect group and $\mu_n$ is commutative. Hence ${\mathfrak{s}}_r|_{{\operatorname{SL}}_{r+1}({\mathcal{O}_F})}$ is unique. (See also [@KP p. 43] for this matter.) We choose $s_r$ so that $$\label{E:canonical_section} s_r|_{{\operatorname{GL}}_r({\mathcal{O}_F})}={\mathfrak{s}}_r|_{l({\operatorname{GL}}_r({\mathcal{O}_F}))}.$$ With this choice, we have the commutative diagram $$\label{E:canonical_diagram} \xymatrix{ {{^{\sigma}\widetilde{\operatorname{GL}}}}_r({\mathcal{O}_F})\ar[r]&{\widetilde{\operatorname{SL}}}_{r+1}({\mathcal{O}_F})\\ K\ar[r]\ar[u]^{k\mapsto (k,\;s_r(k))}&{\operatorname{SL}}_{r+1}({\mathcal{O}_F}),\ar[u]_{k\mapsto(k,\;{\mathfrak{s}}_r(k))} }$$ where the top arrow is $(g,\xi)\mapsto (\l(g),\xi)$, the bottom arrow is $l$, and all the arrows can be seen to be homomorphisms. This choice of $s_r$ will be crucial for constructing the metaplectic tensor product of automorphic representations. Also note that the left vertical arrow in the above diagram is what is called the canonical lift in [@KP] and denoted by $\kappa^\ast$ there. (Although we do not need this fact in this paper, if $r=2$ one can show that $\tau_r$ can be chosen to be block compatible, which is the cocycle used in [@F].) Using $\tau_r$, we realize ${\widetilde{\operatorname{GL}}}_r(F)$ as $${\widetilde{\operatorname{GL}}}_r(F)={\operatorname{GL}}_r(F)\times\mu_n,$$ as a set and the group law is given by $$(g,\xi)\cdot(g',\xi')=(gg', \tau_r(g,g')\xi\xi').$$ Note that we have the exact sequence $$\xymatrix{ 0\ar[r]&\mu_n\ar[r]&{\widetilde{\operatorname{GL}}}_r(F)\ar[r]^{p}&{\operatorname{GL}}_r(F)\ar[r]& 0 }$$ given by the obvious maps, where we call $p$ the canonical projection. We define a set theoretic section $$\kappa:{\operatorname{GL}}_r(F)\rightarrow{\widetilde{\operatorname{GL}}}_r(F),\; g\mapsto (g,1).$$ Note that $\kappa$ is not a homomorphism. But by our construction of the cocycle $\tau_r$, $\kappa|_K$ is a homomorphism if $F$ is non-archimedean and $K$ is a sufficiently small open compact subgroup, and moreover if $|n|_F=1$, one has $K={\operatorname{GL}}_r({\mathcal{O}_F})$. Also we define another set theoretic section $${\mathbf{s}}_r:{\operatorname{GL}}_r(F)\rightarrow{\widetilde{\operatorname{GL}}}_r(F),\; g\mapsto (g,s_r(g)^{-1})$$ where $s_r(g)$ is as above, and then we have the isomorphism $${\widetilde{\operatorname{GL}}}_r(F)\rightarrow{{^{\sigma}\widetilde{\operatorname{GL}}}}_r(F),\quad (g,\xi)\mapsto (g,s_r(g)\xi),$$ which gives rise to the commutative diagram $$\xymatrix{ {\widetilde{\operatorname{GL}}}_r(F)\ar[rr]&&{{^{\sigma}\widetilde{\operatorname{GL}}}}_r(F)\\ &{\operatorname{GL}}_r(F)\ar[ul]^{{\mathbf{s}}_r}\ar[ur]_{g\mapsto (g,1)}& }$$ of set theoretic maps. Also note that the elements in the image ${\mathbf{s}}_r({\operatorname{GL}}_r(F))$ “multiply via $\sigma_r$” in the sense that for $g,g'\in{\operatorname{GL}}_r(F)$, we have $$\label{E:convenient} (g,s_r(g)^{-1}) (g',s_r(g')^{-1})=(gg', \sigma_r(g,g')s_r(gg')^{-1}).$$ Let us mention Assume $F$ is non-archimedean with $|n|_F=1$. We have $$\label{E:kappa_and_s} \kappa|_{T\cap K}={\mathbf{s}}_r|_{T\cap K},\quad \kappa|_{W}={\mathbf{s}}_r|_{W},\quad \kappa|_{N_B\cap K}={\mathbf{s}}_r|_{N_B\cap K},$$ where $W$ is the Weyl group and $K={\operatorname{GL}}_r({\mathcal{O}_F})$. In particular, this implies $s_r|_{T\cap K}=s_r|_{W}=s_r|_{N_B\cap K}=1$. See [@KP Proposition 0.I.3]. Though we do not need this fact in this paper, it should be noted that ${\mathbf{s}}_r$ splits the Weyl group $W$ if and only if $(-1,-1)_F=1$. So in particular it splits $W$ if $|n|_F=1$. See [@BLS §5]. If $P$ is a parabolic subgroup of ${\operatorname{GL}}_r$ whose Levi is $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$, we often write $${\widetilde{M}}(F)={\widetilde{\operatorname{GL}}}_{r_1}(F){\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{\operatorname{GL}}}_{r_k}(F)$$ for the metaplectic preimage of $M(F)$. Next let $${\operatorname{GL}}_r^{(n)}(F)=\{g\in{\operatorname{GL}}_r(F):\det g\in F^{\times n}\},$$ and ${\widetilde{\operatorname{GL}}^{(n)}}_r(F)$ its metaplectic preimage. Also we define $$M^{(n)}(F)=\{(g_1,\dots,g_k)\in M(F): \det g_i\in F^{\times n}\}$$ and often denote its preimage by $${\widetilde{M}^{(n)}}(F)={\widetilde{\operatorname{GL}}^{(n)}}_{r_1}(F){\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}(F).$$ The group ${\widetilde{M}^{(n)}}(F)$ is a normal subgroup of finite index. Indeed, we have the exact sequence $$\label{E:finite_quotient} 1\rightarrow{\widetilde{M}^{(n)}}(F)\rightarrow{\widetilde{M}}(F)\rightarrow \underbrace{F^{\times n}\backslash F^\times\times\cdots\times F^{\times n}\backslash F^\times}_{\text{$k$ times}} \rightarrow 1,$$ where the third map is given by $({\operatorname{diag}}(g_1,\dots,g_k),\xi)\mapsto(\det(g_1),\dots,\det(g_k))$. We should mention the explicit isomorphism $F^{\times n}\backslash F^\times\times\cdots\times F^{\times n}\backslash F^\times\rightarrow{\widetilde{M}^{(n)}}(F)\backslash{\widetilde{M}}(F)$ defined as follows: First for each $i\in\{1,\dots,k\}$, define a map $\iota_i:F^\times\rightarrow{\operatorname{GL}}_{r_i}$ by $$\label{E:iota} \iota_i(a)=\begin{pmatrix}a&\\ &I_{r_i-1}\end{pmatrix}.$$ Then the map given by $$(a_1,\dots,a_k)\mapsto(\begin{pmatrix}\iota_1(a_1)&&\\ &\ddots& \\ &&\iota_k(a_k)\end{pmatrix}, 1)$$ is a homomorphism. Clearly the map is well-defined and 1-1. Moreover this is surjective because each element $g_i\in{\operatorname{GL}}_{r_i}$ is written as $$g_i=g_i\iota_i(\det(g_i)^{n-1})\iota_i(\det(g_i)^{1-n})$$ and $g_i\iota_i(\det(g_i)^{n-1})\in{\operatorname{GL}}_{r_i}^{(n)}$. The following should be mentioned. \[L:closed\_subgroup\_local\] The groups $F^{\times n}, M^{(n)}(F)$ and ${\widetilde{M}^{(n)}}(F)$ are closed subgroups of $F^\times, M(F)$ and ${\widetilde{M}}(F)$, respectively. It is well-known that $F^{\times n}$ is closed and of finite index in $F^\times$. Hence the group $F^{\times n}\backslash F^\times\times\cdots\times F^{\times n}\backslash F^\times$ is discrete, in particular Hausdorff. But both ${\widetilde{M}^{(n)}}(F)\backslash{\widetilde{M}}(F)$ and $M^{(n)}(F)\backslash M(F)$ are, as topological groups, isomorphic to this Hausdorff space. This completes the proof. \[R:archimedean1\] If $F={\mathbb C}$, clearly ${\widetilde{M}^{(n)}}(F)={\widetilde{M}}(F)$. If $F={\mathbb{R}}$, then necessarily $n=2$ and ${\operatorname{GL}}_r^{(2)}({\mathbb{R}})$ consists of the elements of positive determinants, which is usually denote by ${\operatorname{GL}}_r^+({\mathbb{R}})$. Accordingly one may denote ${\widetilde{\operatorname{GL}}^{(n)}}_r({\mathbb{R}})$ and ${\widetilde{M}^{(n)}}({\mathbb{R}})$ by ${\widetilde{\operatorname{GL}}}_r^+({\mathbb{R}})$ and ${{\widetilde{M}}}^+({\mathbb{R}})$ respectively. Both ${\widetilde{\operatorname{GL}}}_r^+({\mathbb{R}})$ and ${\widetilde{\operatorname{GL}}}_r({\mathbb{R}})$ share the identity component, and hence they have the same Lie algebra. The same applies to ${\widetilde{M}}^+({\mathbb{R}})$ and ${\widetilde{M}}({\mathbb{R}})$. Let us mention the following important fact. Let $Z_{{\operatorname{GL}}_r}(F)\subseteq{\operatorname{GL}}_r(F)$ be the center of ${\operatorname{GL}}_r(F)$. Then its metaplectic preimage $\widetilde{Z_{{\operatorname{GL}}_r}}(F)$ is not the center of ${\widetilde{\operatorname{GL}}}_r(F)$ in general. (It might not be even commutative for $n>2$.) The center, which we denote by $Z_{{\widetilde{\operatorname{GL}}}_r}(F)$, is $$\begin{aligned} \label{E:center_GLt} Z_{{\widetilde{\operatorname{GL}}}_r}(F)&=\{(aI_r, \xi):a^{r-1+2rc}\in F^{\times n}, \xi\in\mu_n\}\\ \notag&=\{(aI_r, \xi):a\in F^{\times \frac{n}{d}}, \xi\in\mu_n\},\end{aligned}$$ where $d=\gcd(r-1+2c, n)$. (The second equality is proven in [@GO Lemma 1].) Note that $Z_{{\widetilde{\operatorname{GL}}}_r}(F)$ is a closed subgroup. Let $\pi$ be an admissible representation of a subgroup $\widetilde{H}\subseteq {\widetilde{\operatorname{GL}}}_r(F)$, where ${\widetilde{H}}$ is the metaplectic preimage of a subgroup $H\subseteq{\operatorname{GL}}_r(F)$. We say $\pi$ is “genuine” if each element $(1,\xi)\in\widetilde{H}$ acts as multiplication by $\xi$, where we view $\xi$ as an element of ${\mathbb C}$ in the natural way. **The global metaplectic cover ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$** {#S:group} ---------------------------------------------------------------------------------- In this subsection we consider the global metaplectic group. So we let $F$ be a number field which contains all the $n^\text{th}$ roots of unity and ${\mathbb{A}}$ the ring of adeles. Note that if $n>2$, then $F$ must be totally complex. We shall define the $n$-fold metaplectic cover ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ of ${\operatorname{GL}}_r({\mathbb{A}})$. (Just like the local case, we write ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ even though it is not the adelic points of an algebraic group.) The construction of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ has been done in various places such as [@KP; @FK]. First define the adelic 2-cocycle $\tau_r$ by $$\tau_r(g, g'):=\prod_v\tau_{r,v}({g}_v, g'_v),$$ for $g, g'\in{\operatorname{GL}}_r({\mathbb{A}})$, where $\tau_{r,v}$ is the local cocycle defined in the previous subsection. By definition of $\tau_{r,v}$, we have $\tau_{r,v}(g_v, g'_v)=1$ for almost all $v$, and hence the product is well-defined. We define ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ to be the group whose underlying set is ${\operatorname{GL}}_r({\mathbb{A}})\times\mu_n$ and the group structure is defined via $\tau_r$ as in the local case, [[*i.e.* ]{}]{}$$(g, \xi)\cdot(g', \xi')=(gg', \tau_r(g, g')\xi\xi'),$$ for $g, g'\in{\operatorname{GL}}_r({\mathbb{A}})$, and $\xi, \xi'\in\mu_n$. Just as the local case, we have $$\xymatrix{ 0\ar[r]&\{\pm1\}\ar[r]&{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})\ar[r]^{p}&{\operatorname{GL}}_r({\mathbb{A}})\ar[r]&0, }$$ where we call $p$ the canonical projection. Define a set theoretic section $\kappa:{\operatorname{GL}}_r({\mathbb{A}})\rightarrow{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ by $g\mapsto(g,1)$. It is well-known that ${\operatorname{GL}}_r(F)$ splits in ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$. However the splitting is not via $\kappa$. In what follows, we will see that the splitting is via the product of all the local ${\mathbf{s}}_r$. Let us start with the following “product formula” of $\sigma_r$. \[P:product\_formula\] For $g, g'\in{\operatorname{GL}}_r(F)$, we have $\sigma_{r,v}(g, g')=1$ for almost all $v$, and further $$\prod_v\sigma_{r,v}(g, g')=1.$$ From the explicit description of the cocycle $\sigma_{r,v}(g, g')$ given at the end of $\S 4$ of [@BLS], one can see that $\sigma_{r,v}(g, g')$ is written as a product of Hilbert symbols of the form $(t, t')_{F_v}$ for $t, t'\in F^\times$. This proves the first part of the proposition. The second part follows from the product formula for the Hilbert symbol. If $g\in{\operatorname{GL}}_r(F)$, then we have $s_{r,v}(g)=1$ for almost all $v$, where $s_{r,v}$ is the map $s_{r,v}:{\operatorname{GL}}(F_v)\rightarrow\mu_n$ defining the local section ${\mathbf{s}}_r:{\operatorname{GL}}(F_v)\rightarrow{\widetilde{\operatorname{GL}}}_r(F_v)$. By the Bruhat decomposition we have $g=bwb'$ for some $b, b'\in B(F)$ and $w\in W$. Then for each place $v$ $$\begin{aligned} s_{r,v}(g) &=s_{r,v}(bwb')\\ &=\sigma_{r,v}(b, wb')s_{r,v}(b)s_{r,v}(wb')/\tau_{r,v}(b,wb')\quad\text{by (\ref{E:tau_sigma})}\\ &=\sigma_{r,v}(b, wb')s_{r,v}(b) \sigma_{r,v}(w, b')s_{r,v}(w)s_{r,v}(b')/\tau_{r,v}(w,b')\tau_{r,v}(b,wb') \quad\text{again by (\ref{E:tau_sigma})}.\end{aligned}$$ By the previous proposition, $\sigma_{r,v}(b, wb')=\sigma_{r,v}(w, b')=1$ for almost all $v$. By (\[E:kappa\_and\_s\]) we know $s_{r,v}(b)=s_{r,v}(w)=s_{r,v}(b')=1$ for almost all $v$. Finally by definition of $\tau_{r,v}$, $\tau_{r,v}(w,b')=\tau_{r,v}(b,wb')=1$ for almost all $v$. This proposition implies that the expression $$s_r(g):=\prod_vs_{r,v}(g)$$ makes sense for all $g\in{\operatorname{GL}}_r(F)$, and one can define the map $${\mathbf{s}}_r:{\operatorname{GL}}_r(F)\rightarrow{\widetilde{\operatorname{GL}}}_r({\mathbb{A}}),\quad g\mapsto (g, s_r(g)^{-1}).$$ Moreover, this is a homomorphism because of Proposition \[P:product\_formula\] and (\[E:convenient\]). Unfortunately, however, the expression $\prod_vs_{r,v}(g_v)$ does not make sense for every $g\in{\operatorname{GL}}_r({\mathbb{A}})$ because one does not know whether $s_{r,v}(g_v)=1$ for almsot all $v$. Yet, we have \[P:s\_split\] The expression $s_r(g)=\prod_vs_{r,v}(g_v)$ makes sense when $g$ is in ${\operatorname{GL}}_r(F)$ or $N_B({\mathbb{A}})$, so ${\mathbf{s}}_r$ is defined on ${\operatorname{GL}}_r(F)$ and $N_B({\mathbb{A}})$. Moreover, ${\mathbf{s}}_r$ is indeed a homomorphism on ${\operatorname{GL}}_r(F)$ and $N_B({\mathbb{A}})$. Also if $g\in{\operatorname{GL}}_r(F)$ and $n\in N_B({\mathbb{A}})$, both $s_r(gn)$ and $s_r(ng)$ make sense and further we have ${\mathbf{s}}_r(gn)={\mathbf{s}}_r(g){\mathbf{s}}_r(n)$ and ${\mathbf{s}}_r(ng)={\mathbf{s}}_r(n){\mathbf{s}}_r(g)$. We already know $s_r(g)$ is defined and ${\mathbf{s}}_r$ is a homomorphism on ${\operatorname{GL}}_r(F)$. Also $s_r(n)$ is defined thanks to (\[E:kappa\_and\_s\]) and ${\mathbf{s}}_r$ is a homomorphism on $N_B({\mathbb{A}})$ thanks to Proposition \[P:BLS\] (1). Moreover for all places $v$, we have $\sigma_{r,v}(g_v, n_v)=1$ again by Proposition \[P:BLS\] (1). Hence for all $v$, $s_{r,v}(gn_v)=s_{r,v}(g)s_{r,v}(n_v)/\tau_{r,v}(g,n_v)$. For almost all $v$, the right hand side is $1$. Hence the global $s_r(gn)$ is defined. Also this equality shows that ${\mathbf{s}}_r(gn)={\mathbf{s}}_r(g){\mathbf{s}}_r(n)$. The same argument works for $ng$. If $H\subseteq{\operatorname{GL}}_r({\mathbb{A}})$ is a subgroup on which ${\mathbf{s}}_r$ is not only defined but also a group homomorphism, we write $H^\ast:={\mathbf{s}}_r(H)$. In particular we have $$\label{E:star} {\operatorname{GL}}_r(F)^\ast:={\mathbf{s}}_r({\operatorname{GL}}_r(F))\quad\text{and}\quad N_B({\mathbb{A}})^\ast:={\mathbf{s}}_r(N_B({\mathbb{A}})).$$ We define the groups like ${\widetilde{\operatorname{GL}}^{(n)}}_r({\mathbb{A}})$, ${\widetilde{M}}({\mathbb{A}})$, ${\widetilde{M}^{(n)}}({\mathbb{A}})$, etc completely analogously to the local case. Let us mention \[L:closed\_subgroup\_global\] The groups ${\mathbb{A}}^{\times n}, M^{(n)}({\mathbb{A}})$ and ${\widetilde{M}^{(n)}}({\mathbb{A}})$ are closed subgroups of ${\mathbb{A}}^\times, M({\mathbb{A}})$ and ${\widetilde{M}}({\mathbb{A}})$, respectively. That ${\mathbb{A}}^{\times n}$ and $M^{(n)}({\mathbb{A}})$ are closed follows from the following lemma together with Lemma \[L:closed\_subgroup\_local\]. Once one knows $M^{(n)}({\mathbb{A}})$ is closed, one will know ${\widetilde{M}^{(n)}}({\mathbb{A}})$ is closed because it is the preimage of the closed $M^{(n)}({\mathbb{A}})$ under the canonical projection, which is continuous. \[L:closed\_subgroup\_local\_global\] Let $G$ be an algebraic group over $F$ and $G({\mathbb{A}})$ its adelic points. Let $H\subseteq G({\mathbb{A}})$ be a subgroup such that $H$ is written as $H=\prod'_vH_v$ (algebraically) where for each place $v$, $H_v:=H\cap G(F_v)$ is a closed subgroup of $G(F_v)$. Then $H$ is closed. Let $(x_i)_{i\in I}$ be a net in $H$ that converges in $G({\mathbb{A}})$, where $I$ is some index set. Let $g=\lim_{i\in I} x_i$. Assume $g\notin H$. Then there exists a place $w$ such that $g_w\notin H_w$. Since $H_w$ is closed, the set $U_w:=G(F_w)\backslash H_w$ is open. Then there exists an open neighborhood $U$ of $g$ of the form $U=\prod_v U_v$, where $U_v$ is some open neighborhood of $g_v$ and at $v=w$, $U_v=U_w$. But for any $i\in I$, $x_i\notin U$ because $x_{i, w}\notin U_w$, which contradicts the assumption that $g=\lim_{i\in I} x_i$. Hence $g\in H$, which shows $H$ is closed. Just like the local case, the preimage $\widetilde{Z_{{\operatorname{GL}}_r}}({\mathbb{A}})$ of the center $Z_{{\operatorname{GL}}_r}({\mathbb{A}})$ of ${\operatorname{GL}}_r({\mathbb{A}})$ is in general not the center of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ but the center, which we denote by $Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})$, is $$\begin{aligned} Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})&=\{(aI_r, \xi):a^{r-1+2rc}\in {\mathbb{A}}^{\times n}, \xi\in\mu_n\}\\ &=\{(aI_r, \xi):a\in {\mathbb{A}}^{\times \frac{n}{d}}, \xi\in\mu_n\},\end{aligned}$$ where $d=\gcd(r-1+2c, n)$. The center is a closed subgroup of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$. We can also describe ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ as a quotient of a restricted direct product of the groups ${\widetilde{\operatorname{GL}}}_r(F_v)$ as follows. Consider the restricted direct product $\prod_v'{\widetilde{\operatorname{GL}}}_r(F_v)$ with respect to the groups $\kappa(K_v)=\kappa({\operatorname{GL}}_r(\mathcal{O}_{F_v}))$ for all $v$ with $v\nmid n$ and $v\nmid\infty$. If we denote each element in this restricted direct product by $\Pi'_v(g_v,\xi_v)$ so that $g_v\in K_v$ and $\xi_v=1$ for almost all $v$, we have the surjection $$\label{E:surjection} \rho:{\prod_v}'{\widetilde{\operatorname{GL}}}_r(F_v)\rightarrow{\widetilde{\operatorname{GL}}}_r({\mathbb{A}}),\quad \Pi'_v(g_v,\xi_v)\mapsto (\Pi'_vg_v, \Pi_v\xi_v),$$ where the product $\Pi_v\xi_v$ is literary the product inside $\mu_n$. This is a group homomorphism because $\tau_r=\prod_v\tau_{r,v}$ and the groups ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ and ${\widetilde{\operatorname{GL}}}_r(F_v)$ are defined, respectively, by $\tau_r$ and $\tau_{r,v}$. We have $${\prod_v}'{\widetilde{\operatorname{GL}}}_r(F_v)/\ker\rho\cong {\widetilde{\operatorname{GL}}}_r({\mathbb{A}}),$$ where $\ker\rho$ consists of the elements of the form $(1,\xi)$ with $\xi\in\prod'_v\mu_n$ and $\Pi_v\xi_v=1$. Let $\pi$ be a representation of $\widetilde{H}\subseteq {\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ where ${\widetilde{H}}$ is the metaplectic preimage of a subgroup $H\subseteq{\operatorname{GL}}_r({\mathbb{A}})$. Just like the local case, we call $\pi$ genuine if $(1,\xi)\in\widetilde{H}({\mathbb{A}})$ acts as multiplication by $\xi$ for all $\xi\in\mu_n$. Also we have the notion of automorphic representation as well as automorphic form on ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ or ${\widetilde{M}}({\mathbb{A}})$. In this paper, by an automorphic form, we mean a smooth automorphic form instead of a $K$-finite one, namely an automorphic form is $K_f$-finite, ${\mathcal{Z}}$-finite and of uniformly moderate growth. (See [@Cogdell p.17].) Hence if $\pi$ is an automorphic representation of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ (or ${\widetilde{M}}({\mathbb{A}})$), the full group ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ (or ${\widetilde{M}}({\mathbb{A}})$) acts on $\pi$. An automorphic form $f$ on ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ (or ${\widetilde{M}}({\mathbb{A}})$) is said to be genuine if $f(g,\xi)=\xi f(g,1)$ for all $(g,\xi)\in{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ (or ${\widetilde{M}}({\mathbb{A}})$). In particular every automorphic form in the space of a genuine automorphic representation is genuine. Suppose we are given a collection of irreducible admissible representations $\pi_v$ of ${\widetilde{\operatorname{GL}}}_r(F_v)$ such that $\pi_v$ is $\kappa(K_v)$-spherical for almost all $v$. Then we can form an irreducible admissible representation of $\prod_v'{\widetilde{\operatorname{GL}}}_r(F_v)$ by taking a restricted tensor product $\otimes_v'\pi_v$ as usual. Suppose further that $\ker\rho$ acts trivially on $\otimes_v'\pi_v$, which is always the case if each $\pi_v$ is genuine. Then it descends to an irreducible admissible representation of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$, which we denote by ${\widetilde{\otimes}}'_v\pi_v$, and call it the “metaplectic restricted tensor product”. Let us emphasize that the space for ${\widetilde{\otimes}}'_v\pi_v$ is the same as that for $\otimes_v'\pi_v$. Conversely, if $\pi$ is an irreducible admissible representation of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$, it is written as ${\widetilde{\otimes}}'_v\pi_v$ where $\pi_v$ is an irreducible admissible representation of ${\widetilde{\operatorname{GL}}}_r(F_v)$, and for almost all $v$, $\pi_v$ is $\kappa(K_v)$-spherical. (To see it, view $\pi$ as a representation of the restricted product $\prod_v'{\widetilde{\operatorname{GL}}}_r(F_v)$ by pulling it back by $\rho$ as in (\[E:surjection\]) and apply the usual tensor product theorem for the restricted direct product. This gives the restricted tensor product $\otimes_v'\pi_v$, where each $\pi_v$ is genuine, and hence it descends to ${\widetilde{\otimes}}_v'\pi_v$.) Finally in this section, let us mention that we define $$\label{L:Hasse} {\operatorname{GL}}_r^{(n)}(F):={\operatorname{GL}}_r(F)\cap{\operatorname{GL}}_r^{(n)}({\mathbb{A}}),$$ namely ${\operatorname{GL}}_r^{(n)}(F)=\{g\in{\operatorname{GL}}_r(F):\det g\in {\mathbb{A}}^{\times n}\}$. The author does not know if this is equal to $\{g\in{\operatorname{GL}}_r(F):\det g\in F^{\times n}\}$ unless $n=2$, in which case the Hasse-Minkowski theorem implies those two coincide. Similarly we define $$M^{(n)}(F)=M(F)\cap M^{(n)}({\mathbb{A}}).$$ **The metaplectic cover ${\widetilde{M}}$ of the Levi $M$** {#S:Levi} =========================================================== Both locally and globally, one cannot show the cocycle $\tau_r$ has the block-compatibility as in (\[E:compatibility\]) (except when $r=2$). Yet, in order to define the metaplectic tensor product, it seems to be necessary to have the block-compatibility of the cocycle. To get round it, we will introduce another cocycle $\tau_M$, but this time it is a cocycle only on the Levi $M$, and will show that $\tau_M$ is cohomologous to the restriction $\tau_r|_{M\times M}$ of $\tau_r$ to $M\times M$ both for the local and global cases. **The cocycle $\tau_M$** ------------------------ In this subsection, we assume that all the groups are over $F$ if $F$ is local and over ${\mathbb{A}}$ if $F$ is global, and suppress it from our notation. We define the cocycle $$\tau_M: M\times M\rightarrow\mu_n,$$ by $$\tau_M(\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}, \begin{pmatrix}g'_1&&\\ &\ddots&\\ &&g'_k\end{pmatrix}) =\prod_{i=1}^k\tau_{r_i}(g_i,g_i')\prod_{1\leq i<j\leq k}(\det(g_i), \det(g_j')) \prod_{i\neq j}(\det(g_i), \det(g_j'))^c,$$ where $(-,-)$ is the local or global Hilbert symbol. Note that the definition makes sense both locally and globally. Moreover the global $\tau_M$ is the product of the local ones. We define the group ${{^c\widetilde{M}}}$ to be $${{^c\widetilde{M}}}=M\times\mu_n$$ as a set and the group structure is given by $\tau_M$. The superscript $^c$ is for “compatible”. One advantage to work with ${{^c\widetilde{M}}}$ is that each ${\widetilde{\operatorname{GL}}}_{r_i}$ embeds into ${{^c\widetilde{M}}}$ via the natural map $$(g_i,\xi)\mapsto(\begin{pmatrix}I_{r_1+\cdots+r_{i-1}}&&\\ &g_i&\\ &&I_{r_{i+1}+\cdots+r_k}\end{pmatrix}, \xi).$$ Indeed, the cocycle $\tau_M$ is so chosen that we have this embedding. Also recall our notation $$M^{(n)}={\operatorname{GL}}_{r_1}^{(n)}\times\cdots\times{\operatorname{GL}}_{r_k}^{(n)},$$ and $${\widetilde{M}^{(n)}}={\widetilde{\operatorname{GL}}^{(n)}}_{r_1}{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}.$$ We define ${{^c\widetilde{M}^{(n)}}}$ analogously to ${{^c\widetilde{M}}}$, namely the group structure of ${{^c\widetilde{M}^{(n)}}}$ is defined via the cocycle $\tau_M$. Of course, ${{^c\widetilde{M}^{(n)}}}$ is a subgroup of ${{^c\widetilde{M}}}$. Note that each ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}$ naturally embeds into ${{^c\widetilde{M}^{(n)}}}$ as above. The subgroups ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}$ and ${\widetilde{\operatorname{GL}}^{(n)}}_{r_j}$ in ${{^c\widetilde{M}^{(n)}}}$ commute pointwise for $i\neq j$. Locally or globally, it suffices to show $\tau_M(g_i,g_j)=\tau_M(g_j,g_i)$ for $g_i\in{\operatorname{GL}}^{(n)}_{r_i}$ and $g_j\in{\operatorname{GL}}^{(n)}_{r_j}$. But the block-compatibility of the 2-cocycle $\tau_M$, we have $\tau_M(g_i,g_j)=\tau_{r_i}(g_i, I_{r_j})\tau_{r_j}(I_{r_j}, g_j)=1$, and similarly have $\tau_M(g_j,g_i)=1$. There is a surjection $${\widetilde{\operatorname{GL}}^{(n)}}_{r_1}\times\cdots\times{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}\rightarrow\; {{^c\widetilde{M}^{(n)}}}$$ given by the map $$((g_1,\xi_1),\dots,(g_k,\xi_k))\mapsto (\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}, \xi_1\cdots\xi_k),$$ whose kernel is $$\mathcal{K}_P:=\{((1,\xi_1),\dots,(1,\xi_k)):\xi_1\cdots\xi_k=1\},$$ so that ${{^c\widetilde{M}^{(n)}}}\cong {\widetilde{\operatorname{GL}}^{(n)}}_{r_1}\times\cdots\times{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}/\mathcal{K}_P$. The block-compatibility of $\tau_M$ guarantees that the map is indeed a group homomorphism. The description of the kernel is immediate. **The relation between $\tau_M$ and $\tau_r$** ---------------------------------------------- Note that for the group ${\widetilde{M}}$ (instead of ${{^c\widetilde{M}}}$), the group structure is defined by the restriction of $\tau_r$ to $M\times M$, and hence each ${\widetilde{\operatorname{GL}}}_{r_i}$ might not embed into ${\widetilde{\operatorname{GL}}}_r$ in the natural way because of the possible failure of the block-compatibility of $\tau_r$ unless $r=2$. To make explicit the relation between ${{^c\widetilde{M}}}$ and ${\widetilde{M}}$, the discrepancy between $\tau_M$ and $\tau_r|_{M\times M}$ (which we denote simply by $\tau_r$) has to be clarified. [**Local case:**]{} Assume $F$ is local. Then we have $$\begin{aligned} &\tau_M(\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}, \begin{pmatrix}g'_1&&\\ &\ddots&\\ &&g'_k\end{pmatrix})\\ =&\sigma_r(\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}, \begin{pmatrix}g'_1&&\\ &\ddots&\\ &&g'_k\end{pmatrix}) \prod_{i=1}^k\frac{s_{r_i}(g_i)s_{r_i}(g_i')}{s_{r_i}(g_ig_i')},\end{aligned}$$ so $\tau_M$ and $\sigma_r|_{M\times M}$ are cohomologous via the function $\prod_{i=1}^ks_{r_i}$. Here recall from Section \[S:group\] that the map $s_{r_i}:{\operatorname{GL}}_{r_i}\rightarrow\mu_n$ relates $\tau_{r_i}$ with $\sigma_{r_i}$ by $$\sigma_{r_i}(g_i,g_i')=\tau_{r_i}(g_i,g_i')\cdot\frac{s_{r_i}(g_i,g_i')}{s_{r_i}(g_i)s_{r_i}(g_i')},$$ for $g_i,g_i'\in{\operatorname{GL}}_{r_i}$. Moreover if $|n|_F=1$, $s_{r_i}$ is chosen to be “canonical” in the sense that (\[E:canonical\_section\]) is satisfied. The block-compatibility of $\sigma_r$ implies $$\tau_r(m, m')\cdot\frac{s_r(mm')}{s_r(m)s_r(m')} =\sigma_r(m,m') =\tau_M(m,m')\cdot\prod_{i=1}^k\frac{s_{r_i}(g_ig_i')}{s_{r_i}(g_i)s_{r_i}(g_i')},$$ for $m=\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}$ and $m'=\begin{pmatrix}g_1'&&\\ &\ddots&\\ &&g_k'\end{pmatrix}$. Hence if we define $\hat{s}_M:M\rightarrow\mu_n$ by $$\label{E:s_hat_M} \hat{s}_M(m)=\frac{\prod_{i=1}^ks_{r_i}(g_i)}{s_r(m)},$$ we have $$\label{E:s_hat} \tau_M(m, m')=\tau_r(m, m')\cdot \frac{\hat{s}_M(m)\hat{s}_M(m')}{\hat{s}_M(mm')},$$ namely $\tau_r$ and $\tau_M$ are cohomologous via $\hat{s}_M$. Therefore we have the isomorphism $$\alpha_M:{{^c\widetilde{M}}}\rightarrow{\widetilde{M}},\quad (m,\xi)\mapsto (m, \hat{s}_M(m)\xi).$$ The following lemma will be crucial later for showing that the global $\tau_M$ is also cohomologous to $\tau_r|_{M({\mathbb{A}})\times M({\mathbb{A}})}$. \[L:s\_hat\_M\] Assume $F$ is such that $|n|_F=1$. Then for all $k\in M({\mathcal{O}_F})$, we have $\hat{s}_M(k)=1$. First note that if $k,k'\in M({\mathcal{O}_F})$, then $\tau_r(k,k')=\tau_M(k, k')=1$ and so by (\[E:s\_hat\]) we have $$\hat{s}_M(kk')=\hat{s}_M(k)\hat{s}_M(k'),$$ [[*i.e.* ]{}]{}$\hat{s}_M$ is a homomorphism on $M_M({\mathcal{O}_F})$. Hence it suffices to prove the lemma only for the elements $k\in M({\mathcal{O}_F})$ of the form $$k=\begin{pmatrix} I_{r_1+\cdots+r_{i-1}}&&\\ &k_i&\\ && I_{r_{i+1}+\cdots+r_k}\end{pmatrix}$$ where $k_i\in{\operatorname{GL}}_{r_i}$ is in the $i^{\text{th}}$ place on the diagonal. Namely we need to prove $$\frac{s_{r_i}(k_i)}{s_r(k)}=1.$$ In what follows, we will show that this follows from the “canonicality” of $s_r$ and $s_{r_i}$, and the fact that the cocycle for ${\operatorname{SL}}_{r+1}$ is block-compatible in a very strong sense as in [@BLS Lemma 5, Theorem 7 §2, p.145]. Recall from (\[E:canonical\_section\]) that $s_r$ has been chosen to satisfy $s_r|_{{\operatorname{GL}}_r({\mathcal{O}_F})}={{\mathfrak{s}}_r}|_{l({\operatorname{GL}}_r({\mathcal{O}_F}))}$, where ${\mathfrak{s}}_r$ is the map on ${\operatorname{SL}}_{r+1}(F)$ that makes the diagram (\[E:canonical\_diagram\]) commute. Similarly for $s_{r_i}$ with $r$ replaced by $r_i$. Let us write $$l_i:{\operatorname{GL}}_{r_i}(F)\rightarrow {\operatorname{SL}}_{r_i+1}(F),\quad g_i\mapsto\begin{pmatrix}g_i&\\ &\det(g_i)^{-1}\end{pmatrix}$$ for the embedding that is used to define the cocycle $\sigma_{r_i}$. Define the embedding $$F:{\operatorname{SL}}_{r_i+1}(F)\rightarrow{\operatorname{SL}}_{r+1}(F),\quad \begin{pmatrix} A&b\\ c&d\end{pmatrix}\mapsto \begin{pmatrix} I_{r_1+\cdots+r_{i-1}}&&&\\ & A&&b\\ &&I_{r_{i+1}+\cdots+r_{k}}&\\ &c&&d\end{pmatrix},$$ where $A$ is a $r_i\times r_i$-block and accordingly $b$ is $r_i\times 1$, $c$ is $1\times r_i$ and d is $1\times 1$. Note that this embedding is chosen so that we have $$\label{E:F_and_l} F(l_i(k_i))=l(k).$$ By the block compatibility of $\sigma_{{\operatorname{SL}}_{r+1}}$ we have $$\sigma_{{\operatorname{SL}}_{r+1}}|_{F({\operatorname{SL}}_{r_i+1})\times F({\operatorname{SL}}_{r_i+1})}=\sigma_{{\operatorname{SL}}_{r_i+1}}.$$ This is nothing but [@BLS Lemma 5, §2]. (The reader has to be careful in that the image $F({\operatorname{SL}}_{r_i+1})$ is not a standard subgroup in the sense defined in [@BLS p.143] if one chooses the set $\Delta$ of simple roots of ${\operatorname{SL}}_{r+1}$ in the usual way. One can, however, choose $\Delta$ differently so that $F({\operatorname{SL}}_{r_i+1})$ is indeed a standard subgroup. And all the results of [@BLS §2] are totally independent of the choice of $\Delta$.) This implies the map $(g_i, \xi)\mapsto (F(g_i),\xi)$ for $(g_i,\xi)\in{\widetilde{\operatorname{SL}}}_{r_i+1}$ is a homomorphism. Hence the canonical section ${\operatorname{SL}}_{r+1}({\mathcal{O}_F})\rightarrow {\widetilde{\operatorname{SL}}}_{r+1}(F)$, which is given by $g\mapsto (g, {\mathfrak{s}}_r(g))$, restricts to the canonical section ${\operatorname{SL}}_{r_i+1}({\mathcal{O}_F})\rightarrow {\widetilde{\operatorname{SL}}}_{r_i+1}(F)$, which is given by $g_i\mapsto (g_i,{\mathfrak{s}}_{r_i}(g_i))$. Namely we have the commutative diagram $$\xymatrix{ {\widetilde{\operatorname{SL}}}_{r_i+1}({\mathcal{O}_F})\ar[rrr]^{(g,\;\xi)\rightarrow( F(g),\; \xi)}&&&{\widetilde{\operatorname{SL}}}_{r+1}({\mathcal{O}_F})\\ {\operatorname{SL}}_{r_i+1}({\mathcal{O}_F})\ar[rrr]^F\ar[u]^{g_i\mapsto(g_i,\; {\mathfrak{s}}_{r_i}(g_i))}&&&{\operatorname{SL}}_{r+1}({\mathcal{O}_F})\ar[u]_{g\mapsto(g,\;{\mathfrak{s}}_r(g))}, }$$ where all the maps are homomorphisms. In particular, we have $$\label{E:sss} {\mathfrak{s}}_r(F(g_i))={\mathfrak{s}}_{r_i}(g_i),$$ for all $g_i\in{\operatorname{SL}}_{r_i+1}({\mathcal{O}_F})$. Thus $$\begin{aligned} s_r(k)&={\mathfrak{s}}_r(l(k))\quad\text{by (\ref{E:canonical_section})}\\ &={\mathfrak{s}}_r(F(l_i(k_i)))\quad\text{by (\ref{E:F_and_l})}\\ &={\mathfrak{s}}_{r_i}(l_i(k_i))\quad\text{by (\ref{E:sss})}\\ &=s_{r_i}(k_i)\quad\text{by (\ref{E:canonical_section}) with $r$ replaced by $r_i$}.\end{aligned}$$ The lemma has been proven. [**Global case:**]{} Assume $F$ is a number field. We define $\hat{s}_M: M({\mathbb{A}})\rightarrow\mu_n$ by $$\hat{s}_M(\prod_vm_v):=\prod_v{\hat{s}_{M_v}}(m_v)$$ for $\prod_vm_v\in M({\mathbb{A}})$. The product is finite thanks to Lemma \[L:s\_hat\_M\]. Since both of the cocycles $\tau_r$ and $\tau_M$ are the products of the corresponding local ones, one can see that the relation (\[E:s\_hat\]) holds globally as well. Thus analogously to the local case, we have the isomorphism $$\alpha_M:\; {{^c\widetilde{M}}}({\mathbb{A}})\rightarrow{\widetilde{M}}({\mathbb{A}}),\quad (m,\xi)\mapsto (m, \hat{s}_M(m)\xi).$$ \[L:splitting\_cMPt\] The splitting of $M(F)$ into ${{^c\widetilde{M}}}({\mathbb{A}})$ is given by $${\mathbf{s}}_M:M(F)\rightarrow\;{{^c\widetilde{M}}}({\mathbb{A}}),\quad \begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}\mapsto (\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix},\; \prod_{i=1}^k s_i(g_i)^{-1}).$$ For each $i$ the splitting ${\mathbf{s}}_{r_i}:{\operatorname{GL}}_{r_i}(F)\rightarrow{\widetilde{\operatorname{GL}}}_{r_i}({\mathbb{A}})$ is given by $g_i\mapsto(g_i,\;s_{r_i}(g_i)^{-1})$, where ${\widetilde{\operatorname{GL}}}_{r_i}({\mathbb{A}})$ is defined via the cocycle $\tau_{r_i}$. The lemma follows by the block-compatibility of $\tau_M$ and the product formula for the Hilbert symbol. Just like the case of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$, the section ${\mathbf{s}}_M$ as in this lemma cannot be defined on all of $M({\mathbb{A}})$ even set theoretically because the expression $\prod_is_{r_i}(g_i)$ does not make sense to all ${\operatorname{diag}}(g_1,\dots,g_k)\in M({\mathbb{A}})$. So we only have a partial set theoretic section $${\mathbf{s}}_M:M({\mathbb{A}})\rightarrow{{^c\widetilde{M}}}({\mathbb{A}}).$$ But analogously to Proposition \[P:s\_split\], we have \[P:s\_split\_M\] The partial section ${\mathbf{s}}_M$ is defined on both $M(F)$ and $N_M({\mathbb{A}})$, where $N_M({\mathbb{A}})$ is the unipotent radical of the Borel subgroup of $M$, and moreover it gives rise to a group homomorphism on each of these subgroups. Also for $m\in M(F)$ and $n\in N_M({\mathbb{A}})$, both ${\mathbf{s}}_M(mn)$ and ${\mathbf{s}}_M(nm)$ are defined and further ${\mathbf{s}}_M(mn)={\mathbf{s}}_M(m){\mathbf{s}}_M(n)$ and ${\mathbf{s}}_M(nm)={\mathbf{s}}_M(n){\mathbf{s}}_M(m)$. This follows from Proposition \[P:s\_split\] applied to each ${\widetilde{\operatorname{GL}}}_{r_i}({\mathbb{A}})$ together with the block-compatibility of the cocycle $\tau_M$. (Note that one also needs to use the fact that for all $g, g'$ in the subgroup generated by $M(F)$ and $N_M({\mathbb{A}})$, we have $(\det(g), \det(g'))_{\mathbb{A}}=1$.) This splitting is related to the splitting ${\mathbf{s}}_r:{\operatorname{GL}}_r(F)\rightarrow{\operatorname{GL}}_r({\mathbb{A}})$ by \[P:diagram\] We have the following commutative diagram: $$\xymatrix{{{^c\widetilde{M}}}({\mathbb{A}})\; \ar@{^{(}->}[r]^{\alpha_M}&{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})\\ M(F)\;\ar@{^{(}->}[r]\; \ar[u]^{{\mathbf{s}}_M}&{\operatorname{GL}}_r(F)\ar[u]_{{\mathbf{s}}_r}. }$$ For $m=\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix} \in M(F)$, we have $$\alpha_M({\mathbf{s}}_M(m))=\alpha_M(m, \prod_{i=1}^ks_{r_i}(g_i)^{-1}) =(m, \hat{s}_M(m)\prod_{i=1}^ks_{r_i}(g_i)^{-1}) =(m, s_r(m)^{-1})={\mathbf{s}}_r(m),$$ where for the elements in $M(F)$, all of $s_{r_i}$ and $s_r$ are defined globally, and the second equality follows from the definition of $\hat{s}_M$ as in (\[E:s\_hat\_M\]). This proposition implies \[C:diagram\] Assume $\pi$ is an automorphic subrepresentation of ${{^c\widetilde{M}}}({\mathbb{A}})$. The representation of ${\widetilde{M}}({\mathbb{A}})$ defined by $\pi\circ\alpha_M^{-1}$ is also automorphic. If $\pi$ is realized in a space $V$ of automorphic forms on ${{^c\widetilde{M}}}({\mathbb{A}})$, then $\pi\circ\alpha_M^{-1}$ is realized in the space of functions of the form $f\circ\alpha_M^{-1}$ for $f\in V$. The automorphy follows from the commutativity of the diagram in the above lemma. The following remark should be kept in mind for the rest of the paper. The results of this subsection essentially show that we may identify ${{^c\widetilde{M}}}$ (locally or globally) with ${\widetilde{M}}$. We may even “pretend” that the cocycle $\tau_r$ has the block-compatibility property. We need to make the distinction between ${{^c\widetilde{M}}}$ and ${\widetilde{M}}$ only when we would like to view the group ${\widetilde{M}}$ as a subgroup of ${\widetilde{\operatorname{GL}}}_r$. For most part of this paper, however, we will not have to view ${\widetilde{M}}$ as a subgroup of ${\widetilde{\operatorname{GL}}}_r$. Hence we suppress the superscript $^c$ from the notation and always denote ${{^c\widetilde{M}}}$ simply by ${\widetilde{M}}$, when there is no danger of confusion. Accordingly, we denote the partial section ${\mathbf{s}}_M$ simply by ${\mathbf{s}}$. **The center $Z_{{\widetilde{M}}}$ of ${\widetilde{M}}$** --------------------------------------------------------- In this subsection $F$ is either local or global, and accordingly we let $R=F$ or ${\mathbb{A}}$ as in the notation section. And all the groups are over $R$. For any group $H$ (metaplectic or not), we denote its center by $Z_H$. In particular for each group $\widetilde{H}\subseteq{\widetilde{\operatorname{GL}}}_r$, we let $$Z_{\widetilde{H}}=\text{center of $\widetilde{H}$}.$$ For the Levi part $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_2}\subseteq{\operatorname{GL}}_r$, we of course have $$Z_{M}=\{\begin{pmatrix}a_1I_{r_1}&&\\ &\ddots&\\ &&a_{k}I_{r_k}\end{pmatrix}:a_i\in R^\times\}.$$ But for the center $Z_{{\widetilde{M}}}$ of ${\widetilde{M}}$, we have $$Z_{{\widetilde{M}}}\subsetneq \widetilde{Z_{M}},$$ in general, and indeed $\widetilde{Z_{M}}$ might not be even commutative. In what follows, we will describe $Z_{{\widetilde{M}}}$ in detail. For this purpose, let us start with \[L:to\_compute\_center\] Assume $F$ is local. Then for each $g\in{\operatorname{GL}}_r(F)$ and $a\in F^\times$, we have $$\sigma_r(g, aI_r)\sigma_r(aI_r, g)^{-1}=(\det(g), a^{r-1+2cr}).$$ First let us note that if we write $\sigma_r=\sigma_r^{(c)}$ to emphasize the parameter $c$, then $$\sigma_r^{(c)}(g, aI_r)\sigma_r^{(c)}(aI_r, g)^{-1} =\sigma_r^{(0)}(g, aI_r)\sigma_r^{(0)}(aI_r, g)^{-1}(\det(g), a^r)^{2c}$$ because $(a^r,\det(g))^{-1}=(\det(g), a^r)$. Hence it suffices to show the lemma for the case $c=0$. But this can be done by using the recipe provided by [@BLS]. Namely let $g=nt\eta n'$ for $n, n'\in N_B$, $t\in T$ and $\eta\in{\mathfrak{M}}$. Then $$\begin{aligned} \sigma_r(g, aI_r)&=\sigma_r(nt\eta n', aI_r)\\ &=\sigma_r(t\eta, n' aI_r)\quad\text{by Proposition \ref{P:BLS} (1) and (2)}\\ &=\sigma_r(t\eta, aI_r)\quad\text{by $n'aI_r=aI_rn'$ and Proposition \ref{P:BLS} (1)}\\ &=\sigma_r(t, \eta aI_r)\sigma_r(\eta, aI_r)\sigma_r(t, \eta)^{-1}\quad\text{by Proposition \ref{P:BLS} (0)}\\ &=\sigma_r(t, aI_r\eta)\sigma_r(\eta, aI_r)\quad\text{by Proposition \ref{P:BLS} (5)}\\ &=\sigma_r(taI_r, \eta)\sigma_r(t, aI_r)\sigma_r(aI_r, \eta)^{-1} \sigma_r(\eta, aI_r)\quad\text{by Proposition \ref{P:BLS} (0)}\\ &=\sigma_r(t, aI_r)\sigma_r(\eta, aI_r)\quad\text{by Proposition \ref{P:BLS} (5)}.\end{aligned}$$ Now by Proposition \[P:BLS\] (3), $\sigma(\eta, aI_r)$ is a product of $(-a, a)$’s, which is $1$. Hence by using Proposition \[P:BLS\] (4), we have $$\sigma_r(g, aI_r)=\sigma_r(t, aI_r) =\prod_{i=1}^{r}(t_i, a)^{r-i}.$$ By an analogous computation, one can see $$\sigma_r(aI_r, g)=\sigma_r(aI_r, t)=\prod_{i=1}^{r}(a, t_i)^{i-1}.$$ Using $(a, t_i)^{-1}=(t_i, a)$, one can see $$\sigma_r(g, aI_r) \sigma_r(aI_r, g)^{-1}=\prod_{i=1}^r(t_i, a)^{r-1}.$$ But this is equal to $(\det(g), a^{r-1})$ because $\det(g)=\prod_{i=1}^rt_i$. Note that this lemma immediately implies that the center $Z_{{\widetilde{\operatorname{GL}}}_r}$ of ${\widetilde{\operatorname{GL}}}_r$ is indeed as in (\[E:center\_GLt\]), though a different proof is provided in [@KP]. Also with this lemma, we can prove Both locally and globally, the center $Z_{{\widetilde{M}}}$ is described as $$Z_{{\widetilde{M}}}=\{\begin{pmatrix}a_1I_{r_1}&&\\ &\ddots&\\ &&a_kI_{r_k}\end{pmatrix}: a_i^{r-1+2cr}\in R^{\times n}\;\text{ and }\;a_1\equiv\cdots\equiv a_r\mod{R^{\times n}}\}.$$ First assume $F$ is local. Let $m={\operatorname{diag}}(g_1,\dots,g_k)\in M$ and $a={\operatorname{diag}}(a_1I_{r_1},\dots,a_kI_{r_k})$. It suffices to show $\sigma_r(m,a)\sigma_r(a, m)^{-1}=1$ if and only if all $a_i$ are as in the proposition. But $$\begin{aligned} &\sigma_r(m,a) \sigma_r(a, m)^{-1}\\ =&\prod_{i=1}^r\sigma_{r_i}(g_i, a_iI_{r_i})\sigma_{r_i}(a_iI_{r_i}, g_i)^{-1}\prod_{1\leq i<j\leq r}(\det(g_i), a_j^{r_j})\prod_{i\neq j}(\det(g_i), a_j^{r_j})^c\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad \cdot\prod_{1\leq i<j\leq r}(a_i^{r_i},\det(g_j))^{-1}\prod_{i\neq j}(a_i^{r_i},\det(g_j))^{-c}\\ =&\prod_{i=1}^r\sigma_{r_i}(g_i, a_iI_{r_i})\sigma_{r_i}(a_iI_{r_i}, g_i)^{-1}\prod_{i\neq j}(\det(g_i), a_j^{r_j})^{1+2c}\\ =&\prod_{i=1}^r (\det(g_i), a_i^{r_i-1+2cr_i})\prod_{i\neq j}(\det(g_i), a_j^{r_j+2cr_j})\\ =&\prod_{i=1}^r (\det(g_i),\, a_i^{-1}\prod_{j=1}^ra_j^{r_j+2cr_j}),\end{aligned}$$ where for the third equality we used the above lemma with $r$ replaced by $r_i$. Now assume $a$ is such that $(a,1)\in Z_{{\widetilde{M}}}$. Then the above product must be 1 for any $m$. In particular, choose $m$ so that $g_j=1$ for all $i\neq j$. Then we must have $(\det(g_i),\, a_i^{-1}\prod_{j=1}^ra_j^{r_j+2cr_j})=1$ for all $g_i\in {\operatorname{GL}}_{r_i}$. This implies $$a_i^{-1}\prod_{j=1}^ra_j^{r_j+2cr_j}\in F^{\times n}$$ for all $i$. Since this holds for all $i$, one can see $a_i^{-1}a_j\in F^{\times n}$ for all $i\neq j$, which implies $\;a_1\equiv\cdots\equiv a_r\mod{F^{\times n}}$. But if $\;a_1\equiv\cdots\equiv a_r\mod{F^{\times n}}$, then $$\begin{aligned} \prod_{i=1}^r (\det(g_i),\, a_i^{-1}\prod_{j=1}^ra_j^{r_j+2cr_j}) =&\prod_{i=1}^r(\det(g_i), a_i^{-1}\prod_{j=1}^ra_i^{r_j+2cr_j})\\ =&\prod_{i=1}^r(\det(g_i), a_i^{r-1+2cr}).\end{aligned}$$ This must be equal to 1 for any choice of $g_i$, which gives $a_i^{r-1+2cr}\in F^{\times n}$. Conversely if $a$ is of the form as in the proposition, one can see that $\sigma_r(m,a) \sigma_r(a, m)^{-1}=\prod_{i=1}^r (\det(g_i),\, a_i^{-1}\prod_{j=1}^ra_j^{r_j+2cr_j})=1$ for any $m$. The global case follows from the local one because locally by using (\[E:tau\_sigma\]) and $am=ma$, one can see $\sigma_r(m,a) \sigma_r(a, m)^{-1}=1$ if and only if $\tau_r(m,a) \tau_r(a, m)^{-1}=1$, and the global $\tau_r$ is the product of local ones. Lemma \[L:to\_compute\_center\] also implies \[L:center\_GLtt\] Both locally and globally, $\widetilde{Z_{{\operatorname{GL}}_r}}$ commutes with ${\widetilde{\operatorname{GL}}^{(n)}}_r$ pointwise. The local case is an immediate corollary of Lemma \[L:to\_compute\_center\] because if $g\in{\operatorname{GL}}_r^{(n)}$ the lemma implies $\sigma_r(g, aI_r)=\sigma_r(aI_r, g)$. Hence by (\[E:tau\_sigma\]), locally $\tau_r(g, aI_r)=\tau_r(aI_r, g)$ for all $g\in{\operatorname{GL}}_r^{(n)}$ and $a\in F^\times$. Since the global $\tau_r$ is the product of the local ones, the global case also follows. Let us mention that in particular, if $n=2$ and $r=\text{even}$, then $\widetilde{Z_{{\operatorname{GL}}_r}}\subseteq{\widetilde{\operatorname{GL}}^{(n)}}_r$ and $\widetilde{Z_{{\operatorname{GL}}_r}}$ is the center of ${\widetilde{\operatorname{GL}}^{(n)}}_r$. This fact is used crucially in [@Takeda1]. It should be mentioned that this description of the center $Z_{{\widetilde{M}}}$ easily implies $$\label{E:center=center} Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}=Z_{{\widetilde{M}}}{\widetilde{M}^{(n)}}.$$ Also we have \[P:Z\_M\_commute\_Mtn\] Both locally and globally, the groups $\widetilde{Z_M}$ and ${\widetilde{M}^{(n)}}$ commute pointwise, which gives $$\label{E:center_Mtn} Z_{{\widetilde{M}^{(n)}}}=\widetilde{Z_{M}}\cap {{\widetilde{M}^{(n)}}},$$ and hence $$\label{E:center_Mtn2} Z_{{\widetilde{\operatorname{GL}}}_r}Z_{{\widetilde{M}^{(n)}}}=Z_{{\widetilde{\operatorname{GL}}}_r}(\widetilde{Z_{M}}\cap {{\widetilde{M}^{(n)}}})=\widetilde{Z_{M}}\cap( Z_{{\widetilde{\operatorname{GL}}}_r} {\widetilde{M}^{(n)}}).$$ By the block compatibility of the cocycle $\tau_M$, one can see that an element of the form $(\begin{pmatrix}a_1I_{r_1}&&\\ &\ddots&\\ &&a_{k}I_{r_k}\end{pmatrix}, \xi)$ commutes with all the elements in ${\widetilde{M}^{(n)}}$ if and only if each $(a_iI_{r_i},\xi)$ commutes with all the elements in ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}$. But this is always the case by the above lemme (with $r$ replaced by $r_i$). This proves the proposition. If $F$ is global, we define $$Z_{{\widetilde{M}}}(F)=Z_{{\widetilde{M}}}({\mathbb{A}})\cap{\mathbf{s}}(M(F)),$$ where recall that ${\mathbf{s}}:M(F)\rightarrow{\widetilde{M}}({\mathbb{A}})$ is the section that splits $M(F)$. Similarly we define groups like $Z_{{\widetilde{\operatorname{GL}}}_r}(F), {\widetilde{M}^{(n)}}(F)$, etc. Namely in general for any subgroup ${\widetilde{H}}\subseteq {\widetilde{M}}({\mathbb{A}})$, we define the “$F$-rational points” ${\widetilde{H}}(F)$ of ${\widetilde{H}}$ by $$\label{E:H(F)} {\widetilde{H}}(F):={\widetilde{H}}\cap{\mathbf{s}}(M(F)).$$ **The abelian subgroup $A_{{\widetilde{M}}}$** {#S:abelian} ---------------------------------------------- Again in this subsection, $F$ is local or global, and $R=F$ or ${\mathbb{A}}$. As we mentioned above, the preimage $\widetilde{Z_{M}}$ of the center $Z_M$ of the Levi $M$ might not be even commutative. For later purposes, we let $A_{{\widetilde{M}}}$ be a closed abelian subgroup of $\widetilde{Z_M}$ containing the center $Z_{{\widetilde{\operatorname{GL}}}_r}$. Namely $A_{{\widetilde{M}}}$ is a closed abelian subgroup such that $$Z_{{\widetilde{\operatorname{GL}}}_r}\subseteq A_{{\widetilde{M}}}\subseteq \widetilde{Z_{M}}.$$ We let $$A_M:=p(A_{{\widetilde{M}}}),$$ where $p$ is the canonical projection. If $F$ is global, we always assume $A_{{\widetilde{M}}}({\mathbb{A}})$ is chosen compatibly with the local $A_{{\widetilde{M}}}(F_v)$ in the sense that we have $$A_{M}({\mathbb{A}})={\prod_v}'A_{M}(F_v).$$ Note that if $A_{M}(F_v)$ (hence $A_{{\widetilde{M}}}(F_v)$) is closed, then $A_{M}({\mathbb{A}})$ (hence $A_{{\widetilde{M}}}({\mathbb{A}})$) is closed by Lemma \[L:closed\_subgroup\_local\_global\]. Of course there are many different choices for $A_{{\widetilde{M}}}$. But we would like to choose $A_{{\widetilde{M}}}$ so that the following hypothesis is satisfied: Assume $F$ is global. The image of $M(F)$ in the quotient $A_M({\mathbb{A}})M^{(n)}({\mathbb{A}})\backslash M({\mathbb{A}})$ is discrete in the quotient topology. The author does not know if one can always find such $A_{{\widetilde{M}}}$ for general $n$. But at least we have \[P:hypothesis\] If $n=2$, the above hypothesis is satisfied for a suitable choice of $A_{{\widetilde{M}}}$. For $n>2$, if $d=\gcd(n, r-1+2cr)$ is such that $n$ divides $nr_i/d$ for all $i=1,\dots,k$, (which is the case, for example, if $d=1$,) then the above hypothesis is satisfied with $A_{{\widetilde{M}}}=Z_{{\widetilde{M}}}$. This is proven in Appendix \[A:topology\]. We believe that for any reasonable choice of $A_{{\widetilde{M}}}$ the above hypothesis is always satisfied, but the author does not know how to prove it at this moment. This is a bit unfortunate in that this subtle technical issue makes the main theorem of the paper conditional when $n>2$. However if $n=2$, our main results are complete, and this is the only case we need for our applications to symmetric square $L$-functions in [@Takeda1; @Takeda2], which is the main motivation for the present work. Let us mention that the group $A_M({\mathbb{A}})M^{(n)}({\mathbb{A}})$ (for any choice of $A_M$) is a normal subgroup of $M({\mathbb{A}})$, and hence the quotient $A_M({\mathbb{A}})M^{(n)}({\mathbb{A}})\backslash M({\mathbb{A}})$ is a group. Accordingly, if the hypothesis is satisfied, the image of $M(F)$ in the quotient is a discrete subgroup and hence closed. Also we have $$A_{{\widetilde{M}}}(F)=A_{{\widetilde{M}}}({\mathbb{A}})\cap{\mathbf{s}}(M(F)).$$ following the convention as in (\[E:H(F)\]), and we set $$A_M(F)=p(A_{{\widetilde{M}}}(F)).$$ **On the local metaplectic tensor product** {#S:Mezo} =========================================== In this section we first review the local metaplectic tensor product of Mezo [@Mezo] and then extend his theory further, first by proving that the metaplectic tensor product behaves in the expected way under the Weyl group action, and second by establishing the compatibility of the metaplectic tensor product with parabolic inductions. Hence in this section, all the groups are over a local (not necessarily non-archimedean) field $F$ unless otherwise stated. Accordingly, we assume that our metaplectic group is defined by the block-compatible cocycle $\sigma_r$ of [@BLS], and hence by ${\widetilde{\operatorname{GL}}}_r$ we actually mean ${{^{\sigma}\widetilde{\operatorname{GL}}}}_r$. **Mezo’s metaplectic tensor product** {#SS:Mezo} ------------------------------------- Let $\pi_1,\cdots,\pi_k$ be irreducible genuine representations of ${\widetilde{\operatorname{GL}}}_{r_1},\dots,{\widetilde{\operatorname{GL}}}_{r_k}$, respectively. The construction of the metaplectic tensor product takes several steps. First of all, for each $i$, fix an irreducible constituent ${\pi^{(n)}}_i$ of the restriction $\pi_i|_{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}} $ of $\pi_i$ to ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}$. Then we have $$\pi_i|_{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}}=\sum_{g}m_i\,^g({\pi^{(n)}}_i),$$ where $g$ runs through a finite subset of ${\widetilde{\operatorname{GL}}}_{r_i}$, $m_i$ is a positive multiplicity and $^g({\pi^{(n)}}_i)$ is the representation twisted by $g$. Then we construct the tenor product representation $${\pi^{(n)}}_1\otimes\cdots\otimes{\pi^{(n)}}_k$$ of the group ${\widetilde{\operatorname{GL}}^{(n)}}_{r_1}\times\cdots\times{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}$. Note that this group is merely the direct product of the groups ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}$. The genuineness of the representations ${\pi^{(n)}}_1,\dots,{\pi^{(n)}}_k$ implies that this tensor product representation descends to a representation of the group ${\widetilde{\operatorname{GL}}^{(n)}}_{r_1}{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}$, [[*i.e.* ]{}]{}the representation factors through the natural surjection $${\widetilde{\operatorname{GL}}^{(n)}}_{r_1}\times\cdots\times{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}\twoheadrightarrow {\widetilde{\operatorname{GL}}^{(n)}}_{r_1}{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}={\widetilde{M}^{(n)}}.$$ We denote this representation of ${\widetilde{M}^{(n)}}$ by $${\pi^{(n)}}:={\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k,$$ and call it the metaplectic tensor product of ${\pi^{(n)}}_1,\dots,{\pi^{(n)}}_k$. Let us note that the space $V_{{\pi^{(n)}}}$ of ${\pi^{(n)}}$ is simply the tensor product $V_{{\pi^{(n)}}_1}\otimes\cdots\otimes V_{{\pi^{(n)}}_k}$ of the spaces of ${\pi^{(n)}}_i$. Let $\omega$ be a character on $Z_{{\widetilde{\operatorname{GL}}}_r}$ such that for all $(aI_{r},\xi)\in Z_{{\widetilde{\operatorname{GL}}}_r}\cap{\widetilde{M}^{(n)}}$ where $a\in F^\times$ we have $$\omega(aI_r,\xi)={\pi^{(n)}}(aI_r,\xi)=\xi{\pi^{(n)}}_1(aI_{r_1},1)\cdots{\pi^{(n)}}_k(aI_{r_k},1).$$ Namely $\omega$ agrees with ${\pi^{(n)}}$ on the intersection $Z_{{\widetilde{\operatorname{GL}}}_r}\cap{\widetilde{M}^{(n)}}$. We can extend ${\pi^{(n)}}$ to the representation $${\pi^{(n)}}_\omega:=\omega{\pi^{(n)}}$$ of $Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}$ by letting $Z_{{\widetilde{\operatorname{GL}}}_r}$ act by $\omega$. Now extend the representation ${\pi^{(n)}}_\omega$ to a representation $\rho_\omega$ of a subgroup ${\widetilde{H}}$ of ${\widetilde{M}}$ so that $\rho_\omega$ satisfies Mackey’s irreducibility criterion and so the induced representation $$\label{E:Mezo_tensor} \pi_\omega:={\operatorname{Ind}}_{{\widetilde{H}}}^{{\widetilde{M}}}\rho_\omega$$ is irreducible. It is always possible to find such ${\widetilde{H}}$ and moreover ${\widetilde{H}}$ can be chosen to be normal. Mezo shows in [@Mezo] that $\pi_\omega$ is dependent only on $\omega$ and is independent of the other choices made throughout, namely the choices of ${\pi^{(n)}}_i$, ${\widetilde{H}}$ and $\rho_\omega$. We write $$\pi_\omega=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$$ and call it the metaplectic tensor product of $\pi_1,\dots,\pi_k$ with the character $\omega$. Mezo also shows that the metaplectic tensor product $\pi_\omega$ is unique up to twist. Namely \[P:local\_uniqueness\] Let $\pi_1,\dots,\pi_k$ and $\pi'_1,\dots,\pi'_k$ be representations of ${\widetilde{\operatorname{GL}}}_{r_1},\dots,{\widetilde{\operatorname{GL}}}_{r_k}$. They give rise to isomorphic metaplectic tensor products with a character $\omega$, [[*i.e.* ]{}]{}$$(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega\cong (\pi'_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi'_k)_\omega,$$ if and only if for each $i$ there exists a character $\omega_i$ of ${\widetilde{\operatorname{GL}}}_{r_i}$ trivial on ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}$ such that $\pi_i\cong\omega_i\otimes\pi'_i$. This is [@Mezo Lemma 5.1]. \[R:dependence\_on\_omega\] Though the metaplectic tensor product generally depends on the choice of $\omega$, if the center $Z_{{\widetilde{\operatorname{GL}}}_r}$ is already contained in ${\widetilde{M}^{(n)}}$, we have ${\pi^{(n)}}_\omega={\pi^{(n)}}$ and hence there is no actual choice for $\omega$ and the metaplectic tensor product is canonical. This is the case, for example, when $n=2$ and $r$ is even, which is one of the important cases we consider in our applications in [@Takeda1; @Takeda2]. The equality (\[E:center=center\]) implies that extending a representation ${\pi^{(n)}}$ of ${\widetilde{M}^{(n)}}$ to ${\pi^{(n)}}_\omega$ multiplying the character $\omega$ on $Z_{{\widetilde{\operatorname{GL}}}_r}$ is the same as extending it by multiplying an appropriate character on $Z_{{\widetilde{M}}}$. Let us mention the following, which is not explicitly mentioned in [@Mezo]. \[L:always\_tensor\_product\] Let $\pi_\omega$ be an irreducible admissible representation of ${\widetilde{M}}$ where $\omega$ is the character on $Z_{{\widetilde{\operatorname{GL}}}_r}$ defined by $\omega=\pi_\omega|_{Z_{{\widetilde{\operatorname{GL}}}_r}}$. Then there exist irreducible admissible representations $\pi_1,\dots\pi_k$ of ${\widetilde{\operatorname{GL}}}_{r_1},\dots,{\widetilde{\operatorname{GL}}}_{r_k}$, respectively, such that $$\pi_\omega=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega.$$ Namely a representation of ${\widetilde{M}}$ is always a metaplectic tensor product. The restriction $\pi_\omega|_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}$ contains a representation of the form $\omega ({\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)$ for some representations ${\pi^{(n)}}_i$ of ${\widetilde{\operatorname{GL}}}_{r_i}$. Let $\pi_i$ be an irreducible constituent of ${\operatorname{Ind}}_{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}}^{{\widetilde{\operatorname{GL}}}_{r_i}}{\pi^{(n)}}_i$. Then one can see that $\pi_\omega$ is $(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$. From Mezo’s construction, one can tell that essentially the representation theory of the group ${\widetilde{M}}$ is determined by that of $Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}$. Let us briefly explain why this is so. Let $\pi$ be an irreducible admissible representation of ${\widetilde{M}}$, and $\chi_\pi:{\widetilde{M}}\rightarrow{\mathbb C}$ be the distribution character. If $\pi$ is genuine, so is $\chi_\pi$. Namely $\chi_\pi((1,\xi){\tilde{m}})=\xi\chi_\pi({\tilde{m}})$ for all $\xi\in\mu_n$ and ${\tilde{m}}\in{\widetilde{M}}$. But if ${\tilde{m}}\in{\widetilde{M}}$ is a regular element but not in $Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}$, then one can find $\xi\in\mu_n$ with $\xi\neq 1$ such that $(1,\xi){\tilde{m}}$ is conjugate to ${\tilde{m}}$. This is proven in the same way as [@KP Proposition 0.1.4]. (The only modification one needs is to choose $A\subset M_r(F)$ in their proof so that $A\subset M_{r_1}(F)\times\cdots\times M_{r_k}(F)$.) Therefore for such ${\tilde{m}}$, one has $\chi_\pi({\tilde{m}})=0$. Namely, the support of $\chi_\pi$ is contained in $Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}$. (Indeed, this argument by the distribution character is crucially used in [@Mezo Lemma 4.2]. ) This explains why $\pi$ is essentially determined by the restriction $\pi|_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}$. This idea can be observed in \[L:equivalent\_tensor\_product\] Let $\pi$ and $\pi'$ be irreducible admissible representations of ${\widetilde{M}}$. Then $\pi$ and $\pi'$ are equivalent if and only if $\pi|_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}$ and $\pi'|_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}$ have an equivalent constituent. This follows from Proposition \[P:local\_uniqueness\] and Lemma \[L:always\_tensor\_product\]. Also let us mention \[P:Mezo\] we have $${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_\omega=m\pi_\omega$$ for some finite multiplicity $m$, so every constituent of ${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_\omega$ is isomorphic to $\pi_\omega$. By inducting in stages, we have ${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_\omega ={\operatorname{Ind}}_{{\widetilde{H}}}^{{\widetilde{M}}}{\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{H}}}{\pi^{(n)}}_\omega$, where ${\widetilde{H}}$ is as in (\[E:Mezo\_tensor\]), and by [@Mezo Lemma 4.1] we have $${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{H}}}{\pi^{(n)}}_\omega=\bigoplus_{\chi}\chi\otimes\rho_\omega$$ where $\chi$ runs over the finite set of characters of ${\widetilde{H}}$ that are trivial on $Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}$. Moreover it is shown in [@Mezo Lemma 4.1] that any extension of ${\pi^{(n)}}_\omega$ to ${\widetilde{H}}$ is of the form $\chi\otimes\rho_\omega$ and ${\operatorname{Ind}}^{{\widetilde{M}}}_{{\widetilde{H}}}\chi\otimes\rho_\omega=\pi_\omega$ for all $\chi$ by [@Mezo Lemma 4.2]. Hence we have $${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_\omega =\bigoplus_\chi {\operatorname{Ind}}_{{\widetilde{H}}}^{{\widetilde{M}}}\chi\otimes\rho_\omega =m\pi_\omega.$$ Let $\omega$ be as above and $A_{{\widetilde{M}}}$ as in Section \[S:abelian\]. The restriction ${\pi^{(n)}}|_{A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}}}$ gives a character on $A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}}$ because ${A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}}}$ is contained in the center of ${\widetilde{M}^{(n)}}$ by (\[E:center\_Mtn\]). The product $\omega ({\pi^{(n)}}|_{A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}}})$ of $\omega$ and ${\pi^{(n)}}|_{A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}}} $ defines a character on $Z_{{\widetilde{\operatorname{GL}}}_r}(A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}})$ because the two characters agree on $Z_{{\widetilde{\operatorname{GL}}}_r}\cap (A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}})$. Since the Pontryagin dual is an exact functor, one can extend it to a character on $A_{{\widetilde{M}}}$, which we denote again by $\omega$. Namely $\omega$ is a character on $A_{{\widetilde{M}}}$ extending $\omega$ such that $\omega(a)={\pi^{(n)}}(a)$ for all $a\in A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}}$. With this said, we have \[C:local\_tensor\] Let $\omega$ be the character on $A_{{\widetilde{M}}}$ described above, and let ${\pi^{(n)}}_{\omega}:=\omega{\pi^{(n)}}$ be the representation of $A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}$ extending ${\pi^{(n)}}$ by letting $A_{{\widetilde{M}}}$ act as $\omega$. Then $${\operatorname{Ind}}_{A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_{\omega}=m'\pi_\omega$$ where $m'$ is some finite multiplicity. This follows from the previous proposition because we have the inclusion ${\operatorname{Ind}}_{A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_{\omega}\hookrightarrow {\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_{\omega}$. **The archimedean case** {#S:archimedean} ------------------------ Let us make some remarks when $F$ is archimedean. Strictly speaking, Mezo assumes that the field $F$ is non-archimedean. If $F={\mathbb C}$, then ${\widetilde{M}^{(n)}}={\widetilde{M}}$. Indeed, ${\widetilde{M}}({\mathbb C})=M({\mathbb C})\times\mu_n$ (direct product), and the metaplectic tensor product is obtained simply by taking the tensor product $\pi_1\otimes\cdots\otimes\pi_k$ and descending it to ${\widetilde{M}}({\mathbb C})$. Hence there is essentially no discrepancy between the metaplectic case and the non-metaplectic one. If $F={\mathbb{R}}$ (so necessarily $n=2$), one can trace the argument of Mezo and make sure the construction works for this case as well, with the proviso that equivalence has to be considered as infinitesimal equivalence. However, it has been communicated to the author by J. Adams that for this case, the induced representation ${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_\omega$ is always irreducible. (See [@Adams]). Hence one can simply define the metaplectic tensor product to be this induced representation. **Twists by Weyl group elements** {#S:Weyl_group_local} --------------------------------- As in the notation section, we let $W_M$ be the subset of the Weyl group $W_{{\operatorname{GL}}_r}$ consisting of only those elements which permute the ${\operatorname{GL}}_{r_i}$-factors of $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$. Though $W_M$ is not a group in general, it is identified with the group $S_k$ of permutations of $k$ letters. Assume $w\in W_M$ is such that $$M':=wMw^{-1}={\operatorname{GL}}_{r_{\sigma(1)}}\times\cdots\times{\operatorname{GL}}_{r_{\sigma(k)}}$$ for a permutation $\sigma\in S_k$, and so $w(g_1,\dots,g_k)w^{-1}=(g_{\sigma(1)},\dots,g_{\sigma(k)})$ for each $(g_1,\dots,g_k)\in M$. Namely $w$ corresponds to the permutation $\sigma^{-1}$. Then we have $${\widetilde{M'}}={\mathbf{s}}(w){\widetilde{M}}{\mathbf{s}}(w)^{-1}.$$ Let $\pi=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$ be an irreducible admissible representation of ${\widetilde{M}}$. As in the notation section one can define the twist $^{{\mathbf{s}}(w)}\pi$ of $\pi$ by ${\mathbf{s}}(w)$ to be the representation of ${\widetilde{M'}}$ on the space $V_\pi$ given by $^{{\mathbf{s}}(w)}\pi({\tilde{m}}')=\pi({\mathbf{s}}(w)^{-1}{\tilde{m}}'{\mathbf{s}}(w))$ for ${\tilde{m}}'\in{\widetilde{M'}}$. To ease the notation we simply write $$^w\pi:=\,^{{\mathbf{s}}(w)}\pi.$$ Actually since $\mu_n\subseteq{\widetilde{M}}$ is in the center, for any preimage $\tilde{w}$ of $w$, we have $\,^{{\mathbf{s}}(w)}\pi=\,^{\tilde{w}}\pi$, and hence the notation $^w\pi$ is not ambiguous. The goal of this subsection is to show that the metaplectic tensor product behaves in the expected way under the Weyl group action. Namely, we will prove \[T:Weyl\_group\_local\] With the above notations, we have $$\label{E:Weyl_group_local} ^w(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega \cong(\pi_{\sigma(1)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_{\sigma(k)})_\omega.$$ To prove this, we first need For each $(m,1)\in{\widetilde{M}^{(n)}}$ and $w\in W_M$, where $m\in M^{(n)}$, we have $${\mathbf{s}}(w)(m,1){\mathbf{s}}(w)^{-1}=(wmw^{-1},1),$$ namely ${\mathbf{s}}(w){\mathbf{s}}(m){\mathbf{s}}(w)^{-1}={\mathbf{s}}(wmw^{-1})$. Note that ${\mathbf{s}}(w)=(w,1)$ and ${\mathbf{s}}(w)^{-1}=(w^{-1},\sigma_r(w,w^{-1})^{-1})$ because we are using ${{^{\sigma}\widetilde{\operatorname{GL}}}}_r$, and hence $${\mathbf{s}}(w)(m,1){\mathbf{s}}(w)^{-1} =(wmw^{-1},\sigma_r(w, mw^{-1})\sigma_r(m, w^{-1}) \sigma_r(w,w^{-1})^{-1}).$$ Let $$\varphi_w(m):=\sigma_r(w, mw^{-1})\sigma_r(m, w^{-1}) \sigma_r(w,w^{-1})^{-1}.$$ We need to show $\varphi_w(m)=1$ for all $m\in M^{(n)}$. Let us first show that the map $m\mapsto\varphi_w(m)$ is a homomorphism on $M^{(n)}$. To see it, for $m, m'\in M^{(n)}$ we have $$\begin{aligned} {\mathbf{s}}(w)(m,1)(m',1){\mathbf{s}}(w)^{-1}&={\mathbf{s}}(w)(mm',\sigma_r(m,m')){\mathbf{s}}(w)^{-1}\\ &=(wmm'w^{-1},\sigma_r(m,m')\varphi_w(mm')).\end{aligned}$$ On the other hand, we have $$\begin{aligned} {\mathbf{s}}(w)(m,1)(m',1){\mathbf{s}}(w)^{-1}&={\mathbf{s}}(w)(m,1){\mathbf{s}}(w)^{-1}{\mathbf{s}}(w)(m',1){\mathbf{s}}(w)^{-1}\\ &=(wmw^{-1},\varphi_w(m))(wm'w^{-1},\varphi_w(m'))\\ &=(wmm'w^{-1}, \sigma_r(wmw^{-1},wm'w^{-1}) \varphi_w(m)\varphi_w(m'))\\ &=(wmm'w^{-1}, \sigma_r(m,m') \varphi_w(m)\varphi_w(m')),\end{aligned}$$ where the last equality follows because $\sigma_r(wmw^{-1},wm'w^{-1})=\sigma_r(m,m')$ by the block-compatibility of $\sigma_r$. Hence by comparing those two, one obtains $\varphi_w(mm')=\varphi_w(m)\varphi_w(m')$. Therefore to show $\varphi_w(m)=1$, it suffices to show it for the elmenets of the form $$\label{E:form_of_m} m={\operatorname{diag}}(I_{r_1},\dots,I_{r_{i-1}},g_i,I_{r_{i+1}},\dots, I_{r_k})$$ for $g_i\in{\operatorname{GL}}_{r_i}^{(n)}$. Then one can rewrite $\varphi_w(m)$ as follows: $$\begin{aligned} \varphi_w(m)&=\sigma_r(w, mw^{-1})\sigma_r(m, w^{-1}) \sigma_r(w,w^{-1})^{-1}\\ &=\sigma_r(w, w^{-1}wmw^{-1})\sigma_r(m, w^{-1}) \sigma_r(w,w^{-1})^{-1}\\ &=\sigma_r(ww^{-1}, wmw^{-1})\sigma_r(w, w^{-1})\sigma_r(w^{-1}, wmw^{-1})^{-1}\sigma_r(m, w^{-1})\sigma_r(w,w^{-1})^{-1}\\ &=\sigma_r(w^{-1}, wmw^{-1})^{-1}\sigma_r(m, w^{-1}),\end{aligned}$$ where for the third equality we used Proposition \[P:BLS\] (0). So we only have to show $$\label{E:cocycle_computation1} \sigma_r(w^{-1}, wmw^{-1})^{-1}\sigma_r(m, w^{-1})=1.$$ This can be shown by using the algorithm computing the cocycle $\sigma_r$ given by [@BLS]. To use the results of [@BLS], it should be mentioned that one needs to use the set ${\mathfrak{M}}$ for a set of representatives of the Weyl group of ${\operatorname{GL}}_r$ as defined in the notation section. Also let us recall the following notation from [@BLS]: For each $g\in{\operatorname{GL}}_r$, the “torus part function” ${\mathbf{t}}:{\operatorname{GL}}_r\rightarrow T$ is the unique map such that $${\mathbf{t}}(nt\eta n')=t,$$ where $n, n'\in N_B$, $t\in T$ and $\eta\in\mathfrak{M}$ when ${\operatorname{GL}}_r$ is written as $${\operatorname{GL}}_r=\coprod_{\eta\in\mathfrak{M}}N_BT\eta N_B$$ by the Bruhat decomposition. Namely ${\mathbf{t}}(g)$ is the “torus part” of $g$. Using this language, each $w\in W_M$ is written as $$w={\mathbf{t}}(w)\eta_w$$ where $\eta_w\in\mathfrak{M}$, and ${\mathbf{t}}(w)\in{\operatorname{GL}}_{r_{\sigma(1)}}\times\cdots\times{\operatorname{GL}}_{r_{\sigma(k)}}$ is of the form $${\mathbf{t}}(w)=(\varepsilon_{\sigma(1)}I_{\sigma(1)},\dots,\varepsilon_{\sigma(k)}I_{\sigma(k)}),$$ where $\varepsilon_i\in\{\pm 1\}$. We are now ready to carry out our cocycle computations for (\[E:cocycle\_computation1\]). Let us deal with $\sigma_r(m, w^{-1})$ first. Write $m=nt\eta n'$ by the Bruhat decomposition, so ${\mathbf{t}}(m)=t$. But recall that we are assuming $m$ is of the form as in (\[E:form\_of\_m\]), so the decomposition $nt\eta n'$ takes place essentially inside the ${\operatorname{GL}}_{r_i}$-block. In particular, we can write $$m={\operatorname{diag}}(I_{r_1},\dots,I_{r_{i-1}},n_it_i\eta_i n_i',I_{r_{i+1}}\dots,I_{r_k}),$$ where $t_i\in{\operatorname{GL}}^{(n)}_{r_i}$. (Note that $\det(t_i)\in F^{\times n}$.) Then one can compute $\sigma_r(m, w^{-1})$ as follows: $$\begin{aligned} \sigma_r(m, w^{-1})&=\sigma_r(nt\eta n', w^{-1})\\ &=\sigma_r(t\eta, n'w^{-1})\quad\text{by Proposition \ref{P:BLS} (1), (2)}\\ &=\sigma_r(t\eta, w^{-1}wn'w^{-1})\\ &=\sigma_r(t\eta, w^{-1})\quad\text{because $wn'w^{-1}\in N_B$ and by Proposition \ref{P:BLS} (1)}\\ &=\sigma_r(t\eta, {\mathbf{t}}(w^{-1})\eta_{w^{-1}}).\end{aligned}$$ Now since $\eta$ is essentially inside the ${\operatorname{GL}}_{r_i}$-factor of $M$ and $\eta_{w^{-1}}$ only permutes the ${\operatorname{GL}}_{r_j}$-factors of $M$, we have $l(\eta\eta_{w^{-1}})=l(\eta)+l(\eta_{w^{-1}})$, where $l$ is the length function. Hence by applying [@BLS Lemma 10, p.155], we have $$\label{E:cocycle_computation2} \sigma_r(t\eta, {\mathbf{t}}(w^{-1})\eta_{w^{-1}}) =\sigma_r(t,\eta{\mathbf{t}}(w^{-1})\eta^{-1})\sigma_r(\eta, {\mathbf{t}}(w^{-1})).$$ Here note that ${\mathbf{t}}(w^{-1})\in M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$ is of the form $(\varepsilon_1 I_{r_1},\dots, \varepsilon_k I_{r_k})$ and $\eta$ is in the ${\operatorname{GL}}_{r_i}$-block. Hence $\eta{\mathbf{t}}(w^{-1})\eta^{-1}={\mathbf{t}}(w^{-1})$. Thus by the block-compatibility of $\sigma_r$, (\[E:cocycle\_computation2\]) is written as $$\sigma_{r_i}(t_i, \varepsilon_i I_{r_i})\sigma_{r_i}(\eta_i, \varepsilon_i I_{r_i}).$$ Clearly, if $\varepsilon_i=1$, then both $\sigma_{r_i}(t_i, \varepsilon_i I_{r_i})$ and $\sigma_{r_i}(\eta_i, \varepsilon_i I_{r_i})$ are $1$. If $\varepsilon_i=-1$, then by Proposition \[P:BLS\] (3), one can see that $\sigma_{r_i}(\eta_i, \varepsilon_i I_{r_i})=1$. Hence in either case, one has $$\label{E:cocycle_computation3} \sigma_r(m,w^{-1}) =\sigma_{r_i}(t_i, \varepsilon_i I_{r_i}).$$ Next let us deal with $\sigma_r(w^{-1}, wmw^{-1})$ in (\[E:cocycle\_computation1\]). First by the analogous computation to what we did for $\sigma(m, w^{-1})$, one can write $$\label{E:cocycle_computation4} \sigma_r(w^{-1}, wmw^{-1}) =\sigma_r(w^{-1}, wt\eta w^{-1}) =\sigma_r({\mathbf{t}}(w^{-1})\eta_{w^{-1}}, wt\eta w^{-1}).$$ Since $w$ corresponds to the permutation $\sigma^{-1}$, if we let $$\tau_i:{\operatorname{GL}}_{r_i}\rightarrow wMw^{-1}={\operatorname{GL}}_{r_{\sigma(1)}}\times\cdots\times{\operatorname{GL}}_{r_{\sigma(k)}}$$ be the embedding of ${\operatorname{GL}}_{r_i}$ into the corresponding ${\operatorname{GL}}_{r_i}$-factor of $wMw^{-1}$, then (\[E:cocycle\_computation4\]) is written as $$\sigma_r({\mathbf{t}}(w^{-1})\eta_{w^{-1}}, \tau_i(t_i)\tau_i(\eta_i)).$$ Note that $\tau_i(\eta_i)\in\mathfrak{M}$ and $l(\eta_{w^{-1}}\tau_i(\eta_i))=l(\eta_{w^{-1}})+l(\tau_i(\eta_i))$. Hence by using [@BLS Lemma 10, p.155], this is written as $$\label{E:cocycle_computation5} \sigma_r({\mathbf{t}}(w^{-1}), \eta_{w^{-1}}\tau_i(t_i)\eta_{w^{-1}}^{-1}) \sigma_r(\eta_{w^{-1}}, \tau_i(t_i)).$$ By the block compatibility of $\sigma_r$, one can see $$\sigma_r({\mathbf{t}}(w^{-1}), \eta_{w^{-1}}\tau_i(t_i)\eta_{w^{-1}}^{-1}) =\sigma_{r_i}(\varepsilon_iI_{r_i}, t_i).$$ Also to compute $\sigma_r(\eta_{w^{-1}}, \tau_i(t_i))$, one needs to use Proposition \[P:BLS\] (3). For this purpose, let us write $$t_i=\begin{pmatrix}a_1&&\\ &\ddots&\\ &&a_{r_i}\end{pmatrix}\in{\operatorname{GL}}_{r_i}$$ where $\det(t_i)=a_1\cdots a_{r_i}\in F^{\times n}$. By looking at the formula in Proposition \[P:BLS\] (3), one can see that $\sigma_r(\eta_{w^{-1}}, \tau_i(t_i))$ is a power of $$(-1, a_1)\cdots(-1,a_{r_i}),$$ which is equal to $(-1, a_1\cdots a_{r_i})=1$ because $\det(t_i)=a_1\cdots a_{r_i}\in F^{\times n}$. Hence (\[E:cocycle\_computation5\]), which is the same as (\[E:cocycle\_computation4\]), becomes $$\sigma_{r_i}(\varepsilon_iI_{r_i}, t_i).$$ Hence the left hand side of (\[E:cocycle\_computation1\]) is written as $$\sigma_{r_i}(\varepsilon_iI_{r_i}, t_i)^{-1}\sigma_{r_i}(t_i,\varepsilon_iI_{r_i}).$$ We need to show this is $1$. But clearly this is the case if $\varepsilon_i=1$. So let us assume $\varepsilon_i=-1$. Namely we will show $\sigma_{r_i}(-I_{r_i}, t_i)^{-1}\sigma_{r_i}(t_i,-I_{r_i})=1$. But by Proposition \[P:BLS\] (4), one can compute $$\sigma_{r_i}(-I_{r_i}, t_i)=(-1, a_2)(-1, a_3)^2(-1,a_4)^3\cdots (-1, a_r)^{r-1+2c}$$ and $$\sigma_{r_i}(t_i, -I_{r_i})=(a_1, -1)^{r-1}(a_2,-1)^{r-2}(a_3,-1)^{r-3}\cdots (-1, a_{r-1}).$$ Noting that $(-1,a_i)^{-1}=(a_i, -1)$, we have $$\sigma_{r_i}(-I_{r_i}, t_i)^{-1}\sigma_{r_i}(t_i, -I_{r_i})=\prod_{i=1}^r(a_i, -1)^{r-1+2c}=(\prod_{i=1}^ra_i, -1)^{r-1+2c}=1,$$ where the last equality follows because $\det(t_i)=\prod_{i=1}^ra_i\in F^{\times n}$. This completes the proof. Now we are ready to prove Theorem \[T:Weyl\_group\_local\]. By restricting to ${\widetilde{M'}^{(n)}}$, one can see that the left hand side of (\[E:Weyl\_group\_local\]) contains the representation $^w({\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)$ and the right hand side of (\[E:Weyl\_group\_local\]) contains ${\pi^{(n)}}_{\sigma(1)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_{\sigma(k)}$, where $^w({\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)$ is the representation of ${\widetilde{M'}^{(n)}}={\mathbf{s}}(w){\widetilde{M}^{(n)}}{\mathbf{s}}(w)^{-1}$ whose space is the space of ${\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k$. Hence by Lemma \[L:equivalent\_tensor\_product\], it suffices to show that $$^w({\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)\cong {\pi^{(n)}}_{\sigma(1)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_{\sigma(k)}.$$ But this can be seen from the commutative diagram $$\xymatrix{ &{\widetilde{\operatorname{GL}}^{(n)}}_{r_{\sigma(1)}}\times\cdots\times{\widetilde{\operatorname{GL}}^{(n)}}_{r_{\sigma(k)}}\ar[ddl]\ar[ddr]\ar[d]&\\ &{\widetilde{M'}^{(n)}}\ar[dl]\ar[dr]&\\ {\operatorname{Aut}}(V_{{\pi^{(n)}}_1}\otimes\cdots\otimes V_{{\pi^{(n)}}_k})\ar[rr]^{\sim}&& {\operatorname{Aut}}(V_{{\pi^{(n)}}_{\sigma(1)}}\otimes\cdots\otimes V_{{\pi^{(n)}}_{\sigma(k)}}), }$$ where the left most arrow is the representation of ${\widetilde{\operatorname{GL}}^{(n)}}_{r_\sigma(1)}\times\cdots\times{\widetilde{\operatorname{GL}}^{(n)}}_{r_\sigma(k)}$ (direct product) acting on the space of ${\pi^{(n)}}_1\otimes\cdots\otimes{\pi^{(n)}}_k$ by permuting each factor by $\sigma^{-1}$, which descends to the representation $^w({\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)$ of ${\widetilde{M}^{(n)}}$. To see this indeed descends to $^w({\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)$, one needs the above lemma. **Compatibility with parabolic induction** {#S:parabolic_induction_local} ------------------------------------------ We will show the compatibility of the metaplectic tensor product with parabolic induction. Hence we consider the standard parabolic subgroup $P=MN\subseteq{\operatorname{GL}}_r$ where $M$ is the Levi part and $N$ the unipotent radical. First let us mention \[L:normalizer\_local\] The image $N^\ast$ of the unipotent radical $N$ via the section ${\mathbf{s}}:{\operatorname{GL}}_r\rightarrow{\widetilde{\operatorname{GL}}}_r$ is normalized by the metaplectic preimage ${\widetilde{M}}$ of the Levi part $M$. Though this seems to be well-known, we will give a proof here. Let ${\tilde{m}}\in{\widetilde{M}}$ and $(n,1)\in N^\ast$, where $n\in N$. (Note that since we are assuming the group ${\widetilde{\operatorname{GL}}}_r$ is defined by $\sigma_r$, each element in $N^\ast$ is written as $(n,1)$.) We may assume ${\tilde{m}}=(m,1)$ for $m\in M$. Noting that ${\tilde{m}}^{-1}=(m^{-1},\sigma_r(m,m^{-1})^{-1})$, we compute $$\begin{aligned} {\tilde{m}}(n,1){\tilde{m}}^{-1}&=(m,1)(n,1)(m^{-1}, \sigma_r(m,m^{-1})^{-1})\\ &=(mn, \sigma_r(m,n)) (m^{-1}, \sigma_r(m,m^{-1})^{-1})\\ &=(mnm^{-1},\sigma_r(mn, m^{-1})\sigma_r(m,n) \sigma_r(m,m^{-1})^{-1}).\end{aligned}$$ By Proposition \[P:BLS\] (1), $\sigma_r(m,n)=1$. Also since $mnm^{-1}\in N$, we have $\sigma_r(mn, m^{-1})=\sigma_r(mnm^{-1}m, m^{-1})=\sigma_r(m,m^{-1})$ again by Proposition \[P:BLS\] (1). Thus we have ${\tilde{m}}(n,1){\tilde{m}}^{-1}=(mnm^{-1},1)\in N^\ast$. By this lemma, we can write $${\widetilde{P}}={\widetilde{M}}N^\ast$$ where ${\widetilde{M}}$ normalizes $N^\ast$ and hence for a representation $\pi$ of ${\widetilde{M}}$ one can form the induced representation $${\operatorname{Ind}}_{{\widetilde{M}}N^\ast}^{{\widetilde{\operatorname{GL}}}_r}\pi$$ by letting $N^\ast$ act trivially. \[T:induction\_local\] Let $P=MN\subseteq{\operatorname{GL}}_r$ be the standard parabolic subgroup whose Levi part is $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$. Further for each $i=1,\dots,k$ let $P_i=M_iN_i\subseteq{\operatorname{GL}}_{r_i}$ be the standard parabolic of ${\operatorname{GL}}_{r_i}$ whose Levi part is $M_i={\operatorname{GL}}_{r_{i,1}}\times\cdots\times{\operatorname{GL}}_{r_{i, l_i}}$. For each $i$, we are given a representation $$\sigma_i:=(\tau_{i,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{i,l_i})_{\omega_i}$$ of ${\widetilde{M}}_i$, which is given as the metaplectic tensor product of the representations $\tau_{i,1},\dots,\tau_{i,l_i}$ of ${\widetilde{\operatorname{GL}}}_{r_{i,1}},\dots,{\widetilde{\operatorname{GL}}}_{r_{i, l_i}}$. Assume that $\pi_i$ is an irreducible constituent of the induced representation ${\operatorname{Ind}}_{{\widetilde{P}}_i}^{{\widetilde{\operatorname{GL}}}_{r_i}}\sigma_i$. Then the metaplectic tensor product $$\pi_\omega:=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$$ is an irreducible constituent of the induced representation $${\operatorname{Ind}}_{{\widetilde{Q}}}^{{\widetilde{M}}}(\tau_{1,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{1, l_1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{k,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{k,l_k})_\omega,$$ where $Q$ is the standard parabolic of $M$ whose Levi part is $M_1\times\cdots\times M_k$. First we need For a genuine representation $\pi$ of a Levi part ${\widetilde{M}}$, the map $${\operatorname{Ind}}_{{\widetilde{M}}N^\ast}^{{\widetilde{\operatorname{GL}}}_r}\pi\rightarrow{\operatorname{Ind}}_{{(\widetilde{M})^{(n)}}N^\ast}^{{\widetilde{\operatorname{GL}}^{(n)}}_r}\pi|_{{(\widetilde{M})^{(n)}}}$$ given by the restriction $\varphi\mapsto \varphi|_{{\widetilde{\operatorname{GL}}^{(n)}}_r}$ for $\varphi\in {\operatorname{Ind}}_{{\widetilde{M}}N^\ast}^{{\widetilde{\operatorname{GL}}}_r}\pi$ is an isomorphism, where $${(\widetilde{M})^{(n)}}={\widetilde{M}}\cap{\widetilde{\operatorname{GL}}^{(n)}}_r.$$ Hence in particular $$\left({\operatorname{Ind}}_{{\widetilde{M}}N^\ast}^{{\widetilde{\operatorname{GL}}}_r}\pi\right)|_{{\widetilde{\operatorname{GL}}^{(n)}}_r}\cong \left({\operatorname{Ind}}_{{\widetilde{M}}N^\ast}^{{\widetilde{\operatorname{GL}}}_r}\pi\right)\|_{{\widetilde{\operatorname{GL}}^{(n)}}_r}\cong{\operatorname{Ind}}_{{(\widetilde{M})^{(n)}}N^\ast}^{{\widetilde{\operatorname{GL}}^{(n)}}_r}\pi|_{{(\widetilde{M})^{(n)}}}$$ as representations of ${\widetilde{\operatorname{GL}}^{(n)}}_r$. To show it is one-to-one, assume $\varphi|_{{\widetilde{\operatorname{GL}}^{(n)}}_r}=0$. We need to show $\varphi=0$. But for any $g\in{\operatorname{GL}}_r$, one can write $g=\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g$, where $\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}\in M$ and $\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g\in{\operatorname{GL}}_r^{(n)}$. Hence any ${\tilde{g}}\in{\widetilde{\operatorname{GL}}}_r$ is written as ${\tilde{g}}={\tilde{m}}{\tilde{g}}'$ for some ${\tilde{m}}\in{\widetilde{M}}$ and ${\tilde{g}}'\in{\widetilde{\operatorname{GL}}^{(n)}}_r$. Hence $\varphi({\tilde{g}})=\pi({\tilde{m}})\varphi({\tilde{g}}')$. But $\varphi({\tilde{g}}')=0$. Hence $\varphi({\tilde{g}})=0$. To show it is onto, let $\varphi\in{\operatorname{Ind}}_{{(\widetilde{M})^{(n)}}N^\ast}^{{\widetilde{\operatorname{GL}}^{(n)}}_r}\pi|_{{(\widetilde{M})^{(n)}}}$. Define $\widetilde{\varphi}:{\widetilde{\operatorname{GL}}}_r\rightarrow\pi$ by $$\widetilde{\varphi}(g,\xi)=\xi\pi(\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta)\varphi(\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g, 1),$$ where $\eta$ is chosen to be such that $(\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta)(\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g, 1)=(g,1)$. Namely $\eta$ is given by the cocycle as $$\eta=\sigma_r(\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g)^{-1}.$$ That $\widetilde{\varphi}|_{{\widetilde{\operatorname{GL}}^{(n)}}_r}=\varphi$ follows because if $g\in{\operatorname{GL}}_r^{(n)}$ then $(\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta)\in{(\widetilde{M})^{(n)}}$. Also one can check $\widetilde{\varphi}\in {\operatorname{Ind}}_{{\widetilde{M}}N^\ast}^{{\widetilde{\operatorname{GL}}}_r}\pi$ as follows: We need to check $\varphi({\tilde{m}}(g,\xi))=\pi({\tilde{m}})\varphi(g,\xi)$ for all ${\tilde{m}}\in{\widetilde{M}}$. But since $\pi$ (and hence $\varphi$) is genuine, we may assume ${\tilde{m}}$ is of the form $(m,1)$ for $m\in M$ and $\xi=1$. Then $$\begin{aligned} \notag\widetilde{\varphi}((m,1)(g,1))&=\widetilde{\varphi}(mg,\sigma_r(m,g))\\ \label{E:F_tilde}&=\sigma_r(m,g)\pi(\begin{pmatrix}\det (mg)^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta_1)\varphi(\begin{pmatrix}\det (mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}mg, 1),\end{aligned}$$ where $$\eta_1=\sigma_r(\begin{pmatrix}\det (mg)^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \begin{pmatrix}\det (mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}mg)^{-1}.$$ Now $$\begin{aligned} &(\begin{pmatrix}\det(mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}mg, 1)\\ =&(\begin{pmatrix}\det(mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}m\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta_2) (\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g, 1),\end{aligned}$$ where $$\eta_2=\sigma_r(\begin{pmatrix}\det(mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}m\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g)^{-1}.$$ Since $(\begin{pmatrix}\det(mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}m\begin{pmatrix}\det g^{-n-1}&\\ & I_{r-1}\end{pmatrix}, \eta_2)\in{(\widetilde{M})^{(n)}}$, the right hand side of (\[E:F\_tilde\]) becomes $$\begin{aligned} &\sigma_r(m,g)\pi\Big((\begin{pmatrix}\det (mg)^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta_1) (\begin{pmatrix}\det(mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}m\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta_2)\Big)\\ &\qquad\qquad \varphi(\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g, 1)\\ &=\sigma_r(m,g)\pi(m\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix},\eta_1\eta_2\eta_3) \varphi(\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g, 1)\\ &=\sigma_r(m,g)\pi(m,\eta_1\eta_2\eta_3\eta_4)\pi(\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, 1) \varphi(\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g, 1),\end{aligned}$$ where $$\eta_3=\sigma_r(\begin{pmatrix}\det (mg)^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \begin{pmatrix}\det(mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}m\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}),$$ and $$\eta_4=\sigma_r(m, \begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix})^{-1}.$$ Then one can compute $$\sigma_r(m,g)\eta_1\eta_2\eta_3\eta_4=\eta$$ by using Proposition \[P:BLS\] (0). Hence (\[E:F\_tilde\]) is written as $$\pi(m,1)\pi(\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta) \varphi(\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g, 1) =\pi(m,1)\widetilde{\varphi}(g,1).$$ This completes the proof. With this lemma, one can prove the theorem. Let $\pi_i^{(n)}$ be an irreducible constituent of the restriction $\pi_i|_{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}}$. By the above lemma, it is an irreducible constituent of $${\operatorname{Ind}}_{{(\widetilde{M_i})^{(n)}}N_i^\ast}^{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}}\sigma_i|_{{(\widetilde{M_i})^{(n)}}}.$$ Noting that ${\widetilde{M}^{(n)}}_i\subseteq{(\widetilde{M_i})^{(n)}}$, we have the inclusion $${\operatorname{Ind}}_{{(\widetilde{M_i})^{(n)}}N_i^\ast}^{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}}\sigma_i|_{{(\widetilde{M_i})^{(n)}}} \hookrightarrow {\operatorname{Ind}}_{{\widetilde{M}^{(n)}}_i N_i^\ast}^{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}}\sigma_i|_{{\widetilde{M}^{(n)}}_i}.$$ But since $\sigma_i$ is a metaplectic tensor product of $\tau_{i,1},\dots,{\tau_{i,l_i}}$, the restriction $\sigma_i|_{{\widetilde{M}^{(n)}}_i}$ is a sum of representations of the form $$\tau_{i,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{i,l_i}^{(n)}$$ where each $\tau_{i,t}^{(n)}$ is an irreducible constituent of the restriction $\tau_{i,t}|_{{\widetilde{\operatorname{GL}}^{(n)}}_{^ir_t}}$ of $\tau_{i,t}$ to ${\widetilde{\operatorname{GL}}^{(n)}}_{r_{i,t}}$. Note that this is a metaplectic tensor product representation of ${\widetilde{M}^{(n)}}_i$. Hence the metaplectic tensor product $$\pi^{(n)}:={\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k$$ is an irreducible constituent of $$\label{E:tensor_of_tensor} \widetilde{\bigotimes}_{i=1}^k{\operatorname{Ind}}_{{\widetilde{M}^{(n)}}_i N_i^\ast}^{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}}\,\tau_{i,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\, {\tau_{i,l_i}^{(n)}}.$$ Note that the metaplectic tensor product for the group ${\widetilde{M}^{(n)}}$ can be defined for reducible representations, and hence $\widetilde{\bigotimes}_{i=1}^k$ is defined and the space of the representation is the same as the one for the usual tensor product. In particular, the space of the representation (\[E:tensor\_of\_tensor\]) is the usual tensor product. Then one can see that (\[E:tensor\_of\_tensor\]) is equivalent to $$\label{E:tensor_of_tensor2} {\operatorname{Ind}}_{{\widetilde{M}^{(n)}}_1{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{M}^{(n)}}_k (N_1\times\cdots\times N_k)^\ast}^{{\widetilde{M}^{(n)}}}{\widetilde{\otimes}}_{i=1}^k \,\tau_{i,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\, {\tau_{i,l_i}^{(n)}}.$$ (To see this one can define a map from (\[E:tensor\_of\_tensor\]) to (\[E:tensor\_of\_tensor2\]) by $\varphi_1\,\otimes\cdots\otimes\,\varphi_k\mapsto \varphi_1\cdots\varphi_k$ where $\varphi_1\cdots\varphi_k$ is the product of functions which can be naturally viewed as a function on ${\widetilde{M}^{(n)}}$.) Now let $\omega$ be a character on $Z_{{\widetilde{\operatorname{GL}}}_r}$ that agrees with ${\pi^{(n)}}$ on $Z_{{\widetilde{\operatorname{GL}}}_r}\cap{\widetilde{M}^{(n)}}$, so that the product $${\pi^{(n)}}_\omega:=\omega\cdot{\pi^{(n)}}_n$$ is a well-defined representation of $Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}$. Now all the constituents of the representation (\[E:tensor\_of\_tensor2\]) have the same central character, and hence $\omega$ agrees with (\[E:tensor\_of\_tensor2\]) on $Z_{{\widetilde{\operatorname{GL}}}_r}\cap{\widetilde{M}^{(n)}}$, and hence ${\pi^{(n)}}_\omega$ is a constituent of $${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}_1{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{M}^{(n)}}_k (N_1\times\cdots\times N_k)^\ast}^{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}\omega\cdot{\widetilde{\otimes}}_{i=1}^k \,\tau_{i,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\tau_{i,l_i}^{(n)}}.$$ Recall that the metaplectic tensor product $\pi_\omega$ is a constituent of $${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_\omega$$ and hence a constituent of $${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}_1{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{M}^{(n)}}_k (N_1\times\cdots\times N_k)^\ast}^{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}\omega\cdot{\widetilde{\otimes}}_{i=1}^k \,\tau_{i,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\tau_{i,l_i}^{(n)}},$$ which is $$\label{E:tensor_of_tensor3} {\operatorname{Ind}}_{{\widetilde{Q}}}^{{\widetilde{M}}}{\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}_1{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{M}^{(n)}}_k (N_1\times\cdots\times N_k)^\ast}^{{\widetilde{Q}}}\omega\cdot{\widetilde{\otimes}}_{i=1}^k \,\tau_{i,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\tau_{i,l_i}^{(n)}}$$ by inducing in stages. Now one can see that the inner induced representation in (\[E:tensor\_of\_tensor3\]) is equal to $$\label{E:tensor_of_tensor4} {\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M_Q}}^{(n)}}^{{\widetilde{M_Q}}}\omega\cdot{\widetilde{\otimes}}_{i=1}^k \,\tau_{i,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\tau_{i,l_i}^{(n)}}$$ where the unipotent group $(N_1\times\cdots\times N_k)^\ast$ acts trivially and ${\widetilde{M_Q}}$ is the Levi part of ${\widetilde{Q}}$, namely $${\widetilde{M_Q}}={\widetilde{M}}_1{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{M}}_k.$$ By Proposition \[P:Mezo\] applied to the Levi ${\widetilde{M_Q}}$, the representation (\[E:tensor\_of\_tensor4\]) is a sum of the metaplectic tensor product $$(\tau_{1,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\, {\tau_{1,l_1}^{(n)}}\,{\widetilde{\otimes}}\,\cdots{\widetilde{\otimes}}\, \tau_{k,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\, {\tau_{k,l_k}^{(n)}})_\omega.$$ Hence $\pi_\omega$ is a constituent of ${\operatorname{Ind}}_{{\widetilde{Q}}}^{{\widetilde{M}}}(\tau_{1,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\, {\tau_{1,l_1}^{(n)}}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{k,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\, {\tau_{k,l_k}^{(n)}})_\omega$ as claimed. \[R:induction\_local\] In the statement of Theorem \[T:induction\_local\], one can replace “constituent” by “irreducible subrepresentation” or “irreducible quotient”, and the analogous statement is still true. Namely if each $\pi_i$ is an irreducible subrepresentation (resp. quotient) of the induced representation in the theorem, then the metaplectic tensor product $(\pi_1\,\otimes\cdots\otimes\,\pi_k)_\omega$ is also an irreducible subrepresentation (resp. quotient) of the corresponding induced representation. To prove it, one can simply replace all the occurrences of “constituent” by “irreducible subrepresentation” or “irreducible quotient” in the above proof. **The global metaplectic tensor product** ========================================= Starting from this section, we will show how to construct the metaplectic tensor product of unitary automorphic subrepresentations. Hence all the groups are over the ring of adeles unless otherwise stated, and it should be recalled here that as in the group ${\operatorname{GL}}_r(F)^\ast$ is the image of ${\operatorname{GL}}_r(F)$ under the partial map ${\mathbf{s}}:{\operatorname{GL}}_r({\mathbb{A}})\rightarrow{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$, and we simply write ${\operatorname{GL}}_r(F)$ for ${\operatorname{GL}}_r(F)^\ast$, when there is no danger of confusion. Also throughout the section the group $A_{{\widetilde{M}}}({\mathbb{A}})$ is an abelian group that satisfies Hypothesis ($\ast$). **The construction** -------------------- The construction is similar to the local case in that first we consider the restriction to ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}})$, though we need an extra care to ensure the automorphy. Let us start with the following. \[L:unitary\_restriction1\] Let $\pi$ be a genuine irreducible automorphic unitary subrepresentation of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$. Then the restriction $\pi|_{{\widetilde{\operatorname{GL}}^{(n)}}_r({\mathbb{A}})}$ is completely reducible, namely $$\pi|_{{\widetilde{\operatorname{GL}}^{(n)}}_r{{\mathbb{A}}}}=\bigoplus{\pi^{(n)}}_i$$ where $\pi_i$ is an irreducible unitary representation of ${\widetilde{\operatorname{GL}}^{(n)}}_r({\mathbb{A}})$. This follows from the admissibility and unitarity of $\pi|_{{\widetilde{\operatorname{GL}}^{(n)}}_r{{\mathbb{A}}}}$. The lemma implies that the restriction $\pi_i\|_{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}})}$ is also completely reducible. (See the notation section for the notation $\|$.) Hence each irreducible constituent of $\pi_i\|_{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}})}$ is a subrepresentation. Let $${\pi^{(n)}}_i\subseteq\pi_i$$ be an irreducible subrepresentation. Then each vector $f\in{\pi^{(n)}}_i$ is the restriction to ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}})$ of an automorphic form on ${\widetilde{\operatorname{GL}}}_{r_i}({\mathbb{A}})$. Hence one can naturally view each vector $f\in{\pi^{(n)}}_i$ as a function on the group $$H_i:={\operatorname{GL}}_{r_i}(F){\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}}).$$ Namely the representation ${\pi^{(n)}}_{i}$ is an irreducible representation of the group ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}})$ realized in a space of “automorphic forms on $H_i$”. Note that $H_i$ is indeed a group and moreover it is closed in ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$, which can be shown by using Lemma \[L:discrete\_sub\]. Also note that each element in $H_i$ is of the form $(h_i,\xi_i)$ for $h_i\in{\operatorname{GL}}_{r_i}(F){\operatorname{GL}}_{r_i}({\mathbb{A}})$ and $\xi_i\in\mu_n$. By the product formula for the Hilbert symbol and the block-compatibility of the cocycle $\tau_M$, we have the natural surjection $$\label{E:global_surjection} H_1\times\cdots\times H_k\rightarrow M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$$ given by the map $((h_1,\xi_1),\dots,(h_k,\xi_k))\mapsto (h_1\cdots h_k,\xi_1\cdots\xi_k)$ because $(\det(h_i),\det(h_j))_{\mathbb{A}}=1$ for all $i, j=1,\dots, k$. Now we can construct a metaplectic tensor product of ${\pi^{(n)}}_1,\dots,{\pi^{(n)}}_k$, which is an “automorphic representation” of ${\widetilde{M}^{(n)}}({\mathbb{A}})$ realized in a space of “automorphic forms on $M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$” as follows. \[P:tensor\_product\_Mtn\] Let $$V_{{\pi^{(n)}}_1}\otimes\cdots\otimes V_{{\pi^{(n)}}_k}$$ be the space of functions on the direct product $H_1\times\dots\times H_k$ which gives rise to an irreducible representation of ${\widetilde{\operatorname{GL}}^{(n)}}_{r_1}({\mathbb{A}})\times\dots\times {\widetilde{\operatorname{GL}}^{(n)}}_{r_i}(A)$ which acts by right translation. Then each function in this space can be viewed as a function on the group $M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$, namely it factors through the surjection as in (\[E:global\_surjection\]) and thus gives rise to an representation of ${\widetilde{M}^{(n)}}({\mathbb{A}})$, which we denote by $${\pi^{(n)}}:={\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k.$$ Moreover each function in $V_{{\pi^{(n)}}}=V_{{\pi^{(n)}}_1}\otimes\cdots\otimes V_{{\pi^{(n)}}_k}$ is “automorphic” in the sense that it is left invariant on $M(F)$. Since $\pi_i$ is genuine, for each $f_i\in V_{{\pi^{(n)}}_i}$ and $g\in H_i$, we have $f_i(g(1,\xi))=f_i((1,\xi)g)=\xi f_i(g)$ for all $\xi\in\mu_n$. Now the kernel of the map (\[E:global\_surjection\]) consists of the elements of the form $((I_{r_1},\xi_1),\dots,(I_{r_k},\xi_k))$ with $\xi_1\cdots\xi_k=1$. Hence each $f_1\otimes\cdots\otimes f_k\in V_{{\pi^{(n)}}_1}\otimes\cdots\otimes V_{{\pi^{(n)}}_k}$, viewed as a function on the direct product $H_1\times\cdots\times H_k$, factors through the map (\[E:global\_surjection\]), which we denote by $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k$. Namely we can naturally define a function $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k $ on $M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$ by $$(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k)( \begin{pmatrix}h_1&&\\ &\ddots&\\ &&h_k\end{pmatrix},\xi) =\xi f_1(h_1,1)\cdots f_k(h_k,1).$$ One can see each function $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k$ is “automorphic” as follows: For $\begin{pmatrix}\gamma_1&&\\ &\ddots&\\ &&\gamma_k\end{pmatrix}\in M(F)$ and $\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}\in M(F)M^{(n)}({\mathbb{A}})$, we have $$\begin{aligned} &(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k)\Big({\mathbf{s}}\begin{pmatrix}\gamma_1&&\\ &\ddots&\\ &&\gamma_k\end{pmatrix} (\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix},\xi)\Big)\\ =&(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k)\Big((\begin{pmatrix}\gamma_1&&\\ &\ddots&\\ &&\gamma_k\end{pmatrix},\prod_{i=1}^ks_{r_i}(\gamma_i)^{-1}) (\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix},\xi)\Big)\quad\text{by definition of ${\mathbf{s}}$}\\ =&(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k)\Big(\begin{pmatrix}\gamma_1g_1&&\\ &\ddots&\\ &&\gamma_kg_k\end{pmatrix},\xi\prod_{i=1}^ks_{r_i}(\gamma_i)^{-1} \tau_M(\begin{pmatrix}\gamma_1&&\\ &\ddots&\\ &&\gamma_k\end{pmatrix}, \begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix})\Big)\\ =&(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k)\Big(\begin{pmatrix}\gamma_1g_1&&\\ &\ddots&\\ &&\gamma_kg_k\end{pmatrix},\xi\prod_{i=1}^ks_{r_i}(\gamma_i)^{-1}\tau_{r_i}(\gamma_i,g_i)\Big) \quad\text{by block-compatibility of $\tau_M$}\\ =&\left(\xi\prod_{i=1}^ks_{r_i}(\gamma_i)^{-1}\tau_{r_i}(\gamma_i,g_i)\right) \left(\prod_{i=1}^k f_i(\gamma_ig_i,1)\right) \quad\text{by definition of $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k$}\\ =&\xi\prod_{i=1}^kf_i\Big(\gamma_ig_i,s_{r_i}(\gamma_i)^{-1}\tau_{r_i}(\gamma_i,g_i)\Big) \quad\text{because each $f_i$ is genuine}\\ =&\xi\prod_{i=1}^kf_i\Big((\gamma_i, s_{r_i}(\gamma_i)^{-1})(g_i,1)\Big) \quad\text{by definition of $\tau_{r_i}$}\\ =&\xi\prod_{i=1}^kf_i({\mathbf{s}}_{r_i}(\gamma_i)(g_i,1)) \quad\text{by definition of ${\mathbf{s}}_{r_i}$}\\ =&\xi\prod_{i=1}^kf_i(g_i,1) \quad\text{by automorphy of $f_i$}\\ =&(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k) (\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix},\xi)\quad\text{by definition of $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k$} .\end{aligned}$$ Just like the local case, we would like to extend the representation ${\pi^{(n)}}$ to a representation of $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$ by letting $A_{{\widetilde{M}}}({\mathbb{A}})$ act as a character. This is certainly possible by choosing an appropriate character because $A_{{\widetilde{M}}}({\mathbb{A}})\cap{\widetilde{M}^{(n)}}({\mathbb{A}})$ is in the center of ${\widetilde{M}^{(n)}}({\mathbb{A}})$. To ensure the resulting representation is automorphic, however, one needs extra steps to do it. For this purpose, let us, first, define $$\label{E:F_points_of_AM} A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F):=A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})\cap{\mathbf{s}}(M(F)).$$ Note that this is not necessarily the same as $A_{{\widetilde{M}}}(F){\widetilde{M}^{(n)}}(F)$. Also let $$H:=A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F){\widetilde{M}^{(n)}}({\mathbb{A}}).$$ By our assumption on $A_{{\widetilde{M}}}$ (Hypothesis ($\ast$)), the image of ${\mathbf{s}}(M(F))$ (and hence $A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F)$) in the quotient ${\widetilde{M}^{(n)}}({\mathbb{A}})\backslash{\widetilde{M}}({\mathbb{A}})$ is discrete. Hence by Lemma \[L:discrete\_sub\], $H$ is a closed (and hence locally compact) subgroup of ${\widetilde{M}}({\mathbb{A}})$. Also note that the group $A_{{\widetilde{M}}}({\mathbb{A}})$ commutes pointwise with the group $H$ by Proposition \[P:Z\_M\_commute\_Mtn\] and hence $A_{{\widetilde{M}}}({\mathbb{A}})\cap H$ is in the center of $H$. We need the following subtle but important lemma. \[H:central\_character\_on\_H\] There exists a character $\chi$ on the center $Z_H$ of $H$ such that $f(ah)=\chi(a)f(h)$ for $a\in Z_H$, $h\in H$ and $f\in{\pi^{(n)}}$. (Note that each $f\in{\pi^{(n)}}$ is a function on $M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$ and hence can be viewed as a function on $H$.) Let ${\pi^{(n)}}_{H_i}$ be an irreducible subrepresentation of $\pi_i\|_{H_i}$ such that ${\pi^{(n)}}\subseteq{\pi^{(n)}}_{H_i}\|_{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}})}$. Analogously to the construction of ${\pi^{(n)}}={\pi^{(n)}}_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_k$, one can construct the representation ${\pi^{(n)}}_{H_1}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{H_k}$ of $M(F){\widetilde{M}}({\mathbb{A}})$. (The space of this representation is again a space of “automorphic forms on $M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$” but this time it is an irreducible representation of the group $M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$, rather than just ${\widetilde{M}^{(n)}}({\mathbb{A}})$. The construction is completely the same as ${\pi^{(n)}}$ and one can just modify the proof of Proposition \[P:tensor\_product\_Mtn\].) Then one can see $$V_{{\pi^{(n)}}}\subseteq V_{{\pi^{(n)}}_{H_1}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{H_k}},$$ and $$({\pi^{(n)}}_{H_1}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{H_k})\|_{{\widetilde{M}^{(n)}}({\mathbb{A}})}\cong ({\pi^{(n)}}_{H_1}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{H_k})|_{{\widetilde{M}^{(n)}}({\mathbb{A}})}.$$ Let ${\pi^{(n)}}_H$ be an irreducible subrepresentation of $({\pi^{(n)}}_{H_1}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{H_k})|_H$ such that $$V_{{\pi^{(n)}}}\subseteq V_{{\pi^{(n)}}_H},$$ where both sides are spaces of functions on $M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$. Such ${\pi^{(n)}}_H$ certainly exists, since each $\pi_i$ is unitary and the unitary structure descends to ${\pi^{(n)}}_{H_1}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{H_k}$ making it unitary. Now since ${\pi^{(n)}}_H$ is unitary and $H$ is locally compact, ${\pi^{(n)}}_H$ admits a central character $\chi$. Thus for each $f\in V_{{\pi^{(n)}}_H}$ and [*a fortiori*]{} each $f\in V_{{\pi^{(n)}}}$, we have $f(ah)=\chi(a)f(h)$ for $a\in Z_H$ and $h\in H$. In the above lemma, if $a\in Z_H\cap{\mathbf{s}}(M(F))$, we have $\chi (a)=1$ by the automorphy of $f$, namely $\chi$ is a “Hecke character on $Z_H$”. Now define a character $\omega$ on $A_{{\widetilde{M}}}({\mathbb{A}})$ such that $\omega$ is trivial on $A_{{\widetilde{M}}}(F)$ and $$\omega|_{A_{{\widetilde{M}}}({\mathbb{A}})\cap H}=\chi|_{A_{{\widetilde{M}}}({\mathbb{A}})\cap H}.$$ Such $\omega$ certainly exists because $\chi|_{A_{{\widetilde{M}}}({\mathbb{A}})\cap H}$ is viewed as a character on the group ${\mathbf{s}}(M(F))\cap( A_{{\widetilde{M}}}({\mathbb{A}})\cap H)\backslash A_{{\widetilde{M}}}({\mathbb{A}})\cap H$, which is a locally compact abelian group naturally viewed as a closed subgroup of the locally compact abelian group $A_{{\widetilde{M}}}(F)\backslash A_{{\widetilde{M}}}({\mathbb{A}})$, and thus it can be extended to $A_{{\widetilde{M}}}(F)\backslash A_{{\widetilde{M}}}({\mathbb{A}})$. For each $f\in {{\pi^{(n)}}}$, viewed as a function on $H=A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F){\widetilde{M}^{(n)}}({\mathbb{A}})$, we extend it to a function $f_\omega:A_{{\widetilde{M}}}({\mathbb{A}})H\rightarrow{\mathbb C}$ by $$f_\omega(ah)=\omega(a)f(h),\quad\text{for all $a\in A_{{\widetilde{M}}}({\mathbb{A}})$ and $h\in H$}.$$ This is well-defined because of our choice of $\omega$, and \[L:automorphy\_of\_f\_omega\] The function $f_\omega$ is a function on $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$ such that $$f_\omega(\gamma m)=f_\omega(m)$$ for all $\gamma\in A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F)$ and $m\in{\widetilde{M}^{(n)}}({\mathbb{A}})$. Namely $f_\omega$ is an “automorphic form on $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$”. The lemma follows from the definition of $f_\omega$ and the obvious equality $A_{{\widetilde{M}}}({\mathbb{A}})H=A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$. The group $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}}) $ acts on the space of functions of the form $f_\omega$, giving rise to an “automorphic representation” ${\pi^{(n)}}_\omega$ of $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}}) $, namely $$V_{{\pi^{(n)}}_\omega}:=\{f_\omega:f\in{\pi^{(n)}}\}$$ and $A_{{\widetilde{M}}}({\mathbb{A}})$ acts as the character $\omega$. As abstract representations, we have $$\label{E:pin_omega} {\pi^{(n)}}_\omega\cong\omega\cdot{\pi^{(n)}}$$ where by $\omega\cdot{\pi^{(n)}}$ is the representation of the group $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$ extended from ${\pi^{(n)}}$ by letting $A_{{\widetilde{M}}}({\mathbb{A}})$ act via the character $\omega$. We need to make sure the relation between ${\pi^{(n)}}_\omega$ and its local analogue we constructed in the previous section. For this, let us start with \[L:local\_global\] Let $\pi\cong{\widetilde{\otimes}}'_v\pi_v$ be a genuine admissible representation of ${\widetilde{M}}({\mathbb{A}})$. Let ${\pi^{(n)}}$ be an irreducible quotient of the restriction $\pi|_{{\widetilde{M}^{(n)}}({\mathbb{A}})}$. If we write $${\pi^{(n)}}\cong\underset{v}{{\widetilde{\otimes}}}'{\pi^{(n)}}_v,$$ then each ${\pi^{(n)}}_v$ is an irreducible constituent of the restriction $\pi_v|_{{\widetilde{M}^{(n)}}_r(F_v)}$. Since ${\pi^{(n)}}$ is an irreducible quotient, there is a surjective ${\widetilde{M}^{(n)}}({\mathbb{A}})$ map $$T:{\underset{v}{{\widetilde{\otimes}}'}\pi_v}\rightarrow {\underset{v}{{\widetilde{\otimes}}}'{\pi^{(n)}}_v}.$$ Fix a place $v_0$. Since $T\neq0$, there exists a pure tensor $\otimes w_v\in {{{\widetilde{\otimes}}}'\pi_v}$ such that $T(\otimes w_v)\neq 0$. (Note that, as we have seen, the space of ${{\widetilde{\otimes}}'\pi_v}$ is the space of the usual restricted tensor product $\otimes'_v\pi_v$.) Define $$i:{\pi_{v_0}}\rightarrow {{\widetilde{\otimes}}' \pi_v}$$ by $$i(w)=w\otimes(\otimes_{v\neq v_0} w_v)$$ for $w\in V_{\pi_{v_0}}$. Then the composite $T\circ i: {\pi_{v_0}}\rightarrow {\otimes'_v{\pi^{(n)}}_v}$ is a non-zero ${\widetilde{M}^{(n)}}(F_{v_0})$ intertwining. Let $w\in {\pi_{v_0}}$ be such that $T\circ i(w)\neq 0$. Then $T\circ i(w)$ is a finite linear combination of pure tensors, and indeed it is written as $$T\circ i(w)=x_1\otimes y_1+\cdots+x_t\otimes y_t,$$ where $x_i\in {{\pi^{(n)}}_{v_0}}$ and $y_i\in\otimes'_{v\neq v_0} {{\pi^{(n)}}_v}$. Here one can assume that $y_1,\dots,y_t$ are linearly independent. Let $\lambda: \otimes_{v\neq v_0} {{\pi^{(n)}}_v}\rightarrow{\mathbb C}$ be a linear functional such that $\lambda (y_1)\neq 0$ and $\lambda (y_2)=\cdots=\lambda (y_t)=0$. (Such $\lambda$ certainly exits because $y_1,\dots,y_t$ are linearly independent.) Consider the map $$U:{{\widetilde{\otimes}}'{\pi^{(n)}}_v}\rightarrow {{\pi^{(n)}}_{v_0}}$$ defined on pure tensors by $$U(\otimes x_v)=\lambda (\otimes_{v\neq v_0} x)x_{v_0}.$$ This is a non-zero ${\widetilde{M}^{(n)}}(F_v)$ intertwining map. Moreover the composite $U\circ T\circ i$ gives a non-zero ${\widetilde{M}^{(n)}}(F_v)$ intertwining map from ${\pi_{v_0}}$ to ${{\pi^{(n)}}_{v_0}}$. Hence ${\pi^{(n)}}_{v_0}$ is an irreducible constituent of the restriction $\pi_{v_0}|_{{\widetilde{M}^{(n)}}(F_{v_0})}$. By taking $k=1$ in the above lemma, one can see that if one writes $${\pi^{(n)}}_i\cong\underset{v}{{\widetilde{\otimes}}'}{\pi^{(n)}}_{i,v}$$ then each local component ${\pi^{(n)}}_{i,v}$ is an irreducible constituent of $\pi_{i,v}|_{{\widetilde{\operatorname{GL}}}_{r_i}(F_v)}$ where $\pi_{i,v}$ is the $v$-component of $\pi_i\cong\underset{v}{{\widetilde{\otimes}}'}\pi_{i,v}$. Then one can see that for ${\pi^{(n)}}={\pi^{(n)}}_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_k$, if we write ${\pi^{(n)}}\cong\underset{v}{{\widetilde{\otimes}}'}{\pi^{(n)}}_v$, we have $${\pi^{(n)}}_v\cong{\pi^{(n)}}_{1,v}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{k,v}$$ where the right hand side is the local metaplectic tensor product representation of ${\widetilde{M}^{(n)}}(F_v)$. Also one can see that the character $\omega$ decomposes as $\omega=\underset{v}{{\widetilde{\otimes}}'}\omega_v$ where $\omega_v$ is a character on $A_{{\widetilde{M}}(F_v)}$. Hence by (\[E:pin\_omega\]) we have \[P:global\_local\_pin\_omega\] As abstract representations of $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$, we have $${\pi^{(n)}}_\omega\cong\underset{v}{{\widetilde{\otimes}}'}{\pi^{(n)}}_{\omega_v},$$ where $${\pi^{(n)}}_{\omega_v}=\omega_v\cdot {\pi^{(n)}}_{1,v}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{k,v}$$ is the representation of $A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)$ as defined in the previous section. Now that we have constructed the representation ${\pi^{(n)}}_\omega$ of ${\widetilde{M}^{(n)}}({\mathbb{A}})$, we can construct an automorphic representation of ${\widetilde{M}}({\mathbb{A}})$ analogously to the local case by inducing it to ${\widetilde{M}}({\mathbb{A}})$, though we need extra care for the global case. First consider the compactly induced representation $${\operatorname{c-Ind}}_{A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})}^{{\widetilde{M}}({\mathbb{A}})}{\pi^{(n)}}_\omega =\{\varphi:{\widetilde{M}}({\mathbb{A}})\rightarrow{\pi^{(n)}}_\omega\}$$ where $\varphi$ is such that $\varphi(hm)={\pi^{(n)}}_\omega(h)\varphi(m)$ for all $ h\in A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$ and $m\in{\widetilde{M}}({\mathbb{A}})$, and the map $m\mapsto \varphi(m;1)$ is a smooth function on ${\widetilde{M}}({\mathbb{A}})$ whose support is compact modulo $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$. (Note here that for each $\varphi\in{\operatorname{Ind}}_{A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})}^{{\widetilde{M}}({\mathbb{A}})}{\pi^{(n)}}_\omega$ and $m\in{\widetilde{M}}({\mathbb{A}})$, $\varphi(m)\in V_{{\pi^{(n)}}_\omega}$ is a function on $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$. For $m'\in A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$, we use the notation $\varphi(m;m')$ for the value of $\varphi(m)$ at $m'$ instead of writing $\varphi(m)(m')$.) Also consider the metaplectic restricted tensor product $$\underset{v}{{\widetilde{\otimes}}'}{\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v},$$ where for almost all $v$ at which all the data defining ${\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v}$ are unramified, we choose the spherical vector $\varphi_v^\circ\in {\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v}$ to be the one defined by $$\varphi_v^\circ(m)=\begin{cases}{\pi^{(n)}}_{\omega_v}(h)f_v^\circ&\text{if $m=h(k,1)$ for $h\in A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)$ and $(k,1)\in{\widetilde{M}}({\mathcal{O}_{F_v}})$};\\ 0&\text{otherwise}, \end{cases}$$ where $f_v^\circ\in{\pi^{(n)}}_{\omega_v}$ is the spherical vector defining the restricted metaplectic tensor product ${\pi^{(n)}}_\omega={\widetilde{\otimes}}_v'{\pi^{(n)}}_{\omega_v}$. (Let us mention that we do not know if the dimension of the spherical vectors in ${\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v}$ is one or not.) One has the injection $$T:\underset{v}{{\widetilde{\otimes}}'}{\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v} {\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}}{\operatorname{c-Ind}}_{A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})}^{{\widetilde{M}}({\mathbb{A}})}{\pi^{(n)}}_\omega$$ given by $T(\otimes_v\varphi_v)(m)=\otimes_v \varphi_v(m_v)\in{\widetilde{\otimes}}_v'{\pi^{(n)}}_{\omega_v}$. The reason the image of $T$ lies in the compactly induced space is because for almost all $v$, the support of $\varphi^\circ$ is $A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v){\widetilde{M}}({\mathcal{O}_{F_v}})$ and for all $v$ the index of $A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)$ in ${\widetilde{M}}(F_v)$ is finite by (\[E:finite\_quotient\]). (Indeed, the support property and the finiteness of this index imply that $T$ is actually onto as well, though we do not use this fact.) Let $$V({\pi^{(n)}}_\omega)= T\left(\underset{v}{{\widetilde{\otimes}}'}{\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v}\right),$$ namely $V({\pi^{(n)}}_\omega) $ is the image of $T$. For each $\varphi\in V({\pi^{(n)}}_\omega)$, define ${\widetilde{\varphi}}:{\widetilde{M}}({\mathbb{A}})\rightarrow{\mathbb C}$ by $$\label{E:definition} {\widetilde{\varphi}}(m)=\sum_{\gamma\in A_{M}M^{(n)}(F)\backslash M(F)}\varphi({\mathbf{s}}(\gamma) m;1).$$ Let us note that by $A_{M}M^{(n)}(F)$ we mean $p(A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F))$, which is not necessarily the same as $A_{M}(F){M^{(n)}}(F)$, and $${\mathbf{s}}(A_{M}M^{(n)}(F))=A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F)\subseteq A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}}).$$ By the automorphy of ${\pi^{(n)}}_\omega$, $\varphi$ is left invariant on ${\mathbf{s}}(A_{M}M^{(n)}(F))$ and hence the sum is well-defined. Also note that for each fixed $m\in{\widetilde{M}}({\mathbb{A}})$ the map $m'\mapsto\varphi(m'm;1)$ is compactly supported modulo $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$. Now by our assumption on $A_{{\widetilde{M}}}$ (Hypothesis ($\ast$)), the image of $M(F)$ is discrete in $A_M({\mathbb{A}}){M^{(n)}}({\mathbb{A}})\backslash M({\mathbb{A}})$, and hence the group $A_{M}M^{(n)}(F)\backslash M(F)$ naturally viewed as a subgroup of $A_M({\mathbb{A}}){M^{(n)}}({\mathbb{A}})\backslash M({\mathbb{A}})$ is discrete. Now a discrete subgroup is always closed by [@Deitmar Lemma 9.1.3 (b)]. Thus the above sum is a finite sum, and in particular the sum is convergent. Moreover one can find $\varphi$ with the property that the support of the map $m'\mapsto\varphi(m';1)$ is small enough so that if $\gamma\in A_{M}{M^{(n)}}(F)\backslash M(F)$, then $\varphi(\gamma;1)\neq 0$ only at $\gamma=1$. Thus the map $\varphi\mapsto{\widetilde{\varphi}}$ is not identically zero. \[R:hypothesis\] It should be mentioned here that Hypothesis ($\ast$) is needed to make sure that the sum in (\[E:definition\]) is convergent and not identically zero. The author suspects that either one can always find $A_{{\widetilde{M}}}$ so that Hypothesis ($\ast$) is satisfied (which is the case if $n=2$), or even without Hypothesis ($\ast$) one can show that the sum in (\[E:definition\]) is convergent and not identically zero. But the thrust of this paper is our application to symmetric square $L$-functions ([@Takeda1; @Takeda2]) for which we only need the case for $n=2$. One can verify that ${\widetilde{\varphi}}$ is a smooth automorphic form on ${\widetilde{M}}({\mathbb{A}})$: The automorphy is clear. The smootheness and $K_f$-finiteness follows from the fact that at each non-archimedean $v$, the induced representation ${\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v}$ is smooth and admissible. That ${\widetilde{\varphi}}$ is ${\mathcal{Z}}$-finiteness and of uniform moderate growth follows from the analogous property of $\varphi({\mathbf{s}}(\gamma)m)$, because the Lie algebra of ${\widetilde{M}}(F_v)$ at archimedean $v$ is the same as that of ${\widetilde{M}^{(n)}}(F_v)$. As we already mentioned, the sum in (\[E:definition\]) is finite. But [*a priori*]{} which $\gamma\in A_{M}M^{(n)}(F)\backslash M(F)$ contributes to the above sum depends on $m$. Yet, one can show that as $m$ runs through ${\widetilde{M}}({\mathbb{A}})$, only finitely many $\gamma$’s contribute. To show it, we need the following lemma. \[L:strong\_approximation\] Let $S$ be a finite set of places including all the infinite places and let $\mathcal{O}_S=\prod_{v\notin S}{{\mathcal{O}_F}}_v$. Then the group $$F^\times{\mathbb{A}}^{\times n}\mathcal{O}_S^\times\backslash{\mathbb{A}}^\times$$ is finite. It is well-known that the strong approximation theorem implies $${\mathbb{A}}^\times=F^\times \prod_{v\in S}F_v^\times\mathcal{O}_S^\times,$$ and the index of $(F_v^\times)^n$ in $F_v^\times$ is finite. This proves the lemma. Occasionally, $F^\times{\mathbb{A}}^{\times n}\;\mathcal{O}_S^\times\backslash{\mathbb{A}}^\times$ can be shown to be the trivial group. This is the case for example if $n=2$ and $F={\mathbb{Q}}$. But there are cases where it is not trivial even when $n=2$. An interested reader might want to look at [@Kable Appendix]. \[L:finite\_sum\] For each $\varphi\in V({\pi^{(n)}}_\omega)$, there exist finitely many $\gamma_1,\dots,\gamma_N\in A_{M}M^{(n)}(F)\backslash M(F)$, depending only on $\varphi$, such that $${\widetilde{\varphi}}(m)=\sum_{i=1}^N\varphi({\mathbf{s}}(\gamma_i) m;1).$$ We may assume that $\varphi$ is of the form $T(\otimes_v\varphi_v)$ for a simple tensor $\otimes_v\varphi_v$. Then there exists a finite set $S$ of places such that for all $k\in\kappa(K_S)$, we have $k\cdot\varphi=\varphi$, where $K_S=\prod_{v\notin S}M({{\mathcal{O}_F}}_v)\subset M({\mathbb{A}})$ and $\kappa: M({\mathbb{A}})\rightarrow{\widetilde{M}}({\mathbb{A}})$ is the section $m\mapsto(m,1)$. Namely the stabilizer of $\varphi$ contains $\kappa(K_S)$. Then one can see that $${\operatorname{supp}}(\varphi)={\operatorname{supp}}(m\cdot\varphi)$$ for all $m\in {\widetilde{M}^{(n)}}({\mathbb{A}})\kappa(K_S)$. Also we have ${\widetilde{\varphi}}({\mathbf{s}}(\gamma)m)={\widetilde{\varphi}}(m)$. Hence, noting that ${\widetilde{\varphi}}(m)=\widetilde{m\cdot\varphi}(1)$, we see that the $\gamma$’s that contribute to the sum in (\[E:definition\]) depend only on the class in $${\mathbf{s}}(M(F)){\widetilde{M}^{(n)}}({\mathbb{A}})\kappa(K_S)\backslash{\widetilde{M}}({\mathbb{A}}).$$ But one can see that this set can be identified with the product of $k$ copies of $$F^\times{\mathbb{A}}^{\times n}\;\mathcal{O}_S^\times\backslash{\mathbb{A}}^\times,$$ where $\mathcal{O}_S=\prod_{v\notin S}{{\mathcal{O}_F}}_v$, and this set is finite by Lemma \[L:strong\_approximation\]. This implies that there are only finitely many $\gamma_1,\dots,\gamma_N\in A_{M}(F)M^{(n)}(F)\backslash M(F)$ such that $\varphi({\mathbf{s}}(\gamma_i)m;1)\neq 0$ for at least some $m\in{\widetilde{M}}({\mathbb{A}})$. This completes the proof. \[T:main\] Let $${\widetilde{V}}({\pi^{(n)}}_\omega)=\{{\widetilde{\varphi}}:\varphi\in V({\pi^{(n)}}_\omega)\}$$ and $\pi_\omega$ an irreducible constituent of ${\widetilde{V}}({\pi^{(n)}}_\omega)$. Then it is an irreducible automorphic representation of ${\widetilde{M}}({\mathbb{A}})$ and $$\pi_\omega\cong\underset{v}{{\widetilde{\otimes}}}'\pi_{\omega_v},$$ where $\pi_{\omega_v} $ is the local metaplectic tensor product of Mezo. Also the isomorphism class of $\pi_\omega$ depends only on the choice of the character $\omega|_{Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})}$. Since the map $\varphi\mapsto{\widetilde{\varphi}}$ is ${\widetilde{M}}({\mathbb{A}})$-intertwining, the space ${\widetilde{V}}({\pi^{(n)}}_\omega)$ provides a space of (possibly reducible) automorphic representation of ${\widetilde{M}}({\mathbb{A}})$. Hence $\pi_\omega$ is an automorphic representation of ${\widetilde{M}}({\mathbb{A}})$. Since each $\pi_i$ is unitary, so is each ${\pi^{(n)}}_i$, from which one can see that ${\pi^{(n)}}_\omega$ is unitary. Since $V({\pi^{(n)}}_\omega)$ is a subrepresentation of the compactly induced representation induced from the unitary ${\pi^{(n)}}_\omega$, $V({\pi^{(n)}}_\omega)$ is unitary. Hence $\pi_\omega$, which is a subquotient of $V({\pi^{(n)}}_\omega)\cong\underset{v}{{\widetilde{\otimes}}}'{\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v}$, is actually a quotient of $\underset{v}{{\widetilde{\otimes}}}'{\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v}$ by admissibility. With this said, one can derive the isomorphism $\pi_\omega\cong\underset{v}{{\widetilde{\otimes}}}'\pi_{\omega_v}$ from Lemma \[L:local\_global\]. Since the local $\pi_{\omega_v}$ depends only on the choice of $\omega_v|_{Z_{{\widetilde{\operatorname{GL}}}_r}(F_v)}$, the global $\pi_\omega$ depends only on $\omega|_{Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})}$ up to equivalence. We call the above constructed $\pi_\omega$ the global metaplectic tensor product of $\pi_1,\dots,\pi_k$ (with respect to $\omega$) and write $$\pi_\omega=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega.$$ We do not know if the multiplicity one theorem holds for the group ${\widetilde{M}}({\mathbb{A}})$, and hence do not know if the space ${\widetilde{V}}({\pi^{(n)}}_\omega)$ has only one irreducible constituent. In this sense, the definition of $\pi_\omega$ depends on the choice of the irreducible constituent. For this reason, the metaplectic tensor product should be construed as an equivalence class of automorphic representations, although we know more or less explicit way of expressing automorphic forms in $\pi_\omega$. **The uniqueness** ------------------ Just like the local case, the metaplectic tensor product of automorphic representations is unique up to twist. \[P:global\_uniqueness\] Let $\pi_1,\dots,\pi_k$ and $\pi'_1,\dots,\pi'_k$ be unitary automorphic subrepresentations of ${\widetilde{\operatorname{GL}}}_{r_1}({\mathbb{A}}),\dots,{\widetilde{\operatorname{GL}}}_{r_k}({\mathbb{A}})$. They give rise to isomorphic metaplectic tensor products with a character $\omega$, [[*i.e.* ]{}]{}$$(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega\cong (\pi'_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi'_k)_\omega,$$ if and only if for each $i$ there exists an automorphic character $\omega_i$ of ${\widetilde{\operatorname{GL}}}_{r_i}({\mathbb{A}})$ trivial on ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}})$ such that $\pi_i\cong\omega_i\otimes\pi'_i$. By Theorem \[T:main\], the global metaplectic tensor product is written as the metaplectic restricted tensor product of the local metaplectic tensor products of Mezo. Hence by Proposition \[P:local\_uniqueness\] for each $i$ and each place $v$, there is a character $\omega_{i,v}$ on ${\widetilde{\operatorname{GL}}}_{r_i}(F_v)$ trivial on ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}(F_v)$ such that $\pi_{i, v}\cong\omega_{i, v}\otimes\pi'_{i, v}$. Let $\omega_i={\widetilde{\otimes}}_v'\omega_{i,v}$. Then $\pi_i\cong\omega_i\otimes\pi'_i$. The automorphy of $\omega$ follows from that of $\pi_i$ and $\pi'_i$. This proves the only if part. The if part follows similarly. **Cuspidality and square-integrability** {#S:cuspidality} ---------------------------------------- In this subsection, we will show that the cuspidality and square-integrability are preserved for the metaplectic tensor product. \[T:cuspidal\] Assume $\pi_1,\dots,\pi_k$ are all cuspidal. Then the metaplectic tensor product $\pi_\omega=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$ is cuspidal. Assume $\pi_1,\dots,\pi_k$ are all cuspidal. It suffices to show that for each $\varphi\in V({\pi^{(n)}}_\omega)$ $$\int_{U(F)\backslash U({\mathbb{A}})}{\widetilde{\varphi}}({\mathbf{s}}(u))\,du=0$$ for all unipotent radical $U$ of the standard proper parabolic subgroup of $M$, where recall from Proposition \[P:s\_split\_M\] that the partial set theoretic section ${\mathbf{s}}:M({\mathbb{A}})\rightarrow{\widetilde{M}}({\mathbb{A}})$ is defined (and a group homomorphism) on the groups $M(F)$ and $U({\mathbb{A}})$. We know that ${\widetilde{\varphi}}({\mathbf{s}}(u))$ is a finite sum of $\varphi({\mathbf{s}}(\gamma){\mathbf{s}}(u);1)$ for $\gamma\in A_{M}M^{(n)}(F)\backslash M(F)$. Hence it suffices to show $\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma){\mathbf{s}}(u);1)\,du=0$. Note that for each $\gamma={\operatorname{diag}}(\gamma_1,\dots,\gamma_k)$ with $\gamma_i\in{\operatorname{GL}}_{r_i}(F)$, we have $$\gamma_i=\gamma_i\begin{pmatrix}\det(\gamma_i)^{n-1}&\\ &I_{r_i-1}\end{pmatrix}\begin{pmatrix}\det(\gamma_i)^{-n+1}&\\ &I_{r_i-1}\end{pmatrix},$$ where $\gamma_i\begin{pmatrix}\det(\gamma_i)^{n-1}&\\ &I_{r_i-1}\end{pmatrix}\in {\operatorname{GL}}_r^{(n)}(F)$. Hence we may assume $\gamma\in M(F)$ is a diagonal matrix, and so $\gamma u\gamma^{-1}\in U({\mathbb{A}})$. Then we have $$\begin{aligned} \allowdisplaybreaks &\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma){\mathbf{s}}(u);1)\,du\\ =&\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma) {\mathbf{s}}(u){\mathbf{s}}(\gamma^{-1}){\mathbf{s}}(\gamma);1)\,du\\ =&\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma);{\mathbf{s}}(\gamma){\mathbf{s}}( u){\mathbf{s}}(\gamma^{-1}))\,du\quad\text{because ${\mathbf{s}}(\gamma){\mathbf{s}}( u){\mathbf{s}}(\gamma^{-1})\in\widetilde{U}({\mathbb{A}})$}\\ =&\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma);{\mathbf{s}}(\gamma){\mathbf{s}}( u\gamma^{-1}))\,du\quad\text{by Proposition \ref{P:s_split_M}}\\ =&\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma);{\mathbf{s}}(\gamma){\mathbf{s}}( \gamma^{-1}u)\,du\quad\text{by change of variables $\gamma u\gamma^{-1}\mapsto u$}\\ =&\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma);{\mathbf{s}}(\gamma){\mathbf{s}}( \gamma^{-1}){\mathbf{s}}(u)\,du\quad\text{by Proposition \ref{P:s_split_M}}\\ =&\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma);{\mathbf{s}}(u))\,du \quad\text{by Proposition \ref{P:s_split_M}}.\end{aligned}$$ We would like to show this is equal to zero. For this purpose, recall that for each $\gamma$, $\varphi({\mathbf{s}}(\gamma))$ is in the space $V_{{\pi^{(n)}}_\omega}$ and hence is (a finite sum of functions) of the form $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k$ with $f_i\in V_{\pi_i}$ and each $f_i$ is a cusp form. We may assume $\varphi({\mathbf{s}}(\gamma))$ is a simple tensor $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k$. Now we can write $U=U_1\times\cdots\times U_k$, where each $U_i$ is a unipotent subgroup of ${\operatorname{GL}}_{r_i}$ with at least one of $U_i$ non-trivial, and accordingly we denote each element $u\in U$ by $u={\operatorname{diag}}(u_1,\dots,u_k)$. Then by definition of ${\mathbf{s}}$, we have $${\mathbf{s}}(u)=(u, \prod_is_{r_i}(u_i)^{-1}),$$ and $$\begin{aligned} \varphi({\mathbf{s}}(\gamma);{\mathbf{s}}(u))&=(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k)({\mathbf{s}}(u))\\ &=(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k)(u, \prod_is_{r_i}(u_i)^{-1})\\ &=\left(\prod_is_{r_i}(u_i)^{-1}\right)f_1(u_1,1)\cdots f_k(u_k,1)\quad\text{by definition of $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k$}\\ &=f_1(u_1,s_{r_i}(u_i)^{-1})\cdots f_k(u_k,s_{r_k}(u_k)^{-1})\quad \text{because each $f_i$ is genuine}\\ &=f_1({\mathbf{s}}_{r_1}(u_1))\cdots f_k({\mathbf{s}}_{r_k}(u_k))\quad \text{by definition of ${\mathbf{s}}_{r_i}$}.\end{aligned}$$ Hence $$\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma);{\mathbf{s}}(u))\,du =\prod_{i=1}^k\int_{U_i(F)\backslash U_i({\mathbb{A}})}f_i({\mathbf{s}}_{r_i}(u_i))\,du_i.$$ This is equal to zero because each $f_i$ is cuspidal and at least one of $U_i$ is non-trivial. Next let us take care of the square-integrability. \[T:square\_integrable\] Assume $\pi_1,\dots,\pi_k$ are all square-integrable modulo center. Then the metaplectic tensor product $\pi_\omega=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$ is square-integrable modulo center. We need a few lemmas for the proof of this theorem. \[L:quotient\_measure\] Let $G$ be a locally compact group and $H, N\subset G$ be closed subgroups such that $NH$ is a closed subgroup. Further assume that the quotient measures for $N\backslash G, H\backslash NH$ and $NH\backslash G$ all exist. (Recall that in general the quotient measure for $N\backslash G$ exists if the modular characters of $G$ and $N$ agree on $N$.) Then $$\begin{aligned} \int_{N\backslash G}f(g)\;dg&=\int_{NH\backslash G}\int_{N\backslash NH}\;f(hg)\;dh\;dg\\ &=\int_{NH\backslash G}\int_{N\cap H\backslash H}\;f(hg)\;dh\;dg.\end{aligned}$$ for all $f\in L^1(N\backslash G)$. The first equality is [@Bourbaki Cor. 1 VII 47], and the second equality follows from the natural identification $N\backslash NH\cong N\cap H\backslash H$. Now let $f:{\widetilde{M}}({\mathbb{A}})\rightarrow{\mathbb C}$ be any function. Then the absolute value $|f|$ is non-genuine in the sense that it factors through $M({\mathbb{A}})$. Also we let $$Z^{(n)}_M({\mathbb{A}}):= \{\begin{pmatrix}a_1^nI_{r_1}&&\\ &\ddots&\\ &&a_k^nI_{r_k}\end{pmatrix}:a_i\in{\mathbb{A}}^\times\}.$$ This is a closed subgroup by Lemma \[L:closed\_subgroup\_local\] and \[L:closed\_subgroup\_local\_global\]. Note the inclusions $$Z_M^{(n)}({\mathbb{A}})\subseteq p(Z_{{\widetilde{M}}}({\mathbb{A}}))\subseteq Z_M({\mathbb{A}}),$$ where all the groups are closed subgroups of $M({\mathbb{A}})$. Then we have \[L:M\_square\_integrable\] Let $f:M(F)\backslash {\widetilde{M}}({\mathbb{A}})\rightarrow{\mathbb C}$ be an automorphic form with a unitary central character. Then $f$ is square-integrable modulo the center $Z_{{\widetilde{M}}}({\mathbb{A}})$ if and only if $|f|\in L^2( Z^{(n)}_{M}({\mathbb{A}}) M(F)\backslash M({\mathbb{A}}))$ where $|f|$ is viewed as a function on $M({\mathbb{A}})$ as noted above. Let $f$ be an automorphic form on ${\widetilde{M}}({\mathbb{A}})$ with a unitary central character. Since $|f|$ is non-genuine, we have $$\int_{Z_{{\widetilde{M}}}({\mathbb{A}}) M(F)\backslash{\widetilde{M}}({\mathbb{A}})}|f(\tilde{m})|^2\;d\tilde{m} =\int_{p(Z_{{\widetilde{M}}}({\mathbb{A}})) M(F)\backslash M({\mathbb{A}})}|f(\kappa(m))|^2\;dm,$$ where recall that $p:{\widetilde{M}}({\mathbb{A}})\rightarrow M({\mathbb{A}})$ is the canonical projection. Note that the quotient measure on the right hand side exists because the group $p(Z_{{\widetilde{M}}}({\mathbb{A}}))M(F)$ is closed by [@MW Lemma I.1.5, p.8] and is unimodular because $p(Z_{{\widetilde{M}}}({\mathbb{A}}))$ is unimodular and $M(F)$ is discrete and countable. By Lemma \[L:quotient\_measure\], we have $$\int_{Z^{(n)}_{M}({\mathbb{A}}) M(F)\backslash M({\mathbb{A}})}|f(\kappa(m))|^2\;dm =\int_{p(Z_{{\widetilde{M}}}({\mathbb{A}})) M(F)\backslash M({\mathbb{A}})} \int_{Z_M^{(n)}({\mathbb{A}}) p(Z_{{\widetilde{M}}}(F))\backslash p(Z_{{\widetilde{M}}}({\mathbb{A}}))}|f(\kappa(zm))|^2\;dz\;dm.$$ Since for each fixed $m\in M({\mathbb{A}})$, the function $z\mapsto f(\kappa(zm))$ is a smooth function on $p(Z_{{\widetilde{M}}}({\mathbb{A}}))$, there exists a finite set $S$ of places such that for all $z'\in p(Z_{{\widetilde{M}}}(\mathcal{O}_S))=Z_M(\mathcal{O}_S)\cap p(Z_{{\widetilde{M}}}({\mathbb{A}}))$ we have $f(\kappa(z'zm))=f(\kappa(zm))$. Hence the inner integral of the above integral is written as $$\label{E:square_integrable} \int_{Z^{(n)}_M({\mathbb{A}})p(Z_{{\widetilde{M}}}(\mathcal{O}_S)) p(Z_{{\widetilde{M}}}(F))\backslash p(Z_{{\widetilde{M}}}({\mathbb{A}}))}|f(\kappa(zm))|^2\;dz.$$ Note that we have the inclusion $$Z^{(n)}_M({\mathbb{A}})p(Z_{{\widetilde{M}}}(\mathcal{O}_S)) p(Z_{{\widetilde{M}}}(F))\backslash p(Z_{{\widetilde{M}}}({\mathbb{A}})) \subseteq Z^{(n)}_M({\mathbb{A}})Z_M(\mathcal{O}_S)Z_M(F)\backslash Z_M({\mathbb{A}}),$$ because $p(Z_{{\widetilde{M}}}(\mathcal{O}_S))\cap p(Z_{{\widetilde{M}}}(F))=Z_M(\mathcal{O}_S)\cap Z_M(F)=1$, and note that $Z^{(n)}_M({\mathbb{A}})Z_M(\mathcal{O}_S)Z_M(F)\backslash Z_M({\mathbb{A}})$ can be identified with the product of $k$ copies of $$F^\times{\mathbb{A}}^{\times n}\;\mathcal{O}_S^\times\backslash{\mathbb{A}}^\times.$$ By Lemma \[L:strong\_approximation\], we know that this is a finite group, and hence the integral in (\[E:square\_integrable\]) is just a finite sum. Thus for some finite $z_1,\dots,z_N\in p(Z_{{\widetilde{M}}}({\mathbb{A}}))$, we have $$\begin{aligned} \int_{Z^{(n)}_{M}({\mathbb{A}}) M(F)\backslash M({\mathbb{A}})}|f(\kappa(m))|^2\;dm &=\int_{p(Z_{{\widetilde{M}}}({\mathbb{A}})) M(F)\backslash M({\mathbb{A}})} \sum_{i=1}^N |f(\kappa(z_im))|^2\;dm\\ &=\sum_{i=1}^N \int_{p(Z_{{\widetilde{M}}}({\mathbb{A}})) M(F)\backslash M({\mathbb{A}})} |f(\kappa(m))|^2\;dm\\ &=N\int_{p(Z_{{\widetilde{M}}}({\mathbb{A}})) M(F)\backslash M({\mathbb{A}})} |f(\kappa(m))|^2\;dm,\end{aligned}$$ where for the second equality we used $$|f(\kappa(z_im))|=|f((\kappa(z_i)\kappa(m)))| =|\omega(\kappa(z_1))||f(\kappa(m))|=|f(\kappa(m))|$$ where $\omega$ is the central character of $f$ which is assumed to be unitary. The lemma follows from this. \[L:pin\_square\_integrable\] Assume $\pi_1,\dots,\pi_k$ are as in Theorem \[T:square\_integrable\]. Let $\varphi_i\in{\pi^{(n)}}_i$ for $i=1,\dots,k$ and $\varphi=\varphi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\varphi_k\in{\pi^{(n)}}$, which is a function on ${\widetilde{M}^{(n)}}({\mathbb{A}})$. Then $$\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}(F)\backslash M^{(n)}({\mathbb{A}})}|\varphi(\kappa(m))|^2\;dm<\infty.$$ Write each element $m\in M({\mathbb{A}})$ as $m={\operatorname{diag}}(g_1,\dots,g_k)$ where $g_i\in {\operatorname{GL}}_{r_i}({\mathbb{A}})$. Then ${\operatorname{diag}}(g_1,\dots,g_k)\in M^{(n)}({\mathbb{A}})$ if and only if $g_i\in {\operatorname{GL}}_{r_i}^{(n)}({\mathbb{A}})$ for all $i$. Hence the integral in the lemma is the product of integrals $$\int_{Z_{{\operatorname{GL}}_{r_i}}^{(n)}({\mathbb{A}}){\operatorname{GL}}_{r_i}^{(n)}(F)\backslash {\operatorname{GL}}_{r_i}^{(n)}({\mathbb{A}})}|\varphi_i(\kappa(g_i))|^2\;dg_i,$$ where $Z_{{\operatorname{GL}}_{r_i}}^{(n)}({\mathbb{A}})$ consists of the elements of the form $a_iI_{r_i}$ with $a_i\in{\mathbb{A}}^{\times n}$. So we have to show that this integral converges. But with Lemma \[L:M\_square\_integrable\] applied to $M={\operatorname{GL}}_{r_i}$, we know $$\int_{Z^{(n)}_{{\operatorname{GL}}_{r_i}}({\mathbb{A}}){\operatorname{GL}}_{r_i}(F)\backslash{\operatorname{GL}}_{r_i}({\mathbb{A}})}|\varphi_i(\kappa(g_i))|^2\;dg_i<\infty,$$ because each $\varphi_i$ is square-integrable modulo center. By Lemmas \[L:quotient\_measure\] and \[L:discrete\_sub\], this is written as $$\int_{Z^{(n)}_{{\operatorname{GL}}_{r_i}}({\mathbb{A}}){\operatorname{GL}}^{(n)}_{r_i}({\mathbb{A}}){\operatorname{GL}}_{r_i}(F)\backslash{\operatorname{GL}}_{r_i}({\mathbb{A}})} \int_{Z^{(n)}_{{\operatorname{GL}}_{r_i}}({\mathbb{A}}){\operatorname{GL}}^{(n)}_{r_i}(F)\backslash{\operatorname{GL}}^{(n)}_{r_i}({\mathbb{A}})} |\varphi_i(\kappa(m_i'm_i))|^2\;dm_i'\;dm_i<\infty.$$ In particular the inner integral converges, which proves the lemma. Now we are ready to prove Theorem \[T:square\_integrable\]. By Lemma \[L:M\_square\_integrable\], we have only to show $$\int_{Z_M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})}|{\widetilde{\varphi}}(\kappa(m))|^2\;dm<\infty.$$ By Lemma \[L:quotient\_measure\], we have $$\begin{aligned} \label{E:outer_integral} \notag&\int_{Z_M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})}|{\widetilde{\varphi}}(\kappa(m))|^2\;dm\\ =&\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})}\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}(F)\backslash M^{(n)}({\mathbb{A}})}|{\widetilde{\varphi}}(\kappa(m'm))|^2\;dm'\;dm\\ \notag =&\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})}\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}(F)\backslash M^{(n)}({\mathbb{A}})}\left|\sum_{\gamma}\varphi(\kappa(\gamma m'm);1)\right|^2\;dm'\;dm\\ \notag =&\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})}\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}(F)\backslash M^{(n)}({\mathbb{A}})}\left|\sum_{\gamma}\varphi(\kappa(\gamma m);\kappa(\gamma m'\gamma^{-1}))\right|^2\;dm'\;dm.\end{aligned}$$ Note that by Lemma \[L:finite\_sum\], the sum of the integrand is finite. Also the map $m'\mapsto \left|\varphi(\kappa(\gamma m);\kappa(\gamma m'\gamma^{-1}))\right|^2$ is invariant under $Z_M^{(n)}({\mathbb{A}})M^{(n)}(F)$ on the left. Hence to show the inner integral converges, it suffices to show the integral $$\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}(F)\backslash M^{(n)}({\mathbb{A}})} \left|\varphi(\kappa(\gamma m);\kappa(\gamma m'\gamma^{-1}))\right|^2\;dm'$$ converges. But this follows from Lemma \[L:pin\_square\_integrable\]. To show the outer integral converges, note that the map $m\mapsto |{\widetilde{\varphi}}(\kappa(m'm))|^2$ is smooth and hence there exists a finite set of places $S$ so that ${\widetilde{\varphi}}(\kappa(m'm k))={\widetilde{\varphi}}(\kappa(m'm k))$ for all $k\in M(\mathcal{O}_S)$. Thus the integral in is (a scalar multiple of) $$\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})/M(\mathcal{O}_S)}\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}(F)\backslash M^{(n)}({\mathbb{A}})}|{\widetilde{\varphi}}(\kappa(m'm))|^2\;dm'\;dm.$$ Now the set theoretic map $$F^\times{\mathbb{A}}^{\times n}\underbrace{\mathcal{O}^\times_S\backslash{\mathbb{A}}^\times \times\cdots\times F^\times}_{\text{$k$ copies}}{\mathbb{A}}^{\times n}\mathcal{O}^\times_S \backslash{\mathbb{A}}^\times \rightarrow Z_M^{(n)}({\mathbb{A}})M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})/M(\mathcal{O}_S)$$ given by $$(a_1,\dots,a_k)\mapsto\begin{pmatrix}\iota_1(a_1)&&\\ &\ddots&\\ &&\iota_k(a_k)\end{pmatrix}$$ where $\iota_i$ is as in (\[E:iota\]) is a well-defined surjection. Hence Lemma \[L:strong\_approximation\] implies that the set $$Z_M^{(n)}({\mathbb{A}})M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})/M(\mathcal{O}_S)$$ is a finite set. Therefore the outer integral of the above integral is a finite sum and hence converges. This completes the proof. **Twists by Weyl group elements** {#S:Weyl_group_global} --------------------------------- Just as we saw in Section \[S:Weyl\_group\_global\] for the local case, the global metaplectic tensor product behaves in the expected way under the action of the Weyl group elements in $W_M$. Namely \[T:Weyl\_group\_global\] Let $w\in W_M$ be such that $^w({\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}) ={\operatorname{GL}}_{r_{\sigma(1)}}\times\cdots\times{\operatorname{GL}}_{r_{\sigma(k)}}$. Then we have $$^w(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega \cong(\pi_{\sigma(1)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_{\sigma(k)})_\omega,$$ where $w$ is viewed as an element in ${\operatorname{GL}}_r(F)$. Note that each ${\mathbf{s}}(w)\in{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ is written as $\prod_v(w, s_{r,v}(w))$, where we view $(w, s_{r,v}(w))\in{\widetilde{\operatorname{GL}}}_r(F_v)$ as an element of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ view the natural embedding ${\widetilde{\operatorname{GL}}}_r(F_v)\hookrightarrow{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$, and the product $\prod_v$ is literally the product inside ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$. Then one can see that $$^w(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega={\widetilde{\otimes}}'_v\, ^w(\pi_{1,v}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_{k,v})_{\omega_v}.$$ Hence the theorem follows from the local counter part (Theorem \[T:Weyl\_group\_local\]). The following is immediate: Let $\pi_\omega=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$. For $w$ as in the theorem and each automorphic form ${\widetilde{\varphi}}\in\pi_\omega$, define $^w{\widetilde{\varphi}}:\;^w{\widetilde{M}}(A)\rightarrow{\mathbb C}$ by $$^w{\widetilde{\varphi}}(m)={\widetilde{\varphi}}({\mathbf{s}}(w)^{-1}m{\mathbf{s}}(w))$$ for $m\in\;^w{\widetilde{M}}({\mathbb{A}})$. Then the representation $^w\pi_\omega$ is realized in the space $$\{^w{\widetilde{\varphi}}:{\widetilde{\varphi}}\in V_{\pi_\omega}\}.$$ Let us mention the following subtle point. Here we have (at least) two different realizations of $^w\pi_\omega$ in a space of automorphic forms on $^w{\widetilde{M}}({\mathbb{A}})$, the one is in the space $\{^w{\widetilde{\varphi}}:{\widetilde{\varphi}}\in V_{\pi_\omega}\}$ as in the proposition and the other as in the definition of the metaplectic tensor product $(\pi_{\sigma(1)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_{\sigma(k)})_\omega$ by choosing an appropriate $A_{^w{\widetilde{M}}}$ that satisfies Hypothesis ($\ast$) with respect to the Levi $^w{\widetilde{M}}$ (if possible at all). Without the multiplicity one property for the group $^w{\widetilde{M}}$, we do not know if they coincide. But one can see that if $A_{{\widetilde{M}}}$ satisfies Hypothesis ($\ast$) with respect to ${\widetilde{M}}$, then the group $^wA_{{\widetilde{M}}}:=wA_{{\widetilde{M}}}w^{-1}$ satisfies Hypothesis ($\ast$) with respect to $^w{\widetilde{M}}$. Then if we define $(\pi_{\sigma(1)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_{\sigma(k)})_\omega$ by choosing $A_{^w{\widetilde{M}}}=\,^wA_{{\widetilde{M}}}$, one can see from the construction of our metaplectic tensor product that the space of $(\pi_{\sigma(1)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_{\sigma(k)})_\omega$ is indeed a space of automorphic forms of the form $^w{\widetilde{\varphi}}$ for ${\widetilde{\varphi}}\in V_{\pi_\omega}$. **Compatibility with parabolic induction** {#S:parabolic_induction_global} ------------------------------------------ Just as the local case, we have the compatibility with parabolic inductions. But before stating the theorem, let us mention Let $P=MN$ be the standard parabolic subgroup of ${\operatorname{GL}}_r$. Then ${\widetilde{M}}({\mathbb{A}})$ normalizes $N({\mathbb{A}})^\ast$, where $N({\mathbb{A}})^\ast$ is the image of $N({\mathbb{A}})$ under the partial section ${\mathbf{s}}:{\operatorname{GL}}_r({\mathbb{A}})\rightarrow{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$. \[L:normalizer\_global\] One can prove it by using the local analogue (Lemma \[L:normalizer\_local\]). Namely let ${\tilde{m}}=(m,1)\in{\widetilde{M}}({\mathbb{A}})$, so ${\tilde{m}}^{-1}=(m^{-1},\tau_r(m,m^{-1})^{-1})$. Also let $n^\ast=(n,s_r(n)^{-1})\in N({\mathbb{A}})^\ast$. Then $$\begin{aligned} {\tilde{m}}n^\ast{\tilde{m}}^{-1}&=(m,1)(n,s_r(n)^{-1})(m^{-1},\tau_r(m,m^{-1})^{-1})\\ &=(mnm^{-1}, s_r(n)^{-1}\tau_r(m,n)\tau(m,m^{-1})^{-1}\tau_r(mn,m^{-1})).\end{aligned}$$ Then one needs to show $$s_r(n)^{-1}\tau_r(m,n)\tau_r(m,m^{-1})^{-1}\tau_r(mn,m^{-1}) =s(mnm^{-1})^{-1},$$ so that ${\tilde{m}}n^\ast{\tilde{m}}^{-1}=(mnm^{-1})^\ast\in N({\mathbb{A}})^\ast$. But one can show it by arguing “semi-locally”. Namely for a sufficiently large finite set $S$ of places, we have $$\begin{aligned} &s_r(n)^{-1}\tau_r(m,n)\tau_r(m,m^{-1})^{-1}\tau_r(mn,m^{-1})\\ =&\prod_{v\in S}s_r(n_v)^{-1}\tau_r(m_v,n_v)\tau_r(m_v,m_v^{-1})^{-1}\tau_r(m_vn_v,m_v^{-1})\\ =&\prod_{v\in S}s_r(n_v)^{-1}\sigma_r(m_v,n_v)\frac{s_r(m_v)s_r(n_v)}{s_r(m_vn_v)}\\ &\qquad\qquad\cdot\sigma_r(m_v,m_v^{-1})^{-1}\frac{s_r(m_vm_v^{-1})}{s_r(m_v)s_r(m_v^{-1})} \sigma_r(m_vn_v,m_v^{-1})\frac{s_r(m_vn_v)s_r(m_v^{-1})}{s_r(m_vn_vm_v^{-1})}\\ =&\prod_{v\in S}s_r(m_vn_vm_v^{-1})^{-1}\\ =&s_r(mnm^{-1})^{-1},\end{aligned}$$ where for the second equality we used (\[E:tau\_sigma\]), for the third equality we used the same cocycle computation as in the proof of Lemma \[L:normalizer\_local\] and finally for the last equality we used $s_r(m_vn_vm_v^{-1})=1$ for all $v\notin S$. Let us mention that for the case at hand one can prove this lemma as we did here. However this lemma holds not just for our ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ but for covering groups in general . (See [@MW I.1.3(4), p.4].) At any rate, this lemma allows one to form the global induced representation $${\operatorname{Ind}}_{{\widetilde{M}}({\mathbb{A}})N({\mathbb{A}})^\ast}^{{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})}\pi$$ for an automorphic representation $\pi$ of ${\widetilde{M}}({\mathbb{A}})$, and hence one can form Eisenstein series on ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ just like the non-metaplectic case. With this said, we have \[T:induction\_global\] Let $P=MN\subseteq{\operatorname{GL}}_r$ be the standard parabolic subgroup whose Levi part is $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$. Further for each $i=1,\dots,k$ let $P_i=M_iN_i\subseteq{\operatorname{GL}}_{r_i}$ be the standard parabolic of ${\operatorname{GL}}_{r_i}$ whose Levi part is $M_i={\operatorname{GL}}_{r_{i,1}}\times\cdots\times{\operatorname{GL}}_{r_{i,l_i}}$. For each $i$, assume we can find $A_{{\widetilde{M}}_i}$ that satisfies Hypothesis ($\ast$) with respect to $M_i$ (which is always the case if $n=2$), and we are given an automorphic representation $$\sigma_i:=(\tau_{i,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{i,l_i})_{\omega_i}$$ of ${\widetilde{M}}_i({\mathbb{A}})$, which is given as the metaplectic tensor product of the unitary automorphic subrepresentations $\tau_{i,1},\dots,\tau_{i,l_i}$ of ${\widetilde{\operatorname{GL}}}_{r_{i,1}}({\mathbb{A}}),\dots,{\widetilde{\operatorname{GL}}}_{r_{i,l_i}}({\mathbb{A}})$, respectively. Assume that $\pi_i$ is an irreducible constituent of the induced representation ${\operatorname{Ind}}_{{\widetilde{P}}_i({\mathbb{A}})}^{{\widetilde{\operatorname{GL}}}_{r_i}({\mathbb{A}})}\sigma_i$ and is realized as an automorphic subrepresentation. Then the metaplectic tensor product $$\pi_\omega:=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$$ is an irreducible constituent of the induced representation $${\operatorname{Ind}}_{{\widetilde{Q}}({\mathbb{A}})}^{{\widetilde{M}}({\mathbb{A}})}(\tau_{1,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{1, l_1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\, \tau_{k,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{k, l_k})_\omega,$$ where $Q$ is the standard parabolic of $M$ whose Levi part is $M_1\times\cdots\times M_k$, where $M_i\subseteq{\operatorname{GL}}_{r_i}$ for each $i$. This follows from its local analogue (Theorem \[T:induction\_local\]) and the local-global compatibility of the metaplectic tensor product $\pi_\omega\cong{\widetilde{\otimes}}'\pi_{\omega_v}$. Just as we mentioned in Remark \[R:induction\_local\] for the local case, in the above theorem one may replace “constituent” by “irreducible subrepresentation” or “irreducible quotient”, and the analogous statement still holds. **Restriction to a smaller Levi** {#S:restriction} --------------------------------- As the last thing in this paper, let us mention an important property of the metaplectic tensor product which one needs when one computes constant terms of metaplectic Eisenstein series. (See [@Takeda2].) Both locally and globally, let $$M_2={\operatorname{GL}}_{r_2}\times\cdots\times{\operatorname{GL}}_{r_k} =\{\begin{pmatrix}I_{r_1}&&&\\ &g_2&&\\ &&\ddots&\\ &&&g_k\end{pmatrix}\in M: g_i\in{\operatorname{GL}}_{r_i}\}$$ be viewed as a subgroup of $M$ in the obvious way. We view ${\operatorname{GL}}_{r-r_1}$ as a subgroup of ${\operatorname{GL}}_r$ embedded in the right lower corner, and so $M_2$ can be also viewed as a Levi subgroup of ${\operatorname{GL}}_{r-r_1}$ embedded in this way. Both locally and globally, we let $$\tau_{M_2}:M_2\times M_2\rightarrow\mu_n$$ be the block-compatible 2-cocycle on $M_2$ defined analogously to $\tau_M$. One can see that the block-compatibility of $\tau_M$ and $\tau_{M_2}$ implies $$\label{E:tau_M2} \tau_{M_2}={\tau_M}|_{M_2\times M_2},$$ which gives the embeddings $$\begin{aligned} {\widetilde{M}}_2\subseteq{\widetilde{M}}\hookrightarrow{\widetilde{\operatorname{GL}}}_r.\end{aligned}$$ (Note that the last map is not the natural inclusion because here ${\widetilde{M}}$ is actually ${{^c\widetilde{M}}}$, and that is why we use $\hookrightarrow$ instead of $\subseteq$.) For each automorphic form ${\widetilde{\varphi}}\in V_{\pi_\omega}$ in the space of the metaplectic tensor product, one would like to know which space the restriction ${\widetilde{\varphi}}|_{{\widetilde{M}}_2({\mathbb{A}})}$ belongs to. Just like the non-metaplectic case, it would be nice if this restriction is simply in the space of the metaplectic tensor product of $\pi_2,\dots,\pi_k$ with respect to the character $\omega$ restricted to, say, $A_{{\widetilde{M}}}\cap{\widetilde{M}}_2$. But as we will see, this is not necessarily the case. The metaplectic tensor product is more subtle. Let us first introduce the subgroup $A_{{\widetilde{M}}_2}$ of ${\widetilde{M}}_2$ which plays the role analogous to that of $A_{{\widetilde{M}}}$: $$A_{{\widetilde{M}}_2}(R):=\{(\begin{pmatrix}I_{r_1}&\\ &A_2\end{pmatrix},\xi): (\begin{pmatrix}a_1I_{r_1}&\\ &A_2\end{pmatrix},\xi)\in A_{{\widetilde{M}}}(R) \text{ for some $a_1\in R^{\times n}$}\}.$$ Note that $A_{{\widetilde{M}}}(R)\cap{\widetilde{M}}_2(R) \subseteq A_{{\widetilde{M}}_2}(R)$, but the equality might not hold in general. The following lemma implies that $A_{{\widetilde{M}}_2}$ is abelian. Let $(\begin{pmatrix}I_{r_1}&\\ &A_2\end{pmatrix}, \xi), (\begin{pmatrix}I_{r_1}&\\ &A'_2\end{pmatrix}, \xi')\in A_{{\widetilde{M}}_2}(R)$. Then $$\tau_{M_2}(A_2,A'_2)=\tau_{M_2}(A'_2,A_2).$$ This follows by the block-compatibility of $\tau_M$ and the fact that $A_{{\widetilde{M}}}(R)$ is abelian. Also one can see that the image of $A_{{\widetilde{M}}_2}(R)$ under the canonical projection is closed, and hence $A_{{\widetilde{M}}_2}(R)$ is closed. Another property to be mentioned is \[L:M\_2\] For $R={\mathbb{A}}$ or $F_v$, we have $$A_{{\widetilde{M}}_2}(R){\widetilde{M}^{(n)}}_2(R)=A_{{\widetilde{M}}}(R){\widetilde{M}^{(n)}}(R)\cap {\widetilde{M}}_2(R).$$ Also for global $F$ we have $$A_{{\widetilde{M}}_2}{\widetilde{M}^{(n)}}_2(F)=A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F)\cap {\mathbf{s}}(M_2(F)),$$ where by definition $$A_{{\widetilde{M}}_2}{\widetilde{M}^{(n)}}_2(F):=A_{{\widetilde{M}}_2}({\mathbb{A}}){\widetilde{M}^{(n)}}_2({\mathbb{A}})\cap{\mathbf{s}}(M(F)),$$ which is not necessarily the same as $A_{{\widetilde{M}}_2}(F){\widetilde{M}^{(n)}}_2(F)$. The lemma can be verified by direct computations. Note that for both cases, the inclusion $\subseteq$ is immediate. To show the reverse inclusion, we need that if $a\in A_{{\widetilde{M}}}(R)$ and $m\in {\widetilde{M}^{(n)}}(R)$ are such that $am\in A_{{\widetilde{M}}}(R){\widetilde{M}^{(n)}}(R)\cap {\widetilde{M}}_2(R)$, one can always write $a=a_2a_1$ with $a_2\in A_{{\widetilde{M}}_2}(R)$ such that $a_1m\in{\widetilde{M}^{(n)}}_2(R)$, and hence $am=a_2(a_1m)\in A_{{\widetilde{M}}_2}(R){\widetilde{M}^{(n)}}_2(R)\subseteq A_{{\widetilde{M}}_2}{\widetilde{M}^{(n)}}_2(F)$. Now assume that our group $A_{{\widetilde{M}}}$ satisfies the following: The group $A_{{\widetilde{M}}}$ satisfies: 1. $A_{{\widetilde{M}}}$ satisfies Hypothesis ($\ast$) 2. $A_{{\widetilde{M}}_2}$ as defined above contains the center $Z_{{\widetilde{\operatorname{GL}}}_{r-r_1}}$. 3. $A_{{\widetilde{M}}_2}$ satisfies Hypothesis ($\ast$) with respect to ${\widetilde{M}}_2$. As an example of $A_{{\widetilde{M}}}$ satisfying the above hypothesis, we have If $n=2$, the choice of $A_{{\widetilde{M}}}$ as in Proposition \[P:A\_M\_for\_n=2\] satisfies this hypothesis. Moreover, one has $$A_{{\widetilde{M}}_2}=A_{{\widetilde{M}}}\cap{\widetilde{M}}_2$$ both locally and globally. This can be merely checked case-by-case. Next for each $\delta\in{\operatorname{GL}}_{r_1}(F)$, define $\omega_{\delta}:A_{{\widetilde{M}}_2}(F)\backslash A_{{\widetilde{M}}_2}({\mathbb{A}})\rightarrow{\mathbb C}^1$ by $$\omega_{\delta}(a)=\omega({\mathbf{s}}(\delta) a {\mathbf{s}}(\delta^{-1})).$$ Since ${\mathbf{s}}(\delta) A_{{\widetilde{M}}_2}({\mathbb{A}}){\mathbf{s}}(\delta^{-1})=A_{{\widetilde{M}}_2}({\mathbb{A}})$ and $A_{{\widetilde{M}}_2}({\mathbb{A}})\subseteq A_{{\widetilde{M}}}({\mathbb{A}})$, this is well-defined, and since ${\mathbf{s}}$ is a homomorphism on $M(F)$, $\omega_\delta$ is a character. Indeed, one can compute $$\label{E:omega_delta} \omega_{\delta}(a)=(\det\delta,\det a)^{1+2c}\omega(a)$$ because one can see $${\mathbf{s}}(\delta) a {\mathbf{s}}(\delta^{-1})=(1,(\det\delta,\det a)^{1+2c})a$$ and $\omega$ is genuine. Hence for each $a\in A_{{\widetilde{M}}_2}({\mathbb{A}})\cap A_{{\widetilde{M}}_2}{\widetilde{M}^{(n)}}_2(F){\widetilde{M}^{(n)}}_2({\mathbb{A}})$ we have $\omega_\delta(a)=\omega(a)$ because $(\det\delta,\det a)=1$, namely $$\omega_\delta|_{A_{{\widetilde{M}}_2}({\mathbb{A}})\cap A_{{\widetilde{M}}_2}{\widetilde{M}^{(n)}}_2(F){\widetilde{M}^{(n)}}_2({\mathbb{A}})} =\omega|_{A_{{\widetilde{M}}_2}({\mathbb{A}})\cap A_{{\widetilde{M}}_2}{\widetilde{M}^{(n)}}_2(F){\widetilde{M}^{(n)}}_2({\mathbb{A}})}.$$ Therefore using $\pi_2,\dots,\pi_k$ and $\omega_\delta$, one can construct the metaplectic tensor product representation of ${\widetilde{M}}_2({\mathbb{A}})$ with respect to $A_{{\widetilde{M}}_2}$, namely $$\label{E:pi_restricted_to_M_2} \pi_{\omega_\delta}:=(\pi_2\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_{\omega_\delta}.$$ Then we have \[P:restriction\] Assume $A_{{\widetilde{M}}}$ satisfies Hypothesis ($\ast\ast$). For each ${\widetilde{\varphi}}\in\pi_\omega=(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega}$, $${\widetilde{\varphi}}|_{{\widetilde{M}}_2({\mathbb{A}})}\in\bigoplus_\delta m_\delta\pi_{\omega_\delta},$$ where $\pi_{\omega_\delta}=(\pi_2{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega_\delta}$ as in (\[E:pi\_restricted\_to\_M\_2\]) and $\delta$ runs through a finite subset of ${\operatorname{GL}}_{r_1}(F)$, and $m_\delta\in{\mathbb{Z}}^{>0}$ is a multiplicity. (Note that which $\delta$ appears in the sum could depend on $\varphi$.) Recall that $${\widetilde{\varphi}}(m)=\sum_{\gamma\in A_{M}M^{(n)}(F)\backslash M(F)}\varphi({\mathbf{s}}(\gamma) m;1),$$ where the sum is finite by Lemma \[L:finite\_sum\]. Note that $A_M{M^{(n)}}(F)$ is a normal subgroup of $M(F)$, and hence $A_{M}{M^{(n)}}(F)\backslash M(F)$ is a group, which is actually an abelian group because it is a subgroup of the abelian group $A_{M}({\mathbb{A}}){M^{(n)}}({\mathbb{A}})\backslash M({\mathbb{A}})$. By Lemma \[L:M\_2\] we have the inclusion $$A_{M_2}{M^{(n)}}_2(F)\backslash M_2(F) \hookrightarrow A_{M}{M^{(n)}}(F)\backslash M(F).$$ Hence we have $$\begin{aligned} {\widetilde{\varphi}}(m)&=\sum_{\gamma\in A_{M}{M^{(n)}}(F)\backslash M(F)}\varphi({\mathbf{s}}(\gamma) m;1)\\ &=\sum_{\gamma\in M_2(F)A_{M}{M^{(n)}}(F)\backslash M(F)}\; \sum_{\mu\in A_{M_2}{M^{(n)}}_2(F)\backslash M_2(F)}\varphi({\mathbf{s}}(\mu){\mathbf{s}}(\gamma) m;1).\end{aligned}$$ By using Lemma \[L:M\_2\], one can see that the map on ${\widetilde{M}}_2({\mathbb{A}})$ defined by $m_2\mapsto \varphi(m_2{\mathbf{s}}(\gamma)m)$ is in the induced space ${\operatorname{c-Ind}}_{A_{{\widetilde{M}}_2}({\mathbb{A}}){\widetilde{M}^{(n)}}_2({\mathbb{A}})}^{{\widetilde{M}}_2({\mathbb{A}})}{\pi^{(n)}}_{\omega, 2}$, where ${\pi^{(n)}}_{\omega,2}:=\omega({\pi^{(n)}}_2\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)$ and $\omega$ is actually the restriction of $\omega$ to $A_{{\widetilde{M}}_2}({\mathbb{A}})$. Now since we are assuming that $A_{{\widetilde{M}}}$ satisfies Hypothesis ($\ast\ast$), the inner sum is finite. Since the sum over $\gamma\in A_{M}{M^{(n)}}(F)\backslash M(F)$ is finite, the outer sum is also finite. Thus there are only a finitely many $\gamma_1,\dots,\gamma_N$ that contribute to the sum over $\gamma\in M_2(F)A_{M}{M^{(n)}}(F)\backslash M(F)$ and so we have $${\widetilde{\varphi}}(m)=\sum_{i=1}^N\sum_{\mu\in A_{M_2}M_2^{(n)}(F)\backslash M_2(F)}\varphi({\mathbf{s}}(\mu\gamma_i) m;1).$$ Note that one can choose $\gamma_i$ to be in ${\operatorname{GL}}_{r_1}(F)$, so ${\mathbf{s}}(\mu\gamma_i)={\mathbf{s}}(\gamma_i){\mathbf{s}}(\mu)$. So we have $${\widetilde{\varphi}}(m)=\sum_{i=1}^N\sum_{\mu\in A_{M_2}M_2^{(n)}(F)\backslash M_2(F)}\varphi({\mathbf{s}}(\gamma_i){\mathbf{s}}(\mu) m;1).$$ One can see by using Lemma \[L:M\_2\] that the map on ${\widetilde{M}}_2({\mathbb{A}})$ defined by $$m_2\mapsto \varphi({\mathbf{s}}(\gamma_i)m;1)$$ is in the induced space ${\operatorname{c-Ind}}_{A_{{\widetilde{M}}_2}({\mathbb{A}}){\widetilde{M}^{(n)}}_2({\mathbb{A}})}^{{\widetilde{M}}_2({\mathbb{A}})}{\pi^{(n)}}_{\omega_{\gamma_i}}$, where ${\pi^{(n)}}_{\omega_{\gamma_i}}=\omega_{\gamma_i}({\pi^{(n)}}_2\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)$. Hence the function on ${\widetilde{M}}_2({\mathbb{A}})$ defined by $$m_2\mapsto \sum_{\mu\in A_{M_2}M_2^{(n)}(F)\backslash M_2(F)}\varphi({\mathbf{s}}(\gamma_i){\mathbf{s}}(\mu) m_2;1)$$ belongs to $\pi_{\omega_{\gamma_i}}$. Since we do not know the multiplicity one property for the group ${\widetilde{M}}_2$, we might have a possible multiplicity $m_\delta$. This completes the proof. \[T:restriction\] Assume that the metaplectic tensor product $(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega} $ is realized with the group $A_{{\widetilde{M}}}$ which satisfies Hypothesis ($\ast\ast$). Then we have $$(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega}\|_{{\widetilde{M}}_2({\mathbb{A}})}\subseteq \bigoplus_{\delta\in{\operatorname{GL}}_{r_1}(F)} m_\delta(\pi_2{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega_\delta},$$ where $m_\delta\in {\mathbb{Z}}^{\geq 0}$ This is immediate from the above proposition. Now we can restrict the metaplectic tensor product “from the bottom”, and get the same result. Let $$M_{k-1}={\operatorname{GL}}_{r_1}\times{\operatorname{GL}}_{r_{k-1}}= \{\begin{pmatrix}g_1&&&\\ &\ddots&&\\ &&g_{k-1}&\\ &&&I_{r_k}\end{pmatrix}\in M: g_i\in{\operatorname{GL}}_{r_i}\},$$ and embed $M_{k-1}$ in ${\operatorname{GL}}_r$ in the upper left corner. Then define $A_{{\widetilde{M}}_{k-1}}$ and the character $\omega_\delta$ analogously. Also consider the analogue of Hypothesis ($\ast\ast$), namely The group $A_{{\widetilde{M}}}$ satisfies: 1. $A_{{\widetilde{M}}}$ satisfies Hypothesis ($\ast$) 2. $A_{{\widetilde{M}}_{k-1}}$ as defined above contains the center $Z_{{\widetilde{\operatorname{GL}}}_{r-r_k}}$. 3. $A_{{\widetilde{M}}_{k-1}}$ satisfies Hypothesis ($\ast$) with respect to ${\widetilde{M}}_{k-1}$. Then we have \[T:restriction\] Assume that the metaplectic tensor product $(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega} $ is realized with the group $A_{{\widetilde{M}}}$ which satisfies Hypothesis ($\ast\ast\ast$). Then we have $$(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega}\|_{{\widetilde{M}}_{k-1}({\mathbb{A}})}\subseteq \bigoplus_{\delta\in{\operatorname{GL}}_{r_k}(F)} m_\delta(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_{k-1})_{\omega_\delta},$$ where $m_\delta\in {\mathbb{Z}}^{>0}$ The proof is essentially the same as the case for the restriction to ${\widetilde{M}}_2$. We will leave the verification to the reader. Also for the case $n=2$, we can do even better. \[T:restriction\_n=2\] Assume $n=2$. (a) Choose $A_{{\widetilde{M}}}$ to be as in Proposition \[P:A\_M\_for\_n=2\]. For $j=2,\dots,k$, let $M_j={\operatorname{GL}}_{r_j}\times\cdots\times{\operatorname{GL}}_{r_k}\subseteq M$ embedded into the right lower corner. Then $$(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega}\|_{{\widetilde{M}}_j({\mathbb{A}})}\subseteq \bigoplus_{\omega'} m_{\omega'}(\pi_j{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega'},$$ where $\omega'$ runs through a countable number of characters on $A_{{\widetilde{M}}_j}=A_{{\widetilde{M}}}\cap{\widetilde{M}}_j$. (b) Choose $A_{{\widetilde{M}}}$ to be as in Proposition \[P:A\_M\_for\_n=2\_2\]. For $j=1,\dots,k-1$, let $M_{k-j}={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_{k-j}}\subseteq M$ embedded into the left upper corner. Then $$(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega}\|_{{\widetilde{M}}_{k-j}({\mathbb{A}})}\subseteq \bigoplus_{\omega'} m_{\omega'}(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_{k-j})_{\omega'},$$ where $\omega'$ runs through a countable number of characters on $A_{{\widetilde{M}}_{k-j}}=A_{{\widetilde{M}}}\cap{\widetilde{M}}_{k-j}$. For (a), one can inductively show that $A_{{\widetilde{M}}_j}=A_{{\widetilde{M}}_{j-1}}\cap{\widetilde{M}}_{j-1}$ satisfies both Hypotheses ($\ast$) and ($\ast\ast$) for the Levi $M_j$. Thus one can successively apply the above theorem for $j=2,\dots,k$, which proves the theorem. The case (b) can be treated similarly. In the above theorem, we choose different $A_{{\widetilde{M}}}$ for the two cases to define $(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega}$. They are, however, equivalent, because, though the character $\omega$ is a character on $A_{{\widetilde{M}}}$, the metaplectic tensor product is dependent only on the restriction $\omega|_{Z_{{\widetilde{\operatorname{GL}}}_r}}$ to the center. **On the discreteness of the group $A_{M} M^{(n)}(F)\backslash M (F)$** {#A:topology} ======================================================================= In this appendix, we will discuss the issue of when $A_{{\widetilde{M}}}$ can be chosen so that the group $A_{M} M^{(n)}(F)\backslash M (F)$ is a discrete subgroup of $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})\backslash{\widetilde{M}}({\mathbb{A}})$, and hence the metaplectic tensor product can be defined. In particular, we will show that if $n=2$, one can always choose such $A_{{\widetilde{M}}}$, and hence all the global results hold without any condition. If $n>2$, the author does not know if it is always possible to chose such nice $A_{{\widetilde{M}}}$, though he suspects that this is always the case. Throughout this appendix the field $F$ is a number field. Also for topological groups $H\subseteq G$, we always assume $H\backslash G$ is equipped with the quotient topology. The crucial fact is \[P:F\_is\_discrete\] For any positive integer $m$, the image of $F^\times$ in ${\mathbb{A}}^{\times m}\backslash {\mathbb{A}}^\times$ is discrete in the quotient topology. Let $K=\prod_{v}K_v\subseteq{\mathbb{A}}^\times$ be the open neighborhood of the identity defined by $K_v=\mathcal{O}_{F_v}^\times$ for all finite $v$ and $K_v=F_v^\times$ for all infinite $v$. To show the discreteness of the image of $F^\times$, it suffices to show that the set ${\mathbb{A}}^{\times m} K\cap {\mathbb{A}}^{\times m}F^\times$ has only finitely many points modulo $A^{\times m}$. This is because the image of $F^\times$ in ${\mathbb{A}}^{\times m}\backslash {\mathbb{A}}^\times$ will then have an open neighborhood of the identity in the subspace topology for ${\mathbb{A}}^{\times m}\backslash {\mathbb{A}}^{\times m}F^\times$ containing finitely many points, and the quotient ${\mathbb{A}}^{\times m}\backslash{\mathbb{A}}^\times$ is Hausdorf since ${\mathbb{A}}^{\times m}$ is closed. Now let $a^m\in{\mathbb{A}}^{\times m}$ and $u\in F^\times$ be such that $a^mu\in {\mathbb{A}}^{\times m} K\cap {\mathbb{A}}^{\times m}F^\times$. Then $u\in{\mathbb{A}}^{\times m}K$, and so for each finite $v$, we have $u_v\in F_v^{\times m}K_v$, which implies the fractional ideal $(u)$ generated by $u$ is $m^{\text{th}}$ power in the group $I_F$ of fractional ideals of $F$. Namely $(u)\in P_F\cap I_F^m$, where $P_F$ the group of principal fractional ideals. On the other hand for any $(u)\in P_F\cap I_F^m$, one can see that $u\in{\mathbb{A}}^{\times m}K$. Accordingly, if we define $$G:=\{u\in F^\times: (u)\in P_F\cap I_F^m\},$$ we have the surjection $$F^{\times m}\backslash G\rightarrow {\mathbb{A}}^{\times m}\backslash( {\mathbb{A}}^{\times m} K\cap {\mathbb{A}}^{\times m}F^\times),$$ given by $u\mapsto{\mathbb{A}}^{\times m}u$. So we have only to show that the group $F^{\times m}\backslash G$ is finite. But note that the map $u\mapsto (u)$ gives rise to the short exact sequence $$0\rightarrow U_F^{m}\backslash U_F\rightarrow F^{\times m}\backslash G\rightarrow P_F^m\backslash P_F\cap I_F^m\rightarrow 0,$$ where $U_K$ is the group of units for $F$. Now the group $U_F^{m}\backslash U_F$ is finite by Dirichlet’s unit theorem. The group $P_F^m\backslash P_F\cap I_F^m$ is isomorphic to the group of $m$-torsions in the class group of $F$ via the map $$P_F^m\backslash P_F\cap I_F^m\rightarrow P_F\backslash I_F,\quad \mathfrak{A}^m\mapsto\mathfrak{A}$$ for each fractional ideal $\mathfrak{A}^m \in I_F^m$, and hence finite. Therefore $F^{\times m}\backslash G$ is finite. As a first consequence of this, we have \[P:M(F)\_is\_discrete\] The image of $M(F)$ in $M^{(n)}({\mathbb{A}})\backslash M({\mathbb{A}})$ is discrete. Let $${\operatorname{Det}}_M:M({\mathbb{A}})\rightarrow \underbrace{{\mathbb{A}}^{\times n}\backslash{\mathbb{A}}^\times\times\cdots\times{\mathbb{A}}^{\times n}\backslash {\mathbb{A}}^\times}_{k-\text{times}}$$ be the map defined by ${\operatorname{Det}}_M({\operatorname{diag}}(g_1,\dots,g_k))=(\det(g_1),\dots,\det(g_k))$. Then $\ker({\operatorname{Det}}_M)=M^{(n)}({\mathbb{A}})$. Moreover the map ${\operatorname{Det}}_M$ is continuous. Hence we have a continuous group isomorphism $$M^{(n)}({\mathbb{A}})\backslash M({\mathbb{A}})\rightarrow {\mathbb{A}}^{\times n}\backslash{\mathbb{A}}^\times\times\cdots\times{\mathbb{A}}^{\times n}\backslash {\mathbb{A}}^\times.$$ Moreover, one can construct the continuous inverse by sending each $a_i\in {\mathbb{A}}^{\times n}\backslash{\mathbb{A}}^\times$ to the first entry of the $i^{\text{th}}$ block ${\operatorname{GL}}_{r_i}({\mathbb{A}})$. But the image of $M(F)$ in ${\mathbb{A}}^{\times n}\backslash{\mathbb{A}}^\times\times\cdots\times{\mathbb{A}}^{\times n}\backslash {\mathbb{A}}^\times$ under ${\operatorname{Det}}_M$ is discrete by the above proposition. The proposition follows. As a corollary, \[C:GCD\] If the center $Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})$ is contained in ${\widetilde{M}^{(n)}}({\mathbb{A}})$, which is the case if $n$ divides $nr_i/d$ for all $i=1,\dots,k$ where $d=\gcd(n, r-1+2cr)$, then Hypothesis ($\ast$) is satisfied and the metaplectic tensor product can be defined. If the center is already in ${\widetilde{M}^{(n)}}({\mathbb{A}})$, one can choose $A_{{\widetilde{M}}}({\mathbb{A}})=Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})$ and then $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})={\widetilde{M}^{(n)}}({\mathbb{A}})$, and so $A_MM^{(n)}(F)=M^{(n)}(F)$. Then by the above proposition, $A_MM^{(n)}(F)\backslash M(F)$ is discrete in $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})\backslash{\widetilde{M}}({\mathbb{A}})$. Proposition \[P:M(F)\_is\_discrete\] also implies The group $M(F)M^{(n)}({\mathbb{A}})$ (resp. $M(F)^\ast{\widetilde{M}^{(n)}}({\mathbb{A}})$) is a closed subgroup of $M({\mathbb{A}})$ (resp. ${\widetilde{M}}({\mathbb{A}})$). It suffices to show it for $M(F)M^{(n)}({\mathbb{A}})$ because the canonical projection is continuous. But for this, one can apply the following lemma with $G=M({\mathbb{A}}), Y=M^{(n)}({\mathbb{A}})$ and $\Gamma=M(F)$, which will complete the proof. \[L:discrete\_sub\] Let $G$ be a Hausdorf topological group. If $\Gamma\subset G$ is a discrete subgroup and $Y\subset G$ a closed normal subgroup such that the image of $\Gamma$ in $G\slash Y$ is discrete in the quotient topology, then the group $\Gamma Y$ is closed in $G$. Let $p:G\rightarrow G/Y$ be the canonical projection. By our assumption, the image $p(\Gamma)$ of $\Gamma$ is discrete in the quotient topology. Now since $Y$ is closed, the quotient $G/Y$ is a Hausdorf topological group. Hence $p(\Gamma)$ is closed by [@Deitmar Lemma 9.1.3 (b)]. To show $\Gamma Y$ is closed, it suffices to show every net $\{\gamma_iy_i\}_{i\in I}$ that converges in $G$, where $\gamma_i\in\Gamma$ and $y_i\in Y$, converges in $\Gamma Y$. But since $p$ is continuous, the net $\{p(\gamma_iy_i)\}$ converges in $G/Y$. But $p(\gamma_iy_i)=p(\gamma_i)$ and $p(\gamma_i)\in p(\Gamma)$. Since $p(\Gamma)$ is closed and discrete, in order for the net $\{p(\gamma_i)\}$ to converge, there exists $\gamma\in\Gamma$ such that $p(\gamma_i)=p(\gamma)$ for all sufficiently large $i\in I$, namely, the net $\{p(\gamma_i)\}$ is eventually constant. Hence for sufficiently large $i$, we have $\gamma_iy_i=\gamma y_i'$ for some $y_i'\in Y$. This means that the net $\{\gamma_iy_i\}$ is eventually in the set $\gamma Y$. But since $Y$ is closed, so is $\gamma Y$, which implies that the net $\{\gamma_iy_i\}$ converges in $\gamma Y\subset \Gamma Y$. Finally in this appendix, we will show that if $n=2$, one can always choose $A_{{\widetilde{M}}}$ so that the group $A_MM^{(n)}(F)\backslash M(F)$ is discrete and hence the metaplectic tensor product is defined, and moreover the metaplectic tensor product can be realized in such a way that it behaves nicely with the restriction to the smaller rank groups. First let us note that for any $r$, the center $Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})$ is given by $$Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})=\{(aI_r, \xi): a\in {\mathbb{A}}^{\times \varepsilon}\},\quad \varepsilon= \begin{cases} 1\quad\text{if $r$ is odd};\\ 2\quad\text{if $r$ is even}. \end{cases}\\$$ Accordingly, one can see $$Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}}){\widetilde{\operatorname{GL}}^{(2)}}_r({\mathbb{A}})= \begin{cases} {\widetilde{\operatorname{GL}}}_r({\mathbb{A}})\quad\text{if $r$ is odd};\\ {\widetilde{\operatorname{GL}}^{(2)}}_r({\mathbb{A}})\quad\text{if $r$ is even}. \end{cases}$$ With this said, one can see \[P:A\_M\_for\_n=2\] Assume $n=2$. Let $${\widetilde{Z}}_i({\mathbb{A}})=Z_{{\widetilde{\operatorname{GL}}}_{r_i+\cdots+r_k}}({\mathbb{A}})\subseteq {\widetilde{\operatorname{GL}}}_{r_i}({\mathbb{A}}){\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{\operatorname{GL}}}_{r_k}({\mathbb{A}}) \subseteq{\widetilde{M}}({\mathbb{A}}),$$ and $$A_{{\widetilde{M}}}({\mathbb{A}})={\widetilde{Z}}_1({\mathbb{A}}){\widetilde{Z}}_2({\mathbb{A}})\cdots{\widetilde{Z}}_k({\mathbb{A}}).$$ Then $A_{{\widetilde{M}}}({\mathbb{A}})$ is a closed abelian subgroup of $\widetilde{Z_M}({\mathbb{A}})$ and further the group $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(2)}}({\mathbb{A}})$ is closed and the image of $M(F)$ in $A_M({\mathbb{A}})M^{(2)}({\mathbb{A}})\backslash M({\mathbb{A}})$ as well as in $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(2)}}({\mathbb{A}})\backslash{\widetilde{M}}({\mathbb{A}})$ is discrete. It is clear that $A_{{\widetilde{M}}}({\mathbb{A}})$ is abelian since for each $i=1,\dots, k$, ${\widetilde{Z}}_i$ is the center of ${\widetilde{\operatorname{GL}}}_{r_i+\cdots+r_k}({\mathbb{A}})$, and hence commutes pointwise with ${\widetilde{Z}}_j({\mathbb{A}})\subseteq {\widetilde{\operatorname{GL}}}_{r_i+\cdots+r_k}({\mathbb{A}})$ for all $j\geq i$. To show $A_{{\widetilde{M}}}({\mathbb{A}})$ is closed, it suffices to show $A_{M}({\mathbb{A}}):=p(A_{{\widetilde{M}}}({\mathbb{A}}))$ is closed. Now one can write $A_{M}({\mathbb{A}})=\prod_v'A_M(F_v)$, where $A_M(F_v)$ is defined analogously to the global case. Then one can see that $Z^{(2)}_M(F_v)\subseteq A_M(F_v)\subseteq Z_M(F_v)$, and since $Z^{(2)}_M(F_v)$ is closed and of finite index in $Z_M(F_v)$, so is $A_M(F_v)$. But $Z_M(F_v)$ is closed in $M(F_v)$ and so $A_M(F_v)$ is closed in $M(F_v)$. Then one can show that $A_M({\mathbb{A}})$ is closed in $M({\mathbb{A}})$ by Lemma \[L:closed\_subgroup\_local\_global\]. Now one can show by induction on $k$ that the group $A_{M}({\mathbb{A}})M^{(2)}({\mathbb{A}})$ is the kernel of the map $${\operatorname{Det}}_M:M({\mathbb{A}})\rightarrow {\mathbb{A}}^{\times \varepsilon_1}\backslash{\mathbb{A}}^\times\times\cdots\times{\mathbb{A}}^{\times \varepsilon_k}\backslash {\mathbb{A}}^\times,$$ where $\varepsilon_i$ is either $1$ or $2$. Hence one has a continuous group isomorphism $$A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(2)}}({\mathbb{A}})\backslash{\widetilde{M}}({\mathbb{A}})\rightarrow {\mathbb{A}}^{\times \varepsilon_1}\backslash{\mathbb{A}}^\times\times\cdots\times{\mathbb{A}}^{\times \varepsilon_k}\backslash {\mathbb{A}}^\times,$$ where the space on the right is Hausdorff. Hence the space on the left is Hausdorf as well, which shows $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(2)}}({\mathbb{A}})$ is closed. Also one can show that the image of $M(F)$ is discrete as we did for Proposition \[P:M(F)\_is\_discrete\]. \[P:A\_M\_for\_n=2\_2\] Assume $n=2$. Let $${\widetilde{Z}}_j({\mathbb{A}})=Z_{{\widetilde{\operatorname{GL}}}_{r_1+\cdots+r_{k-j}}}({\mathbb{A}})\subseteq {\widetilde{\operatorname{GL}}}_{r_1}({\mathbb{A}}){\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{\operatorname{GL}}}_{r_{k-j}}({\mathbb{A}}) \subseteq{\widetilde{M}}({\mathbb{A}}),$$ and $$A_{{\widetilde{M}}}({\mathbb{A}})={\widetilde{Z}}_1({\mathbb{A}}){\widetilde{Z}}_2({\mathbb{A}})\cdots{\widetilde{Z}}_k({\mathbb{A}}).$$ Then $A_{{\widetilde{M}}}({\mathbb{A}})$ is a closed abelian subgroup of $\widetilde{Z_M}({\mathbb{A}})$ and further the group $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(2)}}({\mathbb{A}})$ is closed and the image of $M(F)$ in $A_M({\mathbb{A}})M^{(2)}({\mathbb{A}})\backslash M({\mathbb{A}})$ as well as in $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(2)}}({\mathbb{A}})\backslash{\widetilde{M}}({\mathbb{A}})$ is discrete. Identical to the previous proposition. Let us make the following final remark. The above proposition and Corollary \[C:GCD\] imply Proposition \[P:hypothesis\]. Also for $n>2$, if $n$ and $r=r_1+\cdots+r_k$ are such that $n$ divides $nr_i/d$ for all $i=1\cdots k$ where $d=\gcd(n, r-1+2cr)$ and $n$ divides $ nr_i/d_2$ for all $i=2\cdots k$ where $d_2=\gcd(n, r-r_1-1+2c(r-r_2)$, then $A_{{\widetilde{M}}}=Z_{{\widetilde{\operatorname{GL}}}_r}$ satisfies Hypothesis ($\ast\ast$), and hence one has the restriction property to the smaller rank group. Moreover this is always the case, for example, if $\gcd(n, r-1+2cr)=\gcd(n, r-r_1-1+2c(r-r_1))=1$. Similarly one can satisfy Hypothesis ($\ast\ast\ast$) if $n$ divides $nr_i/d$ for all $i=1\cdots k$ and divides $ nr_i/d_{k-1}$ for all $i=1\cdots k-1$ where $d_{k-1}=\gcd(n, r-r_{k-1}-1+2c(r-r_{k-1}))$. Those conditions are indeed often satisfied especially when $n$ is a prime. [999999]{} J. Adams, [*Non-linear real groups*]{}, (preprint). W. Banks, [*Twisted symmetric-square $L$-functions and the nonexistence of Siegel zeros on ${{\operatorname{GL}}}(3)$*]{}, Duke Math. J. 87 (1997), 343–353. W. Banks, D. Bump and D. Lieman, [*Whittaker-Fourier coefficients of metaplectic Eisenstein series*]{}, Compositio Math. 135 (2003), 153–178. W. Banks, J. Levy, M. Sepanski, [*Block-compatible metaplectic cocycles*]{}, J. Reine Angew. Math. 507 (1999), 131–163. N. Bourbaki, Integration. II, Chapters 7–9, translated from the 1963 and 1969 French originals by Sterling K. Berberian. Elements of Mathematics, Springer-Verlag, Berlin, 2004. D. Bump, S. Friedberg and J. Hoffstein, [*p-adic Whittaker functions on the metaplectic group*]{}, Duke Math. J. 63 (1991), 379–397. D. Bump and D. Ginzburg, [*Symmetric square $L$-functions on ${\rm GL}(r)$*]{}, Ann. of Math. 136 (1992), 137–205. D. Bump and J. Hoffstein, [*On Shimura’s correspondence*]{}, Duke Math. J. 55 (1987), 661–691. J. Cogdell, [*Lectures on $L$-functions, converse theorems, and functoriality of ${\operatorname{GL}}(n)$*]{}, in Lectures on Automorphic $L$-functions, Fields Institute Monographs, AMS (2004), 5–100. A. Deitmar and S. Echterhoff, Principles of harmonic analysis, Springer, New York, 2009. Y. Flicker, [*Automorphic forms on covering groups of ${{\operatorname{GL}}}(2)$*]{}, Invent. Math. 57 (1980), 119–182. Y. Flicker and D. Kazhdan, [*Metaplectic correspondence*]{}, Inst. Hautes Etudes Sci. Publ. Math. No. 64 (1986), 53–110 G. Chinta and O. Offen, [*A metaplectic Casselman-Shalika formula for ${\operatorname{GL}}_r$*]{}, to appear in Amer. J. Math. A. Kable, [*Exceptional representations of the metaplectic double cover of the general linear group*]{}, PH.D thesis, Oklahoma State University (1997). A. Kable, [*The tensor product of exceptional representations on the general linear group*]{}, Ann. Sci. École Norm. Sup. (4) 34 (2001), 741–769. D. A. Kazhdan and S. J. Patterson, [*Metaplectic forms*]{}, Inst. Hautes Etudes Sci. Publ. Math. No. 59 (1984), 35–142. T. Kubota, [*On automorphic functions and the reciprocity law in a number field*]{}, Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 2 Kinokuniya Book-Store Co., Ltd., Tokyo 1969 iii+65 pp. H. Matsumoto, [*Sur les sous-groupes arithmétiques des groupes semi-simples déployés*]{}, Ann. Sci. École Norm. Sup. 2 (1969), 1–62. P. Mezo, [*Metaplectic tensor products for irreducible representations*]{}, Pacific J. Math. 215 (2004), 85–96. C. Moeglin and J.-L. Waldspurger, [*Spectral decomposition and Eisenstein series*]{}, Cambridge Tracts in Mathematics, 113. Cambridge University Press, Cambridge, 1995. T. Suzuki, [*Metaplectic Eisenstein series and the Bump-Hoffstein conjecture*]{}, Duke Math. J. 90 (1997), 577–630. S. Takeda, [*The twisted symmetric square $L$-function of ${\operatorname{GL}}(r)$*]{}, Duke Math. J. (to appear). S. Takeda, [*On a certain metaplectic Eisenstein series and the twisted symmetric square $L$-function*]{}, preprint.

--- abstract: '[The aim of this study is to investigate systematic chemical differentiation of molecules in regions of high mass star formation.]{}[We observed five prominent sites of high mass star formation in HCN, HNC, HCO$^+$, their isotopes, C$^{18}$O, C$^{34}$S and some other molecular lines, for some sources both at 3 and 1.3 mm and in continuum at 1.3 mm. Taking into account earlier obtained data for N$_2$H$^+$ we derive molecular abundances and physical parameters of the sources (mass, density, ionization fraction, etc.). The kinetic temperature is estimated from CH$_3$C$_2$H observations. Then we analyze correlations between molecular abundances and physical parameters and discuss chemical models applicable to these species.]{}[The typical physical parameters for the sources in our sample are the following: kinetic temperature in the range $\sim 30-50$ K (it is systematically higher than that obtained from ammonia observations and is rather close to dust temperature), masses from tens to hundreds solar masses, gas densities $\sim 10^5$ cm$^{-3}$, ionization fraction $\sim 10^{-7}$. In most cases the ionization fraction slightly (a few times) increases towards the embedded YSOs. The observed clumps are close to gravitational equilibrium. There are systematic differences in distributions of various molecules. The abundances of CO, CS and HCN are more or less constant. There is no sign of CO and/or CS depletion as in cold cores. At the same time the abundances of HCO$^+$, HNC and especially N$_2$H$^+$ strongly vary in these objects. They anti-correlate with the ionization fraction and as a result decrease towards the embedded YSOs. For N$_2$H$^+$ this can be explained by dissociative recombination to be the dominant destroying process. ]{}[N$_2$H$^+$, HCO$^+$, and HNC are valuable indicators of massive protostars.]{}' author: - | I. Zinchenko$^{1,2,3}$[^1], P. Caselli$^{4}$ and L. Pirogov$^{1}$\ $^1$Institute of Applied Physics of the Russian Academy of Sciences, Ulyanova 46, 603950 Nizhny Novgorod, Russia\ $^2$Nizhny Novgorod University , Gagarin av. 23, 603950 Nizhny Novgorod, Russia\ $^3$Helsinki University Observatory, Tähtitorninmäki, P.O. Box 14, FIN-00014 University of Helsinki, Finland\ $^4$School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK date: title: 'Chemical differentiation in regions of high mass star formation II. Molecular multiline and dust continuum studies of selected objects' --- \[firstpage\] Astrochemistry – Stars: formation – ISM: clouds – ISM: molecules – Radio lines: ISM Introduction ============ It is now well established that the central parts of dense low mass cloud cores suffer strong depletion of molecules onto dust grains. Best studied is CO, which has been shown to be depleted in, e.g., L1544 [@Caselli99], IC 5146 [@Kramer99], L1498 [@Willacy98], and L1689B [@Jessop01]. Species related to CO, such as HCO$^+$, are also expected to disappear at gas densities above $\sim 10^5$ cm$^{-3}$ [@Caselli02]. Moreover, @Tafalla02 and @Bergin01 have shown that CS also depletes out in the central parts of dense cores, suggesting that CS (so far considered a high density tracer) does not actually probe the central core regions. On the other hand, N$_2$H$^+$ is an excellent tracer of dust continuum emission [@Caselli02], implying that this species does not deplete out (due to the volatility of the parent species N$_2$). In more massive cores, depletion is probably active in dense regions away from star forming sites where dust temperatures may be low enough ($T < 20$ K) for CO and CS abundances to drop and cause chemical differentiation [@Fontani06]. Several years ago we mapped several tens of dense cores towards water masers in CS(2–1) with the SEST-15m and Onsala 20m radio telescopes [@Zin95; @Zin98]. In 2000 many of them were mapped in N$_2$H$^+$ [@Pirogov03]. The goal was to identify dense clumps as local maxima in N$_2$H$^+$ maps and to further investigate their properties. However, large differences between the N$_2$H$^+$ and CS distributions have been found. In this situation it is important to understand which species better trace the total gas distribution: is it N$_2$H$^+$ as in low mass cores? What is the reason for this differentiation in warm clouds, where freeze-out is hardly effective? To answer these questions we observed dust continuum emission and several additional molecular lines towards selected sources which show significant differences between the CS and N$_2$H$^+$ maps. The results for the southern sources (where we observed only N$_2$H$^+$ $J=1-0$, CS $J=2-1$ and $J=5-4$ and dust continuum) have been published separately [@Pirogov07 hereafter Paper I]. In that paper we have shown that the differences in the CS and N$_2$H$^+$ maps cannot be explained by molecular excitation and/or line opacity effects but are caused by chemical differentiation of these species. We found that N$_2$H$^+$ abundance in many cases drops significantly towards embedded luminous YSOs. However, the reasons for this behavior were not clear. Two possible explanations were mentioned: an accelerated collapse model suggested by @Lintott05 and dissociative recombination of N$_2$H$^+$. Here we present and discuss the results for the northern sample where we observed also HCN, HNC, HCO$^+$, their isotopes, C$^{18}$O and some other molecular lines, for some sources both at 3 and 1.3 mm. These data help to understand better the chemical differentiation in these objects. In addition they give important information on their physical properties. Observations ============ Sources ------- The sources for this investigation were selected from the sample of massive cores studied by us earlier in various lines [@Zin98; @Zin00; @Pirogov03]. The main criterium for this selection was a presence of significant differences in the CS and N$_2$H$^+$ maps. The list of the sources is given in Table \[table:sources\]. For S187, W3 and S140, the source coordinates correspond to water masers. For S255, the IRAS position is used. In the DR-21 area we used the coordinates of an ammonia core as given by @Jijina99 which practically coincide (within a few arcsec) with the position of DR 21 (OH). In the last column the distances to the sources are indicated. The data sets obtained for these sources are different. [llll]{} Source & $\alpha$(2000) & $\delta$(2000) & $D$\ & (${\rm ^h\ ^m\ ^s }$) &($\degr$ $\arcmin$ $\arcsec$) & (kpc)\ S 187 &01 23 15.0 &61 48 47 & 1.0 $^a$\ W3 &02 25 28.2 &62 06 58 & 2.1 $^b$\ S 255 &06 12 53.3 &17 59 22 & 2.5 $^b$\ DR-21 NH$_3$ &20 39 00.4 &42 22 53 & 3.0 $^c$\ S 140 &22 19 18.2 &63 18 49 & 0.9 $^b$\ $^a$@fich, $^b$@blitz, $^c$@harvey1 \[table:sources\] Instruments and frequencies --------------------------- The sources were observed with the 20-m Onsala, 12-m NRAO (which belongs now to the Arizona Radio Observatory) and 30-m IRAM radio telescopes in the 3 mm and 1.3 mm wavebands. Several molecular transitions were observed in each waveband. At IRAM 30m also the dust continuum emission was mapped at 1.2 mm. The details of observations at each instrument are given below. A part of these observations has been already published (some of the CS and C$^{34}$S $J=2-1$ data by @Zin98 and the N$_2$H$^+$ $J=1-0$ data by @Pirogov03). We express the results of the line observations in units of main beam brightness temperature ($T_{\rm mb}$) assuming the main beam efficiencies ($\eta_{\rm mb}$) as provided by the telescope documentation. ### Onsala observations The observations were performed with SIS receiver in a single-sideband (SSB) mode using either dual beam switching with a beam throw of 115 or frequency switching. As a backend until the year 2000 we used 2 filter spectrometers (usually in parallel): a 256 channel filterbank with 250 kHz resolution and a 512 channel filterbank with 1 MHz resolution. Since 2000 we used mainly the autocorrelator spectrometer tuned to 50 kHz resolution. Pointing was checked periodically by observations of nearby SiO masers; the pointing accuracy was typically $\la 5$. The half-power beam width (HPBW) was from about 35 at the highest frequencies to about 40 at the lowest frequencies. The standard chopper-wheel technique was used for the calibration. The system temperature varied in a wide range depending on the weather and observing frequency, from $\sim 200$ K at lower frequencies in good weather to $\sim 1000$ K and more at higher frequencies and under cloudy conditions. ### NRAO 12m observations At the NRAO 12-m telescope only two sources from those listed in Table \[table:sources\] were observed: S187 and S255 (in the C$^{18}$O, CS, C$^{34}$S, SiO and methanol lines at 1.3 mm). As a result, the data sets for these sources are the most complete ones. The observations were perfomed in 2000 with the SIS receiver and two backends in parallel: the filter bank with the 0.5 MHz resolution and MAC autocorrelator with the 100 kHz resolution. We used frequency switching and position switching observing modes. The pointing and focus were checked periodically by observations of planets. The HPBW was 26–27. The system temperature varied from $\sim 300$ K in good weather conditions to $\sim 1500$ K with increasing humidity. ### IRAM 30m observations At the IRAM 30m telescope we obtained maps of the continuum emission from our sources at 1.2 mm and their maps in several components of the CH$_3$C$_2$H $J=13-12$ transition. The continuum observations were performed with the MAMBO bolometer array (the details of this instrument are available at the IRAM Web site) and reduced with the MOPSIC package. The spectral line observations were done with the HERA (Heterodyne Receiver Array) and VESPA autocorrelator backend. The pointing and focus were checked periodically on nearby strong continuum sources. The antenna HPBW is about 12 at these frequencies. The typical system temperatures for HERA observations were in the range $\sim 200-400$ K. Observational results ===================== We present the observational results in the form of maps and tables where the line parameters at selected positions are given. These positions correspond to different emission peaks in the sources which are identified in the maps of molecular emission and in most cases can be seen in the continuum maps presented in Fig. \[fig:cont-maps\]. These maps are plotted using logarithmic scale for intensity in order to emphasize weak features. The continuum brightness, fluxes and angular sizes of the emission clumps towards selected positions are summarized in Table \[table:cont-res\]. The sources are rather extended in continuum. In order to provide a better comparison with the molecular data, most of which were obtained with $\sim 30''$ (HPBW) beams, we give the fluxes integrated over 1 circles centered at the selected positions. These positions are labelled as “CS” and “N$_2$H$^+$” emission peaks according to our previous CS and N$_2$H$^+$ surveys [@Zin98; @Pirogov03] and other available data. The sizes are derived from these fluxes and brightness as $\theta = \sqrt{4F/\pi B}$. For Gaussian brightness distribution this corresponds to size at the $1/e$ level which is about 20% larger than the size at the half maximum level ($\theta_{0.5}$). In S140 it is hard to see a distinct continuum clump at the N$_2$H$^+$ emission peak, one can see rather an extended filament here. Nevertheless, we provide brightness and flux for this position, too. The specific features of every source are briefly described below. The spectral data reduction was performed with the GILDAS and XSpec (Onsala data) software packages. \ [lrrllll]{} Source &$\Delta\alpha$ &$\Delta\delta$ &$B$ &$F$ &$\theta$ &Peak\ &($''$) &($''$) &(Jy/beam) &(Jy) &($''$)\ S 187 &+160 &0 &0.15 &2.1 &45 &CS\ &0 &+80&0.09 &0.46 &28 &N$_2$H$^+$\ W3 &20 &$-$40&2.03 &19 &38 &CS\ &+160&$-$160&0.70 &2.23 &22 &N$_2$H$^+$\ S 255 &0 &0 &1.35 &6.4 &27 &CS\ &0 &+60 &1.24 &6.4 &28 &N$_2$H$^+$\ S 140 &0 &0 &1.58 &15.9 &39 &CS\ &+40 &+20 &0.34 &5.3 &49 &N$_2$H$^+$\ \[table:cont-res\] Molecular maps and line parameters ---------------------------------- ### S187 Maps of S187 in various lines overlaid on the greyscale map of the 1.2 mm continuum emission are presented in Fig. \[fig:maps-s187\]. The CS(2–1) and N$_2$H$^+$(1–0) maps have been published earlier [@Zin98; @Pirogov03]. There is a striking difference between the various maps. Two separated clumps are clearly seen in N$_2$H$^+$(1–0), which peaks North-West of the IRAS 01202+6133 source. The two peaks are still visible in the HNC(1–0) map, although they are not as well separated as in the N$_2$H$^+$(1–0) map. All the other maps have a cometary or more irregular morphology, with peaks all shifted from the continuum peak. The gaussian line parameters at the CS and N$_2$H$^+$ emission peaks are summarized in Table \[table:lines-s187\]. In C$^{18}$O $J=1-0$ and $J=2-1$ emission towards the CS peak there is an additional narrow ($\sim 0.7$ km/s) component at about $-$15.5 km/s. However, it is not pronounced in the lines of other high density tracers and we do not include it in Table \[table:lines-s187\]. The molecular data indicate the presence of at least 3 clumps in the area. There are several IRAS point sources and molecular masers here. The strongest IRAS point source, IRAS 01202+6133, is located at about 2 to the east from our central position and coincides with the main 1.2 mm continuum peak. Here an OH maser and UC  region are present [@Argon00]. The secondary N$_2$H$^+$ peak coincides with this IRAS position. Also a weaker CS clump is located here as is clearly seen from the CS $J=5-4$ data. The main CS, HCO$^+$ and HCN emission peaks are shifted by about 1 further to the east. No IR sources or masers are known in this area. It is worth noting that the methanol emission peak is shifted still further to the east. At the same time C$^{18}$O emission peaks near IRAS 01202+6133. The strongest N$_2$H$^+$ peak coincides with a relatively weak 1.2 mm continuum clump. It is shifted by about 05 from the strong near IR source NIRS 60 [@Salas98]. \ \ --------------------- -------------- --------------- ------------ -- -------------- --------------- ------------ Line $T_{\rm mb}$ $V_{\rm LSR}$ $\Delta V$ $T_{\rm mb}$ $V_{\rm LSR}$ $\Delta V$ (K) (km/s) (km/s) (K) (km/s) (km/s) C$^{18}$O(1–0) 5.47(18) $-$13.92(03) 1.71(09) 3.24(18) $-$13.78(04) 1.67(10) C$^{18}$O(2–1) 5.85(19) $-$14.05(02) 1.69(05) CS(2–1) 4.29(10) $-$14.26(03) 2.30(07) 1.20(20) $-$13.63(27) 3.24(70) C$^{34}$S(2–1) 1.00(04) $-$14.05(04) 2.25(11) CS(5–4) 3.58(06) $-$14.09(01) 1.86(03) C$^{34}$S(5–4) $<$0.4 HCN(1–0) 5.14(07) $-$14.31(02) 2.33(04) 1.92(08) $-$13.81(04) 1.63(07) H$^{13}$CN(1–0) 0.50(05) $-$14.17(10) 1.83(21) HCO$^+$(1–0) 5.54(08) $-$14.49(02) 2.22(04) 2.41(08) $-$14.03(03) 1.85(08) H$^{13}$CO$^+$(1–0) 0.58(07) $-$14.15(09) 1.56(22) 1.28(10) $-$13.51(03) 0.83(07) HNC(1–0) 5.44(12) $-$14.29(02) 2.27(06) 5.70(15) $-$13.58(02) 1.39(04) HN$^{13}$C(1–0) 0.33(04) $-$13.88(09) 1.66(20) 0.63(05) $-$13.35(03) 0.76(07) N$_2$H$^+$(1–0) 0.61(08) $-$14.09(07) 1.45(20) 1.59(12) $-$13.33(03) 0.90(06) --------------------- -------------- --------------- ------------ -- -------------- --------------- ------------ ### W3 The molecular line maps of W3 are shown in Fig. \[fig:maps-w3\] overlaid with the 1.2 mm continuum emission map. The CS(2–1) and N$_2$H$^+$(1–0) maps have been published earlier [@Zin98; @Pirogov03]. The brightest continuum peak is attributed to free-free emission from a compact  region [e.g. @Tieftrunk97]. There are two main molecular emission peaks in this area associated with two dust clumps. As in the S187 case, the N$_2$H$^+$ map looks very different from most other maps. The N$_2$H$^+$ peak coincides with a relatively weak south-east (SE) clump at the approximately (160$''$,$-$160$''$) position. The gaussian line parameters at the CS and N$_2$H$^+$ emission peaks are summarized in Table \[table:lines-w3\]. Some spectra clearly show non-gaussian features: broad wings at the CS peak position and red-shifted self-absorption in HCO$^+$ at the N$_2$H$^+$ peak. \ --------------------- -------------- --------------- ------------ -- -------------- --------------- ------------ Line $T_{\rm mb}$ $V_{\rm LSR}$ $\Delta V$ $T_{\rm mb}$ $V_{\rm LSR}$ $\Delta V$ (K) (km/s) (km/s) (K) (km/s) (km/s) C$^{18}$O(1–0) 2.05(20) $-$42.70(27) 5.32(71) 2.36(37) $-$38.58(11) 1.49(27) CS(2–1) 8.82(10) $-$43.09(03) 4.76(06) 3.19(03) $-$38.66(19) 4.43(53) HCN(1–0) 7.96(11) $-$43.08(02) 3.41(05) 1.65(13) $-$38.55(40) 8.08(38) H$^{13}$CN(1–0) 1.51(07) $-$42.75(12) 4.62(20) HCO$^+$(1–0) 16.26(07) $-$43.40(01) 3.86(02) 4.05(06) $-$38.97(04) 4.73(09) H$^{13}$CO$^+$(1–0) 0.88(17) $-$43.11(21) 2.44(51) 0.95(18) $-$38.33(19) 2.02(45) HNC(1–0) 5.36(11) $-$42.92(04) 3.83(09) 4.41(12) $-$38.74(04) 3.12(10) HN$^{13}$C(1–0) 0.37(03) $-$42.63(09) 2.80(21) 0.17(01) $-$38.38(05) 2.95(23) N$_2$H$^+$(1–0) 0.66(09) $-$42.22(11) 2.04(30) 1.37(10) $-$38.76(10) 1.75(14) --------------------- -------------- --------------- ------------ -- -------------- --------------- ------------ ### S255 For S255 we have the most complete data set. In addition to the common set of molecular transitions it includes SO($3_2-2_1$), SiO (2–1) and (5–4), several methanol 2–1 and 5–4 series lines. Earlier we mapped it also in ammomia (1,1) and (2,2) [@Zin97]. The molecular maps overlaid on the greyscale map of the 1.2 mm continuum emission are presented in Fig. \[fig:maps-s255\]. This source was mapped at 1.2 mm in continuum at the IRAM 30m telescope by @Mezger88. Our observations with the new array receiver provide a better sensitivity and a wider map area, although the basic features of our map are consistent with those previous results. The maps show two main peaks of molecular and dust continuum emission, around the (0,0) and (0,+60$''$) positions. The commonly accepted names for these clumps are S255IR and S255N. There is also a third (southern) peak at about (+20$''$,$-$50$''$) noticeable in the continuum map and in several molecular maps, at least in N$_2$H$^+$(1–0) and HCO$^+$(1–0). N$_2$H$^+$ and ammonia are significantly stronger at the northern peak (S255N) while most other species are either stronger at the central position (S255IR) or comparable at both central and northern clumps. The dust emission is almost equal for these clumps. The nature of these two components is different. The central one is associated with a luminous cluster of IR sources, whereas toward the northern one an ultracompact  region (G192.58-0.04) was detected. Recently several compact submillimetre continuum clumps were detected there with SMA [@Cyganowski07]. This object is extremely red in the mid-IR band [@Crowther03]. @Mezger88 derived almost exactly the same dust masses and temperatures for both components from their 1.2 mm and $350\mu$m observations. The gaussian line parameters at the central and northern emission peaks are summarized in Table \[table:lines-s255\]. \ \ --------------------- -------------- --------------- ------------ -- -------------- --------------- ------------ Line $T_{\rm mb}$ $V_{\rm LSR}$ $\Delta V$ $T_{\rm mb}$ $V_{\rm LSR}$ $\Delta V$ (K) (km/s) (km/s) (K) (km/s) (km/s) C$^{18}$O(1–0) 1.99(12) 6.84(17) 4.31(43) 1.67(13) 8.74(14) 3.71(34) C$^{18}$O(2–1) 6.45(08) 7.18(01) 2.88(03) 5.89(08) 8.82(01) 2.95(03) CS(2–1) 13.36(09) 7.46(01) 2.78(02) 9.08(08) 8.40(01) 3.16(03) C$^{34}$S(2–1) 1.93(11) 7.46(08) 2.78(18) 1.33(10) 8.56(12) 3.20(29) CS(5–4) 10.94(12) 7.49(01) 2.73(03) 7.46(12) 8.55(02) 3.32(05) C$^{34}$S(5–4) 1.13(06) 7.52(05) 2.33(12) 0.75(08) 9.10(13) 3.57(26) HCN(1–0) 13.79(11) 7.33(01) 3.25(03) 11.52(11) 8.40(02) 3.60(03) H$^{13}$CN(1–0) 1.41(07) 7.57(06) 2.58(12) 1.37(07) 8.91(06) 2.12(11) HCO$^+$(1–0) 7.34(10) 7.53(03) 3.90(06) 11.27(11) 8.83(02) 3.21(04) H$^{13}$CO$^+$(1–0) 0.74(10) 7.51(11) 1.69(26) 1.39(08) 8.97(07) 2.79(18) HNC(1–0) 11.87(18) 7.30(02) 2.73(05) 11.30(17) 8.55(02) 3.32(06) HN$^{13}$C(1–0) 0.18(02) 7.29(20) 3.30(47) 0.28(02) 8.87(16) 4.78(37) N$_2$H$^+$(1–0) 1.46(06) 7.44(04) 2.25(10) 2.58(05) 8.97(02) 2.61(06) SiO(2–1) 0.18(06) 7.96(52) 3.32(122) 0.20(05) 8.06(61) 5.24(144) --------------------- -------------- --------------- ------------ -- -------------- --------------- ------------ ### DR-21 NH$_3$ This is another well known site of active star formation. The CS, HCN and HCO$^+$ spectra here suffer from a very strong red-shifted self-absorption (Fig. \[fig:dr21-spectra\]), which makes an analysis of corresponding maps almost senseless. For this reason we present here and discuss only rarer isotopologue and N$_2$H$^+$(1–0) maps (Fig. \[fig:maps-dr21\]). We did not map the dust continuum emission in this area, given that it has been already observed by @Chandler93. It is easy to see that the N$_2$H$^+$ and HN$^{13}$C peaks are shifted by about 05 to the south from the emission peaks of other species. The latter peaks practically coincide with the main dust emission peak DR21(OH)M in the notation introduced by @Mangum91. The N$_2$H$^+$ and HN$^{13}$C peaks lie near a weaker dust peak DR21(OH)S. The gaussian line parameters are given in Table \[table:lines-dr21\]. \ --------------------- -------------- --------------- ------------ -- -------------- --------------- ------------ Line $T_{\rm mb}$ $V_{\rm LSR}$ $\Delta V$ $T_{\rm mb}$ $V_{\rm LSR}$ $\Delta V$ (K) (km/s) (km/s) (K) (km/s) (km/s) C$^{18}$O(1–0) 4.99(14) $-$2.99(05) 3.84(13) 4.69(15) $-$3.37(05) 3.28(12) C$^{34}$S(2–1) 2.62(06) $-$3.04(05) 4.68(12) 2.89(06) $-$3.01(04) 3.83(10) H$^{13}$CN(1–0) 2.34(06) $-$3.24(06) 4.07(10) 2.34(06) $-$3.47(05) 3.58(09) H$^{13}$CO$^+$(1–0) 3.17(07) $-$3.46(04) 3.67(09) 2.73(07) $-$3.57(04) 3.25(10) HN$^{13}$C(1–0) 1.34(09) $-$3.05(14) 4.38(32) 1.70(09) $-$3.25(10) 3.56(23) N$_2$H$^+$(1–0) 4.99(04) $-$3.11(02) 4.04(03) 6.93(04) $-$3.25(01) 3.51(02) --------------------- -------------- --------------- ------------ -- -------------- --------------- ------------ ### S140 S140 is one of the best studied sites of active star formation. In particular it was observed at various instruments in the HCN, HCO$^+$ and CO isotopic lines [e.g. @Park95]. Nevertheless, we present here our own data on these lines too, to better compare with other molecular maps (all maps have been obtained at Onsala). The N$_2$H$^+$ results have been published earlier [@Pirogov03]. The maps (Fig. \[fig:maps-s140\]) clearly show that HCN and HCO$^+$ emissions peak near the (0,0) position while the N$_2$H$^+$ peak is shifted to about (+50$''$,+20$''$). A multi-transitional CS study [@Zhou94] shows that CS emission peak is near the (0,0) position. HNC is an intermediate case and there is a secondary HNC peak near the N$_2$H$^+$ peak. These differences in emission distributions are not caused by opacity effects because the maps in the lines of rarer isotopic modifications of these molecules show the same features (the N$_2$H$^+$ optical depth is rather small, $\sim 0.4$, as shown by @Pirogov03). The gaussian line parameters are presented in Table \[table:lines-s140\]. \ --------------------- -------------- --------------- ------------ -- -------------- --------------- ------------ Line $T_{\rm mb}$ $V_{\rm LSR}$ $\Delta V$ $T_{\rm mb}$ $V_{\rm LSR}$ $\Delta V$ (K) (km/s) (km/s) (K) (km/s) (km/s) C$^{18}$O(1–0) 3.19(16) $-$6.83(09) 3.46(22) 2.53(18) $-$7.51(10) 2.95(24) HCN(1–0) 18.37(11) $-$6.80(01) 3.05(02) 11.43(11) $-$6.92(01) 2.84(03) H$^{13}$CN(1–0) 1.50(07) $-$7.00(05) 2.22(10) 0.79(08) $-$6.96(08) 1.53(16) HCO$^+$(1–0) 25.63(09) $-$6.91(01) 3.05(01) 15.13(09) $-$6.88(01) 2.86(02) H$^{13}$CO$^+$(1–0) 2.64(10) $-$7.18(05) 2.58(11) 1.94(10) $-$7.21(06) 2.26(14) HNC(1–0) 16.62(17) $-$6.77(01) 2.83(03) 14.95(18) $-$7.03(02) 2.51(04) HN$^{13}$C(1–0) 0.68(05) $-$6.42(12) 3.27(27) 1.24(07) $-$6.94(05) 1.75(11) N$_2$H$^+$(1–0) 3.41(07) $-$6.90(02) 2.20(05) 6.33(09) $-$7.00(01) 1.87(03) --------------------- -------------- --------------- ------------ -- -------------- --------------- ------------ Physical parameters of the sources and molecular abundances =========================================================== Kinetic temperatures -------------------- The kinetic temperatures of the sources have been estimated from the CH$_3$C$_2$H observations. As shown e.g. by @Bergin94 this symmetric-top molecule is a good “thermometer” for gas densities $n\ga 10^4$ cm$^{-3}$. Our main goal is to compare the temperatures at the peaks of the CS and N$_2$H$^+$ emission. At first, we derived the temperatures from the CH$_3$C$_2$H $J=6-5$ data obtained at Onsala. These measurements were done only at the peak positions. Later the CH$_3$C$_2$H $J=13-12$ emission was mapped at IRAM with the multibeam receiver. The details of the temperature estimates are published elsewhere [@Malafeev05 Zinchenko et al., in preparation]. Here we present the main results. In S187 the CH$_3$C$_2$H emission was too weak. The kinetic temperatures at the CS and N$_2$H$^+$ peaks, derived from the Onsala and IRAM CH$_3$C$_2$H data, are listed in Table \[table:Tkin\]. It is easy to see that in most cases there is no significant temperature difference between the CS and N$_2$H$^+$ emission peaks, although on average the N$_2$H$^+$ peaks are somewhat colder than the CS ones. -------- ----------------- ----------------- --------------------- ----------------------- ------------ Source $\Delta \alpha$ $\Delta \delta$ $T_{\rm kin}^{6-5}$ $T_{\rm kin}^{13-12}$ Emission $('')$ $('')$ (K) (K) peak W3 20 -40 $52.6\pm 3.1$ $58.4\pm 1.5$ CS 160 -160 $30.7\pm 0.8$ $37.3\pm 2.2$ N$_2$H$^+$ S255 0 60 $34.9\pm 1.4$ $39.6\pm 0.4$ N$_2$H$^+$ 0 0 $34.5\pm 1.0$ $38.8\pm 0.9$ CS DR21 0 -40 $28.8\pm 0.1$ $34.8\pm 1.1$ N$_2$H$^+$ 0 0 $33.4\pm 2.7$ $44.0\pm 1.4$ CS S140 40 20 $27.8\pm 1.6$ $37.7\pm 2.2$ N$_2$H$^+$ 0 0 $30.6\pm 0.7$ $42.4\pm 1.0$ CS -------- ----------------- ----------------- --------------------- ----------------------- ------------ : Kinetic temperatures at the CS and N$_2$H$^+$ peaks derived from the Onsala and IRAM CH$_3$C$_2$H data ($T_{\rm kin}^{6-5}$ and $T_{\rm kin}^{13-12}$, respectively).[]{data-label="table:Tkin"} CH$_3$C$_2$H maps of sufficiently high quality were obtained at IRAM for DR21, S140 and the CS peak in W3. The IRAM data indicate somewhat higher peak temperatures than obtained from the Onsala data, as expected due to a higher angular resolution and higher excitation requirements for the $J=13-12$ transition if temperature increases towards an embedded heating source. Our data do show such temperature gradients consistent with theoretical expectations [@Zin05] but we do not discuss them here. At the same time the IRAM data confirm our main conclusion that there is no significant temperature difference between the CS and N$_2$H$^+$ peaks in most cases, although the N$_2$H$^+$ peaks are somewhat colder than the CS ones, on average. It is interesting to compare our estimates of kinetic temperature with other available data for these sources. Most of them have been actively studied. Such comparison was made by @Malafeev05 and here we repeat the main points. ### W3 Based on NH$_3$(1,1) and (2,2) lines, @Tieftrunk98 derived kinetic temperatures of 44 K and 25 K toward our CS and N2H+ peaks, respectively. A comparison with Table \[table:Tkin\] shows that these temperatures are somewhat lower than those obtained from the CH$_3$C$_2$H data, although the relation is the same. ### S255 From our ammonia observations [@Zin97] the kinetic temperature at the (0,+80$''$) position (near the NH$_3$ and N$_2$H$^+$ peaks) is $23\pm 1$ K. At the central position the uncertainty in the kinetic temperature is too high. @Schreyer96 obtained the kinetic temperature in the centre from their ammonia data of 30.8 K and the dust temperature from the IRAS data of 31.1 K. @Mezger88 derived almost the same dust temperatures ($\sim 30$ K) for the two peaks discussed here from their 1.3 mm and 350 $\mu$m continuum observations. ### DR21 NH$_3$ This source was observed (without mapping) in the CH$_3$C$_2$H lines at the 11-m NRAO telescope [@Kuiper84]. They observed the same $J=6-5$ transition and derived practically the same kinetic temperature at the centre as in our present work, 33.0 K. There are VLA ammonia observations by @Mangum92. They derived kinetic temperatures of 32 K at the (0,0) position and 25 K near the (0,$-$40$''$) position. This is rather close to our estimates. ### S140 S140 was also observed in CH$_3$C$_2$H at the 11-m NRAO telescope by @Kuiper84. They obtained for the cloud centre the kinetic temperature of $32.1\pm 6.7$ K which coincides with our $J=6-5$ estimate within the uncertainties. From ammonia observations at Effelsberg, @Ungerechts86 found a temperature of about 20 K, once again a value significantly lower than found with CH$_3$C$_2$H. The dust temperature from IRAS data is 34 K [@Zhou94], more similar to our findings. Thus, CH$_3$C$_2$H appears to better trace the dust. Masses and densities -------------------- There are several ways to estimate masses and densities of interstellar clouds. Here we derive the source masses and column densities primarily from the dust continuum observations. This is considered to be one of the most reliable methods, partly due to the fact that dust/gas mass ratio is rather constant. At the same time the uncertainties in the dust opacities are still rather high. Additional uncertainties are related to uncertainties in the dust temperature. Here we use the temperature estimates obtained from our methyl acetylene observations, found to be similar to the dust temperature. It is known that at the typical densities of dense cores the dust and gas kinetic temperatures are close to each other. The results of these estimates are presented in Table \[table:mass\]. It is worth noting that the masses here are not really total masses of the sources because they are derived from the fluxes integrated over 1 diameter circle around the indicated positions. The dust opacities were adopted from @Ossenkopf94. The gas/dust mass ratio was assumed to be equal to 100. In addition we present also estimates of the average volume density along the line of sight obtained as $\bar{n}=N_{\rm L}/L$ where the size $L$ is derived from the angular size (Table \[table:cont-res\]) and distance to the source (Table \[table:sources\]). In the last column we give estimates of the virial masses of the clumps based on these sizes and widths of optically thin lines at these positions. These estimates are rather uncertain, at least by a factor of 2 due to large uncertainties in the parameters. Nevertheless, within this factor they agree well with the masses derived from continuum observations. This shows that the clumps are close to gravitational equilibrium. [lcccll]{} Source &Peak &$N_{\rm L}\times 10^{-23}$ &$\bar{n}\times10^{-5}$ &$M$ &$M_{\mathrm{vir}}$\ & &(cm$^{-2}$) &(cm$^{-3}$) &(M$_{\sun}$) &(M$_{\sun}$)\ S 187 &CS &0.4 &0.7 &30 &50\ &N$_2$H$^+$ &0.5 &1.0 &14 &10\ W3 &CS &2.0 &1.6 &405 &310\ &N$_2$H$^+$ &1.2 &1.5 &74 &90\ S 255 &CS &2.0 &1.9 &290 &210\ &N$_2$H$^+$ &1.9 &1.6 &280 &220\ S 140 &CS &2.8 &3.9 &86 &90\ &N$_2$H$^+$ &0.6 &0.7 &32 &70\ \[table:mass\] The average gas volume densities are $\bar{n} \sim 10^5$ cm$^{-3}$ in all cases. However, it is well known that such mean densities can be significantly lower than densities found from molecular excitation analysis [e.g. @Zin98]. This discrepancy is most probably explained by small-scale gas clumpiness. Such clumpiness is indicated by many studies [e.g. @Bergin96]. At the same time our data do not provide sufficient material for excitation analysis in most cases. Only for S255 we have 3 mm and 1.3 mm data in C$^{18}$O, CS, C$^{34}$S and methanol transitions towards both CS and N$_2$H$^+$ peaks which can be used for modeling. In particular, our LVG estimates of gas density from the C$^{34}$S data give values of about $n\sim 5\times 10^5$ cm$^{-3}$ for both components. Methanol data modeling gives $n\sim 3\times 10^5$ cm$^{-3}$ also for both peaks (S. Salii, private communication). Estimates based on C$^{18}$O are highly uncertain but are consistent with these values and show similar densities for both components because the line intensity ratios are similar. These estimates are rather close to the average densities presented in Table \[table:mass\], although somewhat higher as expected. This shows that the volume filling factor in this density range is rather high. The continuum data also indicate practically equal mean gas densities for the components. Abundances ---------- We derive here abundances of several species which were observed in all our sources and are the most informative ones for our purposes: C$^{18}$O, H$^{13}$CN, H$^{13}$CO$^+$, HN$^{13}$C, N$_2$H$^+$ and C$^{34}$S. In most cases we use only rare isotope data which are presumably not affected by optical depth effects. The optical depth in the N$_2$H$^+$ lines is also small or moderate as shown by @Pirogov03. The column densities are estimated from integrated line intensities in the LTE approximation assuming the excitation temperatures equal to the kinetic temperatures as given in Table \[table:Tkin\] (approximate values close to $T_{\rm kin}^{6-5}$ and $T_{\rm kin}^{13-12}$ were used). For S187 we assume $T_{\rm kin}$ of 20 K for the CS peak and $T_{\rm kin}$ of 10 K for the N$_2$H$^+$ peak (taking into account narrow line widths and very weak CH$_3$C$_2$H emission). Then, we derive abundances from comparison of the molecular column densities and total gas column densities obtained from dust continuum observations (Table \[table:mass\]). In order to take into account different beam sizes we correct the abundance estimates assuming Gaussian beams and Gaussian brightness distributions. For DR21 where we do not have our own dust continuum data, the abundances were derived assuming $X(\mathrm{C^{18}O})= 2\times 10^{-7}$. The results are summarized in Table \[table:abundances\]. @Pirogov03 derived N$_2$H$^+$ abundances in a different way: from the N$_2$H$^+$ column densities and virial masses of the clouds. Nevertheless, their estimates are very close to those presented in Table \[table:abundances\] when the positions coincide (within a factor of 2). -------- ----------------- ----------------- ----------------------- ----------------------- ----------------- --------------------- ----------------- ---------------- ------------------- ---------------- Source $\Delta \alpha$ $\Delta \delta$ $X$(C$^{18}$O)$_{10}$ $X$(C$^{18}$O)$_{21}$ $X$(H$^{13}$CN) $X$(H$^{13}$CO$^+$) $X$(HN$^{13}$C) $X$(C$^{34}$S) $X$(N$_{2}$H$^+$) $X_\mathrm{e}$ $('')$ $('')$ S187 160 0 4.2E$-$07 1.5E$-$07 1.0E$-$10 5.8E$-$11 6.5E$-$11 6.0E$-$10 6.5E-11 1.7E-07 0 80 2.5E$-$07 7.6E$-$11 6.2E$-$11 1.2E-10 1.3E-07 W3 0 $-$40 2.9E$-$07 4.8E$-$10 8.6E$-$11 7.6E$-$11 6.5E-11 1.2E-07 160 $-$160 2.0E$-$07 1.9E$-$10 8.8E$-$11 2.7E-10 5.4E-08 S255 0 0 2.4E$-$07 1.4E$-$07 2.8E$-$10 5.7E$-$11 4.9E$-$11 9.2E$-$10 1.7E-10 1.8E-07 0 60 1.8E$-$07 1.3E$-$07 2.3E$-$10 1.9E$-$10 1.2E$-$10 7.7E$-$10 3.7E-10 5.4E-08 DR21 0 $-$20 2.2E$-$10 1.6E$-$10 1.5E$-$10 6.3E$-$10 3.2E-10 6.3E-08 0 $-$40 2.4E$-$10 1.5E$-$10 1.9E$-$10 7.2E$-$10 4.7E-10 6.7E-08 S140 0 0 1.3E$-$07 1.0E$-$10 1.2E$-$10 7.1E$-$11 1.5E-10 8.4E-08 40 20 3.4E$-$07 1.4E$-$10 2.9E$-$10 2.7E$-$10 9.1E-10 3.4E-08 -------- ----------------- ----------------- ----------------------- ----------------------- ----------------- --------------------- ----------------- ---------------- ------------------- ---------------- Non-LTE modeling for some clumps in our sample using the LVG approximation or the RADEX code (based on the escape probability method; @Vandertak07) gives systematically lower (by a factor of 1.5–2) column densities for all species considered here, except C$^{18}$O, assuming temperature and density as described above. This is a natural result because C$^{18}$O is practically thermalized at these conditions while the other species are still far from equilibrium and population of lower levels exceeds that expected in LTE. Nevertheless, we believe that non-LTE modeling is not justified here due to the absence of the necessary data for most of the sources. One can bear in mind that the abundances derived here are somewhat overestimated. Ionization fraction ------------------- The ionization fraction can be estimated from the HCO$^+$ abundance using the results of relevant chemical models. It is known that in dense clouds HCO$^+$ is formed mainly from H$_3^+$ [e.g. @Turner95]: $$\label{eq:H3+} \mathrm{H_3^+} + \mathrm{CO} \rightarrow \mathrm{HCO^+} + \mathrm{H_2}$$ and is destroyed by dissociative recombination: $$\label{eq:HCO+} \mathrm{HCO^+} + e \rightarrow \mathrm{CO} + \mathrm{H}$$ In addition, one has to take into account HCO$^+$ recombination onto negatively charged dust grains [e.g. @Caselli08]. The rate of this reaction is given by the product $k_\mathrm{g} X_\mathrm{g}$ where the rate coefficient $k_\mathrm{g}$ and the fractional abundance of dust grains $X_\mathrm{g}$ are determined by grain size distribution and gas kinetic temperature [@Draine87]. For the typical temperatures in our objects ($\sim 30$ K) and MRN [@Mathis77] grain size distribution $k_\mathrm{g} X_\mathrm{g} \approx 3\times 10^{-15}$ s$^{-1}$. In this model for HCO$^+$ abundance in steady state we can write: $$\label{eq:X_HCO+} X(\mathrm{HCO^+}) = \frac{k_1 X(\mathrm{H_3^+}) X(\mathrm{CO})}{\alpha(\mathrm{HCO^+}) X_e + k_\mathrm{g} X_\mathrm{g}}$$ where $k_1$ is the rate of reaction (\[eq:H3+\]) and $\alpha(\mathrm{HCO^+})$ is the HCO$^+$ dissociative recombination rate. H$_3^+$ is formed by cosmic-ray ionization of molecular hydrogen and is destroyed in dense clouds primarily by reaction (\[eq:H3+\]) [e.g. @Black00]. For regions where CO is not frozen onto dust grains, recombination onto negatively charged grains is a negligible H$_3^+$ destruction process compared with this reaction. Therefore, for H$_3^+$ abundance in this regime we obtain: $$\label{eq:X_H3+} X(\mathrm{H_3^+}) = \frac{\zeta/n}{k_1 X(\mathrm{CO})}$$ where $\zeta$ is the cosmic-ray ionization rate and $n$ is the total gas density. Now combining Eq. (\[eq:X\_HCO+\]) and (\[eq:X\_H3+\]) we come to the following simple expression: $$\label{eq:X_HCO+_2} X(\mathrm{HCO^+}) = \frac{\zeta/n}{\alpha(\mathrm{HCO^+}) X_e + k_\mathrm{g} X_\mathrm{g}}$$ Estimates of the electron abundances based on this formula are presented in the last column of Table \[table:abundances\]. The cosmic-ray ionization rate was assumed to be equal to $\zeta = 3\times 10^{-17}$ s$^{-1}$, $\alpha(\mathrm{HCO^+}) = 7.5\times 10^{-7}$ s$^{-1}$cm$^3$ [@Turner95]. The gas density was assumed to be $n= 10^5$ cm$^{-3}$ and $[\mathrm{HCO^+}]/[\mathrm{H^{13}CO^+}]=40$. The $k_\mathrm{g} X_\mathrm{g}$ term leads to corrections less than 10% in the $X_\mathrm{e}$ values and we neglected it. Strictly speaking, in this way we derive more reliably the parameter $nX_\mathrm{e}$, i.e. the electron density, not abundance. However, for a better comparison with other results we will discuss further mainly the $X_\mathrm{e}$ values. Discussion ========== Variations of molecular abundances ---------------------------------- An inspection of the observational results and estimates of the physical parameters presented above leads to several important conclusions: \(1) There are strong variations of the relative intensities in the lines of different molecular tracers across the investigated high mass star forming regions. In Paper I we mentioned already striking differences between N$_2$H$^+$ and CS maps and the fact that the CS distribution follows that of the dust emission while N$_2$H$^+$ does not. Now we see that C$^{18}$O and HCN, like CS, are good tracers of the dust emission which presumably shows the total mass distribution. At the same time the behavior of HCO$^+$ and HNC resembles that of N$_2$H$^+$. Significant differences in distributions of various species in particular sources have been noticed in some other studies [e.g. @Ungerechts97] but here we see systematic effects common for the sources in the sample. \(2) There is no sign of CO and/or CS depletion in these objects (in contrast to cold dark clouds). The abundances derived for C$^{18}$O from the $J=1-0$ and $J=2-1$ transitions more or less agree with each other. They are practically constant within each source and among the whole sample (with small deviations which can be caused e.g. by temperature uncertainties) and are close to the “canonical” values derived earlier (e.g. $X(\mathrm{C^{18}O}) \approx 1.7\times 10^{-7}$ obtained by @Frerking82). \(3) The N$_2$H$^+$ and HNC abundances significantly decrease with increasing ionization fraction (Fig. \[fig:Xe-abund\]). The best fit gives $X(\mathrm{N_2H^+})\propto X_\mathrm{e}^{-1.3\pm 0.3}$ and $X(\mathrm{HN^{13}C})\propto X_\mathrm{e}^{-0.8\pm 0.2}$ with the correlation coefficients of $\vert\rho \vert \approx 0.85$ for both dependences. At the same time the HCN abundance does not show significant variations. It is interesting that the data for N$_2$H$^+$ and $X_\mathrm{e}$ presented by @Bergin99 are in excellent agreement with the $X(\mathrm{N_2H^+}) - X_\mathrm{e}$ relation obtained here, as one can see in Fig. \[fig:Xe-abund\], where the open symbols are the $X(\mathrm{N_2H^+})$ values derived from the N$_2$H$^+$ and C$^{18}$O column densities presented by @Bergin99, and assuming $X(\mathrm{C^{18}O}) = 1.7\times 10^{-7}$ [@Frerking82]. ![Relative abundances of N$_2$H$^+$, HN$^{13}$C and H$^{13}$CN (from top to bottom) in dependence on the electron abundance. Our estimates are plotted by the filled symbols. The open symbols correspond to the data presented by @Bergin99.[]{data-label="fig:Xe-abund"}](Xe-abund-B-dec08.eps "fig:"){width="\columnwidth"}\ These correlations should not be affected much by the uncertainties in the abundances due to the LTE approximation, given that both variables scale in a similar way if the non-LTE approach is adopted. \(4) The derived ionization fraction in these objects in general increases towards the strong embedded IR sources which coincide with the main peaks of the dust and CS emission (by a factor of 2$-$3). The N$_2$H$^+$, HCO$^+$ and HNC abundances decrease correspondingly in these areas. This is consistent with our results in Paper I where we found that the N$_2$H$^+$ abundance is systematically lower towards the CS/dust emission peaks which coincide with strong IR sources. However, it is important to emphasize that estimates of the electron abundance were performed under the assumption of a constant gas density. Our estimates of the gas density towards S255 IR and N give practically the same values. At the same time it is not unreasonable to expect density variations which can smooth the derived variations of the ionization fraction. It is worth noting that these variations of the ionization fraction refer to the values averaged over the line of sight and an increase of the electron abundance in vicinities of a young star can be significantly larger. @Ungerechts97 found similar variations of molecular abundances in the Orion molecular cloud and suggested that the abundances of molecular ions can be reduced by a higher electron abundance caused by UV radiation propagating in a clumpy photodissociation region. It is well known that in clumpy media UV photons can penetrate deep into dense clouds leading to significantly enhanced ionization [e.g. @Bethell07]. At the same time in their study of the ionization fraction in massive cores, @Bergin99 did not find any noticeable increase of the electron abundance from the edge to the centre of massive cores. This can be caused perhaps by insufficient luminosities of their sample sources. Thus, we believe that UV radiation from young massive stars in clumpy media can be the primary cause of the observed ionization enhancement. In addition, X-rays detected already from many massive stars [e.g. @Zhekov07] can be also responsible for this effect. \(5) We see no dependence of the relative abundances on the velocity dispersion, as derived from the observed line widths. In principle such dependence could be expected in particular for N$_2$H$^+$ if it really escapes perturbed regions as suggested e.g. by @Womack92. \(6) There may be a trend for decreasing N$_2$H$^+$ and increasing H$^{13}$CN abundances with increasing temperature. However, this is based on only one point (W3 data) and cannot be considered reliable. Chemical implications --------------------- Apparently the observed differences in molecular distributions cannot be explained by molecular freeze-out as in low-mass cores. The kinetic (and dust) temperatures at the CS and N$_2$H$^+$ peaks are similar in most cases and rather high, $\ga 30$ K, which probably makes freeze-out ineffective. This is confirmed by the absence of any indication of the CO and CS depletion as mentioned above. One possible explanation for the observed chemical differentiations was proposed by @Lintott05. They suggested that the enhancement of CS and reduction in $\mathrm{N_2H^+}$ abundance found in regions of high-mass star formation may be related to the high dynamical activity in these regions which could enhance the rate of collapse of cores above the free-fall rate. Consequently, high gas densities would be achieved before freeze-out had removed the molecules responsible for $\mathrm{N_2H^+}$ loss, while the high densities promote CS formation. However, this model has several drawbacks. In particular, its predictions for some species (e.g. SO) are not supported by observations. Then, it predicts a decrease in the abundance accompanied by CS enhancement. Our data show no sign of the CS abundance variations. Nothing to say that an accelerated collapse itself has no more or less advanced physical basement. Our results presented in the previous section indicate that ionization fraction is probably more important in establishing the steady-state N$_2$H$^+$ and HNC abundance in massive cores. The only important process which forms N$_2$H$^+$ is [e.g. @Turner95]: $$\label{eq:N2H+} \mathrm{H_3^+} + \mathrm{N_2} \rightarrow \mathrm{N_2H^+} + \mathrm{H_2}$$ There are two main processes which destroy N$_2$H$^+$: the reaction with the CO molecule $$\label{eq:N2H+-CO} \mathrm{N_2H^+} + \mathrm{CO} \rightarrow \mathrm{HCO^+} + \mathrm{N_2}$$ and dissociative recombination $$\begin{aligned} \label{eq:N2H+-e1} \mathrm{N_2H^+} + e &\rightarrow& \mathrm{N_2} + \mathrm{H}\\ \mathrm{N_2H^+} + e &\rightarrow& \mathrm{NH} + \mathrm{N} \label{eq:N2H+-e2}\end{aligned}$$ which mainly lead to the formation of N$_2$ (90% of the total reaction), as recently found by @Molek07. Then, we have to take into account N$_2$H$^+$ recombination onto negatively charged dust grains as in the case of HCO$^+$. The steady state N$_2$H$^+$ abundance in this model will be given by the formula: $$\label{eq:X_N2H+} X(\mathrm{N_2H^+}) = \frac{k_2 X(\mathrm{H_3^+}) X(\mathrm{N_2})}{k_3 X(\mathrm{CO})+\alpha(\mathrm{N_2H^+}) X_e + k_\mathrm{g} X_\mathrm{g}}$$ where $k_2$ is the rate of the reaction (\[eq:N2H+\]), $k_3$ is the rate of the reaction (\[eq:N2H+-CO\]) which is $8.8\times 10^{-10}$ cm$^3$s$^{-1}$ according to the UMIST database and $\alpha(\mathrm{N_2H^+})$ is the summary rate of the reactions (\[eq:N2H+-e1\]), (\[eq:N2H+-e2\]). Its value is significantly different in the UMIST and OSU databases. We accept the value of $8\times 10^{-7}$ cm$^3$s$^{-1}$ at 30 K (E. Herbst, private communication). This means that the dissociative recombination will dominate at $X_\mathrm{e} \ga 10^{-3}X(\mathrm{CO})$, i.e. at $X_\mathrm{e} \ga 10^{-7}$ for the standard CO abundance $X(\mathrm{CO}) \sim 10^{-4}$. This is close to an average electron abundances derived in this work. Therefore, in principle it is possible that the dissociative recombination of N$_2$H$^+$ can dominate at least in part of our objects. Then, if the H$_3^+$ and N$_2$ abundances are more or less constant (as can be expected) we obtain for $X(\mathrm{N_2H^+})$ the dependence on $X_\mathrm{e}$ similar to that presented in Fig. \[fig:Xe-abund\]. It is less clear how to explain the behavior of the HCN and HNC abundances. Their formation pathways are very different. HCN definitely forms from the neutral-neutral processes [@Turner97] $$\mathrm{N} + (\mathrm{CH_2},\mathrm{CH_3}) \rightarrow \mathrm{HCN} \label{eq:HCN}$$ HNC forms via the distinctly independent sequence $$\mathrm{C^+} + \mathrm{NH_3} \rightarrow \mathrm{H_2CN^+} + {e} \rightarrow \mathrm{HNC} \label{eq:HNC}$$ The main destruction processes for HNC are probably reactions with C$^+$ and H$_3^+$ [@Turner97]. Calculations of steady state abundances of these species require additional chemical modeling. We note that the HNC formation is closely linked to NH$_3$. Given that NH$_3$ and N$_2$H$^+$ both form from N$_2$, thus they are chemically related, one expects similar morphologies of HNC and N$_2$H$^+$, as in fact we observe. Chemical indicators of massive protostars ----------------------------------------- One of the most intriguing problems in studies of high mass star formation is the identification of massive protostars at the earliest phases of evolution. Some clumps from our sample represent probably such protostellar objects. For example the SE clump in W3 area is associated with a water maser but has no embedded IR sources and/or UC H II regions. The HCO$^+$ line profile shows a red-shifted self-absorption feature typical for a collapsing cloud. The mass of this clump from continuum data is about 70 M$_\odot$ (Table \[table:mass\]). Therefore, it can be a massive protostar on a rather early evolutionary stage. It is also very pronounced in N$_2$H$^+$ emission. Another example of this kind is the N$_2$H$^+$ emission peak in S187 (although we did not see signs of contraction on the line profiles). In the S255 area, the northern component which is dominant in N$_2$H$^+$, is apparently much younger than S255 IR. These examples show that a relatively strong N$_2$H$^+$ emission can be considered as an indicator of such objects. The species which correlate with N$_2$H$^+$ (HCO$^+$ and HNC) can be also useful in this respect. Conclusions =========== We presented and discussed here observations of five regions of active high mass star formation in various molecular lines (at 3 mm and at 1.3 mm) and in continuum at 1.3 mm. On the basis of these observations we estimated physical parameters of the sources and molecular abundances. The main results of this study can be summarized as follows: 1. The typical physical parameters for the sources in our sample are: kinetic temperature in the range $\sim 30-50$ K, masses from tens to hundreds solar masses, gas densities $\sim 10^5$ cm$^{-3}$, ionization fraction $\sim 10^{-7}$. In most cases the ionization fraction slightly (a few times) increases towards the embedded YSOs. The observed clumps are close to gravitational equilibrium. Our temperature estimates are systematically lower (by a factor of about 1.5–2) compared to those obtained with NH$_3$ observations. However, temperatures measured with CH$_3$C$_2$H are similar to dust temperatures, suggesting that the observed methyl acetylene transitions better trace the dust than ammonia (1,1) and (2,2) lines. 2. There are systematic differences in distributions of various molecules in regions of high mass star formation. The abundances of CO, CS and HCN are more or less constant and optically thin lines of rare isotopes of these species are good tracers of the dense gas distribution in these regions. There is no sign of CO and/or CS depletion as in cold low mass cores. 3. At the same time, the abundances of the high density tracers HCO$^+$, HNC and especially N$_2$H$^+$, strongly vary in these objects. They anti-correlate with the ionization fraction ($X(\mathrm{N_2H^+})\propto X_\mathrm{e}^{-1.3\pm 0.3}$ and $X(\mathrm{HN^{13}C})\propto X_\mathrm{e}^{-0.8\pm 0.2}$) and as a result decrease towards the embedded YSOs. For N$_2$H$^+$ this can be explained by dissociative recombination to be the dominant destroying process. This conclusion is more or less consistent with the data on chemical reaction rates. There is no correlation of these abundances with the line width. 4. The described variations of the HCO$^+$, HNC and N$_2$H$^+$ abundances make them potentially valuable indicators of massive protostars. In our sample there are some clumps which represent probably massive protostars at very early stages of evolution and they are very pronounced in the lines of these species, especially N$_2$H$^+$. Acknowledgements {#acknowledgements .unnumbered} ================ The invaluable contributions to obtaining observational data and to initial discussions were made by Lars E.B. Johansson and Barry Turner who recently passed away. We acknowledge the support from the telescope staff at OSO, NRAO and IRAM. Estimates of the physical parameters from the methanol data were made by Svetlana Salii and kinetic temperature estimates by Sergey Malafeef who also participated actively in the observations at Onsala. We are grateful to Alexander Lapinov for providing his LVG code and to Eric Herbst who provided the rate coefficient for the dissociative recombination of N$_2$H$^+$. The constructive comments by the referee, Ted Bergin, helped to improve the manuscript. The work was supported by Russian Foundation for Basic Research grants 03-02-16307 and 06-02-16317, by the Russian Academy research program “Extended objects in the Universe” and by the INTAS grant 99-1667. The research has made use of the SIMBAD database, operated by CDS, Strasbourg, France. [99]{} Argon A. L., Reid M. J., Menten K. M., 2000, ApJS, 129, 159 Bergin E. A., Goldsmith P. F., Snell R. L., Ungerechts H., 1994, ApJ, 1994, 431, 674 Bergin E. A., Snell R. L., Goldsmith P. F., 1996, ApJ, 460, 343 Bergin E. A., Plume R., Williams J. P., Myers P. C., 1999, ApJ 512, 724 Bergin E. A., Ciardi D. R., Lada C. J., Alves J., Lada E. A., 2001, ApJ, 557, 209 Bethell T. J., Zweibel E. G., Li P. 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--- abstract: 'In this paper we fill in a fundamental gap in the extremal bootstrap percolation literature, by providing the first proof of the fact that for all $d \geq 1$, the size of the smallest percolating sets in $d$-neighbour bootstrap percolation on $[n]^d$, the $d$-dimensional grid of size $n$, is $n^{d-1}$. Additionally, we prove that such sets percolate in time at most $c_d n^2$, for some constant $c_d >0 $ depending on $d$ only.' author: - 'Micha[ł]{} Przykucki[^1] [^2]' - Thomas Shelton title: | Smallest percolating sets\ in bootstrap percolation on grids --- Introduction {#sec:intro} ============ *Bootstrap percolation*, suggested by Chalupa, Leath, and Reich [@bootstrapbethe], is a simple cellular automaton modelling the spread of an infection on the vertex set of a graph $G$. For some positive integer $r$, given a set of initially infected vertices $A \subseteq V(G)$, in consecutive rounds we infect all vertices with at least $r$ already infected neighbours. *Percolation* occurs if every vertex of $G$ is eventually infected. The majority of research into bootstrap percolation processes has been focused on the probabilistic properties of the model. More precisely, if we initially infect every vertex independently at random with some probability $p$, how likely is the system to percolate? The monotonicity of the model (i.e., the fact that infected vertices never heal) makes it reasonable to ask about the value of the *critical probability* $p$, above which percolation becomes more likely to occur than not. This quantity has been analysed for many different families of graphs $G$ and for various infection rules, and often very sharp results have been obtained by, e.g., Aizenman and Lebowitz [@metastabilityeffects], Holroyd [@sharpmetastability], and Balogh, Bollob[á]{}s, Duminil-Copin, and Morris [@sharpbootstrapall]. Another family of questions related to bootstrap percolation that have been studied is concerned with the extremal properties of the model. Morris [@largestgridbootstrap] analysed the size of the largest minimal percolating sets in $2$-neighbour bootstrap percolation on the $n \times n$ square. For the same setup, Benevides and Przykucki [@maxtime] determined the maximum time the process can take until it stabilises. However, the first extremal question that attracted attention in bootstrap percolation was about the size of the smallest percolating sets. For grid graphs, this has been studied by Pete [@diseaseprocesses] (the summary of Pete’s results can be found in Balogh and Pete [@randomdisease]). For the hypercube, the size of the smallest percolating sets for all values of the infection threshold was found by Morrison and Noel [@extremalcube]. Feige, Krivelevich, and Reichman [@contagiousGnp] analysed the size of these sets in random graphs, while Coja-Oghlan, Feige, Krivelevich, and Reichman [@contagiousExpanders] studied such sets in expander graphs. The $d$-neighbour process in $d$ dimensions ------------------------------------------- Let us introduce some notation. For $n \in {\mathbb{N}}$, let $[n] = \{1,2, \ldots, n\}$. The $d$-dimensional grid graph of size $n$ is the graph with vertex set $[n]^d$, in which $u,v \in [n]^d$ are adjacent if and only if they differ by a value of 1 in exactly one coordinate. For $d,r,n \in {\mathbb{N}}$, let $G_{d,r}(n)$ denote the size of the smallest percolating sets in $r$-neighbour bootstrap percolation on $[n]^d$. For a set $A \subset [n]^d$, let $\langle A \rangle_r$ be the *closure* of $A$ in $r$-neighbour bootstrap percolation, i.e., the set of all vertices that become infected in the process that was started from $A$. Among the results stated in [@diseaseprocesses] (see also the Perimeter Lemma in the Appendix to [@randomdisease]) is the following theorem. \[thm:pete\] For all $n, d \in {\mathbb{N}}$, we have $G_{d,d}(n)=n^{d-1}$. This is obviously trivial for $d=1$, and the case when $d=2$ constitutes a lovely and well-known puzzle. Indeed, finding a percolating set of size $n$ is easy: just take one of the diagonals of the square. To show that there is no percolating set of size strictly less than $n$, we can refer to the famous *perimeter argument*: the perimeter of the infected set (understood as the number of edges between an infected and a healthy vertex, if we naturally embed our square $[n]^2$ in the infinite grid ${\mathbb{Z}}^2$) can never grow. Indeed, whenever a new vertex becomes infected, it is by virtue of at least two perimeter edges. Thus at least two edges are removed from the perimeter of the infected set, and at most two new ones are added, and the aforementioned monotonicity of the perimeter follows. Since the whole $n \times n$ grid has perimeter $4n$, and any initially infected vertex contributes at most $4$ edges to the perimeter, we need at least $n$ initially infected vertices to percolate. Somewhat surprisingly, the perimeter argument carries immediately to higher dimensions, giving us the appropriate lower bound $G_{d,d}(n) \geq n^{d-1}$ for all $d \in {\mathbb{N}}$. As for the upper bound, there is a natural candidate, sometimes referred to as a “cyclic combination” of the one-dimensional lower set. More precisely, for $d\leq k\leq dn$, let $V_k=\{v=(v_1,...,v_d)\in [n]^d:\sum_{i=1}^{d} v_i=k\}$. It is then natural to believe that the set $$\label{eqn:initialSet} A = A_d = \bigcup_{i=1}^{d} V_{in}$$ percolates in $d$-neighbour bootstrap percolation on $[n]^d$, and indeed this is the construction that was used to deduce the upper bound in [@diseaseprocesses]. One can imagine how two “neighbouring hyperplanes”, $V_{(i-1)n}$ and $V_{in}$, fill in the space between them with infection until the two growths meet, from which point on the process quickly finishes. The fact that $G_{d,d}(n) = n^{d-1}$ has become a “folklore knowledge” in the area of bootstrap percolation, and has sometimes even been referred to as an “observation”. Up to our best knowledge [@PetePrivate], no formal proof of Theorem \[thm:pete\] was provided in [@diseaseprocesses], and no such proof exists in the literature. However, problems arise quickly when one tries to describe how exactly the space between the two hyperplanes is filled in. Any vertex in $V_{(i-1)n+1}$ with at least one coordinate equal to $1$ has fewer than $d$ infected neighbours in $V_{(i-1)n}$, and consequently does not become infected in step 1. Similarily, after one step, any vertex in $V_{(i-1)n+2}$ with at least one coordinate equal to at most $2$ has fewer than $d$ infected neighbours in $V_{(i-1)n+1}$, and also remains healthy. This problem builds up (analogous constraints can be easily formulated for the layers being infected “from above” by $V_{in}$) and, in fact, the two growths barely meet - two hyperplanes at distance $n+1$ apart would have stayed separated, while hyperplanes at distance $n-1$ would result in some vertices being infected by more than $d$ infected neighbours, and consequently no percolation by the perimeter argument. What is however even more troublesome, describing the growth from the moment of the meeting onwards is where the real challenges occur. By the perimeter argument, we know that we have no elbow room in this description: no proper subset of $A$ percolates, and even a small perturbation of $A$ would not percolate if any vertex ever became infected by virtue of more than $d$ infected neighbours. In Figure \[fig:G\_[3,3]{}(6)\] we present the growth of the infected set, starting from $A$ as defined in , in $3$-neighbour bootstrap percolation on $[6]^3$. Even though we are in just three dimensions, and the size of the grid is very small, the process already feels quite difficult to describe and lasts as many as 14 steps. Consequently, we believe that Theorem \[thm:pete\] requires a proper, formal proof, which we provide as the main result of this paper in Section \[sec:main\]. ![Example showing the spread of infection in $[6]^3$ starting from a set of size $G_{3,3}(6) = 6^2 = 36$.[]{data-label="fig:G_3,3(6)"}](./exampleSet.png) Another reason to convince oneself about the fact that the process of filling in the space between $V_{(i-1)n}$ and $V_{in}$ is nontrivial becomes apparent when we look at the results of numerical simulations, and analyse the time the process takes to terminate. It quickly becomes apparent that, for a fixed $d$, this time grows quadratically with $n$. This should be somewhat surprising, as by averaging there is some $i$ such that the volume between $V_{(i-1)n}$ and $V_{in}$ is of the order $n^d$. For a percolating set $A$, let $T(A)$ be the time (i.e., the number of time steps) the process takes to infect the whole vertex set. Let $$\label{eqn:smallestTimeDefn} m_d(n) = \min \left \{T(A) : \langle A \rangle_d = [n]^d, |A| = n^{d-1} \right \}.$$ In Section \[sec:time\], we come back to the question of percolation time and we prove the following theorem. \[thm:smallestTime\] We have $m_1(n) = \lceil n/2 \rceil$, $m_2(n) = n-1$, and for $d \geq 3$, $$\frac{dn}{2}+O(1) \leq m_d(n) \leq (d+2)n^2+n.$$ Before we proceed to the main part of this work, let us emphasise the importance of the extremal results in bootstrap percolation. The lower bound in [@metastabilityeffects], where the order of magnitude of the critical probability in $2$-neighbour bootstrap percolation on $[n]^2$ was determined, follows very easily from the fact that $G_{2,2}(n) = n$. In [@bootstraphigh], Balogh, Bollob[á]{}s, and Morris used the value of $G_{d,2}(n)$ for arbitrary $d$ as a vital tool to determine the critical probability in $2$-neighbour bootstrap percolation on high-dimensional grids. Finally, we remark that Balister, Bollob[á]{}s, Johnson, and Walters [@randomMajority], and Huang and Lee [@deterministicHigh], independently observed that $G_{d,d}(n) \leq c_d n^{d-1}$, where $c_d > 0$ is some constant depending on $d$ only, as infecting the boundary of $[n]^{d}$ (of size at most $2dn^{d-1}$) gives us a percolating set in the $d$-neighbour bootstrap process. Proof of the main result {#sec:main} ======================== In this section we prove Theorem \[thm:pete\]. The result $G_{d,d}(n)\geq n^{d-1}$ follows from our discussion of the monotonicity of the perimeter of the infected set. Therefore we need to prove that $G_{d,d}(n)\leq n^{d-1}$. Unlike for $d=1,2$, in the general case proving the upper bound turns out to be much more challenging. Let $G=[n]^d$ be the $d$-dimensional grid of size $n$. For $d\leq k\leq dn$, we define $V_k=\{v=(v_1,...,v_d)\in [n]^d:\sum_{i=1}^{d} v_i=k\}$. Note that $\bigcup_{k=d}^{dn} V_k=V([n]^d)$. We will show that the set $A = \bigcup_{i=1}^{d} V_{in}$ percolates in $d$-neighbour bootstrap percolation on $[n]^d$. (We can immediately see that $|A| = n^{d-1}$ as for fixed values of $v_2,...,v_d$, there is exactly one choice of $v_1$ such that $v=(v_1,...,v_d) \in A$.) To do this, we will prove that, for all $1\leq s\leq d$, $$\label{eqn:F_s} F_s=\bigcup_{i=1}^{n-1}V_{(s-1)n+i}\subseteq \langle V_{(s-1)n}\cup V_{sn}\rangle_d.$$ Note that we have $V_0, \ldots, V_{d-1} =\emptyset$, and consequently the sets $V_n, \ldots , V_{\left(\lceil d/n \rceil -1 \right)n}$ are empty. First we deal with the “bottom corner” of the grid. \[claim:bottomCorner\] We have $\bigcup_{j=d}^{\lceil d/n \rceil n-1}V_j \subseteq \langle V_{\lceil d/n \rceil n} \rangle_d$. Any vertex $v \in V_{\lceil d/n \rceil n - 1}$ has $\sum_{i=1}^{d} v_i=\lceil d/n \rceil n - 1$. If there was some $1 \leq i \leq d$ such that $v_i = n$, then the vertex $v - (n-1)e_i$ would lie in $V_{( \lceil d/n \rceil -1 )n}$ which we know is empty, a contradiction. Hence, $v_i < n$ for all $1 \leq i \leq d$. Therefore, again for all $1 \leq i \leq d$, we have $v+e_i \in V_{\lceil d/n \rceil n}$ infected. Therefore $v$ has at least $d$ infected neighbours and itself becomes infected. Since $v \in V_{\lceil d/n \rceil n-1}$ was arbitrary, all of $V_{\lceil d/n \rceil n-1}$ becomes infected. We proceed in this manner, in consecutive rounds infecting all vertices in $V_{\lceil d/n \rceil n-2}, V_{\lceil d/n \rceil n-3}, \ldots, V_d$. This completes the proof of the claim. Observe that $V_{dn} = \{(n,\ldots,n)\} \subset A$, hence we do not need to deal with the “upper corner”. Therefore from now on we shall analyse the dynamics of the process “sandwiched” between two initially infected hyperplanes. Fix $1+\left \lceil \frac{d}{n}\right \rceil \leq s\leq d$ and assume that $V_{(s-1)n}\cup V_{sn}$ is infected. Given $v\in F_s$, let $t_v=\sum_{i=1}^{d}v_i-(s-1)n$. Next, for $v\in F_s$, we define $$\label{eqn:Pre(v)} \Pre(v)=\{v+e_j:v_j\leq t_v\}\cup \{v-e_j:v_j>t_v\}.$$ For all $v\in F_s$, we have $|\Pre(v)|=d$. Therefore, if all vertices in $\Pre(v)$ are infected, then $v$ also becomes infected. We define the *infection witness tree* of $v$, $\IW(v)$, to be a directed labelled $d$-ary tree, with all edges directed away from the root and with vertices labelled with the elements of $F_s\cup V_{(s-1)n}\cup V_{sn}$, where these labels can be repeated in the tree. We construct $\IW(v)$ as follows. We start by declaring the root of the tree active and labelling it with $v$. Then, in consecutive rounds, we select an arbitrary active vertex. If the label $u$ of the vertex belongs to $V_{(s-1)n}\cup V_{sn}$, then this vertex becomes a leaf of $\IW(v)$ and we simply change its status to inactive. Otherwise, if the label $u$ of the vertex belongs to $F_s$, then we attach $d$ active children to this vertex and label them with the elements of $\Pre(u)$. Then, we again declare the selected vertex inactive. See Figure \[fig:IW\] for an example of a tree constructed in our algorithm. child [ node [322/2]{} child [ node [222/1]{} child [ node \[circle, draw\][221/0]{} edge from parent node\[draw=none,left\] [$3$]{} ]{} child \[missing\] child \[missing\] ]{} child [ node [332/3]{} child [ node [432/4]{} child [ node \[circle, draw\][532/5]{} ]{} child [ node \[circle, draw\][442/5]{} ]{} child [ node \[circle, draw\][433/5]{} ]{} edge from parent node\[draw=none,left\] [$2$]{} ]{} child \[missing\] child [ node [333/4]{} child \[missing\] child [ node \[circle, draw\][433/5]{} edge from parent node\[draw=none,left\] [$3$]{}]{} ]{} edge from parent node\[draw=none,left\] [$2$]{} ]{} ]{} child \[missing\] child \[missing\] child [ node [432/4]{} child [ node \[circle, draw\][532/5]{} ]{} child [ node \[circle, draw\][442/5]{} ]{} child [ node \[circle, draw\][433/5]{} ]{} edge from parent node\[draw=none,left, below\] [$2$]{} ]{}; By definition, all leaves of the tree $\IW(v)$ are initially infected. Since $\IW(v)$ is a $d$-ary tree, if $\IW(v)$ is finite then $v$ becomes infected. Since every non-leaf of $\IW(v)$ belongs to $F_s$ which is a finite set, an infinite directed path in $\IW(v)$ would contain infinitely many instances of the same label. Hence, the finiteness of $\IW(v)$ follows immediately from the next lemma. \[lem:noCycles\] For any $v \in F_s$, $\IW(v)$ has no directed path $u^1,...,u^m,u^{m+1}=u^1$, where $u^{i+1}\in \Pre(u^i)$ for all $1\leq i\leq m$. Suppose for a contradiction that the lemma does not hold, so there exists a directed path $u^1,...,u^m,u^{m+1}=u^1$, where $u^{i+1}\in \Pre(u^i)$ for all $1\leq i\leq m$. Then, since the algorithm that we use to construct $\IW(v)$ is deterministic, we know that we have an infinite directed path with labels $(u^{i})_{i\geq 1}$, where $u^{i+1}\in \Pre(u^i)$ for all $i\geq 1$, and there is some $m \geq 1$ (in fact we could only have $m \geq 2$ even) such that $u^{i+m}=u^i$ for all $i\geq 1$. Hence, we can assume without loss of generality that $C=t_{u^1}=\max\limits_{1\leq i\leq m}t_{u^i}$. As we traverse the directed path $(u^{i})_{i\geq 1}$, whenever $u^{i+1}\in \Pre(u^i)$ with $u^{i+1}=u^i+e_j$ for some $1\leq j\leq d$, by  we know that $u_j^i\leq t_{u^i}=t_{u^{i+1}}-1\leq t_{u^1}-1=C-1$. So we deduce that $u_j^{i+1}\leq C$. Now, given a vertex $v\in [n]^d$, we define $$L_C(v) =\sum_{\substack{1\leq j\leq d:\\v_j\geq C+1}}v_j.$$ (I.e., $L_C(v)$ is the sum of all coordinates of $v$ that are larger than $C$.) Therefore, when $t_{u^{i+1}}>t_{u^i}$, we know that $$L_C(u^{i+1})=\sum_{\substack{1\leq j\leq d:\\u_j^{i+1}\geq C+1}}u_j^{i+1}=\sum_{\substack{1\leq j\leq d:\\u_j^i\geq C+1}}u_j^i=L_C(u^i).$$ However, if $u^{i+1}\in \Pre(u^i)$ with $u^{i+1}=u^i-e_j$ for some $1\leq j\leq d$, then it is clear that $L_C(u^{i+1})\leq L_C(u^i)$. Additionally, by the maximality of $t_{u^1}$, we have that $t_{u^2}=t_{u^1}-1$. So we deduce that $u^2=u^1-e_j$ for some $1\leq j\leq d$. Then, by , we have $u_j^1\geq t_{u^1}+1=C+1$. Hence, as $u_j^2=u_j^1-1$, we clearly have $L_C(u^2)<L_C(u^1)$. Thus, following from the fact that $L_C$ never increases as we go along our directed path, $L_C(u^{m+1})<L_C(u^1)$. This implies that $u^{m+1}\neq u^1$, a contradiction to our previous assumption. Hence, $\IW(v)$ has no directed paths on which the same label is repeated more than once and, as discussed earlier, the whole tree is finite. The following corollary is immediate, and concludes the proof of Theorem \[thm:pete\]. For any vertex $v\in F_s$, $\IW(v)$ is finite. Consequently, $v$ becomes infected in finite time and, since $v\in F_s$ was chosen arbitrarily, all of $F_s$ becomes infected. Percolation time {#sec:time} ================ In this section, we exploit the machinery developed in Section \[sec:main\] to prove Theorem \[thm:smallestTime\]. In particular, by tightening our analysis of the height of $\IW(v)$, we will show that the bootstrap percolation process started from the set $A$, defined in , terminates after at most $(d+2)n^2+n$ time steps. The case $d=1$ is trivial; to minimise the percolation time we simply place one infected vertex at $\lceil n/2 \rceil$. The case $d=2$ is an interesting puzzle. As for the upper bound on $m_2(n)$, we can clearly see that a diagonal percolates $[n]^2$ in $n-1$ steps. For the lower bound, we observe that, by the perimeter argument, at least one of the following two neighbouring vertices: $( \lceil n/2 \rceil, \lceil n/2 \rceil )$ and $( \lfloor n/2 \rfloor, \lceil (n+1)/2 \rceil )$, must be initially healthy. (For $n$ even these two vertices are neighbours in the central $2 \times 2$ subsquare, while for $n$ odd the former one is in the very centre of the grid, with the latter one being its neighbour on the left.) Now, we keep applying the perimeter argument: every time a vertex becomes infected, it must be by virtue of exactly $2$ infected neighbours. Moreover, it is an immediate observation that the perimeter of the infected set would also decrease if two neighbouring vertices became infected at the same time step. Hence, only the corner vertices can become infected without having any of their neighbours still healthy after their infection. This means that, for any percolating set of size $n$, we can construct a path of neighbouring vertices, starting at either $( \lceil n/2 \rceil, \lceil n/2 \rceil )$ or $( \lfloor n/2 \rfloor, \lceil (n+1)/2 \rceil )$ and finishing in one of the corners of the grid, such that the consecutive vertices of the path become infected at strictly later time steps. All such paths have length at least $n-1$: for $n$ even we could take a path from $( \lceil n/2 \rceil, \lceil n/2 \rceil )$ to $(1,1)$, while for $n$ odd from $( \lfloor n/2 \rfloor, \lceil (n+1)/2 \rceil )$ to $(1,1)$. This gives us the desired lower bound on $m_2(n)$. (We remark that Benevides and Przykucki [@maxtimeMinsize] showed that the maximum percolation time for a set of size $n$ in $[n]^2$ is equal to the integer nearest to $(5n^2 - 2n)/8$). Hence, let us assume that $d \geq 3$. Here, the lower bound follows by an identical argument to the one we used for $d=2$. Consider the vertices $$(\lceil n/2 \rceil, \lceil n/2 \rceil, \ldots, \lceil n/2 \rceil) \mbox{ and } (\lceil n/2 \rceil+1, \lceil n/2 \rceil, \ldots, \lceil n/2 \rceil),$$ and observe that at least one of them has to be initialy healthy by the perimeter argument. Then, every path from one of these vertices to a corner of the grid has length $dn/2+O(1)$, meaning that $m_d(n) \geq dn/2+O(1)$ as claimed. The upper bound on $m_d(n)$ in Theorem \[thm:smallestTime\] follows immediately from the next lemma, which sharpens the analysis in Lemma \[lem:noCycles\]. \[lem:quadraticHeight\] Let $v \in F_s$ and let $u^1 = v,u_2,\ldots,u^m$ be a directed path in $\IW(v)$, with $u^{i}\in \Pre(u^{i-1})$ for all $2\leq i\leq m$. Then $m \leq (d+2)n^2+n+1$. Given $u \in F_s$, let $h(u) = \sum_{i=1}^d u_i^2$ be the sum of squares of the coordinates of $u$. The idea of the proof is to show that long paths in $\IW(v)$, corresponding to large values of $m$, result in very small values of $h$; we want to show that $m > (d+2)n^2+n+1$ would give $h(u^m) < 0$, which is a clear contradiction. For notational convenience, we shall denote $t_i = t_{u^i}$. Clearly $|t_m-t_1 | \leq n$, since $u^1 \in F_s$, and $ u^m \in F_s \cup V_{(s-1)n} \cup V_{sn}$. Thus, we can find a subset $I \subset \{2,3,\ldots,m\}$ with $|I| \geq m-1-n$ and $|I|$ even, such that we can group the elements of $I$ into pairs $$(i^1, j^1), (i^2, j^2), \ldots, (i^{|I|/2}, j^{|I|/2}),$$ with the following property: for all $1 \leq k \leq |I|/2$, we have 1. $t_{i^k} = t_{j^k} - 1$, 2. $t_{i^k} = t_{i^k-1} - 1$ and $t_{j^k} = t_{j^k-1} + 1$. In other words, all but at most $n$ elements of the subpath $u^2,\ldots,u^m$ can be partitioned into pairs $(u^i, u^j)$ such that $u^i$ lies one level below $u^{i-1}$, as well as one level below $u^j$, which in turn lies one level above $u^{j-1}$ (where the level of a vertex $u$ is equal to $t_u$, see Figure \[fig:longPath\]). (0,1) – (10,1); (0,0) – (10,0); plot \[smooth\] coordinates [(0,0.5) (1,-0.5) (2,1.5) (3,1) ]{}; plot \[smooth\] coordinates [(4,0) (5,0.5) (6,-0.25) (7,0) ]{}; plot \[smooth\] coordinates [(8,1) (9,0.5) (10,2) ]{}; (3,1) – (4,0); (7,0) – (8,1); at (-1.5,0) [$t_{i} = t_{j-1}$]{}; at (-1.5,1) [$t_{j} = t_{i-1}$]{}; (3,1) circle (3pt); (4,0) circle (3pt); (7,0) circle (3pt); (8,1) circle (3pt); (0,0.5) circle (3pt); (10,2) circle (3pt); at (0,-1) [$u^{1} = v$]{}; at (3,-1) [$u^{i-1}$]{}; at (4,-1) [$u^{i}$]{}; at (7,-1) [$u^{j-1}$]{}; at (8,-1) [$u^{j}$]{}; at (10,-1) [$u^{m}$]{}; By a reasoning analogous to the one in the proof of Lemma \[lem:noCycles\] and by the convexity of $x^2$, we have $$h(u^{i-1}) - h(u^i) \geq (t_{i-1}+1)^2 - t_{i-1}^2 = 2 t_{i-1} + 1.$$ On the other hand, we have $$h(u^{j}) - h(u^{j-1}) \leq (t_{j-1}+1)^2 - (t_{j-1})^2 = 2 t_{j-1} + 1.$$ However, by the properties of our pairs, we have $$2 t_{i-1} + 1 = 2 (t_{i}+1) + 1 = 2 t_{j} + 1 = 2 (t_{j-1}+1) + 1 = 2 t_{j-1} + 3.$$ Hence, the sum of the changes in the value of $h$ as we move from $u^{i-1}$ to $u^i$, and from $u^{j-1}$ to $u^j$ (in an arbitrary order), is at most $-2$. We clearly have $h(u^1) \leq dn^2$, and through the at most $n$ unpaired moves we increase the value of $h$ by at most $n(n^2-(n-1)^2) < 2n^2$. Therefore we must have $|I|/2 \leq (d+2)n^2/2$, which gives $m \leq (d+2)n^2+n+1$. This concludes the proof of the lemma. Theorem \[thm:smallestTime\] now follows immediately, as the label $u$ of any vertex of $\IW(v)$ becomes infected at most one step after all the vertices in $\Pre(u)$ are infected. The height of $\IW(v)$, being bounded by $ (d+2)n^2+n$, implies the desired bound on the percolation time. In fact, numerical simulations suggest that, for $d \geq 3$, the percolation time of the process started from $A$ (as given in ) grows quadratically in $n$. For $d=3$ the process terminates after $n^2/2-n+O(1)$ time steps, for $d=4$ it lasts $2n^2/3-2n/3+O(1)$ steps, and for $d=5$ infection takes $n^2-3n+O(1)$ steps. We do not believe that these exact sets $A$ minimise percolation time of a set of size $n^{d-1}$ in $[n]^d$; for example, taking initially infected sets $A' = A'_d = \bigcup_{i=1}^{d} V_{in - \lfloor n/2 \rfloor}$ appears to lead to strictly smaller coefficients of $n^2$. However, motivated by Theorem \[thm:smallestTime\] and the results of our simulations, we expect the answer to the following question to be positive. Is $m_d(n) = \Theta(n^2)$ for all $d \geq 3$? One could also ask about $m({\mathbb{T}}_n^d)$, the size of the smallest percolating sets in $d$-neighbour bootstrap percolation on ${\mathbb{T}}_n^d$, the $d$-dimensional torus of size $n$. It is known that $m({\mathbb{T}}_n^2) = n-1$, but the situation quickly becomes more complicated in higher dimensions. Our result immediately implies that $m({\mathbb{T}}_n^d) \leq n^{d-1}$, but this bound is not sharp. For example, for $d=3$ we could infect an $[n-1]^3$ cube using $(n-1)^2$ initially infected vertices, and then use the boundary conditions of the torus to infect $$([n] \times [n-1] \times [n-1]) \cup ([n-1] \times [n] \times [n-1]) \cup ([n-1] \times [n-1] \times [n])$$ with only three additional initially infected vertices. It is easy to see that this set percolates the torus, giving us $$m({\mathbb{T}}_n^3) \leq (n-1)^{2}+3 = n^2-2n+4 < n^2$$ for all $n \geq 3$. What is the value of $m({\mathbb{T}}_n^d)$ for $d \geq 3$? **Acknowledgement** We would like to thank Gabor Pete for helpful guidance concerning earlier results on the smallest percolating sets, and Ellen Harrison for help in preparing this manuscript. [10]{} M. Aizenman and J. Lebowitz. Metastability effects in bootstrap percolation. , 21:3801–3813, 1988. P. Balister, B. Bollob[á]{}s, R. Johnson, and M. Walters. Random majority percolation. , 36:315–340, 2010. J. Balogh, B. Bollob[á]{}s, H. Duminil-Copin, and R. Morris. 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Sharp metastability threshold for two-dimensional bootstrap percolation. , 125:195–224, 2003. H. Huang and C. Lee. Deterministic bootstrap percolation in high dimensional grids. Available at <https://arxiv.org/abs/1308.6791>. R. Morris. Minimal percolating sets in bootstrap percolation. , 16:1–20, 2009. N. Morrison and J.A. Noel. Extremal bounds for bootstrap percolation inthehypercube. , 156:61–84, 2018. G. Pete. Personal communication. G. Pete. Disease processes and bootstrap percolation. Master’s thesis, Bolyai Institute, József Attila University, Szeged, 1997. [^1]: School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom. E-mail: `m.j.przykucki@bham.ac.uk, tomshelton148@aol.com.` [^2]: Supported by the EPSRC grant EP/P026729/1.

--- abstract: 'It has been proved that almost all $n$-bit Boolean functions have [*exact classical query complexity*]{} $n$. However, the situation seemed to be very different when we deal with [*exact quantum query complexity*]{}. In this paper, we prove that almost all $n$-bit Boolean functions can be computed by an exact quantum algorithm with less than $n$ queries. More exactly, we prove that $\mbox{AND}_n$ is the only $n$-bit Boolean function, up to isomorphism, that requires $n$ queries.' address: | $^{1}$Faculty of Informatics, Masaryk University, Brno 60200, Czech Republic\ $^2$ Faculty of Computing, University of Latvia,Rīga, LV-1586, Latvia\ $^3$ School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA\ author: - 'Andris Ambainis $^{2,3}$' - 'Jozef Gruska$^{1}$' - 'Shenggen Zheng$^{1,}$' title: Exact quantum algorithms have advantage for almost all Boolean functions --- Quantum computing,Quantum query complexity ,Boolean function ,Symmetric Boolean function ,Monotone Boolean function ,Read-once Boolean function Introduction ============ [*Quantum query complexity*]{} is the quantum generalization of classical [*decision tree complexity*]{}. In this complexity model, an algorithm is charged for “queries" to the input bits, while any intermediate computation is considered as free (see [@BdW02]). For many functions one can obtain large quantum speed-ups in this model in the case algorithms are allowed a constant small probability of error (bounded error). As the most famous example, Grover’s algorithm [@Gro96] computes the $n$-bit $\mbox{OR}$ function with $O(\sqrt {n})$ queries in the bounded error mode, while any classical (also exact quantum) algorithm needs $\Omega(n)$ queries. More such cases of polynomial speed-ups are known, see[@Amb07; @Bel12; @DHHM06]. For [*partial functions*]{}, even an exponential speed-up is possible, in case quantum resources are used, see [@Shor97; @Sim97]. In the bounded-error setting, quantum complexity is now relatively well understood. The model of [*exact quantum query complexity*]{}, where the algorithms must output the correct answer with certainty for every input, seems to be more intriguing. It is much more difficult to come up with exact quantum algorithms that outperform, concerning number of queries, classical exact algorithms. Though for partial functions exact quantum algorithms with exponential speed-up are known (for instance in [@AmYa11; @BH97; @DJ92; @GQZ14; @ZQ14; @GQZ14b; @Zhg13]), the results for total functions have been much less spectacular: the best known quantum speed-up was just by a factor of 2 for many years [@CEMM98; @FGGS98]. Recently, in a breakthrough result, Ambainis [@Amb13] has presented the first example of a Boolean function $f:\{0,1\}^n\to \{0,1\}$ for which exact quantum algorithms have superlinear advantage over exact classical algorithms. In exact classical query complexity ([*decision tree complexity*]{}, [*deterministic query complexity*]{}) model, almost all $n$-bit Boolean functions require $n$ queries [@BdW02]. However, the situation seemed very different for the case of exact quantum complexity. Montanaro et al. [@MJM11] proved that $\mbox{AND}_3$ is the only $3$-bit Boolean function, up to isomorphism, that requires 3 queries and using the semidefinite programming approach, they numerically[^1] demonstrated that all $4$-bit Boolean functions, with the exception of functions isomorphic to the $\mbox{AND}_4$ function, have exact quantum query algorithms using at most 3 queries. They also listed their numerical results for all symmetric Boolean functions on 5 and 6 bits, up to isomorphism. In 1998, Beals at al. [@BBC+98] proved, for any $n$, that $\mbox{AND}_n$ has exact quantum complexity $n$. Since that time it was an interesting problem whether $\mbox{AND}_n$ is the only $n$-bit Boolean function, up to isomorphism, that has exact quantum complexity $n$. In this paper we approve that this is indeed the case. As a corollary we get that almost all $n$-bit Boolean functions have exact quantum complexity less than $n$. We prove our main results in four stages. In the first one we give the proof for symmetric Boolean functions, in the second one for monotone Boolean functions and in the third one for the case of read-once Boolean functions. On this basis we prove in the fourth stage the general case. In all four cases proofs used quite different approaches. They are expected to be of a broader interest since all these special classes of Boolean functions are of broad interest. The paper is organized as follows. In Section 2 we introduce some notation concerning Boolean function and query complexity. In Section 3 we investigate symmetric Boolean functions. In section 4 we investigate monotone Boolean functions. In section 5 we investigate read-once Boolean functions. In Section 6 we prove our main result. Finally, Section 7 contains a conclusion. Preliminaries ============= We introduce some basic needed notation in this section. See also [@Gru99; @NC00] for details on quantum computing and see [@BdW02; @BBC+98; @NS94] for more on query complexity models and [*multilinear polynomials*]{}. Boolean functions ----------------- An $n$-bit Boolean function is a function $f:\{0,1\}^n\to \{0,1\}$. We say $f$ is total if $f$ is defined on all inputs. For an input $x\in\{0,1\}^n$, we use $x_i$ to denote its $i$-th bit, so $x=x_1x_2\cdots x_n$. Denote $[n]=\{1,2,\ldots,n\}$. For $i\in[n]$, we write $$f_{x_i=b}(x)=f(x_1,\ldots,x_{i-1},b,x_{i+1},\ldots,x_n),$$ which is an $(n-1)$ bit Boolean function. For any $i\in[n]$, we have $$\label{Eq-df(x)} f(x)=(1-x_i)f_{x_i=0}(x)+x_if_{x_i=1}(x).$$ We say that two Boolean functions $f$ and $g$ are [*query-isomorphic*]{} (by convenience, isomorphic will mean query-isomorphic in this paper) if they are equal up to negations and permutations of the input variables, and negation of the output variable. This relationship is sometimes known as NPN-equivalence [@MJM11]. We will use the sign $(\neg)$ for a possible negation. For example, $\mbox{AND}((\neg)x_1,x_2)$ can denote $x_1\wedge x_2$ or $\neg x_1\wedge x_2$. We use $|x|$ to denote the Hamming weight of $x$ (its number of 1’s). [**Definition 1:**]{} We call a Boolean function $f:\{0,1\}^n\to \{0,1\}$ symmetric if $f(x)$ depends only on $|x|$. An $n$-bit symmetric Boolean function $f$ can be fully described by a vector $(b_0,b_1,\ldots,b_n)\linebreak[0]\in\{0,1\}^{n+1}$, where $f(x)=b_{|x|}$, i.e. $b_k$ is the value of $f(x)$ for $|x|=k$ [@ZGR97]. For $x,y\in\{0,1\}^n$, we will write $x\preceq y$ if $x_i\leq y_i$ for all $i\in[n]$. We will write $x\prec y$ if $x\preceq y$ and $x\neq y$. [**Definition 2:**]{} We call a Boolean function $f:\{0,1\}^n\to \{0,1\}$ monotone if $f(x)\leq f(y)$ holds whenever $x\preceq y$. Monotonic Boolean functions are precisely those that can be defined by an expression combining the input bits (each of them may appear more than once) using only the operators $\wedge$ and $\vee$ (in particular $\neg$ is forbidden). Monotone Boolean functions have many nice properties. For example they have a unique prime conjunctive normal form (CNF) and a unique prime disjunctive normal form (DNF) in which no negation occurs [@EMG08]. Let $f:\{0,1\}^n\to \{0,1\}$ be a monotone Boolean function, $f$ has a prime CNF $$f(x)=\bigwedge_{I\in C}\bigvee_{i\in I} x_i,$$ where $C$ is the set of some $I\subseteq[n]$. Similarly, $f$ has a prime DNF $$f(x)=\bigvee_{J\in D}\bigwedge_{j\in J} x_j,$$ where $D$ is the set of some $J\subseteq[n]$. [**Definition 3:**]{} A read-once Boolean function is a Boolean function that can be represented by a Boolean formula in which each variable appears exactly once. For example $f(x_1,x_2,x_3)=(x_1\vee x_2)\wedge (\neg x_3)$ is a $3$-bit read-once Boolean function and $f'(x_1,x_2,x_3)=(x_1\vee x_2)\wedge (\neg x_1\vee \neg x_3)$ is not read-once. A Boolean formula over the standard basis $\{\wedge,\vee,\neg \}$ can be represented by a binary tree where each internal node is labeled with $\wedge$ or $\vee$, and each leaf is labeled with a literal, that is, a Boolean variable or its negation. The size of a formula is the number of leaves. [**Definition 4:**]{} The formula size of a Boolean function $f$, denoted $L(f)$, is the size of the smallest formula which computes $f$. A read-once Boolean function is a function $f$ such that $L(f)=n$ and $f$ depends on all of its $n$ variables. Exact query complexity models ----------------------------- An exact classical (deterministic) query algorithm for computing a Boolean function $f:\{0,1\}^n\to \{0,1\}$ can be described by a decision tree. A decision tree $T$ is a rooted binary tree where each internal vertex has exactly two children, each internal vertex is labeled with a variable $x_i$ and each leaf is labeled with a value 0 or 1. $T$ computes a Boolean function $f$ as follows: Start at the root. If this is a leaf then stop and the output of the tree is the value of the leaf. Otherwise, query the variable $x_i$ that labels the root. If $x_i=0$, then recursively evaluate the left subtree, if $x_i=1$ then recursively evaluate the right subtree. The output of the tree is the value of the leaf that is reached at the end of this process. The depth of $T$ is the maximal length of a path from the root to a leaf (i.e. the worst-case number of queries used on any input). The [*exact classical query complexity*]{} (deterministic query complexity, decision tree complexity) $D(f)$ is the minimal depth over all decision trees computing $f$. Let $f:\{0,1\}^n\to \{0,1\}$ be a Boolean function and $x = x_1x_2\cdots x_n$ be an input bit string. Each exact quantum query algorithm for $f$ works in a Hilbert space with some fixed basis, called standard. It starts in a fixed starting state, then performs on it a sequence of transformations $U_1$, $Q$, $U_2$, $Q$, …, $U_t$, $Q$, $U_{t+1}$. Unitary transformations $U_i$ do not depend on the input bits, while $Q$, called the [*query transformation*]{}, does, in the following way. Each of the basis states corresponds to either one or none of the input bits. If the basis state $|\psi\rangle$ corresponds to the $i$-th input bit, then $Q|\psi\rangle=(-1)^{x_i}|\psi\rangle$. If it does not correspond to any input bit, then $Q$ leaves it unchanged: $Q|\psi\rangle=|\psi\rangle$. Finally, the algorithm performs a measurement in the standard basis. Depending on the result of the measurement, the algorithm outputs either 0 or 1 which must be equal to $f(x)$. The [*exact quantum query complexity*]{} $Q_E(f)$ is the minimum number of queries used by any quantum algorithm which computes $f(x)$ exactly for all $x$. Note that if Boolean functions $f$ and $g$ are isomorphic, then $D(f)=D(g)$ and $Q_E(f)=Q_E(g)$. According to Eq. (\[Eq-df(x)\]), if we query $x_i$ first, suppose that $x_i=b$, then we can compute $f_{x_i=b}(x)$ further. Therefore, for any $i\in[n]$, we have $$\label{Eq-n-1ton} Q_E(f)\leq \max\{Q_E(f_{x_i=0}),Q_E(f_{x_i=1})\}+1.$$ Some special functions and their exact quantum query complexity --------------------------------------------------------------- Symmetric, monotone and read-once Boolean functions were well studied in query complexity [@BdW02]. The well known Grover’s algorithm [@Gro96] computes $\mbox{OR}_n$, which is symmetric, monotone and read-once. Read-once functions are also well investigated [@BS04; @SW86; @San95]. Some symmetric functions and their exact quantum query complexity that we will refer to in this paper are as follows: 1. $\mbox{OR}_n(x)=1$ iff $|x|\geq 1$. $Q_E(\mbox{OR}_n)=n$ [@BBC+98]. 2. $\mbox{AND}_n(x)=1$ iff $|x|=n$. $Q_E(\mbox{AND}_n)=n$ [@BBC+98]. 3. $\mbox{PARITY}_n(x)=1$ iff $|x|$ is odd. $Q_E(\mbox{PARITY}_n)=\lceil\frac{n}{2}\rceil$ [@CEMM98; @FGGS98]. 4. $\mbox{EXACT}_n^{k}(x)=1$ iff $|x|=k$. $Q_E(\mbox{EXACT}_n^{k})=\max\{k,n-k\}$ [@AISJ13]. 5. $\mbox{Th}_{n}^{k}(x)=1$ iff $|x|\geq k$. $Q_E(\mbox{Th}_n^{k})=\max\{k,n-k+1\}$ [@AISJ13]. $\mbox{OR}_n$ is isomorphic to $\mbox{AND}_n$ since $$\neg\mbox{OR}_n(\neg x_1,\neg x_2,\ldots,\neg x_n)=\mbox{AND}_n(x_1,x_2,\ldots,x_n).$$ Some other functions and their exact quantum query complexity that we will refer to in this paper are as follows: 1. $\mbox{NAE}_{n}(x)=1$ iff there exist $i,j$ such that $x_i\neq x_j$. $Q_E(\mbox{NAE}_{n})\leq n-1$. 2. $f(x_1,x_2,x_3)=x_1\wedge(x_2\vee x_3)$. Its exact quantum query complexity is 2 [@MJM11]. It is easy to prove that $Q_E(\mbox{NAE}_{n})\leq n-1$ since $$\mbox{NAE}_{n}(x_1,\ldots,x_n)=(x_1\oplus x_2)\vee(x_2\oplus x_3)\cdots\vee(x_{n-1}\oplus x_n).$$ Multilinear polynomials ----------------------- Every Boolean function $f:\{0,1\}^n\to \{0,1\}$ has a unique representation as an $n$-variate multilinear polynomial over the reals, i.e., there exist real coefficients $a_S$ such that $$f(x_1,\ldots,x_n)=\sum_{S\subseteq [n]} a_S \prod_{i\in S} x_i.$$ The degree of $f$ is the degree of its largest monomial: $deg(f)=\max\{|S|:a_S\neq 0\}$. For example, $\mbox{AND}_2(x_1,x_2)=x_1\cdot x_2$ and $\mbox{OR}_2(x_1,x_2)=x_1+x_2-x_1\cdot x_2$. $\textrm{deg}(f)$ gives a lower bound on $D(f)$. Indeed, it holds [*[@BdW02]*]{}\[D(f)-geq-Deg(f)\] $D(f)\geq \textrm{deg}(f)$. Symmetric Boolean functions =========================== Let $f:\{0,1\}^n\to \{0,1\}$ be a symmetric Boolean function. $Q_E(f)=n$ iff $f$ is isomorphic to $\mbox{AND}_n$. If $f$ is isomorphic to $\mbox{AND}_n$, then $Q_E(f)=n$ [@BBC+98]. An $n$-bit symmetric Boolean function can be fully described by a vector $(b_0,b_1,\ldots,b_n)\in\{0,1\}^{n+1}$, where $f(x)=b_{|x|}$, i.e. $b_k$ is the value of $f(x)$ for $|x|=k$. $(b_0,b_1,b_2,b_3)$ Type of function Query complexity --------------------- ------------------------------------ ------------------ 0 0 0 0 Constant function 0 0 0 0 1 $\mbox{AND}_3$ 3 0 0 1 0 $\mbox{EXACT}_3^{2}$ 2 0 0 1 1 $\mbox{Th}_3^{2}$ 2 0 1 0 0 $\mbox{EXACT}_3^{1}$ 2 0 1 0 1 $\mbox{PARITY}_3$ 2 0 1 1 0 $\mbox{NAE}_3$ 2 0 1 1 1 Isomorphic to $\mbox{AND}_3$ 3 1 0 0 0 Isomorphic to $\mbox{AND}_3$ 3 1 0 0 1 Isomorphic to $\mbox{NAE}_3$ 2 1 0 1 0 Isomorphic to $\mbox{PARITY}_3$ 2 1 0 1 1 Isomorphic to $\mbox{EXACT}_3^{1}$ 2 1 1 0 0 Isomorphic to $\mbox{Th}_3^{2}$ 2 1 1 0 1 Isomorphic to $\mbox{EXACT}_3^{2}$ 2 1 1 1 0 Isomorphic to $\mbox{AND}_3$ 3 1 1 1 1 Constant function 0 : Exact quantum query complexity for $3$-bit symmetric functions.[]{data-label="T1"} Table \[T1\] contains all 3-bit Boolean functions and their exact quantum query complexity. Four 3-bit Boolean functions that achieve 3 queries are those that can be described by one of the following vectors: $(0,0,0,1),(0,1,1,1),(1,0,0,0), (1,1,1,0)$. They are isomorphic to $\mbox{AND}_3$. We claim that only $n$-bit Boolean functions that can be described by one of the following vectors $(0,\ldots,0,1),\linebreak[0](0,1,\ldots,1),\linebreak[0](1,0,\ldots,0),\linebreak[0](1,\ldots,1,0)$, which are isomorphisms of $\mbox{AND}_n$, that can achieve $n$ queries. We prove this claim by an induction on $n$ as follows: [**BASIS**]{}: The result holds clearly for $n=3$. [**INDUCTION**]{}: Suppose the result holds for $n=k$ ($\geq 3$). We will prove that the result holds also for $n=k+1$. We use vector $(b_0,b_1,\ldots,b_k,b_{k+1})$ to describe the function $f(x_1,\cdots,x_k,x_{k+1})$. Since $$Q_E(f)\leq \max\{Q_E(f_{x_1=0}),Q_E(f_{x_1=1})\}+1,$$ we just need to consider the case that at least one of the functions $f_{x_1=0}$ and $f_{x_1=1}$ is isomorphic to $\mbox{AND}_k$. For other cases we have $Q_E(f)<k+1$. $b_0b_1\ldots,b_k,b_{k+1}$ Type of function Query complexity ---------------------------- ---------------------------------------- ------------------ $(0,|0,\ldots,0,1)$ $\mbox{AND}_{k+1}$ $k+1$ $(0,|0,1,\ldots,1)$ $\mbox{Th}_{k+1}^{2}$ $k$ $(0,|1,0,\ldots,0)$ $\mbox{EXACT}_{k+1}^1$ $k$ $(0,|1,\ldots,1,0)$ $\mbox{NAE}_{k+1}$ $<k+1$ $(1,|0,\ldots,0,1)$ Isomorphic to $\mbox{NAE}_{k+1}$ $<k+1$ $(1,|0,1,\ldots,1)$ Isomorphic to $\mbox{EXACT}_{k+1}^1$ $k$ $(1,|1,0,\ldots,0)$ Isomorphic to $\mbox{Th}_{k+1}^{2}$ $k$ $(1,|1,\ldots,1,0)$ Isomorphic to $\mbox{AND}_{k+1}$ $k+1$ $(0,\ldots,0,1,|0) $ $\mbox{EXACT}_{k+1}^{k}$ $k$ $(0,1,\ldots,1,|0) $ $\mbox{NAE}_{k+1}$ $<k+1$ $(1,0,\ldots,0,|0)$ Isomorphic to $\mbox{AND}_{k+1}$ $k+1$ $(1,\ldots,1,0,|0) $ Isomorphic to $\mbox{Th}_{k+1}^{k}$ $k$ $(0,\ldots,0,1,|1)$ $\mbox{Th}_{k+1}^{k}$ $k$ $(0,1,\ldots,1,|1) $ Isomorphic to $\mbox{AND}_{k+1}$ $k+1$ $(1,0,\ldots,0,|1) $ Isomorphic to $\mbox{NAE}_{k+1}$ $<k+1$ $(1,\ldots,1,0,|1)$ Isomorphic to $\mbox{EXACT}_{k+1}^{k}$ $k$ : Exact quantum query complexity for $(k+1)$-bit symmetric Boolean functions. []{data-label="T2"} There are three cases we have to consider according to the value of $b$. [**Case 1**]{} $b=(0,\ldots,0,1)$. In this case $f=\mbox{AND}_{k+1}$. [**Case 2**]{} $b=(1,0,\ldots,0)$. In this case $f$ is isomorphic to $\mbox{AND}_{k+1}$. [**Case 3**]{} Otherwise, $f_{x_1=0}$ can be described by the vector $(b_0,b_1,\ldots,b_{k})$ and $f_{x_1=1}$ can be described by the vector $(b_1,\ldots,b_{k},b_{k+1})$. Thus we just need to consider Boolean functions that can be described by vector $b=(b_0,b_1,\ldots,b_k,b_{k+1})$ such that one of the following vectors $$(\overbrace{0,\ldots,0}^{k},1),\linebreak[0](0,\overbrace{1,\ldots,1}^{k}),\linebreak[0](1,\overbrace{0,\ldots,0}^k),(\overbrace{1,\ldots,1}^k,0)$$ is its prefix or suffix[^2]. There are 16 such Boolean functions and their query complexity are listed in Table \[T2\]. According to Table \[T2\], only $(k+1)$-bit Boolean functions which are isomorphic to $\mbox{AND}_{k+1}$ require $k+1$ queries. Thus, the theorem has been proved. It is mentioned in [@MJM11; @Aar03] that all non-constant $n$-bit symmetric Boolean functions have exact classical complexity $n$. We give now a rigorous proof of that. If $f:\{0,1\}^n\to \{0,1\}$ is a non-constant symmetric function, then $D(f)=n$. Suppose $f$ can be described by the vector $(b_0,b_1,\ldots,b_n)\in\{0,1\}^{n+1}$. Since $f$ is non-constant, there exists a $k\in[n]$ such that $b_{k-1}\neq b_{k}$. If the first $k-1$ queries return $x_i=1$ and the next $n-k$ queries return $x_i=0$, then we will need to query the last variable as well. Monotone Boolean functions ========================== Let $f:\{0,1\}^n\to \{0,1\}$ be a monotone Boolean function. $Q_E(f)=n$ iff $f$ is isomorphic to $\mbox{AND}_n$. Obviously, $\mbox{AND}_n(x)$ and $\mbox{OR}_n(x)$ are the only two $n$-bit monotone Boolean functions that are isomorphic to $\mbox{AND}_n(x)$. If $f$ is isomorphic to $\mbox{AND}_n(x)$, then $Q_E(f)=n$ [@BBC+98]. We prove the other direction by an induction on $n$. [**BASIS**]{}: Case $n=2$, $\mbox{AND}_2(x_1,x_2)$ is the only $2$-bit function, up to isomorphism, that requires 2 queries. Therefore the result holds for $n=2$. [**INDUCTION**]{}: Suppose the result holds for all $n\leq k$, we prove that the result holds also for $n=k+1$ in the following way. For any $i\in[k+1]$, if $Q_E(f_{x_i=0})<k$ and $Q_E(f_{x_i=1})<k$, then $Q_E(f)\leq \max\{Q_E(f_{x_i=0}),\linebreak[0]Q_E(f_{x_i=1})\}+1<k+1$. Therefore, we need to consider only the case that at least one of functions $f_{x_i=0}$ and $f_{x_i=1}$ requires $k$ quires. There are two such cases: [**Case 1:**]{} $Q_E(f_{x_1=1})=k$. According to the assumption, $f_{x_1=1}$ is isomorphic to $\mbox{AND}_k$. There are now two subcases to consider: [**Case 1a:**]{} $f_{x_1=1}(x)=\mbox{OR}_k(x_2,\cdots,x_{k+1})=\mbox{OR}_k(x_{-1})$ (For convenience, we write $x_{-i}=x_1,\ldots, x_{i-1},x_{i+1},\linebreak[0]\ldots x_{k+1}$). Let us consider the CNF of $f$: $$f(x)=\bigwedge_{I\in C}\bigvee_{i\in I} x_i=\left(\bigwedge_{I\in C,1\in I}\bigvee_{i\in I} x_i\right)\wedge\left(\bigwedge_{I\in C,1\not\in I}\bigvee_{i\in I} x_i\right) .$$ Therefore, $$f(x)=(x_1\vee g_1(x_{-1}))\wedge \mbox{OR}_k(x_{-1}),$$ where $x_1\vee g_1(x_{-1})=\left(\bigwedge_{I\in C,1\in I}\bigvee_{i\in I} x_i\right)$ and $g_1$ is also a monotone function. So we have $f(x)=1$ for any $x$ such that $10\cdots 0\prec x$ and $f(x)=0$ for any $x$ such that $x\preceq 10\cdots 0 $. Let us consider now two subcases. Namely $f_{x_2=1}$ and $f_{x_2=0}$. Since $10\cdots0\preceq 10\cdots 0$, we have $f(10\cdots0)\linebreak[0]=0$ and $f_{x_2=0}(x)\neq \mbox{OR}_k(x_{-2})$. Since $10\cdots 0\prec 1010\cdots 0$, we have $f(1010\cdots 0)\linebreak[0]=1$ and $f_{x_2=0}(x)\neq \mbox{AND}_k(x_{-2})$. Now we have $Q_E(f_{x_2=0})<k$ and therefore $Q_E(f_{x_2=1})\linebreak[0]=k$. Since $10\cdots 0\prec 110\cdots 0$, we have $f(110\cdots 0)=1$ and $f_{x_2=1}(x)\neq \mbox{AND}_k(x_{-2})$. Therefore, $f_{x_2=1}(x)=\mbox{OR}_k(x_{-2})$. Using a similar argument, we can prove that for any $i\geq 2$, $f_{x_i=1}(x)=OR_k(x_{-i})$. Hence, for any $i\in[k+1]$, we have $$f(x)=(x_i\vee g_i(x_{-i}))\wedge \mbox{OR}_k(x_{-i}).$$ So $f(x)=1$ for any $x$ such that $y\prec x$ and $f(x)=0$ for any $x$ such that $x\preceq y$, where $y_i=1$ and $y_j=0$ for any $j\neq i$. It is not hard to see that in this case $f(x)=\mbox{Th}_{k+1}^2(x)$ and therefore $Q_E(f)=k$. [**Case 1b:**]{} $f_{x_1=1}(x)=\mbox{AND}_k(x_{-1})$. Let us consider the CNF of $f$. We have, $$f(x)=(x_1\vee g'(x_{-1}))\wedge \mbox{AND}_k(x_{-1}),$$ where $g'(x_{-1})$ is also a monotone Boolean function. If $g'$ is a constant function and $g'(x_{-1})=0$, we have $f(x)=\mbox{AND}_{k+1}(x_1x_2,\cdots,\linebreak[0] x_{k+1})$ and $Q_E(f)=k+1$. Otherwise, $\mbox{AND}_k(x_{-1})\leq g'(x_{-1})$, then $f(x)=\mbox{AND}_k(x_{-1})$ and therefore $Q_E(f)=k$. [**Case 2:**]{} $Q_E(f_{x_1=0})=k$. There are again two subcases: [**Case 2a:**]{} $f_{x_1=0}(x)=\mbox{OR}_k(x_{-1})$. Let us consider the DNF of $f$: $$f(x)=\bigvee_{I\in D}\bigwedge_{i\in I} x_i=\left(\bigvee_{I\in D, 1\in I} \bigwedge_{i\in I} x_i \right)\vee\left(\bigvee_{I\in D, 1\not\in I}\bigwedge_{i\in I}x_i\right).$$ We have $$f(x)=(x_1\wedge h'(x_{-1}))\vee \mbox{OR}_{n-1}(x_{-1}),$$ where $h'$ is a monotone Boolean function. If $h'$ is a constant function and $h'(x_{-1})=1$, then $f(x)=\mbox{OR}_{k+1}(x_1x_2,\cdots,x_{k+1})$ and $Q_E(f)=k+1$. Otherwise $h'(x_{-1})\leq OR_k(x_{-1})$ and therefore $f(x)=\mbox{OR}_k(x_{-1})$ and $Q_E(f)=k$. [**Case 2b:**]{} $f_{x_1=0}(x)=\mbox{AND}_k(x_{-1})$. Let us consider the DNF of $f$. It has the form $$f(x)=(x_1\wedge h_1(x_{-1}))\vee \mbox{AND}_k(x_{-1}),$$ where $h_1(x_{-1})$ is also a monotone Boolean function. Therefore $f(x)=1$ for any $x$ such that $01\cdots 1\preceq x$ and $f(x)=0$ for any $x$ such that $x\prec 01\cdots 1 $. Let us consider now two subcases: $f_{x_2=1}$ and $f_{x_2=0}$. Since $0110\cdots 0\prec 01\cdots 1$, we have $f(0110\linebreak[0]\cdots 0)\linebreak[0]=0$ and $f_{x_2=1}(x)\neq \mbox{OR}_k(x_{-2})$. Since $01\cdots 1\preceq 01\cdots 1$, we have $f(01\cdots 1)\linebreak[0]=1$ and $f_{x_2=1}(x)\neq \mbox{AND}_k(x_{-2})$. Therefore we have $Q_E(f_{x_2=1})<k$ and $Q_E(f_{x_2=0})\linebreak[0]=k$. Since $0010\cdots 0\prec 01\cdots 1$, we have $f(0010\cdots 0)\linebreak[0]=0$ and $f_{x_2=0}(x)\neq \mbox{OR}_k(x_{-2})$. Therefore, $f_{x_2=0}(x)=\mbox{AND}_k(x_{-2})$. Using a similar argument, we can prove that for any $i\geq 2$, $f_{x_i=0}(x)=\mbox{AND}_k(x_{-i})$. Hence, for any $i\in[k+1]$, we have $$f(x)=(x_i\wedge h_i(x_{-i}))\vee \mbox{AND}_k(x_{-i}).$$ Therefore $f(x)=1$ for any $x$ such that $y\preceq x$ and $f(x)=0$ for any $x$ such that $x\prec y$, where $y_i=0$ and $y_j=1$ for any $j\neq i$. It is now not hard to show that $f(x)=\mbox{Th}_{k+1}^{k}$ and $Q_E(f)=k$. Therefore, the theorem has been proved. Read-once Boolean functions =========================== \[fsize\] If $f:\{0,1\}^n\to \{0,1\}$ is a read-once Boolean function, then $Q_E(f)=n$ iff $f$ is isomorphic to $\mbox{AND}_n$. If $f$ is isomorphic to $\mbox{AND}_n$, then $Q_E(f)=n$ [@BBC+98]. We prove the other direction as follows. Since $f$ is a read-once Boolean function, $f$ depends on all $n$ variables and $L(f)=n$, i.e each $(\neg) x_i$ labels once and only once a leaf variable, where $(\neg)$ denotes a possible negation. We prove the result by an induction. [**BASIS**]{}: $\mbox{AND}_3(x_1,x_2,x_3)$ is the only $3$-bit Boolean function, up to isomorphism, that requires 3 quantum queries [@MJM11]. Therefore the result holds for $n=3$. [**INDUCTION**]{}: We will suppose the result holds for all $n\leq k$ ($k\geq 3$) and we will prove that the result holds also for all $n\leq k+1$. Suppose the root of a formula $F$ is labeled with $\wedge$. Without loss of generality, we assume that there exist Boolean functions $g:\{0,1\}^p\to \{0,1\}$ and $h:\{0,1\}^q\to \{0,1\}$ such that $f(x)=g(y)\wedge h(z)$ and $p+q=k+1$, where $x=yz$. Since $f$ depends on all $k+1$ variables and $L(f)=k+1$, we have $L(g)=p$ and $L(h)=q$, where $g$ depends on all $p$ variables and $h$ depends on all $q$ variables. If $Q_E(g)<p$ or $Q_E(h)<q$, then $Q_E(f)\leq Q_E(g)+Q_E(h)<k+1$. Now suppose $Q_E(g)=p$ and $Q_E(h)=q$. According to the assumption, $g$ is isomorphic to $\mbox{AND}_p$ and $h$ is isomorphic to $\mbox{AND}_q$. There are therefore the following four cases to consider. [**Case 1:**]{} $g(y)=\mbox{AND}_p\left((\neg)x_1,\ldots,(\neg)x_p\right)$ and $h(z)=\mbox{AND}_q\left((\neg)x_{p+1},\linebreak[0]\ldots,\linebreak[0](\neg)x_{k+1}\right)$. Then $f$ is isomorphic to $\mbox{AND}_{k+1}$ and therefore $Q_E(f)=k+1$. [**Case 2:**]{} $g(y)=\mbox{OR}_p\left((\neg)x_1,\ldots,(\neg)x_p\right)$ and $h(z)=\mbox{OR}_q\left((\neg)x_{p+1},\ldots,\linebreak[0](\neg)x_{k+1}\right)$. Therefore $$f(x)=\mbox{OR}_p\left((\neg)x_1,\ldots,(\neg)x_p\right)\wedge \mbox{OR}_q\left((\neg)x_{p+1},\ldots,(\neg)x_{k+1}\right).$$ Without loss of generality, we suppose that $f(x)=\mbox{OR}_p\left(x_1,\ldots,x_p\right)\wedge \mbox{OR}_q\left(x_{p+1},\ldots,x_{k+1}\right)$. Since $p+k-p+1=k+1>3$, we have $p\geq 2$ or $k-p+1\geq 2$. Without loss of generality, we assume that $k-p+1\geq 2$. Let us query $x_2$ to $x_{k-1}$ first. 1. If $x_i=1$ for some $2\leq i\leq p$ and $x_j=1$ for some $p+1\leq j\leq k-1$, then $f_{x_2\cdots x_{k-1}}(x)=1$. 2. If $x_i=1$ for some $2\leq i\leq p$ and $x_{p+1}=\cdots=x_{k-1}=0$, then $f_{x_2\cdots x_{k-1}}(x)=\mbox{OR}_2\left(x_{k},x_{k+1}\right)$. 3. If $x_2=\cdots=x_p=0$ and $x_i=1$ for some $p+1\leq i\leq k-1$, then $f_{x_2\cdots x_{k-1}}(x)=x_1$. 4. Otherwise, $x_2=\cdots=x_{k-1}=0$ and therefore $f_{x_2\cdots x_{k-1}}(x)=x_1\wedge(x_k\vee x_{k+1})$ and $Q_E(f_{x_2\cdots x_{k-1}})=2$. Therefore $Q_E(f)\leq k-2+2<k+1$. [**Case 3:**]{} $g(y)=\mbox{AND}_p\left((\neg)x_1,\ldots,(\neg)x_p\right)$ and $h(z)=\mbox{OR}_q\left((\neg)x_{p+1},\ldots,(\neg)x_{k+1}\right)$. Therefore $f(x)=\mbox{AND}_p\left((\neg)x_1,\ldots,(\neg)x_p\right)\wedge \mbox{OR}_q\left((\neg)x_{p+1},\ldots,(\neg)x_{k+1}\right)$. Without loss of generality, we can now suppose that $$f(x)=\mbox{AND}_p\left(x_1,\ldots,x_p\right)\wedge \mbox{OR}_q\left(x_{p+1},\ldots,x_{k+1}\right).$$ If $p=k$, then $f=\mbox{AND}_{k+1}$ and $Q_E(f)=k+1$. Now we consider the case $p<k$. Let us query $x_2$ to $x_{k-1}$ first. 1. If $x_2\cdots x_{p}\neq 1\cdots1$, then $f(x)=0$. 2. If $x_2\cdots x_{p}= 1\cdots1$ and $x_{p+1}\cdots x_{k-1}\neq 0\cdots0$, then $f_{x_2\cdots x_{k-1}}(x)=x_1$. 3. If $x_2\cdots x_{p}= 1\cdots 1$ and $x_{p+1}\cdots x_{k-1}= 0\cdots0$, then $f_{x_2\cdots x_{k-1}}(x)=x_1\wedge(x_k\vee x_{k+1})$ and $Q_E(f_{x_2\cdots x_{k-1}})=2$. Therefore $Q_E(f)\leq k-2+2<k+1$. [**Case 4:**]{} $g(y)=\mbox{OR}_p\left((\neg)x_1,\ldots,(\neg)x_p\right)$ and $h(z)=\mbox{AND}_q\left((\neg)x_{p+1},\ldots,(\neg)x_{k+1}\right)$. This case is analogous to the [**Case 3**]{}. Symmetrically, we can consider the case that the root of the formula $F$ is labeled with $\vee$. In this case, we will need to deal with functions with the same structure of $f(x_1,x_2,x_3)=x_1\vee (x_2\wedge x_3)$, which is isomorphic to $x_1\wedge(x_2\vee x_3)$. We omit the details here. It is mentioned in [@San95] that all $n$-bit read-once Boolean functions have exact classical quantum complexity $n$. We give now a rigorous proof of that: \[Th-readonce\] If $f:\{0,1\}^n\to \{0,1\}$ is a read-once Boolean function, then $D(f)=n$. Let us consider the multilinear polynomial representation of $f$. It is easy to prove by induction that $\textrm{deg}(f)=n$ and there is just one monomial of $f$ of the degree $n$. [**BASIS**]{}: If $n=1$, then $f(x)=(\neg) x_1$. Therefore, $\textrm{deg}(f)=1$. [**INDUCTION**]{}: Suppose the result holds for all $n\leq k$, we will prove the result holds for all $n\leq k+1$. Without loss of generality, let us assume that three exists an $i\in [n]$ such that $$f(x_1,\ldots,x_{k+1})=g(x_1,\ldots,x_i)\wedge h(x_{i+1},\ldots,x_{k+1})$$ or $$f(x_1,\ldots,x_{k+1})=g(x_1,\ldots,x_i)\vee h(x_{i+1},\ldots,x_{k+1}),$$ where $L(g)=i$, $L(h)=k+1-i$, $g$ and $h$ depend on all their variables. According to assumption of the theorem, we have $\mbox{deg}(g)=i$ and $g(x_1,\ldots,x_i)=(\pm)\prod_{j=1}^i(\neg)x_j+p(x_1,\ldots,x_i)$ where $\mbox{deg}(p)<i$, and $\mbox{deg}(h)=k+1-i$ and $h(x_{i+1},\ldots,x_{k+1})=(\pm)\prod_{j={i+1}}^{k+1}(\neg)x_j+q(x_{i+1},\ldots,x_{k+1})$ where $\mbox{deg}(q)<k+1-i$. Since $$f(x_1,\ldots,x_{k+1})=g(x_1,\ldots,x_i)\wedge h(x_{i+1},\ldots,x_{k+1})=g\cdot h$$ and $$f(x_1,\ldots,x_{k+1})=g(x_1,\ldots,x_i)\vee h(x_{i+1},\ldots,x_{k+1})=g+h-g\cdot h.$$ Therefore $\textrm{deg}(f)=k+1$ and there is just one monomial of $f$ of the degree $k+1$. According to Lemma \[D(f)-geq-Deg(f)\], $D(f)\geq \textrm{deg}(f)=n$. Thus, $D(f)=n$. General $n$-bit Boolean functions ================================= In this section we prove our main result. Without explicitly pointed out, $n>3$ in this section. If $f$ is an $n$-bit Boolean function that is isomorphic to $\mbox{AND}_{n}$, then there must exist $b=b_1\ldots b_n\in\{0,1\}^n$ such that every $f_{x_i=b_i}$ is equivalent to $\mbox{AND}_{n-1}$ ($\mbox{OR}_{n-1}$) up to some negations of variables. Moreover $b$ has to be unique. For example, if $f(x)=\mbox{OR}_n(x_1,x_2,\ldots,x_n)$, then we have $f_{x_i=0}(x)=\mbox{OR}_{n-1}(x_1,\ldots,x_{i-1},x_{i+1},\linebreak[0]\ldots,x_n)$ for $i\in[n]$ and $b=0\ldots 0$. For an $n$-bit Boolean function $f$ that has exact quantum query complexity $n$, we prove the following lemma. \[Lm-c7-1\] Suppose that $\mbox{AND}_{n-1}$ is the only (n-1)-bit Boolean function, up to isomorphism, has exact quantum query complexity $n-1$. Let $f:\{0,1\}^n\to \{0,1\}$ be an $n$-bit Boolean function that has exact quantum query complexity $n$. There exists one and only one $b=b_1\ldots b_n\in\{0,1\}^n$ for every $i\in[n]$ such that $f_{x_i=b_i}$ is equivalent to $\mbox{AND}_{n-1}$ ($\mbox{OR}_{n-1}$) up to some negations of the variables. [**Proof:**]{} In order to prove this lemma, we study some properties of exact quantum query complexity of Boolean functions. According to Eq. (\[Eq-n-1ton\]), we have the following lemma: \[C6-l1\] Let $f:\{0,1\}^n\to \{0,1\}$ be a Boolean function. If there exists an $i\in [n]$ such that both $Q_E(f_{x_i=0})<n-1$ and $Q_E(f_{x_i=1})<n-1$, then $Q_E(f)<n$. We know from [@MJM11] that $\mbox{AND}_3$ is the only $3$-bit Boolean function, up to isomorphism, that has exact quantum query complexity 3. For any $4$-bit function $f$, if there exists $i\in[4]$ such that neither $f_{x_i=0}$ nor $f_{x_i=1}$ is isomorphic to $\mbox{AND}_{n-1}$, then $Q_E(f)<4$. \[Lm-both\] Let $f:\{0,1\}^n\to \{0,1\}$ be a Boolean function. If there exists an $i\in [n]$ such that both $f_{x_i=0}$ and $f_{x_i=1}$ are isomorphic to $\mbox{AND}_{n-1}$, then $Q_E(f)<n$. [**Proof:**]{} Without loss of generality, we can assume that $i=1$. According to Eq. (\[Eq-df(x)\]), we have $$f(x)=\left(\neg x_1\wedge f_{x_1=0}(x_2,\ldots,x_n)\right)\vee\left(x_1\wedge f_{x_1=1}(x_2,\ldots,x_n)\right).$$ Suppose that at least one of the functions $f_{x_1=0}$ and $f_{x_1=1}$ is equivalent to $\mbox{AND}_{n-1}$ up to some negations of the variables. Without loss of generality, we will now assume that $f_{x_1=1}(x)=\mbox{AND}_{n-1}(x_2,\ldots,x_n)$. To prove the theorem, we consider two cases. [**Case 1:**]{} $f_{x_1=0}(x)=\mbox{AND}_{n-1}((\neg)x_2,\ldots,(\neg)x_n)$. In this case we have two subcases. [**Case 1a:**]{} $f_{x_1=0}(x)=\mbox{AND}_{n-1}(\neg x_2,\ldots,\neg x_n)$. We have $$f(x)=\mbox{AND}_{n}(\neg x_1,\neg x_2,\ldots,\neg x_n)\vee \mbox{AND}_{n}\linebreak[0](x_1,\linebreak[0]x_2,\ldots,x_n)=\neg \mbox{NAE}(\linebreak[0]x_1\linebreak[0]x_2,\linebreak[0]\ldots,x_n).$$ Therefore, $Q_E(f)<n$. [**Case 1b:**]{} $f_{x_1=0}(x)\neq \mbox{AND}_{n-1}(\neg x_2,\ldots,\neg x_n)$. Without loss of generality, we can suppose that there exists a $k\in\{2,\ldots,n-1\}$ such that $f_{x_1=0}\linebreak[0](x)=\mbox{AND}_{n-1}(\neg x_2,\linebreak[0]\ldots,\neg x_k,x_{k+1},\ldots,x_n )$. Then $$f(x)=\mbox{AND}_{n}(\neg x_1,\ldots,\neg x_k,x_{k+1}, \ldots,x_n)\vee \mbox{AND}_{n}\linebreak[0](x_1,\linebreak[0]x_2,\ldots,x_n)$$ $$=\left(\mbox{AND}_{k}(\neg x_1,\ldots,\neg x_k)\vee \mbox{AND}_{k}\linebreak[0](x_1,\linebreak[0]\ldots,x_k)\right)\wedge \mbox{AND}_{n-k}(x_{k+1},\ldots,x_n)$$ $$=\neg \mbox{NAE}_k(\neg x_1,\ldots,\neg x_k)\wedge \mbox{AND}_{n-k}(x_{k+1},\ldots,x_n).$$ Therefore, $Q_E(f)< k+n-k=n$. [**Case 2:**]{} $f_{x_1=0}(x)=\mbox{OR}_{n-1}((\neg)x_2,\ldots,(\neg)x_n)$. This means that we have two subcases. [**Case 2a:**]{} $f_{x_1=0}(x)=\mbox{OR}_{n-1}(\neg x_2,\ldots,\neg x_n)$. If $g(y)= \mbox{AND}_{n-1}(x_{2},\ldots,x_n)$, then $$f(x)=\left(\neg x_1\wedge \neg g(y)\right)\vee \left( x_1\wedge g(y)\right)=x_1\oplus g(y).$$ Therefore, $Q_E(f)<n$. [**Case 2b:**]{} $f_{x_1=0}(x)\neq \mbox{OR}_{n-1}(\neg x_2,\ldots,\neg x_n)$. Without loss of generality, we can suppose that $f_{x_1=0}(x)=\mbox{OR}_{n-1}( x_2,(\neg)x_3 \linebreak[0]\ldots,(\neg) x_n )$, then let us query $x_2$ first. If $x_2=0$, then $f_{x_2=0}(x)=\neg x_1\wedge \mbox{OR}_{n-2}( (\neg)x_3 \linebreak[0]\ldots,(\neg) x_n )$. According to Theorem \[fsize\], $Q_E(f_{x_2=0})<n-1$. If $x_2=1$, then $f_{x_2=1}(x)=\neg x_1\vee \mbox{AND}_{n-1}\linebreak[0](x_1,\linebreak[0]x_3,\ldots,x_n)=\neg x_1\vee \mbox{AND}_{n-2}\linebreak[0](x_3,\ldots,x_n)$. According to Theorem \[fsize\], $Q_E(f_{x_2=1})<n-1$. According to Eq. (\[Eq-n-1ton\]), $Q_E(f)< n-1+1=n$. Now we need to consider the case that both $f_{x_1=0}$ and $f_{x_1=1}$ are $\mbox{OR}_{n-1}$ functions. Without loss of generality, we assume that $f_{x_1=1}(x)=\mbox{OR}_{n-1}(x_2,\ldots,x_n)$. This means that we have again two subcases. [**Case 3a:**]{} $f_{x_1=0}(x)=\mbox{OR}_{n-1}(x_2,\ldots,x_n)$. In this case, we have $f(x)=\mbox{OR}_{n-1}(x_2,\ldots,x_n)$ and $Q_E(f)=n-1<n.$ [**Case 3b:**]{} $f_{x_1=0}(x)\neq \mbox{OR}_{n-1}(x_2,\ldots,x_n)$. Without loss of generality generality, let us suppose that there exists a $k\in\{2,\ldots,n\}$ such that $f_{x_1=0}(x)=\mbox{OR}_{n-1}(\neg x_2,\linebreak[0]\ldots,\neg x_k,\linebreak[0]x_{k+1},\ldots,x_n )$. In such a case $$f(x)=\left(\neg x_1\wedge\mbox{OR}_{n-1}(\neg x_2,\ldots,\neg x_k,x_{k+1}, \ldots,x_n)\right)\vee \left( x_1\wedge\mbox{OR}_{n-1}\linebreak[0](x_2,\ldots,x_n)\right)$$ Let us query $x_{k+1}$ to $x_n$ first. If $x_{k+1}=\cdots=x_n=0$, let $g(y)=f(x_1,\ldots,x_k,0,\ldots,0)$, then $$g(y)=\left(\neg x_1\wedge\mbox{OR}_{n-1}(\neg x_2,\ldots,\neg x_k,)\right)\vee \left( x_1\wedge\mbox{OR}_{n-1}\linebreak[0](x_2,\ldots,x_k)\right)$$ $$= \mbox{NAE}_{n}(\neg x_1,x_2,\ldots,x_k).$$ Therefore, $Q_E(g)<k.$ Otherwise, there exists a $j\geq k+1$ such that $x_j=1$. It is now easy to show that $f(x)=\neg x_1\vee x_1=1$. Therefore, $Q_E(f)<n-k+k=n.$ \[Lm-and-or\] Let $f:\{0,1\}^n\to \{0,1\}$ be a Boolean function. If there exist an $i\in[n]$ such that $f_{x_i=b}$ is equivalent to $\mbox{AND}_{n-1}$ ($\mbox{OR}_{n-1}$) up to some negations of the variables, then $f_{x_j=c}$ is not equivalent to $\mbox{OR}_{n-1}$ ($\mbox{AND}_{n-1}$) up to some negations of the variables for $j\neq i$, where $b,c\in\{0,1\}$. [**Proof:**]{} Without loss of generality, we assume that $i=1$, $j=2$ and $f_{x_1=b}(x)=\mbox{AND}_{n-1}(x_2,\linebreak[0]\cdots,x_n)$. In such a case we have $f(bc00*\cdots*)=f(bc01*\cdots*)=0$[^3]. If we fix $c$, then there are more than one inputs such that $f_{x_2=c}(x)=0$. Therefore, $f_{x_2=c}$ is not equivalent to $\mbox{OR}_{n-1}$ up to some negations of the variables. [**Proof of Lemma \[Lm-c7-1\]:**]{} According to Lemma \[C6-l1\], for every $i\in[n]$, there must exist a $b_i\in\{0,1\}$ such that $f_{x_i=b_i}$ is isomorphic to $\mbox{AND}_{n-1}$, otherwise $Q_E(f)<n$. Without loss of generality, we assume that $f_{x_1=b_1}$ is equivalent to $\mbox{AND}_{n-1}$ ($\mbox{OR}_{n-1}$) up to some negations of the variables. According to Lemma \[Lm-and-or\], no $f_{x_i=b_i}$ is equivalent to $\mbox{OR}_{n-1}$ ($\mbox{AND}_{n-1}$) up to some negations of the variables. Therefore, for every $i>1$, $f_{x_i=b_i}$ is equivalent to $\mbox{AND}_{n-1}$ ($\mbox{OR}_{n-1}$) up to some negations of the variables. Now, suppose there exists $c=c_1\ldots c_n\neq b$ for every $i\in[n]$ such that $f_{x_i=c_i}$ is equivalent to $\mbox{AND}_{n-1}$ ($\mbox{OR}_{n-1}$) up to some negations of the variables. Since $c\neq b$, there exist $i\in[n]$ such that $b_i\neq c_i$. We have therefore that both $f_{x_i=b_i}$ and $f_{x_i=c_i}$ are isomorphic to $\mbox{AND}_{n-1}$. According to Lemma \[Lm-both\], we have $Q_E(f)<n$, which is a contradiction. In order to make our main result easier to understand, we consider $4$-bit Boolean functions first. \[Th6\] If $f$ is a 4-bit Boolean function, then $Q_E(f)=4$ iff $f$ is isomorphic to $\mbox{AND}_4$. [**Proof:**]{} If $f$ is isomorphic to $\mbox{AND}_4$, then $Q_E(f)=4$ [@BBC+98]. Assume that a $4$-bit Boolean function $f$ such that $Q_E(f)=4$, we prove that $f$ is isomorphic to $\mbox{AND}_4$ as follows. According to Lemma \[Lm-c7-1\], there exists one and only one $b=b_1b_2b_3b_4$ for every $i\in[4]$ such that $f_{x_i=b_i}$ is equivalent to $\mbox{AND}_{3}$ ($\mbox{OR}_{3}$) up to some negations of the variables. Since for any $4$-bit function $f$ with $b=b_1b_2b_3b_4$, there exists a function $f'$ with $b'=0000$ isomorphic to $f$. We can get $f'$ by some negations of the variables $x_i$ whenever $b_i=1$. Therefore, without loss of generality, we assume that $b=0000$ and for every $i\in[4]$ such that $f_{x_i=0}$ is equivalent to $\mbox{OR}_{3}$ up to some negations of the variables. There are three cases that we need now to consider: $x_1$ $x_2$ $x_3$ $x_4$ $f(x)$: Case 1 Case 2 Case 3 ------- ------- ------- ------- ---------------- -------- -------- 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 0 1 0 1 1 \* 0 0 1 1 1 1 \* 0 1 0 0 1 1 1 0 1 0 1 1 1 1 0 1 1 0 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 1 1 0 1 1 1 0 0 1 1 \* 1 1 0 1 1 0 \* 1 1 1 0 1 0 \* 1 1 1 1 \* \* \* : Values of $4$-bit Boolean functions. []{data-label="T3"} [**Case 1:**]{} For every $i\in[4]$, there is no negation variable occurrence in $f_{x_i=0}$, that is $f_{x_1=0}(x)=\mbox{OR}(x_2,x_3,x_4)$, $f_{x_2=0}(x)=\mbox{OR}(x_1,x_3,x_4)$, $f_{x_3=0}(x)=\mbox{OR}(x_1,x_2,x_4)$ and $f_{x_4=0}(x)=\mbox{OR}(x_1,x_2,x_3)$. See Case 1 in Table \[T3\] for values of $f(x)$. We still do not the value of $f(1111)$. If $f(1111)=1$, then $f(x)=\mbox{OR}(x_1,x_2,x_3,x_4)$, which is isomorphic to $\mbox{AND}_4$. If $f(1111)=0$, then $f(x)=\mbox{NAE}(x_1,x_2,x_3,x_4)$ and $Q_E(f)<4$. [**Case 2:**]{} There are negations of all variables in every $f_{x_i=0}$, that is $f_{x_1=0}(x)=\mbox{OR}(\neg x_2, \neg x_3,\linebreak[0] \neg x_4)$, $f_{x_2=0}(x)=\mbox{OR}(\neg x_1, \neg x_3, \neg x_4)$, $f_{x_3=0}(x)=\mbox{OR}(\neg x_1, \neg x_2,\neg x_4)$ and $f_{x_4=0}(x)=\mbox{OR}(\neg x_1, \linebreak[0]\neg x_2, \neg x_3)$. See Case 2 in Table \[T3\] for values of $f(x)$. If $f(1111)=1$, then $f(x)= \neg \mbox{Th}_4^3$ and $Q_E(f)=3<4$. If $f(1111)=0$, then $f(x)= \neg \mbox{EXACT}_4^3$ and $Q_E(f)=3<4$. [**Case 3:**]{} There is an $i\in[4]$ such that there is at least one negation variable occurrence and one no negation variable occurrence in $f_{x_i=0}$. Without loss of generality, we can now assume that $f_{x_1=0}(x)=\mbox{OR}( x_2, \neg x_3, (\neg) x_4)$. In order to analyse this case, we prove the following two lemmas first. \[lm-7\] Let $f$ be an $n$-bit Boolean function and $f_{x_i=0}$ be equivalent to $\mbox{OR}_{n-1}$ up to some negations of the variables for every $i\in[n]$. If $f_{x_1=0}(x)=\mbox{OR}_{n-1}(x_2,\neg x_3, (\neg)x_4,\ldots)$, then $f_{x_2=0}(x)=\mbox{OR}_{n-1}(x_1,\neg x_3, (\neg)x_4,\ldots)$ and $f_{x_3=0}(x)=\mbox{OR}_{n-1}(\neg x_1,\neg x_2, (\neg)x_4,\ldots)$. [**Proof:**]{} Since $f_{x_1=0}(x)=\mbox{OR}_{n-1}(x_2,\neg x_3, (\neg)x_4,\ldots)$, there exists a $y\in\{0,1\}^{n-3}$ such that $f(001y)=0$. Suppose that $f_{x_2=0}(x)=\mbox{OR}_{n-1}(\neg x_1,(\neg) x_3, (\neg)x_4,\ldots)$ or $f_{x_2=0}(x)=\mbox{OR}_{n-1}((\neg) x_1, x_3, (\neg)x_4,\ldots)$. We have $f(001y)=1$, which is a contradiction. Therefore, $f_{x_2=0}=\mbox{OR}_{n-1}(x_1,\neg x_3, (\neg)x_4,\ldots)$. Now suppose that $f_{x_3=0}(x)=\mbox{OR}_{n-1}(x_1, (\neg) x_2, (\neg)x_4,\ldots)$. There have to exist $c\in\{0,1\}$ and $z\in\{0,1\}^{n-3}$ such that $f(0c0z)=0$. Since $f_{x_1=0}(x)=\mbox{OR}_{n-1}(x_2,\neg x_3, (\neg)x_4,\ldots)$, we have $f(0c0z)=1$, which is a contradiction. Suppose that $f_{x_3=0}(x)=\mbox{OR}_{n-1}((\neg) x_1, x_2, (\neg)x_4,\linebreak[0]\ldots)$. There exist $c\in\{0,1\}$ and $z\in\{0,1\}^{n-3}$ such that $f(c00z)=0$. Since $f_{x_2=0}(x)=\mbox{OR}_{n-1}(x_1,\neg x_3, \linebreak[0](\neg)x_4,\ldots)$, we have $f(c00z)=1$, which is a contradiction. Therefore, $f_{x_3=0}(x)\linebreak[0]=\linebreak[0]\mbox{OR}_{n-1}(\neg x_1,\neg x_2,\linebreak[0] (\neg)x_4,\ldots)$. \[lm-8\] Let $f$ be an $n$-bit Boolean function. If there exist 4 distinct inputs $x,y,u,v\in\{0,1\}^n$ such that $f(x)=f(y)=1$ and $f(u)=f(v)=0$, then $f$ is not isomorphic to $\mbox{AND}_n$. [**Proof:**]{} If $f$ is equivalent to $\mbox{AND}_n$ up to some negations of the variables, then there exists just one $x \in\{0,1\}^n$ such that $f(x)=1$. If $f$ is equivalent to $\mbox{OR}_n$ up to some negations of the variables, then there exists just one $u \in\{0,1\}^n$ such that $f(u)=0$. According to Lemma \[lm-7\], we have $f_{x_2=0}(x)=\mbox{OR}(x_1, \neg x_3, (\neg) x_4)$, and $f_{x_3=0}(x)=\mbox{OR}(\neg x_1,\linebreak[0] \neg x_2, (\neg) x_4)$. See Case 3 in Table \[T3\] for values of $f(x)$. It is easy to see that if $x_1\oplus x_2=1$, then $f(x)=1$. If $x_1\oplus x_2=0$, then $x_1=x_2$ and $f$ can be represented as a $3$-bit Boolean function $g(x_2,x_3,x_4)$, see Table \[T4\] for its values. Since $f_{x_1=0}(x)=\mbox{OR}( x_2, \neg x_3, (\neg) x_4)$, we have either $g(010)=f(0010)=0$ or $g(011)=f(0011)=0$. Since $f_{x_3=0}(x)=\mbox{OR}(\neg x_1, \neg x_2, (\neg) x_4)$, we have either $g(100)=f(1100)=0$ or $g(101)=f(1101)=0$. We also have $g(000)=f(0000)$ and $g(001)=f(0001)=1$. According to Lemma \[lm-8\], $g(x_2,x_3,x_4)$ is not isomorphic to $\mbox{AND}_3$ and $Q_E(g)<3$. $x_2$ $x_3$ $x_4$ $g(x_2,x_3,x_4)$ ------- ------- ------- ------------------ -- -- -- 0 0 0 1 0 0 1 1 0 1 0 \* 0 1 1 \* 1 0 0 \* 1 0 1 \* 1 1 0 \* 1 1 1 \* : Values of $g(x_2,x_3,x_4)$.[]{data-label="T4"} Now we give an exact quantum algorithm for $f$ as follows: 1. Evaluate $x_1\oplus x_2$ with one query. 2. If $x_1\oplus x_2=1$, then $f(x)=1$. 3. If $x_1\oplus x_2=0$, then $f(x)=g(x_2,x_3,x_4)$. Evaluate $g$ with exact quantum algorithm. Therefore, we have $Q_E(f)<1+ Q_E(g)<1+3=4.$ The theorem has been proved. Finally, we prove the most general case. The main idea of the proof is similar to the proof of the previous theorem. If $f$ is an $n$-bit Boolean function, then $Q_E(f)=n$ iff $f$ is isomorphic to $\mbox{AND}_n$. [**Proof:**]{} If $f$ is isomorphic to $\mbox{AND}_n$, then $Q_E(f)=n$ [@BBC+98]. We prove the other direction by an induction on $n$. [**BASIS**]{}: The result holds for $n=3$. [**INDUCTION**]{}: Suppose the result holds for $n-1$, we will prove that the result holds for $n$. According to Lemma \[Lm-c7-1\], there exists one and only one $b=b_1\ldots b_n$ for every $i\in[n]$ such that $f_{x_i=b_i}$ is equivalent to $\mbox{AND}_{n-1}$ ($\mbox{OR}_{n-1}$) up to some negations of the variables. Without loss of generality, we assume that $b=0\ldots0$ and for every $i\in[n]$ such that $f_{x_i=0}$ is equivalent to $\mbox{OR}_{n-1}$ up to some negations of the variables. There are three cases that we need to consider: [**Case 1:**]{} For every $i\in[n]$, there is no negation variable occurrence in $f_{x_i=0}$, that is $f_{x_i=0}(x)=\mbox{OR}_{n-1}(x_1,\ldots,x_{i-1},x_{i+1},\ldots, x_n)$ for $i\in[n]$. It is easy to see that in such a case $f(0\ldots0)=0$, $f(1\ldots1)=*$ and $f(x)=1$ for $x\not\in\{0\ldots0,1\ldots 1\}$. If $f(1\ldots1)=1$, then $f(x)=\mbox{OR}_n(x_1,\ldots,x_n)$, which is isomorphic to $\mbox{AND}_n$. If $f(1\ldots1)=0$, then $f(x)=\mbox{NAE}(x_1,\ldots,x_n)$ and $Q_E(f)<n$. [**Case 2:**]{} There are all negation variable occurrences in every $f_{x_i=0}$, that is $f_{x_i=0}(x)=\mbox{OR}_{n-1}(\neg x_1,\ldots,\linebreak[0]\neg x_{i-1},\neg x_{i+1},\ldots, \neg x_n)$ for $i\in[n]$. It is easy to see that $f(x)=1$ for $|x|<n-1$, $f(x)=0$ for $|x|=n-1$ and $f(x)=*$ for $|x|=n$. If $f(1\ldots1)=1$, then $f(x)= \neg \mbox{Th}_n^{n-1}$ and $Q_E(f)=n-1<n$. If $f(1\ldots1)=0$, then $f(x)= \neg \mbox{EXACT}_n^{n-1}$ and $Q_E(f)=n-1<n$. [**Case 3:**]{} There is an $i\in[n]$ such that there is at least one negation variable occurrence and one no negation variable occurrence $f_{x_i=0}$. Without loss of generality, we assume that $f_{x_1=0}(x)=\mbox{OR}( x_2, \neg x_3, \linebreak[0](\neg) x_4,\ldots)$. According to Lemma \[lm-7\], we have $f_{x_2=0}(x)=\mbox{OR}(x_1, \neg x_3,\linebreak[0] (\neg) x_4, \ldots)$ and $f_{x_3=0}(x)=\mbox{OR}(\neg x_1, \neg x_2, \linebreak[0] (\neg) x_4, \ldots)$. For any $y\in\{0,1\}^{n-2}$, $f(01y)=f(10y)=1$, that is $f(x)=1$ if $x_1\oplus x_2=1$. If $x_1\oplus x_2=0$, then $x_1=x_2$ and $f$ can be represented as an $(n-1)$-bit Boolean function $g(x_2,\ldots,x_n)$. Since $f_{x_1=0}(x)=\mbox{OR}( x_2, \neg x_3, (\neg) x_4,\ldots)$, there must exist a $u\in\{0,1\}^{n-3}$ such that $f(001u)=g(01u)=0$. Since $f_{x_3=0}(x)=\mbox{OR}(\neg x_1, \neg x_2, (\neg) x_4, \ldots)$, there must exist a $v\in\{0,1\}^{n-3}$ such that $f(110v)=g(10v)=0$. We also have $g(00\ldots00)=f(000\ldots00)=1$ and $g(00\ldots01)=f(000\ldots01)=1$. According to Lemma \[lm-8\], we have that $g(x_2,\ldots,x_n)$ is not isomorphic to $\mbox{AND}_{n-1}$ and $Q_E(g)<n-1$. Now we give an exact quantum algorithm for $f$ as follows: 1. Evaluate $x_1\oplus x_2$ with one query. 2. If $x_1\oplus x_2=1$, then $f(x)=1$. 3. If $x_1\oplus x_2=0$, then $f(x)=g(x_2,\ldots,x_n)$. Evaluate $g$ with exact quantum algorithm. Therefore, we have $Q_E(f)<1+ Q_E(g)<1+n-1=n.$ The theorem has been proved. Almost all $n$-bit Boolean functions can be computed by an exact quantum algorithm with less than $n$ queries. [**Proof:**]{} It is easy to see that there are $2\times 2^n$ $n$-bit Boolean functions which are isomorphic to $\mbox{AND}_n$. Since there are $2^{2^n}$ Boolean functions on $n$ variables, we see that the fraction of functions which have exact quantum query complexity $n$ is $o(1)$. Thus almost all $n$-bit Boolean functions can be computed by an exact quantum algorithm with less than $n$ queries. Conclusion ========== We have first shown that $\mbox{AND}_n$ is the only $n$-bit Boolean function in three special classes of Boolean functions, (including symmetric, monotone, read-once functions), up to isomorphism, that has exact quantum query complexity $n$. Finally, we have proved that in general $\mbox{AND}_n$ is the only $n$-bit Boolean function, up to isomorphism, that has exact quantum query complexity $n$. This shows that the advantages for exact quantum query algorithms are more common than previously thought. In the proof for special classes of Boolean functions, we have used their special properties of different types of Boolean functions. Each approach is different from each other. These approaches that we used in each type of Boolean functions may be helpful in analysis of exact quantum complexity for other interesting functions. In the approach for general case, we have used the properties of the true value table of the Boolean functions. Acknowledgements {#acknowledgements .unnumbered} ================ The authors are thankful to the anonymous referees for their comments and suggestions on the early version of this paper. The third author would like to thank Alexander Rivosh for his help while visiting University of Latvia. 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--- abstract: 'Caching at mobile devices, accompanied by device-to-device (D2D) communications, is one promising technique to accommodate the exponentially increasing mobile data traffic. While most previous works ignored user mobility, there are some recent works taking it into account. However, the duration of user contact times has been ignored, making it difficult to explicitly characterize the effect of mobility. In this paper, we adopt the alternating renewal process to model the duration of both the contact and inter-contact times, and investigate how the caching performance is affected by mobility. The *data offloading ratio*, i.e., the proportion of requested data that can be delivered via D2D links, is taken as the performance metric. We first approximate the distribution of the *communication time* for a given user by beta distribution through moment matching. With this approximation, an accurate expression of the data offloading ratio is derived. For the homogeneous case where the average contact and inter-contact times of different user pairs are identical, we prove that the data offloading ratio increases with the user moving speed, assuming that the transmission rate remains the same. Simulation results are provided to show the accuracy of the approximate result, and also validate the effect of user mobility.' author: - '[^1]' bibliography: - 'IEEEabrv.bib' - 'report.bib' title: Mobility Increases the Data Offloading Ratio in D2D Caching Networks --- Introduction ============ The mobile data traffic is growing at an exponential rate, among which mobile video accounts for more than a half [@forecast2016cisco]. Caching popular contents at helper nodes or user devices is a promising approach to reduce the data traffic on the backhaul links, as well as improving the user experience of video streaming applications [@d2d-cache; @jcache]. In comparison with the commonly considered femto-caching system, caching at devices enjoys a unique advantage, i.e., the devices’ aggregate caching capacity grows with the number of devices [@d2d-cache]. Moreover, device caching can promote device-to-device (D2D) communications, where nearby mobile devices may communicate directly rather than being forced to communicate through the base station (BS)[@design]. Recently, caching in D2D networks has attracted lots of attentions. In [@scaling], the scaling behavior of the number of D2D collaborating links was identified. Three concentration regimes, classified by the concentration of the file popularity, were investigated. The outage-throughput tradeoff and optimal scaling laws of both the throughput and outage probability were studied in [@tradeoff]. Two coded caching schemes, i.e., centralized and decentralized, were proposed in [@fundamentallimits], where the contents are delivered via broadcasting. So far, an important characteristic of mobile users, i.e., the user mobility, has been ignored in previous studies of D2D caching networks. There are some works starting to consider the effect of user mobility. Effective methodologies to utilize the user mobility information in caching design were discussed in [@magmobility]. In [@mobilitycodedcaching], the effect of mobility was evaluated in D2D networks with coded caching, with the conclusion that mobility can improve the scaling law of throughput. This result was based on the assumption that the user locations are random and independent in each time slot, which failed to take into account the temporal correlation. The inter-contact model, which considers the temporal correlation of the user mobility, has been widely applied [@exintercontactmodel], where the timeline for an arbitrary pair of mobile users are divided into *contact times* and *inter-contact times*. Specifically, the *contact times* denote the time intervals when the mobile users are located within the transmission range. Correspondingly, the *inter-contact times* denote the time intervals between contact times [@pocket]. This model has been used to develop device caching schemes to exploit the user mobility pattern in [@mobilitycaching]. The throughput-delay scaling law was developed by characterizing the inter-contact pattern of the random walk model [@scalingmobility]. In these works, it was assumed that a fixed amount of data can be delivered within one contact time, while the duration of the contact times was not considered. However, as the user moving speed will affect the durations of both the contact and inter-contact times, it is critical to account for their effects when investigating the impact of user mobility on caching performance. In this paper, we shall analytically evaluate the effect of mobility in D2D caching networks, by adopting an alternating renewal process to model the mobility pattern so that both the contact and inter-contact times are accounted for. The *data offloading ratio*, which is defined as the proportion of data that can be obtained via D2D links, is adopted as the performance metric. The main contribution is an approximate expression for the data offloading ratio, for which the main difficulty is to deal with multiple alternating renewal processes. We tackle it by first deriving the expectation and variance of the *communication time* of a given user, and then use a beta random variable to approximate it by moment matching. Furthermore, we investigate the effect of mobility in a homogeneous case, where the average contact and inter-contact times for all the user pairs are the same. In the low-to-medium mobility scenario, by assuming that the transmission rate is irrelevant to the user speed, it is proved that the data offloading ratio increases with the user speed for any caching strategy that does not cache the same contents at all devices. Simulation results validates the accuracy of the derived expression, as well as the effect of the user mobility. System Model and Performance Metric =================================== In this section, we will first introduce the alternating renewal process to model the user mobility pattern, and discuss the caching and file delivery models. Then, the performance metric, i.e., the data offloading ratio, will be defined. User Mobility Model ------------------- ![The timeline for an arbitrary pair of mobile users.[]{data-label="intercontact"}](intercontact){width="3in"} The inter-contact model, which captures the temporal correlation of the user mobility [@exintercontactmodel], is used to model the user mobility pattern. Specifically, the timeline of each pair of users is divided into *contact times*, i.e, the times when the users are in the transmission range, and *inter-contact times*, i.e., the times between consecutive contact times. Considering that contact times and inter-contact times appear alternatively in the timeline of a pair of users, similar to [@renewalmodel], an alternating renewal process is applied to model the pairwise contact pattern, as defined below [@renewalprocess]. Consider a stochastic process with state space $\{A,B\}$, and the successive durations for the system to be in states $A$ and $B$ are denoted as $\xi_k,k=1,2,\cdots$ and $\eta_k,k=1,2,\cdots$, respectively, which are i.i.d.. Specifically, the system starts at state $A$ and remains for $\xi_1$, then switches to state $B$ for $\eta_1$, then backs to state $A$ for $\xi_2$, and so forth. Let $\psi_k=\xi_k+\eta_k$. The counting process of $\psi_k$ is called as an *alternating renewal process*. As shown in Fig. \[intercontact\], if the pair of users is in contact at $t=0$, $\xi_k$ and $\eta_k$ represent the contact times and inter-contact times, respectively; otherwise, $\xi_k$ and $\eta_k$ represent the inter-contact times and contact times, respectively. It was shown in [@exintercontact] that exponential curves well fit the distribution of inter-contact times, while in [@excontact], it was identified that exponential distribution is a good approximation for the distribution of the contact times. Thus, same as [@renewalmodel], we assume that the contact times and inter-contact times follows independent exponential distributions. For simplicity, the timelines of different user pairs are assumed to be independent. Specifically, we consider $N_u$ users in a network, and the index set of the users is denoted as $\mathcal{S}=\{1,2,\cdots,N_u\}$. The contact times and inter-contact times of users $i \in \mathcal{S}$ and $j \in \mathcal{S} \backslash \{i\}$ follow independent exponential distributions with parameters $\lambda^C_{i,j}$ and $\lambda^I_{i,j}$, respectively. Caching and File Transmission Model ----------------------------------- ![A sample network with three mobile users.[]{data-label="model"}](model){width="2.6in"} There is a library with $N_f$ files, whose index set is denoted as $\mathcal{F}=\{1,2,\cdots,N_f\}$, each with size $C$. Each user device has a limited storage capacity, and each file can be completely cached or not cached at all at each user device. Specifically, the caching placement is denoted as $$x_{j,f}= \begin{cases} 1, \text{if user $j$ caches file $f$}, \\ 0, \text{if user $j$ does not cache file $f$}, \end{cases}$$ where $j \in \mathcal{S}$ and $f \in \mathcal{F}$. User $i \in \mathcal{S}$, is assumed to request a file $f \in \mathcal{F}$ with probability $p^r_{i,f}$, where $\sum \limits_{f \in \mathcal{F} } p^r_{i,f} =1$. When a user requests a file $f$, it will first check its own cache, and then download the file from the users that are in contact and store file $f$, with a fixed transmission rate, denoted as $R$. If the user cannot get the whole file within a certain delay threshold, denoted as $T^d$, it will download the remaining part from the BS. We assume that the delay threshold is larger than the time required to download each content, i.e., $T^d>\frac{C}{R}$. Fig. \[model\] shows a sample network, where user $1$ gets part of the requested file during the contact time with user $2$, then gets the whole file after the contact time with user $3$. Performance metric ------------------ ![An illustration of the communication time.[]{data-label="transmissiontime"}](transmissiontime){width="3.5in"} The *data offloading ratio*, which is defined as the percentage of requested content that can be obtained via D2D links rather than downloading from the BS, is used as the performance metric. Specifically, the data offloading ratio for user $i \in \mathcal{S}$ is defined as $$\mathcal{P}_i=\sum \limits_{f \in \mathcal{F}} p^r_{i,f} \left\{ x_{i,f} + \frac{ (1-x_{i,f})\mathbb{E} _{D_{i,f}}\left[ \min \left( D_{i,f} ,C \right) \right]}{C} \right\},$$ where $D_{i,f}$ denotes the amount of requested data that can be delivered via D2D links when user $i$ requests file $f$. Since a fixed transmission rate is assumed, $D_{i,f}$ can be written as $D_{i,f}=R T^c_{i,f}$, where $T^c_{i,f}$ is the *communication time* for user $i$ to download file $f$ from other users caching file $f$ within time $T^d$. We assume that user $i$ can download file $f$ while at least one user caching file $f$ is in contact, where the handover time is ignored. Fig. \[transmissiontime\] shows the communication time of user $1$ in Fig. \[model\]. Then, the average data offloading ratio is $$\begin{aligned} \label{define_ratio} &\mathcal{P}= \notag \\ &\frac{1}{N_u}\sum \limits_{i \in \mathcal{S}} \sum \limits_{f \in \mathcal{F}} p^r_{i,f} \left\{ x_{i,f} + \frac{ (1-x_{i,f}) \mathbb{E}_{T^c_{i,f} } \left[ \min \left( R T^c_{i,f} ,C \right) \right]}{C} \right\}.\end{aligned}$$ In the following, we will evaluate the data offloading ratio given in (\[define\_ratio\]) for any given caching strategy, and investigate the effect of user mobility on caching performance. Data Offloading Ratio Analysis ============================== The main difficulty of evaluating the data offloading ratio is to find the distribution of the communication time. As this distribution is highly complicated, instead of deriving it directly, we will develop an accurate approximation. In this section, we will first approximate the distribution of the communication time by a beta distribution, and then, an approximation of the data offloading ratio will be obtained. Communication time analysis --------------------------- To help analyze the communication time, we first define some stochastic processes. Define $\mathbf{H}_{i,j}$, where $i \in \mathcal{S}$ and $j \in \mathcal{S} \backslash \{i\}$, as the continuous-time random process, i.e., $\mathbf{H}_{i,j}=\{H_{i,j}(t), t \in (0,\infty)\}$ with state space $\{0,1\}$, where $H_{i,j}(t)=1$ means that users $i$ and $j$ are in contact at the time instant $t$; otherwise $H_{i,j}(t)=0$. The durations of staying in states $1$ and $0$ follow i.i.d. exponential distributions with parameter $\lambda^C_{i,j}$ and $\lambda^I_{i,j}$, respectively. Define $\mathbf{H}_i^f$, where $i \in \mathcal{S}$ and $f \in \mathcal{F}$, as the continuous-time random process, i.e., $\mathbf{H}^f_{i}=\{H^f_{i}(t), t \in (0,\infty)\}$ with state space $\{0,1\}$, where $H^f_{i}(t)=1$ means that users $i$ can download file $f$ from any other user caching file $f$ at time instant $t$; otherwise $H^f_{i}(t)=0$. At time $t$, since user $i$ can download file $f$ when at least one user caching file $f$ is in contact, we get $H^f_i(t)=1- \prod \limits_{j \in \mathcal{S} \backslash \{i\},x_{j,f}=1}\left[1-H_{i,j}(t) \right]$. Similar to [@renewalmodel], it is reasonable to assume that when a user requests a file, the alternating process between each pair of users has been running for a long time. Thus, denote $T^r_{i,f}$, $i \in \mathcal{S}$ and $f \in \mathcal{F}$, as the time of user $i$ requests file $f$, and the communication time $T^c_{i,f}$ can be derived as $T^c_{i,f}= \lim \limits_{T^r_{i,f} \to \infty} \int _{T^r_{i,f}}^{T^r_{i,f}+T^d} H^f_i(t) dt.$ In the following, we will derive the expectation and variance of the communication time. \[ev\] When user $i \in \mathcal{S}$ requests file $f \in \mathcal{F}$, which is not stored at its own cache, the expectation and variance of its communication time is $$\label{expectation_t} \mathbb{E}\left[T^c_{i,f} \right]=T^d\left( 1-\prod \limits_{j \in \mathcal{S}, x_{j,f}=1} \frac{\lambda^C_{i,j}}{\lambda^C_{i,j}+\lambda^I_{i,j}} \right).$$ and $$\begin{aligned} \label{var_t} \mathrm{Var} \left[ T^c_{i,f} \right] = & 2 \int_0^{T^d} (T^d-u) \prod \limits_{j \in \mathcal{S}, x_{j,f}=1} \frac{\lambda^C_{i,j}}{(\lambda^I_{i,j}+\lambda^C_{i,j})^2} \notag \\ &\times \left[ \lambda^C_{i,j} + \lambda^I_{i,j} e^{-u(\lambda_{i,j}^C+\lambda^I_{i,j})} \right] du \notag \\ &-(T^d)^2 \prod \limits_{j \in \mathcal{S}, x_{j,f}=1} \left(\frac{\lambda^C_{i,j}}{\lambda^C_{i,j}+\lambda^I_{i,j}} \right)^2.\end{aligned}$$ See Appendix A. Since the communication time $T^c_{i,f}$ is a bounded random variable, we propose to approximate its distribution by a beta distribution, which is widely used to model the random variables limited to finite ranges. Specifically, we consider $T^c_{i,f} \approx T^d Y_{i,f}$, where $Y_{i,f} \sim B(\alpha_{i,f},\beta_{i,f})$, $i \in \mathcal{S}$ and $f \in \mathcal{F}$, if $\sum_{j \in \mathcal{S} \backslash \{i\}} x_{j,f}>0$, which means that user $i$ may download file $f$ from at least one user; otherwise, $T^c_{i,f}=0$. Let $\mathbb{E}[T^dY_{i,f}]=\mathbb{E}[T^c_{i,f}]$ and $\mathrm{Var}[T^dY_{i,f}]=\mathrm{Var}[T^c_{i,f}]$, and the parameters of the beta distribution to match the first two moments can be derived as[^2] $$\label{beta_p} \begin{cases} \alpha_{i,f}=\frac{\mathbb{E}[T^c_{i,f}]^2 (T^d-\mathbb{E}[T^c_{i,f}])}{ \mathrm{Var}[T^c_{i,f}]T^d}-\frac{\mathbb{E}[T^c_{i,f}]}{T^d} \\ \beta_{i,f}=\frac{T^d-\mathbb{E}[T^c_{i,f}]}{\mathbb{E}[T^c_{i,f}]} \alpha_{i,f} \end{cases}$$ Data offloading ratio approximation ----------------------------------- Based on the above approximation, we get an approximate expression of the data offloading ratio in Proposition \[E\_od\]. Simulations will show that the approximation is quite accurate. \[E\_od\] The data offloading ratio is approximated as $$\begin{aligned} \label{ratio} \mathcal{P} = &\frac{1}{N_u}\sum \limits_{i \in \mathcal{S}} \sum \limits_{f \in \mathcal{F}} p^r_{i,f} \left[ x_{i,f} + (1-x_{i,f}) \mathcal{P}_{i,f} \right],\end{aligned}$$ where $\mathcal{P}_{i,f}$ is the data offloading ratio when user $i$ requests file $f$, which is not in its own cache, approximated by $$\begin{aligned} \label{ex_g} &\mathcal{P}_{i,f} \approx 1-I_{\frac{C}{T^dR}} (\alpha_{i,f},\frac{T^d-\mathbb{E}[T^c_{i,f}]}{\mathbb{E}[T^c_{i,f}]} \alpha_{i,f}) \notag \\ & +\frac{\mathbb{E}[T^c_{i,f}]R}{C} I_{\frac{C}{T^dR}}(\alpha_{i,f}+1,\frac{T^d-\mathbb{E}[T^c_{i,f}]}{\mathbb{E}[T^c_{i,f}]} \alpha_{i,f}) \Big] \Big\},\end{aligned}$$ if $\sum_{j \in \mathcal{S} \backslash \{i\}} x_{j,f}>0$ and $0$ elsewhere, where $I_r(\cdot,\cdot)$ is the incomplete beta function, and $\alpha_{i,f}$ is given in (\[beta\_p\]). Following (\[define\_ratio\]), (\[expectation\_t\]), (\[var\_t\]), and (\[beta\_p\]), the expression in (\[ratio\]) can be obtained. Due to the space limitation, the detail is omitted. Effect of Mobile User Speed =========================== In this section, we will consider a homogeneous case, where the contact and inter-contact parameters among all pairs of users are the same, i.e., $\lambda^C=\lambda^C_{i,j}>0$ and $\lambda^I=\lambda^I_{i,j}>0$, where $i \in \mathcal{S}$ and $j \in \mathcal{S} \backslash \{i\}$. We will investigate how the user moving speed affects the data offloading ratio for a given caching strategy. If all users cache the same contents, the D2D communications will not help the content delivery. Thus, in the following, we assume that the contents cached at different users are not all the same. This investigation will be based on the approximate expression in (\[ratio\]), and simulations will be provided later to verify the results. Communication time analysis --------------------------- Under the above assumptions, the expectation and variance of the communication time can be simplified, as in the following corollary. \[sim\_ev\] When $\lambda^C=\lambda^C_{i,j}$ and $\lambda^I=\lambda^I_{i,j}$, where $i \in \mathcal{S}$ and $j \in \mathcal{S} \backslash \{i\}$, the expectation and variance of a user requests file $f$, which is not stored at its own cache, are given by $$\begin{aligned} &\mathbb{E}[T^c_{i,f}]=T^d\left[ 1-\left( \frac{\lambda^C}{\lambda^C+\lambda^I} \right)^{n_f} \right], \label{expect} \\ &\mathrm{Var}[T^c_{i,f}]= \left[ \frac{\lambda^C}{(\lambda^C+\lambda^I)^2} \right]^{n_f} \sum \limits_{l=1}^{{n_f}} \binom{{n_f}}{l} \frac{(\lambda^C)^{n_f-l} (\lambda^I)^{l}}{l(\lambda^C+\lambda^I)} \notag \\ & \quad \times \left[ T^d-\frac{1}{l(\lambda^C+\lambda^I)}+\frac{e^{-l(\lambda^C+\lambda^I)T}}{l(\lambda^C+\lambda^I)} \right], \label{variance}\end{aligned}$$ where $i \in \mathcal{S}$ and $n_f=\sum \limits_{j \in \mathcal{S}} x_{j,f}$ denotes the number of users caching file $f$. See Appendix A. Mobile user speed ----------------- We first characterize the relationship between the user speed and the parameters $\lambda^C$ and $\lambda^I$ in Lemma \[speed\]. \[speed\] When all the user speeds change by $s$ times, the contact and inter-contact parameters will also change by $s$ times, i.e., from $\lambda^C$ and $\lambda^I$ to $s\lambda^C$ and $s\lambda^I$, respectively. The time for user $i$ to move along a certain path $L_i$ can be given as a curve integral $\int_{L_i} \frac{dz}{v_i(z)}$, where $v_i(z)$ is the speed of user $i$ when passing by a point $z$ on the path $L_i$. When the speed of user $i$ changes by $s$ times, the time for moving along the path $L_i$ changes to $\int_{L_i} \frac{dz}{s v_i(z)}=\frac{1}{s}\int_{L_i} \frac{dz}{ v_i(z)}$, which scales by $\frac{1}{s}$ times. During each contact or inter-contact time, users $i$ and $j$ move along certain paths. When all the user speeds change by $s$ times, each contact or inter-contact time changes by $\frac{1}{s}$ times, and thus, the average ones change by $\frac{1}{s}$ times. Since the contact and inter-contact times are assumed to be exponential distributed with mean $\frac{1}{\lambda^C}$ and $\frac{1}{\lambda^I}$, respectively, the parameters $\lambda^C$ and $\lambda^I$ scale by $s$ times. Considering that a larger $s$ means that users are moving faster, in the following, we will investigate how changing $s$ will affect the data offloading ratio. For simplicity, we assume that the transmission rate is a constant, and will not change with the user speed. This is reasonable in the low-to-medium mobility regime. Firstly, the effect of user speed on the communication time is shown in Lemma \[s\_t\] . \[s\_t\] When $s$ increases, which is equivalent to increasing the user speed, the expectation of the communication time when a user $i \in \mathcal{S}$ requests file $f \in \mathcal{F}$ that is not in its own cache, i.e., $\mathbb{E}[T^c_{i,f}]$, keeps the same, and the corresponding variance, i.e., $\mathrm{Var}[T^c_{i,f}]$, decreases, if the number of users caching file $f$ is larger than $0$, i.e., $n_f>0$. Accordingly, the parameter $\alpha_{i,f}$ of the beta distribution increases. See Appendix B. Then, we evaluate the relationship between $\alpha_{i,f}$ and the data offloading ratio when user $i$ requests file $f$ that is not in its own cache, i.e., $\mathcal{P}_{i,f}$ in (\[ex\_g\]), in Lemma \[s\_g\]. \[s\_g\] When user $i \in \mathcal{S}$ requests file $f \in \mathcal{F}$ and cannot find it in its own cache, the data offloading ratio, i.e., $\mathcal{P}_{i,f}$, increases with $\alpha_{i,f}$. See Appendix C. Base on Lemmas \[s\_t\] and \[s\_g\], we can specify the effect of user speed on the data offloading ratio in Proposition \[s\_d\]. \[s\_d\] If the transmission rate does not change with the user speed, and the average contact and inter-contact times among all the pairs are the same, the data offloading ratio increases with the user moving speed. See Appendix D. The result in Proposition \[s\_d\] is valid for any caching strategy, only excluding the case that all the users have the same cache contents. Simulation results ================== In the simulation, the content request probability follows a Zipf distribution with parameter $\gamma_r$, i.e., $p_f=\frac{f^{-\gamma_r}}{\sum \limits_{i \in \mathcal{F}} i^{-\gamma_r}}$, $f \in \mathcal{F}$ [@d2d-cache]. Meanwhile, each user caches 5 contents, and a random caching strategy is applied [@randomcache], where the probabilities of the contents cached at each user are proportional to the file request probabilities. ![Data offloading ratio with $N_f=100$, $T^d=300s$ and $\gamma_r=0.6$.[]{data-label="fig_scheme1"}](paper1){width="2.6in"} Fig. \[fig\_scheme1\] validates the accuracy of the approximation in (\[ratio\]). The inter-contact parameters $\lambda^I_{i,j}, i \in \mathcal{S}, j \in \mathcal{S} \backslash \{i\}$ are generated according to a gamma distribution as $\Gamma(4.43,1/1088)$ [@aggregate_real]. Similar as [@renewalmodel], we assume the average of the contact parameters are $5$ times larger than the inter-contact parameters. Thus, the contact parameters are generated∂ according to $\Gamma(4.43 \times 25,1/1088/5)$. It is shown from Fig. \[fig\_scheme1\] that the theoretical results are very close to the simulation results, which means the approximate expression (\[ratio\]) is quit accurate. Furthermore, the data offloading ratio increases with the number of users, which is brought by the increasing aggregate caching capacity and the content sharing via D2D links. ![Data offloading ratio with $N_u=15$, $\lambda^C=0.001s$, $\lambda^I=0.0002s$, $N_f=100$, $T^d=300s$ and $\gamma_r=0.6$.[]{data-label="fig_scheme2"}](paper2){width="2.6in"} In Fig. \[fig\_scheme2\], the effect of $s$ is demonstrated, where increasing $s$ is equivalent to increasing the user speed. Firstly, the small gap between the theoretical and simulation results again verifies the accuracy of the approximate expression in (\[ratio\]). It is also shown that the data offloading ratio increases with $s$, which confirms the conclusion in Proposition \[s\_d\]. Moreover, from Fig. \[fig\_scheme2\], the increasing rate of the data offloading ratio is decreases with the user moving speed. conclusions =========== In this paper, we investigated the effect of user mobility on the caching performance in a D2D caching network. The communication time of a given user was firstly approximated by a beta distribution, through matching the first two moments. Then, an approximate expression of the data offloading ratio was derived. For a homogeneous case, where the average contact and inter-contact times are the same for all the user pairs, we evaluated how the user moving speed affects the data offloading ratio. Specifically, it was proved that the data offloading ratio increases with the user speed, assuming that the transmission rate is irrelevant to the user speed. Simulation results validated the accuracy of the approximate expression of the data offloading ratio, and demonstrated that the data offloading ratio increases with the user speed, while the increasing rate decreases with the user speed. Appendix {#appendix .unnumbered} ======== Proof of Lemma \[ev\] and Corollary \[sim\_ev\] ----------------------------------------------- As the timeline of different user pairs are independent, the expectation of the communication time when user $i$ requests file $f$, which is not in its own cache, can be written as $$\mathbb{E} [ T^c_{i,f} ]=\lim \limits_{T^r_{i,f} \to \infty} \int _{T^r_{i,f}}^{T^r_{i,f}+T^d} \left[ 1- \prod \limits_{j \in \mathcal{S},x_{j,f}=1}\left(1- \mathbb{E} H_{i,j}(t) \right) \right] dt.$$ Since the timeline between each pair of users is modeled as an alternating renewal process, according to [@renewalprocess], we have $\lim \limits_{t \to \infty} \Pr[H_{i,j}(t)=1]=\frac{\lambda^I_{i,j}}{\lambda^C_{i,j}+\lambda^I_{i,j}}$. Thus, $\lim \limits_{t \to \infty} \mathbb{E}[ H_{i,j}(t)]=\frac{\lambda^I_{i,j}}{\lambda^C_{i,j}+\lambda^I_{i,j}}$, and then, the expectation in (\[expectation\_t\]) can be obtained. Let $\lambda^C=\lambda^C_{i,j}$ and $\lambda^I=\lambda^I_{i,j}$, and we can get the expression in (\[expect\]). The variance of the communication time is $$\begin{aligned} \label{v} &\mathrm{Var} [ T^c_{i,f} ]= \notag \\ &2 \lim \limits_{T^r_{i,f} \to \infty} \int_{T^r_{i,f}}^{T^r_{i,f}+T^d} \int_{T^r_{i,f}}^{\tau} \Pr[H^f_{i}(t)=1,H^f_{i}(\tau)=1] dtd\tau \notag \\ &-\left( \mathbb{E} [ T^c_{i,f} ] \right)^2\end{aligned}$$ According to [@renewalprocess], $\Pr[H_{i,j}(\tau)=0|H_{i,j}(t)=0]=\frac{\lambda^C_{i,j}}{\lambda^C_{i,j}+\lambda^I_{i,j}}+\frac{\lambda^I_{i,j}}{\lambda^C_{i,j}+\lambda^I_{i,j}}e^{-({\lambda^C_{i,j}+\lambda^I_{i,j}})(\tau-t)}$. Then, when $T_{i,f}^r \to \infty$, we can get $$\begin{aligned} \label{prob} &\Pr[H^f_{i}(\tau)=1,H^f_{i}(t)=1] =1-2 \prod \limits_{j \in \mathcal{S},x_{j,f}=1}\frac{\lambda^C_{i,j}}{\lambda^C_{i,j}+\lambda^I_{i,j}} \notag \\ & \quad + \prod \limits_{j \in \mathcal{S},x_{j,f}=1} \frac{\lambda^C_{i,j}}{(\lambda^I_{i,j}+\lambda^C_{i,j})^2}\left[ \lambda^C_{i,j} + \lambda^I_{i,j} e^{-(\lambda_{i,j}^C+\lambda^I_{i,j})(\tau-t)} \right]\end{aligned}$$ Let $u=\tau-t$ and substitute (\[prob\]) into (\[v\]), and we can get (\[var\_t\]). Let $\lambda^C=\lambda^C_{i,j}$ and $\lambda^I=\lambda^I_{i,j}$, and we can get (\[variance\]) with the binomial theorem. Proof of Lemma \[s\_t\] ----------------------- When the user speed changes by $s$ times, the expectation of the communication time in (\[expect\]) keeps the same, while the variance changes to $$\begin{aligned} &\mathrm{Var}[T^c_{i,f}]= \left[ \frac{\lambda^C}{(\lambda^C+\lambda^I)^2} \right]^{n_f} \sum \limits_{l=1}^{{n_f}} \binom{{n_f}}{l} \frac{(\lambda^C)^{n_f-l} (\lambda^I)^{l}}{sl(\lambda^C+\lambda^I)} \notag \\ & \quad \times \left[ T^d-\frac{1}{sl(\lambda^C+\lambda^I)}+\frac{e^{-sl(\lambda^C+\lambda^I)T^d}}{sl(\lambda^C+\lambda^I)} \right],\end{aligned}$$ To prove that $\mathrm{Var}[T^c_{i,f}]$ decreases with $s$, we will prove that $\frac{ \partial \mathrm{Var}[T^c_{i,f}]}{\partial s}<0$. The partial derivation of $\mathrm{Var}[T^c_{i,f}]$ is $$\begin{aligned} \label{dir} &\frac{\partial \mathrm{Var}[T^c_{i,f}]}{\partial s}= \notag \\ &\left[ \frac{\lambda^C}{(\lambda^C+\lambda^I)^2} \right]^{n_f} \sum \limits_{l=1}^{{n_f}} \binom{{n_f}}{l} \frac{(\lambda^C)^{n_f-l} (\lambda^I)^{l}}{s^3l^2(\lambda^C+\lambda^I)^2} \mathcal{A}_1(x),\end{aligned}$$ where $\mathcal{A}_1(x)=-x-x e^{-x}-2(e^{-x}-1)$ and $x=sl(\lambda^C+\lambda^I)T^d>0$. Since $\mathcal{A}'_1(x)=-1+(1+x) e^{-x} < -1+(1+x) \frac{1}{1+x}=0$, $\mathcal{A}_1(x)$ is a decreasing function of $x$. Thus, $\mathcal{A}_1(x)<\mathcal{A}_1(0)=0$. According to (\[dir\]), when $n_f>0$, we have $\frac{ \partial \mathrm{Var}[T^c_{i,f}]}{\partial s}<0$. The parameter $\alpha_{i,f}$ given in (\[beta\_p\]) is a decreasing function of $\mathrm{Var}[T^c_{i,f}]$, and thus increases with $s$. Proof of Lemma \[s\_g\] ----------------------- To simplify the expression in (\[ex\_g\]), denote $r \triangleq \frac{C}{T^d R} \in (0,1)$, $y \triangleq \frac{T^d-\mathbb{E}[T^c_{i,f}]}{\mathbb{E}[T^c_{i,f}]} \ge 0$, and $\alpha \triangleq \alpha_{i,f}$. The expression in (\[ex\_g\]) can be rewritten as a function of $\alpha$, given as $$\begin{aligned} &\mathcal{P}_{i,f}=1- \frac{\int_0^r (1-\frac{u}{r}) u^{\alpha-1} (1-u)^{y \alpha-1} du }{B(\alpha,y\alpha)}.\end{aligned}$$ Let $g(\alpha)=1-\mathcal{P}_{i,f}$, the derivation of $g(\alpha)$ is $$\begin{aligned} &g'(\alpha)= \notag \\ &\frac{1}{B(\alpha,y\alpha)} \Bigg\{\int_0^r (1-\frac{u}{r}) u^{\alpha-1} (1-u)^{y \alpha-1}[\ln u + y \ln (1-u)] du \notag \\ &- \int_0^r (1-\frac{u}{r}) u^{\alpha-1} (1-u)^{y \alpha-1} du D(y,\alpha) \Bigg\},\end{aligned}$$ where $D(y,\alpha)=\psi(\alpha)+y \psi(y\alpha)-(1+y)\psi[(1+y)\alpha]$ and $\psi(\cdot)$ is the digamma function. If $r=1$, $g'(\alpha)=\frac{\partial [y/(1+y)]}{\partial \alpha}=0$. Denote $\mathcal{A}_2(r)=\frac{B(\alpha,y\alpha)}{r}g'(\alpha)$, $\mathcal{A}_2(1)=0$ and $$\begin{aligned} &\lim \limits_{r \to 0^{+}} \mathcal{A}_2(r)= \notag \\ &\lim \limits_{r \to 0^{+}} \int_0^r (r-u) u^{\alpha-1} (1-u)^{y \alpha-1}[\ln u + y \ln (1-u)] du \notag \\\end{aligned}$$ Since $ r \ge u \ge 0$ and $y \ge 0$, $(r-u) u^{\alpha-1} (1-u)^{y \alpha-1} \ge 0$ and $\ln u + y \ln (1-u) \le 0$, thus, $\lim \limits_{r \to 0^{+}} \mathcal{A}_2(r) \le 0$. The derivation of $\mathcal{A}_2(r)$ is $$\begin{aligned} \mathcal{A}'_2(r)= &\int_0^r u^{\alpha-1} (1-u)^{y \alpha-1}[\ln u + y \ln (1-u)] du \notag \\ &- \int_0^r u^{\alpha-1} (1-u)^{y \alpha-1} du D(y,\alpha).\end{aligned}$$ Thus, $\mathcal{A}'_2(1)=\frac{\partial B(\alpha,y \alpha)}{\partial \alpha}-\frac{\partial B(\alpha,y \alpha)}{\partial \alpha}=0$ and $\lim \limits_{r \to 0^{+}} \mathcal{A}'_2(r) \le 0$. Then, we can get $\mathcal{A}''_2(r)= r^{\alpha-1} (1-r)^{y \alpha-1}[\ln r + y \ln (1-r)-D(y,\alpha)]$. Let $\mathcal{A}_3(r)=r^{1-\alpha} (1-r)^{1-y \alpha} \mathcal{A}''_2(r)$, then, there is one zero point of $\mathcal{A}'_3(r)=\frac{1-(1+y)x}{x(1-x)}$ in $(0,1]$. Thus, there is one inflection point of $\mathcal{A}_3(r)$. Considering that $\lim \limits_{r \to 0^{+}}\mathcal{A}_3(r)=\lim \limits_{r \to 1^{-}}\mathcal{A}_3(r)=-\infty$, the sign of $\mathcal{A}_3(r)$ may be negative, or first negative, then positive, and then negative, while $r$ increases in $(0,1)$. If $\mathcal{A}_3(r)<0$, then $\mathcal{A}''_2(r)<0$ when $r \in (0,1)$. However, we have $\lim \limits_{r \to 0^{+}} \mathcal{A}'_2(r) \le \mathcal{A}'_2(1)$, which means that $\mathcal{A}'_2(r)$ can not be a decreasing function in $(0,1)$. Thus, the sign of $\mathcal{A}_3(r)$ is first negative, then positive, and then negative, while $r$ increases in $(0,1)$. Since $\mathcal{A}''_2(r)$ has the same sign with $\mathcal{A}_3(r)$ in $(0,1)$, $\mathcal{A}'_2(r)$ first decreases, then increases, and then decreases while $r$ increases in $(0,1)$. Considering that $\lim \limits_{r \to 0^{+}} \mathcal{A}'_2(r) \le 0$ and $\mathcal{A}'_2(1)=0$, the sign of $\mathcal{A}'_2(r)$ must be first negative, and then positive in $(0,1)$. Therefore, while $r$ increases in $(0,1)$, $\mathcal{A}_2(r)$ first decreases, and then increases. Considering that $\lim \limits_{r \to 0^{+}} \mathcal{A}_2(r) \le 0$ and $\mathcal{A}_2(1)=0$, we have $\mathcal{A}_2(r)<0$ in $(0,1)$ and $\mathcal{A}_2(r)=0$ when $r=1$. Since $g'(\alpha)=\frac{r}{B(\alpha,y\alpha)} \mathcal{A}_2(r)$, we get $g'(\alpha)<0$ in $(0,1)$. Thus, $g(\alpha)$ decreases with $\alpha$, and $\mathcal{P}_{i,f}=1-g(\alpha)$ increases with $\alpha$. Proof of Proposition \[s\_d\] ----------------------------- The data offloading ratio in (\[ratio\]) increases with the increasing of $\mathcal{P}_{i,f}$ if $x_{i,f}=0$, $i \in \mathcal{S}$, $f \in \mathcal{F}$. Then, based on Lemmas \[s\_t\] and \[s\_g\], we can get that the data offloading ratio when user $i$ requests file $f$ from other users, i.e., $\mathcal{P}_{i,f}$, decreases with the user speed when $n_f>0$, otherwise $\mathcal{P}_{i,f}=0$. Accordingly, the data offloading ratio when user $i$ requests file $f$, i.e., $x_{i,f}+ (1-x_{i,f}) \mathcal{P}_{i,f} $, increases with the user speed when $x_{i,f}=0$ and $n_f>0$; otherwise, it keeps the same, where $i \in \mathcal{S}$, $f \in \mathcal{F}$. Since we consider that not all the users cache the same contents, there must exists $i' \in \mathcal{S}$, $j' \in \mathcal{S}$ and $f' \in \mathcal{F}$, where $x_{i',f'}=0$ and $x_{j',f'}=1$, i.e, $n_{f'}>0$. Thus, the data offloading ratio increases with the user speed. [^1]: This work was supported by the Hong Kong Research Grants Council under Grant No. 610113. [^2]: The parameters of the beta distribution should be positive, and it can be proved that $\alpha_{i,f}>0$ and $\beta_{i,f}>0$, by $e^{-u(\lambda_{i,j}^I+\lambda_{i,j}^C)} \le 1$. The detail is omitted due to the space limitation.

--- abstract: 'With the growth of Internet of Things (IoT) and mobile edge computing, billions of smart devices are interconnected to develop applications used in various domains including smart homes, healthcare and smart manufacturing. Deep learning has been extensively utilized in various IoT applications which require huge amount of data for model training. Due to privacy requirements, smart IoT devices do not release data to a remote third party for their use. To overcome this problem, collaborative approach to deep learning, also known as Collaborative Deep Learning (CDL) has been largely employed in data-driven applications. This approach enables multiple edge IoT devices to train their models locally on mobile edge devices. In this paper, we address IoT device training problem in CDL by analyzing the behavior of mobile edge devices using a game-theoretic model, where each mobile edge device aims at maximizing the accuracy of its local model at the same time limiting the overhead of participating in CDL. We analyze the Nash Equilibrium in an *N*-player static game model. We further present a novel cluster-based fair strategy to approximately solve the CDL game to enforce mobile edge devices for cooperation. Our experimental results and evaluation analysis in a real-world smart home deployment show that 80% mobile edge devices are ready to cooperate in CDL, while 20% of them do not train their local models collaboratively.' author: - 'deepti.gupta@my.utsa.edu, olumide.kayode@utsa.edu, sbhatt@tamusa.edu, mgupta@tntech.edu, tosun@cs.utsa.edu' bibliography: - 'references.bib' title: 'Learner’s Dilemma: IoT Devices Training Strategies in Collaborative Deep Learning' --- Collaborative deep learning, IoT device, Edge computing, Game Theory. Introduction {#sec:introduction} ============ In recent years, Internet of Things (IoT) has grown rapidly and billions of smart devices are expected to be added over next few years. These devices generate a tremendous amount of data, from health information [@celik2018soteria] to social networking [@zeng2017end]. Deep learning models use this data for training and enhancing intelligence of various data driven IoT applications. Most of the IoT devices connect to a central cloud platform to use cloud services. These cloud services are crucial for storage of the datasets and model learning. However, use of cloud services requires additional latency in real time applications. To overcome this issue, edge devices are used for local data training which also safeguards privacy of personal data. Unlike constrained IoT devices, such devices have the capability to support Machine Learning (ML) models and have been used in various applications. For example, video doorbell performs training on its local datasets, and identifies person at the door. Deep learning models are often associated with the size of training dataset. Under a reasonable learning mechanism, more training data will enhance the accuracy and performance of a trained model. However, in the era of big data, data is often distributed and cannot be brought together due to personal privacy constraints. Collaborative Deep Learning (CDL) allows multiple IoT devices to train their models, without revealing associated personal data. CDL offers an attractive trade-off between privacy and utility of data sets. Recent research [@jiang2019lightweight; @chen2019communication] have discussed the privacy issues of local training devices and the impact of communication latency between IoT edge devices and Parameter Server (PS). However, the strategic behavior of the rational local training devices have not been discussed in previous research, i.e., the authors have assumed that all IoT devices are altruistic. Altruistic devices are ones which always follow a suggested protocol (what all devices have decided to follow initially) regardless whether they are benefiting or losing by following this protocol. However, devices are not altruistic in real life, they are rational. Rational devices are the ones which will deviate from suggested protocol if they think that they will be benefited more by following a different protocol. In our proposed system model, we assume that all the mobile edge devices are rational. A mobile edge device, which has low quality data, always wants to be a part of CDL to increase accuracy of its local model. Other mobile edge devices, who have high quality data, do not want to collaborate with low quality data holder mobile edge device. Therefore, there is a dilemma for mobile edge devices to participate or not in CDL. In this paper, we address this research problem of learner’s dilemma by proposing a general system model, a CDL game model, and a novel cluster-based fair strategy which enables each participant to cooperate in CDL based on the clusters formed to achieve overall benefit to itself in training the local ML model. We also evaluate our CDL game model and novel cluster-based strategy in smart home deployment using ARAS dataset[@SmartHome]. The main contributions of this paper are as follows. 1. We identify the problem of unfair cooperation of participants in CDL. A local training device, which has low quality data builds its learning model to take advantage from other device, which has high quality data. 2. We address this research problem by analyzing the behavior of mobile edge devices using a game-theoretic model, where each device aims at maximizing the accuracy of its local model with minimal cost of participating in CDL. 3. We introduce a system model for CDL and propose a solution of above defined problem. 4. We also implement a cluster-based fair algorithm on ARAS dataset [@SmartHome], and the results reflect that proposed solution elicit cooperation in CDL. The rest of paper is organized as follows. Section \[sec:related\] presents relevant work and related background. System model along with rational assumption is discussed in Section \[sec:system-model\]. Game model and game analysis are explained in Section \[sec:The Collaborative Deep Learning Game\] and Section \[sec:game-analysis\] respectively. Section \[sec:num-anal\] presents implementation of proposed system model along with results. Section \[sec:conc\] concludes the paper with future research directions. Related Work {#sec:related} ============ In this section, we describe related work on information leakage on deep learning models, privacy-preserving deep learning and game models. Information leakage on Deep Learning Models ------------------------------------------- Information leakage of individuals’ private data has become a well known problem for deep learning models. Data masking techniques, such as pseudonymize and anonymize are used to prevent this problem. In pseudonymize, data can be traced back into its original state, whereas it becomes impossible to return data into its original state in anonymize. However, indirect re-identification could be possible in anonymize. For example, Netflix released a hundred million anonymized film ratings which was matched with the other dataset Internet Movie Database (IMDb). Cloud platforms such as Google and Amazon offer various services “AI Deep Learning”. Any customer can upload a dataset to use the service and pay to build a prediction model, which works as black-box API. The membership inference attack on black-box API is discussed in [@shokri2017membership; @yeom2018privacy]. An attacker asks queries to target the model and receives the model’s prediction. Rahman et al. [@rahman2018membership] show that differentially private deep model can also fail against membership inference attack. A novel white-box membership inference attack was proposed by Nasr et al. [@nasr2018comprehensive], against deep learning algorithms to measure their training datasets membership leakage. Melis et al. [@melis2019exploiting] demonstrate that the updated parameter leaks information of participants, thus develops passive and active inference attacks to exploit this leakage. Privacy-Preserving Deep Learning -------------------------------- Each participant has its own sensitive datasets, which needs to be protected the dataset from information leakage. Various privacy mechanisms, such as Secure Multi-party Communication (SMC) [@kerschbaum2009practical], Homomorphic Encryption (HE) [@rivest1978data], and Differential Privacy (DP) [@dwork2014algorithmic] have been proposed to protect the datasets in CDL. SMC helps to protect intermediate steps of the computation. Mohassel et al. [@mohassel2017secureml] adopt a two-server model for privacy-preserving training, used by previous work on privacy-preserving deep learning via SMC [@nikolaenko2013privacy1]. However, Aono et al. [@aono2018privacy] pointed out that the local data may be actually leaked to an honest-but-curious server. Using additively HE techniques fix several problems and also have some drawbacks. To obscure an individual’s identity, DP adds mathematical noise to a small sample of the individual’s usage pattern. Prior work [@abadi2016deep; @jiang2019lightweight; @shokri2015privacy; @weng2018deepchain] use differential privacy on privacy-preserving CDL system to protect privacy of training data. However, Hitaj et al. [@hitaj2017deep] pointed out that privacy preserving deep learning approach is failed to protect data privacy and demonstrated that a malicious participant can learn personal information of other participant through Generative Adversarial Network (GAN) learning. The most dominant technique to optimize the loss function is Stochastic Gradient Descent (SGD). SGD is a method to find the optimal parameter configuration for a ML algorithm. SGD is applied in various privacy-preserving deep learning models [@abadi2016deep; @melis2019exploiting; @mohassel2017secureml; @nasr2018comprehensive]. PS receives the gradients from mobile edge devices by using different approaches like round robin, random order [@shokri2015privacy], cosine distance [@chen2018machine], time based [@weng2018deepchain]. The server aggregates the received parameters using Federated Averaging algorithm [@mcmahan2016communication], and weighted aggregation strategy [@chen2018machine]. Federated averaging algorithm introduced for model averaging combines local SGD on each client with a server. It is robust for unbalanced and non-IID data distributions, and reduce rounds of communication needed to train a Deep Learning (DL) model. Game Models ----------- In prior academic research, game theory has been applied into data privacy game to analyze privacy and accuracy. Pejo et al. [@pejo2018price] defined two player game, in which one is privacy concerned and other not. Esposito et al. [@esposito2018securing] proposed a game model to analyze the interaction between a provider (global ML model) and a requester (local ML model) within a CDL model. In this literature, there have been various game models about privacy-accuracy trade-off and energy-efficient solution. However, to the best of our knowledge, there is no prior work to utilize game theory to analyze mobile edge devices’ rational behavior in a selfish environment. Therefore, we construct a game model for rational mobile edge devices in CDL and analyze the game. {width=".5\textwidth" height=".23\textheight"} System Model {#sec:system-model} ============ We first generically outline details of the CDL model, in which all edge computing devices, such as mobile phones and IoT devices are assumed to be altruistic. Then, we clarify the rationality assumptions of mobile edge devices in CDL model. Collaborative Deep Learning Model --------------------------------- Figure 1 illustrates the main components of the system model. Consider there are *N* number of mobile edge devices, and each mobile edge device is connected with multiple IoT devices. These IoT devices generate huge amount of data, which is used for training to build ML model. These devices train their data to build local models in a collaborative manner without compromising data privacy, which is beneficial for mobile edge devices and IoT devices. In our model, we assume that each mobile edge device maintains a local vector of neural-network parameters, $w^i$. The PS maintains a separate parameter vector $w^{global}$. Each edge device can initialize parameters (weights) $w^i$ where *i*=1,2,3,..*N* randomly or by downloading their latest values from PS. Each edge device trains a local model and optimize the loss value using SGD. Here, one weight sample is selected at random in each optimization step. This process continues until SGD converges to a local optimum. Let E be the error or loss value, i.e., the difference between the true value of the objective function and the computed output of the network, it can be based on $L^2$ norm or cross entropy. The back-propagation algorithm computes the partial derivate of E with respect to each parameter in $w^k$ and updates the parameter so as to reduce its gradient. We refer to one full iteration over all available input data as an epoch. All mobile edge devices train their local models simultaneously through PS. $$\Delta w^{i} = \Delta w^{i} - \alpha\frac{\partial E_{i}}{\partial w^{i}}$$ There is no need for any coordination among all local training devices. They can influence each other’s training indirectly, via PS. PS receives local gradients $\Delta{w^i}$ from each edge devices and aggregates them with global parameter *w^global^*. After updating this global parameter, each participant downloads *w^global^* parameter from PS and starts training based on global parameter. There are various scenarios to exchange the parameters from PS to mobile edge device. In this model, PS exchanges the parameters asynchronously, i.e. PS does not wait for all local gradients from all edge devices. When a participant trains his local model, others may update their parameters through PS. This process continues until the model achieves the desired output. $$w\textsuperscript{global} = w\textsuperscript{global} + \Delta w^{i}$$ Mobile Edge Device costs ------------------------ We now characterize the costs (computation and communication costs) borne by mobile edge devices and IoT devices to their participation in CDL system. It should be noted that our goal is not to arrive at a precise quantification of these costs, rather to characterize them such that they could be used to analyze the strategic behavior of the devices while participating in CDL. The CDL system is basically grouped into two phases: (1) Training phase, and (2) Participating phase. During the training phase, each device builds a local model and initialize their weights to train the network. During training, each participant calculates its local gradients to upload on PS. PS aggregates all the local gradients and sends back to each device. The updated parameters are downloaded by each device and training process continues until loss value becomes negligible. Thus, we can characterize the total cost for a mobile edge device to participate in an epoch to build ML model based on the cost for executing the above two phases. For the training phase, a mobile edge device bears a cost *c^plocal^*, which is computation cost to build a local ML model. Another computation cost is *c^pglobal^* for training a local model using updated global parameters. Accordingly, for executing the participation phase, a mobile edge devices bears another costs *c^m’^* and *c^m^*. The cost *c^m^* is communication cost, where a mobile edge device uploads its parameters to PS, the cost *c^m’^* is also communication cost, where a mobile edge device $i$ downloads the updated parameters from PS. The average per mobile edge devices cost $c_{i}^{t}$ for participation in each epoch of collaborative deep learning system can be characterized as $$c_i^t = c^{plocal} + c^{m} + c^{m'} + c^{pglobal}$$ One point that needs further clarification is why a participant may choose not to spend these costs *c^m^*, *c^m’^*, and *c^pglobal^*. Our rationality assumption provides this clarification. Rationality Assumption ---------------------- Prior research in distributed DL [@chen2018machine] have assumed a byzantine adversary where mobile edge devices or IoT devices controlled by adversary can be arbitrarily malicious, i.e. malicious participant could arbitrarily deviate from suggested protocol in CDL or could arbitrarily drop communication between mobile edge device and PS. However, here we assume that mobile edge devices and IoT devices are honest but they are selfish. In this setup, the notion of rationality means that a rational device choose to participate or not to maximize its profit in CDL. **Symbol** **Definition** ------------------------------ ---------------------------------------------------------------------- $N$ Number of mobile edge devices $n$ Total number of IoT devices $K$ Batch size $H$ Numbers of local epoch $D\textsubscript{i}$ Generated data from IoT device i $\Delta w\textsubscript{i}$ Local gradient of participant i $w\textsuperscript{global}$ Global parameter $\alpha$ Learning rate $M$ Privacy mechanism $\theta\textsubscript{i}$ Loss value of participant i, train individually $\phi\textsubscript{i}$ Loss value of participant i,train collaboratively $\tau_i$ Loss value of participant i, train individually on auxiliary dataset $B$ coefficient $c\textsuperscript{plocal}$ Computation cost to build a local model $c\textsuperscript{pglobal}$ Computation cost to build a global model $c\textsuperscript{m}$ Communication cost to upload the parameters to PS $c\textsuperscript{m'}$ Communication cost to download the parameters from PS $c_i\textsuperscript{t}$ Total cost for build a ML model $C_i$ Number of cooperative participants $N-C_i$ Number of defective participants : List of Symbols. \[tab:Symbols\] The Collaborative Deep Learning Game {#sec:The Collaborative Deep Learning Game} ==================================== We present a game model of CDL system with multiple mobile edge devices in a honest but selfish environment. We introduce a game model with $N$-players that we refer to as the collaborative learning game G. In this game, the edge devices send their local gradients to PS to learn a common objective without compromising the privacy of data. PS aggregates the gradients and creates a global model. This updated global model is downloaded by all participating edge devices, where exists a social dilemma for all defection behavior. Game Model ---------- Game theory allows for modeling situations of conflict and for predicting the behavior of participants when they interact with each other. In our CDL game G, mobile edge devices who are connected with multiple IoT devices are participants, they interact with PS simultaneously without having any knowledge about each other. The Game G is a static game, because all participants must choose their strategy simultaneously. The Game G is a tuple $(P,S,U)$, where $P$ is the set of players, $S$ is the set of strategies and $U$ is the set of payoff values. - **Players** ($P$): The set of players $P=\sum_{i=1}^{N} P_{i}$ corresponds to the set of mobile edge devices who received a common objective from PS to build its own local model in CDL game G. - **Strategy** ($S$): Each participant $P_i$ can choose between two actions $s_i$ (i) Cooperative ($CP$) or (ii) Defective ($DF$). Hence the set of strategies in this game is $S$ = {$CP$,$DF$}. Strategy of mobile edge devices $P_i$ determines whether $P_i$ participates in CDL. In particular, if a participant $P_i$ plays $CP$ strategy, i.e., it will send its local gradients to PS and downloads updated parameters from PS to update its local model. Here, the participant pays total cost. In contrast, if a participant $P_i$ neither uploads its local gradients to PS nor downloads the updated global parameters from PS, i.e. the mobile edge devices $P_i$ plays $DF$ strategy. Thus, participants saves its communication costs $c^m$, $c^m{}^{'}$ and global computation cost $c^{pglobal}$. Here, this participant is not part of CDL and trains its local model individually. - **Payoff** ($U$): At a high level, the players’ goal in CDL game G is to maximize their utility, which is a function of the loss value and its costs. In this work, we do not consider the adversarial aspect of players; hence, the gain includes only the accuracy improvements on the model for a particular player as a benefit while the costs are used for training the models and communication between participant and PS. Here, the benefit and the cost are not on the same scale as the first depends on the loss value while the latter on cost. To make them comparable, we introduce a coefficient: the benefit is multiplied with B. Now, we compute the payoff of mobile edge devices $P_i$ in this game. If we assume that the participant $P_i$ is cooperative, i.e. $P_i$ $\in$ $CP$. Similarly, if $P_i$ is defective, i.e. $P_i$ $\in$ $DF$, and these payoff can be defined as follows. $$u_i(CP) = B(\frac{1}{\phi_i}) - (c^{plocal} + c^m + c^{m'} + c^{pglobal})$$ $$u_i(DF) = B(\frac{1}{\theta_i}) - (c^{plocal})$$ Based on the above calculated utilities, we analyze the game G as discussed in the next section. Game Analysis {#sec:game-analysis} ============= In order to get an insight into strategic behavior of participants, we apply the most fundamental game-theoretic concept, known as Nash Equilibrium, introduced by John Nash [@nash1951non]. **Definition 1.** A Nash Equilibrium is a concept of game theory where none of the players can unilaterally change their strategy to increase their payoff. In other words, if in a non-cooperative game all strategies are mutual best responses to each other, then no player has any motivation to deviate unilaterally from the given strategy one Nash Equilibrium strategy profile. For example, in any prisoners’ dilemma game, there is always a cooperative strategy and a defecting strategy. If both players use cooperative strategy, then it yields the best outcome for the players. If the players do not cooperate with one another, then they choose defecting strategy in the hope of attaining individual gain at the rival’s expense. In prisoners’ dilemma defecting strategy strictly dominates the cooperation strategy. Hence, the only Nash Equilibrium in prisoners’ dilemma, is a mutual defection. Based on the cost and benefit of mobile edge devices to learn a neural-network model, we build a one-shot CDL game model G. In the following theorems, we show that the game G is a public good game. **Theorem 1.** *In CDL game G, if a participant builds its local ML model, then G reduces to a public good game.* *Proof.* Let us consider all *N* participants follow defective-$DF$ strategy where all participants neither send their local gradients to PS nor download updated parameters from PS, i.e., no communication between mobile edge device and PS. So, participants do not pay any communication costs $c^m$, $c^{m'}$, and global computation cost $c^{pglobal}$. Now each participant $P_i$ trains local data sets $D_i$ to build its ML model individually and minimizes its loss value $\theta_i$. None of participants cannot change his strategy profile unilaterally. Let us assume if a participant deviates from defect-$DF$ strategy to cooperate-$CP$ strategy unilaterally, then participant will pay all these costs ($c^m$ + $c^{m'}$ + $c^{pglobal}$). The payoff of cooperate-$CP$ strategy is less than defect-$DF$ strategy, so All-$DF$ is a Nash equilibrium profile and G is a public good game. Theorem 2 further shows we can never enforce an all cooperate-$CP$ strategy in game G, and therefore, we could not establish a Nash Equilibrium. **Theorem 2.** *In CDL game G, if a participant builds its local ML model, then we cannot establish All-Cooperation strategy profile as a Nash Equilibrium.* *Proof.* We first assume that all *N* participants have already cooperated in collaborative learning (i.e., all cooperate-$CP$ strategy profile) and payed communication costs as well as global computation cost. We can compute the payoff of each participant $P_i$ by Equation (4). Hence, if a participant deviates from the cooperation and play defection unilaterally, its payoff would be equal to Equation (5), which is always greater than cooperative payoffs at Equation (4). Hence, each participant has incentive to deviate unilaterally and increases its payoff. Then, the All cooperate-$CP$ strategy profile is never a Nash Equilibrium. Cluster-Based Representation ---------------------------- Each participant has loss values of all other participants, which is calculated on auxiliary dataset. Before the start of the game, each participant has to choose his strategy to play the collaborative game G. However, in the beginning of the game, the participant is not sure about his strategy, which will depend on other participant’s strategy. Therefore, all the participants are in dilemma to choose a strategy between $CP$ and $DF$. We solve this problem by proposing the cluster-based fair strategy algorithm. K-means clustering is an unsupervised ML technique, whose purpose is to segment a data set into K clusters. Each participant applies k-means cluster algorithm on all loss values (one-dimensional data). $P_i$ , $P_j$ $\in$ $CP$ $P_i$ $\in$ $DF$ {width=".60\textwidth" height=".30\textheight"} Numerical Analysis {#sec:num-anal} ================== To evaluate our proposed cluster-based fair strategy, we apply this novel strategy on smart home datasets. Datasets -------- We use ARAS datasets to build smart home interaction model, since this data is available publicly [@SmartHome]. ARAS dataset is real world dataset for activity recognition with ambient sensing. The living residents did not follow any specific rules to live in smart homes. This dataset contains two real smart home data with multiple residents for one month. It contains 3000 daily life activities captured by 26 million sensor readings in smart homes. This dataset also has ground truth labels for activities, which enables to develop a new sophisticated ML smart home interaction model. Experimental Setup ------------------ We simulate the proposed system model with *N* number of mobile edge devices associated with smart homes. IoT devices are connected with one mobile edge device in each smart home. We partitioned ARAS dataset unevenly into 10 participants. For unbalanced datasets setting, the data is sorted by class and divided into two cases: (a) low quality dataset, where the participant receives data partition from a single class, and (b) high quality, where participant receives data partition from 27 classes. Figure 2 shows unbalanced partitioning of the dataset, smart home-1 generates high quality data (multiple IoT devices), while smart home-10 generates low quality dataset (one IoT device). The following parameters are used for Algorithm 1 and 2: batch size K = 10 or 100, H = 1 or 3, $\alpha$ = 0.01. Results ------- Our goal in this work is to design a mechanism for eliciting cooperation in CDL. Cluster-based fair strategy enforces participants for cooperation in CDL; however, Theorem 1 and 2 proves that participants defect in CDL. For the unbalanced datasets, the clusters of loss values are shown in figure 3. The results show that 80% participants collaborates with other participants, and 20% participants learns individually by choosing $DF$ strategy in the game G. {width=".5\textwidth"} Conclusion and Future Work {#sec:conc} ========================== In this paper, we presented a system model of CDL, and introduced the problem of strategic behavior of mobile edge devices in CDL system. We evaluated rationality of mobile edge devices in CDL using game theory model, a CDL game. We also evaluated the Nash Equilibrium (NE) strategy profile for each scenario, where the learning mobile edge devices are enforced to cooperate using our cluster-based fair strategy in CDL. We believe that this work is the first step towards a deeper understanding of the effect of non-cooperative behavior in CDL. For future work, we plan to extend the model and evaluation to determine the accuracy of ML/DL model and to train our proposed model with other IoT datasets.

--- abstract: | The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. Let $n$ be an even number and $\mathbb{U}_{n}$ be the set of all conjugated unicyclic graphs of order $n$ with maximum degree at most $3$. Let $S_n^{\frac{n}{2}}$ be the radialene graph obtained by attaching a pendant edge to each vertex of the cycle $C_{\frac{n}{2}}$. In \[Y. Cao et al., On the minimal energy of unicyclic Hückel molecular graphs possessing Kekulé structures, Discrete Appl. Math. 157 (5) (2009), 913–919\], Cao et al. showed that if $n\geq 8$, $S_n^{\frac{n}{2}}\ncong G\in \mathbb{U}_{n}$ and the girth of $G$ is not divisible by $4$, then $E(G)>E(S_n^{\frac{n}{2}})$. Let $A_n$ be the unicyclic graph obtained by attaching a $4$-cycle to one of the two leaf vertices of the path $P_{\frac{n}{2}-1}$ and a pendent edge to each other vertices of $P_{\frac{n}{2}-1}$. In this paper, we prove that $A_n$ is the unique unicyclic graph in $\mathbb{U}_{n}$ with minimal energy.\ [**Keywords:**]{} Minimal energy; Unicyclic graph; Perfect matching; Characteristic polynomial; Degree\ [**AMS Subject Classification 2000:**]{} 15A18; 05C50; 05C90; 92E10 --- =0.30in \[section\] \[lem\][Theorem]{} \[lem\][Corollary]{} \[lem\][Conjecture]{} \[lem\][Remark]{} \[lem\][Definition]{} [**On the minimal energy of conjugated unicyclic graphs with maximum degree at most 3**]{} =0.20in =0.20in [ Hongping Ma$^{1}$, Yongqiang Bai$^{1}$[^1], Shengjin Ji$^{2}$\ $^{1}$ School of Mathematics and Statistics, Jiangsu Normal University,\ Xuzhou 221116, China\ $^{2}$ School of Science, Shangdong University of Technology,\ Zibo 255049, China\ Email: hpma@163.com, bmbai@163.com, jishengjin2013@163.com ]{} =0.27in Introduction ============ Let $G$ be a simple graph with $n$ vertices and $A(G)$ the adjacency matrix of $G$. The eigenvalues $\lambda_{1}, \lambda_{2},\ldots, \lambda_{n}$ of $A(G)$ are said to be the eigenvalues of the graph $G$. The energy of $G$ is defined as $$E=E(G)=\sum_{i=1}^{n}|\lambda_{i}|.$$ This concept was intensively studied in chemistry, since it can be used to approximate the total $\pi$-electron energy of a molecular. Further details on the mathematical properties and chemical applications of $E(G)$, see the recent book [@LSG], reviews [@G2; @GLZ], and papers [@BB; @DM; @DMG; @GFAHG; @LSWL; @MMZ; @Z]. One of the fundamental question that is encountered in the study of graph energy is which graphs (from a given class) have minimal and maximal energies. A large of number of papers were published on such extremal problems, especially for various subclasses of trees and unicyclic graphs, see Chapter 7 in [@LSG]. A conjugated unicyclic graph is a connected graph with one unique cycle that has a perfect matching. The problem of determining the conjugated unicyclic graph with minimal energy has been considered in [@LZZ; @WCL], and Li et al. [@LZZ] proved that the conjugated unicyclic graph of order (even) $n$ with minimal energy is $U_1$ or $U_2$, as shown in Figure \[fig-minimal\]. It has been shown that $E(U_1)<E(U_2)$ by Li and Li [@LL]. Recently, results on ordering of conjugated unicyclic graphs by minimal energies have been extended in [@W; @Z]. In particular, $U_2$ is unique conjugated unicyclic graph of order $n$ with second-minimal energy. (1784.8, 562.2)(0,0) (0,0)[![The conjugated unicyclic graphs with minimal and second-minimal energy.[]{data-label="fig-minimal"}](fig-minimal "fig:")]{} (73.92,38.20) (150.0, 60.0)\[l\] $U_1$ (1294.83,382.88) (450.0, 60.0)\[l\] $\frac{n}{2}-2$ (561.90,383.72) (450.0, 60.0)\[l\] $\frac{n}{2}-3$ (944.44,58.39) (150.0, 60.0)\[l\] $U_2$ The degree of a vertex $v$ in a graph $G$ is denoted by $d_G(v)$. Denote by $\Delta$ the maximum degree of a graph. From now on, let $n$ be an even number. Let $\mathbb{U}_{n}$ be the set of all conjugated unicyclic graphs of order $n$ with $\Delta\leq 3$. Let $G\in \mathbb{U}_{n}$, the length of the unique cycle of $G$ is denoted by $g(G)$, or simply $g$, and the unique cycle of $G$ is denoted by $C_g(G)$, or simply $C_g$. Let $S_n^{\frac{n}{2}}$ be the radialene graph obtained by attaching a pendant edge to each vertex of the cycle $C_{\frac{n}{2}}$. Wang et al. [@WCZL] showed the following results: Assume that $n\geq 6$ and $S_n^{\frac{n}{2}}\ncong G\in \mathbb{U}_{n}$. Then if one of the following conditions holds: (i) $\frac{n}{2}\equiv g\equiv 1$ (mod $2$) and $g\leq\frac{n}{2}$, (ii) $g\not\equiv \frac{n}{2}\equiv 0$ (mod $4$), (iii) $\frac{n}{2}\equiv g\equiv 2$ (mod $4$), and $g\leq\frac{n}{2}$, then $E(G)>E(S_n^{\frac{n}{2}})$. Y. Cao et al. [@CLLZ] improved the above results by proving the following Lemma. [[@CLLZ]]{.nodecor}\[lem g-non-4-multiple\] If $n\geq 8$, $S_n^{\frac{n}{2}}\ncong G\in \mathbb{U}_{n}$ with $g\not\equiv 0$ [(mod $4$)]{.nodecor}, then $E(G)>E(S_n^{\frac{n}{2}})$. Let $A_n$, $B_n$, $D_n$ and $E_n$ be the graphs shown in Figure \[fig-minimal-degree-3\]. (2792.5, 525.4)(0,0) (0,0)[![Four graphs in $\mathbb{U}_{n}$.[]{data-label="fig-minimal-degree-3"}](fig-minimal-degree-3 "fig:")]{} (185.67,39.45) (150.0, 60.0)\[l\] $A_n$ (374.88,419.32) (450.0, 60.0)\[l\] $\frac{n}{2}-2$ (847.63,38.20) (150.0, 60.0)\[l\] $B_n$ (1012.53,423.62) (450.0, 60.0)\[l\] $\frac{n}{2}-3$ (1472.33,42.99) (150.0, 60.0)\[l\] $D_n$ (1578.87,409.06) (450.0, 60.0)\[l\] $\frac{n}{2}-3$ (1578.87,409.06) (450.0, 60.0)\[l\] $\frac{n}{2}-3$ (2196.00,41.96) (150.0, 60.0)\[l\] $E_n$ (2302.54,408.04) (450.0, 60.0)\[l\] $\frac{n}{2}-4$ In this paper, we completely characterize the graph with minimal energy in $\mathbb{U}_{n}$ by showing the following result. \[main thm\] $A_n$ is the unique unicyclic graph in $\mathbb{U}_{n}$ with minimal energy for $n\geq 8$. Preliminaries ============= In this section, we first introduce some notations and properties which are need in the sequel. Then we give some results on the energies of graphs $A_n$, $B_n$, $D_n$, $E_n$ and $S_n^{\frac{n}{2}}$. Let $G$ be a graph of order $k$. The characteristic polynomial of $A(G)$ is also called the characteristic polynomial of $G$, denoted by $\phi(G,x)=\mbox{det}(xI-A(G))=\sum_{i=0}^{k}a_i(G)x^{k-i}$. Using these coefficients of $\phi(G,x)$, the energy of $G$ can be expressed as the Coulson integral formula [@GP]: $$\begin{aligned} E(G)=\frac{1}{2\pi}{\large\int}_{-\infty}^{+\infty}\frac{1}{x^{2}}\ln\left[\left(\sum\limits_{i=0}^{\lfloor \frac{k}{2}\rfloor} (-1)^ia_{2i}(G)x^{2i}\right)^2+\left(\sum\limits_{i=0}^{\lfloor \frac{k}{2}\rfloor} (-1)^ia_{2i+1}(G)x^{2i+1}\right)^2\right]dx. \label{energy-1}\end{aligned}$$ Write $b_i(G)=|a_i(G)|$. Clearly, $b_0(G)=1$, $b_1(G)=0$, and $b_2(G)$ equals the number of edges of $G$. For unicyclic graphs or bipartite graphs, it can be shown [@GP; @H] that $$\begin{aligned} E(G)=\frac{1}{2\pi}{\large\int}_{-\infty}^{+\infty}\frac{1}{x^{2}}\ln\left[\left(\sum\limits_{i=0}^{\lfloor \frac{k}{2}\rfloor} b_{2i}(G)x^{2i}\right)^2+\left(\sum\limits_{i=0}^{\lfloor \frac{k}{2}\rfloor} b_{2i+1}(G)x^{2i+1}\right)^2\right]dx. \label{energy-2}\end{aligned}$$ By formula , it is convenient to introduce the following quasi-order relation [@SSGL]: if $G_1$ and $G_2$ are two unicyclic or bipartite graphs with $k$ vertices, then $$G_1\succeq G_2 \Leftrightarrow b_i(G_1)\geq b_i(G_2) \mbox{ for all } i=2, \ldots, k.$$ If $G_1\succeq G_2$ and there exists some $j$ such that $b_j(G_1) > b_j(G_2)$, then we write $G_1\succ G_2$. Clearly, $G_1\succeq G_2\Rightarrow E(G_1)\geq E(G_2)$, and $G_1\succ G_2\Rightarrow E(G_1)>E(G_2)$. It is known [@CDS] that $b_{2i+1}(G)=0$ for a bipartite graph $G$, and $b_{2i}(G)$ equals the number of $i$-matchings of $G$ if $G$ is a tree. [[@LZ]]{.nodecor}\[lem delete-edge\] Let $G$ be a graph whose components are all trees except at most one being a unicyclic graph. \(1) If $G$ contains exactly one cycle $C_g$, and $uv$ is an edge on this cycle, then $$\begin{array}{lll} b_i(G)=b_i(G-uv) + b_{i-2}(G-u-v)-2b_{i-g}(G-C_g)\ \mbox{if}\ g\equiv 0 \ (\textnormal{mod} \ 4),\\ b_i(G)=b_i(G-uv) + b_{i-2}(G-u-v)+2b_{i-g}(G-C_g)\ \mbox{if}\ g\not\equiv 0 \ (\textnormal{mod} \ 4). \end{array}$$ (2) If $uv$ is a cut edge of $G$, then $$b_i(G)=b_i(G-uv) +b_{i-2}(G-u-v).$$ In particular, if $uv$ is a pendent edge with pendent vertex $u$, then $$b_i(G)=b_i(G-u) + b_{i-2}(G-u-v).$$ [[@G1]]{.nodecor}\[lem tree maximal-minimal\] Let $T$ be a tree on $n$ vertices. If $T$ is different from the path $P_n$ and the star $S_n$, then $P_n\succ T\succ S_n$. Let $T$ be a tree of order $n\geq 3$, $e=uv$ be a non-pendent edge of $T$. Denote by $T_1$ and $T_2$ the two components of $T-e$ with $u\in T_1$ and $v\in T_2$. If $T'$ is the tree obtained from $T$ by contracting the edge $e=uv$ and attaching a pendent vertex to the vertex $u$ ($=v$), we say that $T'$ is obtained from $T$ by edge-growing transformation (on edge $e=uv$), or e.g.t (on edge $e=uv$) for short [@Xu; @LGL]. [[@LGL]]{.nodecor}\[lem edge-growing transformation\] If $T'$ is obtained from $T$ by one step of e.g.t, then $T\succ T'$. [[@SSGL]]{.nodecor}\[lem delete-edge-2\] Let $G$ be a unicyclic or bipartite graph, $uv$ be a cut edge of $G$. Then $G\succ G-uv$. [[@SSGL]]{.nodecor}\[lem delete-vertex\] Let $G$ be a unicyclic or bipartite graph, $v$ be a non-isolated vertex in $G$, and $K_1$ be the trivial graph of order $1$. Then $G\succ (G-v)\cup K_1$. Let $F_n$ and $H_{n+1}$ be the graphs shown in Figure \[fig-minimal-tree-degree-3\]. (1197.3, 437.5)(0,0) (0,0)[![The graphs $F_n$ and $H_{n+1}$.[]{data-label="fig-minimal-tree-degree-3"}](fig-minimal-tree-degree-3 "fig:")]{} (78.18,43.07) (150.0, 60.0)\[l\] $F_n$ (114.75,329.90) (390.0, 60.0)\[l\] $\frac{n}{2}$ (765.29,38.20) (270.0, 60.0)\[l\] $H_{n+1}$ (767.34,335.67) (390.0, 60.0)\[l\] $\frac{n}{2}$ [[@ZL]]{.nodecor}\[lem minimal-tree-degree-3\] Among conjugated trees of order $n$ with $\Delta\leq 3$, $F_n$ has minimal energy. \[lem formular for Bn and An\] For $A_n$, $B_n$, $D_n$ and $E_n$, we have that \(1) $ b_{2i}(A_n) = b_{2i}(F_n) + b_{2i-2}(F_{n-2})-2 b_{2i-4}(F_{n-4}) = b_{2i}(H_{n-1})+2b_{2i-2}(H_{n-3}). $ \(2) $b_{2i}(B_n) = b_{2i}(A_n) + 2b_{2i-6}(F_{n-8}).$ \(3) -1cm -2cm $$\begin{array}{lll} b_{2i}(A_n) = b_{2i}(A_{n-2})+b_{2i-2}(A_{n-2})+b_{2i-2}(A_{n-4}) \ (n\geq 8),\\ b_{2i}(B_n) = b_{2i}(B_{n-2}) + b_{2i-2}(B_{n-2})+ b_{2i-2}(B_{n-4}) \ (n\geq 10),\\ b_{2i}(D_n) = b_{2i}(D_{n-2}) + b_{2i-2}(D_{n-2})+ b_{2i-2}(D_{n-4}) \ (n\geq 10),\\ b_{2i}(E_n) = b_{2i}(E_{n-2}) + b_{2i-2}(E_{n-2})+ b_{2i-2}(E_{n-4}) \ (n\geq 12). \end{array}$$ \(1) By Lemma \[lem delete-edge\] (1) and (2), we can obtain that $$\begin{array}{lll} b_{2i}(A_n) & = & b_{2i}(F_n) + b_{2i-2}(F_{n-2})-2 b_{2i-4}(F_{n-4})\\ & = & b_{2i}(H_{n-1})+2b_{2i-2}(H_{n-3}). \end{array}$$ \(2) By Lemma \[lem delete-edge\] (2), we have $$\begin{array}{lll} b_{2i}(B_n) & = & b_{2i}(K_2 \cup A_{n-2}) + b_{2i-2}(H_{n-3})\\ & = & b_{2i}(K_2 \cup A_{n-2}) + b_{2i-2}(F_{n-4})+ b_{2i-4}(F_{n-6}),\\ b_{2i}(A_n) & = & b_{2i}(K_2 \cup A_{n-2}) + b_{2i-2}(A_{n-4})\\ & = & b_{2i}(K_2 \cup A_{n-2}) + b_{2i-2}(F_{n-4})+ b_{2i-4}(F_{n-6})-2 b_{2i-6}(F_{n-8}). \end{array}$$ Hence $ b_{2i}(B_n) = b_{2i}(A_n)+2 b_{2i-6}(F_{n-8})$. \(3) The results directly follow from Lemma \[lem delete-edge\] (2). \[lem Bn and An\] $B_n\succ A_n$ for $n\geq 8$. It is easy to obtain that $$\begin{array}{lll} \phi(B_8,x) = x^8-8x^6+16x^4-8x^2,\\ \phi(A_8,x) = x^8-8x^6+16x^4-6x^2. \end{array}$$ So $B_8\succ A_8$. Suppose $n> 8$. By Lemma \[lem formular for Bn and An\], $b_{2i}(B_n)-b_{2i}(A_n)=2 b_{2i-6}(F_{n-8})\geq 0$ and $b_{8}(B_n)> b_{8}(A_n)$. So $B_n\succ A_n$ for $n>8$. The proof is thus complete. \[lem En and Dn\] $D_8\succ E_8$ and $E_n\succ D_n$ for $n\geq 10$. It is easy to obtain that $$\begin{array}{lll} \phi(E_{8},x) = x^8-8x^6+14x^4-8x^2+1,\\ \phi(D_{8},x) = x^8-8x^6+15x^4-8x^2+1,\\ \phi(E_{10},x) = x^{10}-10x^8+29x^6-31x^4+12x^2-1,\\ \phi(D_{10},x) = x^{10}-10x^8+29x^6-28x^4+10x^2-1,\\ \phi(E_{12},x) = x^{12}-12x^{10}+47x^8-74x^6+51x^4-14x^2+1,\\ \phi(D_{12},x) = x^{12}-12x^{10}+47x^8-72x^6+46x^4-12x^2+1. \end{array}$$ So $D_{8}\succ E_{8}$, $E_{10}\succ D_{10}$ and $E_{12}\succ D_{12}$. Suppose $n > 12$. By Lemma \[lem delete-edge\], $$\begin{array}{lll} b_{2i}(E_n) = b_{2i}(E_{12}\cup F_{n-12}) + b_{2i-2}(E_{10}\cup F_{n-14}),\\ b_{2i}(D_n) = b_{2i}(D_{12}\cup F_{n-12}) + b_{2i-2}(D_{10}\cup E_{F-14}). \end{array}$$ Hence we have $E_n\succ D_n$ for $n>12$. The proof is thus complete. \[lem Sn and Bn\] $E(S_8^{4})>E(B_8)$ and $S_n^{\frac{n}{2}}\succ B_n$ for $n\geq 10$. It is easy to obtain that $9.65685\doteq E(S_8^{4})>E(B_8)\doteq 9.15298$. Suppose $n\geq 10$. Since $b_{2i+1}(S_n^{\frac{n}{2}})\geq 0=b_{2i+1}(B_n)$, we only need to consider $b_{2i}(S_n^{\frac{n}{2}})$ and $b_{2i}(B_n)$. By Lemmas \[lem delete-edge\] and \[lem formular for Bn and An\], $$b_{2i}(S_n^{\frac{n}{2}}) = \left\{\begin{array}{ll} b_{2i}(F_n) + b_{2i-2}(F_{n-4})-2 b_{2i-\frac{n}{2}}(\frac{n}{2}K_1), & \mbox{if}\ g\equiv 0 \ (\textnormal{mod} \ 4) \\ b_{2i}(F_n) + b_{2i-2}(F_{n-4})+2 b_{2i-\frac{n}{2}}(\frac{n}{2}K_1), & \mbox{if}\ g\not\equiv 0 \ (\textnormal{mod} \ 4) \end{array}\right.,$$ $$\begin{array}{lll} b_{2i}(B_n) & = & b_{2i}(F_n) + b_{2i-2}(F_{n-2})-2b_{2i-4}(F_{n-4})+2b_{2i-6}(F_{n-8})\\ & = & b_{2i}(F_n) + b_{2i-2}(F_{n-4})+b_{2i-4}(F_{n-6})-b_{2i-4}(F_{n-4})+2b_{2i-6}(F_{n-8})\\ & = & b_{2i}(F_n) + b_{2i-2}(F_{n-4})-b_{2i-6}(F_{n-6})+b_{2i-6}(F_{n-8})\\ & = & b_{2i}(F_n) + b_{2i-2}(F_{n-4})-b_{2i-8}(F_{n-8})-b_{2i-8}(F_{n-10}). \end{array}$$ Since $$b_{2i-\frac{n}{2}}(\frac{n}{2}K_1) = \left\{\begin{array}{ll} 1, & \mbox{if} \ 2i=\frac{n}{2}\\ 0, & \mbox{otherwise}\end{array}\right.,$$ we have $b_{2i}(S_n^{\frac{n}{2}})\geq b_{2i}(B_n)$, $b_8(S_{10}^5)> b_8(B_{10})$, and $b_{10}(S_n^{\frac{n}{2}})> b_{10}(B_n)$ when $n\geq 12$. The proof is thus complete. In the following, we will show that $E(A_n)<E(D_n)$ by using the Coulson integral formula method, which had successfully been applied to compare the energy of two given graphs by Huo et al., see [@Huo1]-[@Huo4]. Before proving it, we prepare some results as follows. [[@CDS]]{.nodecor}\[lem characteristic polynomial\] Let $uv$ be an edge of $G$. Then $$\phi(G,x) = \phi(G-uv,x)-\phi(G-u-v,x)-2\sum_{C\in\mathcal {C}(uv)}\phi(G-C,x),$$ where $\mathcal{C}(uv)$ is the set of cycles containing $uv$. In particular, if $uv$ is a pendent edge with pendent vertex $v$, then $\phi(G,x) = x\phi(G-v,x)-\phi(G-u-v,x)$. [[@Z2]]{.nodecor}\[lem log inequality\] For any real number $X > -1$, we have $$\frac{X}{1 + X}\leq \log(1 + X)\leq X.$$ In particular, $\log(1 + X) < 0$ if and only if $X < 0$. [[@G2]]{.nodecor}\[lem energy difference\] If $G_1$ and $G_2$ are two graphs with the same number of vertices, then $$E(G_1)-E(G_2) =\frac{1}{\pi} \int_{-\infty}^{+\infty}\log\left|\frac{\phi(G_1,ix)}{\phi(G_2,ix)}\right|dx.$$ From Lemma \[lem characteristic polynomial\], we can easily obtain the following lemma. \[lem characteristic polynomial-An and Dn\] $ \phi(A_n,x) = (x^2-1)\phi(A_{n-2},x)-x^2\phi(A_{n-4},x)$ for $n\geq 8$, and $\phi(D_n,x) = (x^2-1)\phi(D_{n-2},x)-x^2\phi(D_{n-4},x)$ for $n\geq 10$. By some easy calculations, we have $\phi(A_6,x) = x^6-6x^4+6x^2$, $\phi(A_8,x) = x^8-8x^6+16x^4-6x^2$, $\phi(D_6,x) = x^6-6x^4+5x^2-1$ and $\phi(D_8,x) = x^8-8x^6+15x^4-8x^2+1$. Now for convenience, we define some notations as follows: $$\begin{aligned} Y_1(x)=\frac{x^2-1+\sqrt{x^4-6 x^2+1}}{2}, & & Y_2(x)=\frac{x^2-1-\sqrt{x^4-6 x^2+1}}{2}, \\ Z_1(x)=\frac{-x^2-1+\sqrt{x^4+6 x^2+1}}{2}, & & Z_2(x)=\frac{-x^2-1-\sqrt{x^4+6 x^2+1}}{2}, \\ A_1(x)=\frac{\phi(A_8,x)-Y_2(x)\phi(A_6,x)}{(Y_1(x))^4-(Y_1(x))^2x^2}, & & A_2(x)=\frac{\phi(A_8,x)-Y_1(x)\phi(A_6,x)}{(Y_2(x))^4-(Y_2(x))^2x^2},\\ B_1(x)=\frac{\phi(D_8,x)-Y_2(x)\phi(D_6,x)}{(Y_1(x))^4-(Y_1(x))^2x^2}, & & B_2(x)=\frac{\phi(D_8,x)-Y_1(x)\phi(D_6,x)}{(Y_2(x))^4-(Y_2(x))^2x^2},\end{aligned}$$ $$\begin{aligned} f_6(x)=x^6+6x^4+6x^2, & & f_8(x)=x^8+8x^6+16x^4+6x^2,\\ g_6(x)=x^6+6x^4+5x^2+1, & & g_8(x)=x^8+8x^6+15x^4+8x^2+1.\end{aligned}$$ It is easy to check that $Y_1(ix)=Z_1(x)$, $Y_2(ix)=Z_2(x)$, $$\begin{aligned} A_1(ix)=\frac{f_8(x)+Z_2(x)f_6(x)}{(Z_1(x))^4+(Z_1(x))^2x^2}, & & A_2(ix)=\frac{f_8(x)+Z_1(x)f_6(x)}{(Z_2(x))^4+(Z_2(x))^2x^2},\\ B_1(ix)=\frac{g_8(x)+Z_2(x)g_6(x)}{(Z_1(x))^4+(Z_1(x))^2x^2}, & & B_2(ix)=\frac{g_8(x)+Z_1(x)g_6(x)}{(Z_2(x))^4+(Z_2(x))^2x^2},\end{aligned}$$ $Z_1(x)+Z_2(x)=-x^2-1$ and $Z_1(x)Z_2(x)=-x^2$. In addition, for $x>0$, $0<\frac{Z_1(x)}{x}<1$; for $x<0$, $-1<\frac{Z_1(x)}{x}<0$. \[lem recursive formula of characteristic polynomial-An and Dn\] For $n\geq 6$ and $x\neq 0$, the characteristic polynomials of $A_n$ and $D_n$ have the following forms: $$\phi(A_n,x) = A_1(x)(Y_1(x))^{\frac{n}{2}}+A_2(x)(Y_2(x))^{\frac{n}{2}}$$ and $$\phi(D_n,x) = B_1(x)(Y_1(x))^{\frac{n}{2}}+B_2(x)(Y_2(x))^{\frac{n}{2}}.$$ By Lemma \[lem characteristic polynomial-An and Dn\], we have that $\phi(A_n,x), \phi(D_n,x)$ satisfy the recursive formula $f(n,x)=(x^2-1)f(n-2,x)-x^2f(n-4,x)$. Therefore, the form of the general solution of the linear homogeneous recursive relation is $f(n,x)=C_1(x)(Y_1(x))^{\frac{n}{2}}+C_2(x)(Y_2(x))^{\frac{n}{2}}$. By some simple calculations, together with the initial values $\phi(A_6,x)$ and $\phi(A_8,x)$ ($\phi(D_6,x)$ and $\phi(D_8,x)$, respectively), we can get that $C_i(x)=A_i(x)$ ($C_i(x)=B_i(x)$, respectively), $i=1, 2$. \[lem An and Dn\] $E(A_n) < E(D_n)$ for $n\geq 6$. By Lemma \[lem energy difference\], we have $$E(A_n)-E(D_n) =\frac{1}{\pi} \int_{-\infty}^{+\infty}\log\left|\frac{\phi(A_n,ix)}{\phi(D_n,ix)}\right|dx.$$ From Lemma \[lem recursive formula of characteristic polynomial-An and Dn\], we know that both $\phi(A_n,ix)$ and $\phi(D_n,ix)$ are polynomials of $x$ with all real coefficients. For convenience, we abbreviate $A_k(ix)$, $B_k(ix)$ and $Z_k(x)$ to $A_k$, $B_k$ and $C_k$ for $k=1,2$, and abbreviate $f_k(x)$ and $g_k(x)$ to $f_k$ and $g_k$ for $k=6,8$, respectively. In the following, we assume that $x\neq 0$. We distinguish two cases in terms of the parity of $n/2$. [**Case 1.**]{} $n=4k$ $(k\geq 2)$. Notice that $Z_1Z_2=-x^2$. When $n \rightarrow \infty$, $$\begin{aligned} \frac{\phi(A_n,ix)}{\phi(D_n,ix)} = \frac{A_1Z_1^{\frac{n}{2}}+A_2Z_2^{\frac{n}{2}}}{B_1Z_1^{\frac{n}{2}}+B_2Z_2^{\frac{n}{2}}} = \frac{A_2+A_1(\frac{Z_1}{x})^n}{B_2+B_1(\frac{Z_1}{x})^n} \rightarrow \frac{A_2}{B_2}.\end{aligned}$$ We will show that $$\begin{aligned} \log\left|\frac{\phi(A_n,ix)}{\phi(D_n,ix)}\right|<\log\left|\frac{A_2(ix)}{B_2(ix)}\right|=\log\frac{A_2}{B_2}.\end{aligned}$$ Assume that $$\begin{aligned} 2\log\left|\frac{\phi(A_n,ix)}{\phi(D_n,ix)}\right|-2\log\left|\frac{A_2}{B_2}\right|=\log\left(1+\frac{F_1(n,x)}{G_1(n,x)}\right).\end{aligned}$$ Then we get that $G_1(n,x)=(A_2(ix) \phi(D_n,ix))^2>0$ and $$\begin{aligned} F_1(n,x) & = & B_2^2(A_1Z_1^{\frac{n}{2}}+A_2Z_2^{\frac{n}{2}})^2-A_2^2(B_1Z_1^{\frac{n}{2}}+B_2Z_2^{\frac{n}{2}})^2\\ & = & (A_1^2B_2^2-A_2^2B_1^2)Z_1^n +2A_2B_2(A_1B_2-A_2B_1)(Z_1Z_2)^{\frac{n}{2}}\\ & < & 0.\end{aligned}$$ Since $Z_1^{n}>0$, $(Z_1Z_2)^{\frac{n}{2}}=(-x^2)^{\frac{n}{2}}>0$, $A_2B_2>0$, and by some elementary calculations, we have $$\begin{aligned} A_1B_2-A_2B_1=\frac{(-3x^8-15x^6-8x^4)\sqrt{x^4+6x^2+1}}{x^6(x^4+6x^2+1)}<0,\end{aligned}$$ and $$\begin{aligned} A_1B_2+A_2B_1=\frac{4x^4+17x^6+20x^8+3x^{10}}{x^6(x^4+6x^2+1)}>0.\end{aligned}$$ Hence by Lemma \[lem log inequality\], $$\begin{aligned} \frac{1}{\pi} \int_{-\infty}^{+\infty}\log \frac{A_2(ix)}{B_2(ix)}dx \leq \frac{1}{\pi} \int_{-\infty}^{+\infty} \left(\frac{A_2(ix)}{B_2(ix)}-1\right)dx\doteq \frac{1}{\pi} (-0.8538292323)<0.\end{aligned}$$ Therefore, $$E(A_n)-E(D_n)\leq \frac{1}{\pi} \int_{-\infty}^{+\infty}\log \frac{A_2(ix)}{B_2(ix)}dx<0.$$ [**Case 2.**]{} $n=4k+2$ $(k\geq 1)$. We will show that $\log\left|\frac{\phi(A_n,ix)}{\phi(D_n,ix)}\right|$ is monotonically decreasing in $n$. Assume that $$\begin{aligned} 2\log\left|\frac{\phi(A_{n+4},ix)}{\phi(D_{n+4},ix)}\right|-2\log\left|\frac{\phi(A_{n},ix)}{\phi(D_{n},ix)}\right|=\log\left(1+\frac{F_2(n,x)}{G_2(n,x)}\right).\end{aligned}$$ Then we can obtain that $G_2(n,x)=(\phi(A_n,ix)\phi(D_{n+4},ix))^2>0$ and $$\begin{aligned} F_2(n,x) & = & (\phi(A_{n+4},ix)\phi(B_n,ix))^2-(\phi(A_n,ix)\phi(D_{n+4},ix))^2\\ & = & (A_1Z_1^{\frac{n}{2}+2}+A_2Z_2^{\frac{n}{2}+2})^2(B_1Z_1^{\frac{n}{2}}+B_2Z_2^{\frac{n}{2}})^2 \\ & & - (B_1Z_1^{\frac{n}{2}+2}+B_2Z_2^{\frac{n}{2}+2})^2(A_1Z_1^{\frac{n}{2}}+A_2Z_2^{\frac{n}{2}})^2\\ & = & -(A_1B_2-A_2B_1)x^n(Z_1^2-Z_2^2)\cdot F_3(n,x),\end{aligned}$$ where $ F_3(n,x)=(A_1B_2+A_2B_1)(Z_1Z_2)^{\frac{n}{2}}(Z_1^2+Z_2^2)+2A_1B_1Z_1^{n+2}+2A_2B_2Z_2^{n+2}$. Since $A_1B_2-A_2B_1<0$, $Z_1^2-Z_2^2<0$ and $x^n>0$, to prove $F_2(n,x)<0$, it suffice to show that $F_3(n,x)>0$. By some elementary calculations, we can get that $$\begin{aligned} F_3(n,x)=\frac{2f_8g_8F_4(n,x)+2f_6g_6F_5(n,x)+(f_8g_6+f_6g_8)F_6(n,x)}{x^4+6x^2+1},\end{aligned}$$ where $$\begin{aligned} F_4(n,x) & = & Z_1^{n-4}+Z_2^{n-4}-(Z_1Z_2)^{\frac{n}{2}-3}(Z_1^2+Z_2^2)\\ & = & (Z_1^{\frac{n}{2}-1}-Z_2^{\frac{n}{2}-1})(Z_1^{\frac{n}{2}-3}-Z_2^{\frac{n}{2}-3}),\\ F_5(n,x) & = & Z_2^2Z_1^{n-4}+Z_1^2Z_2^{n-4}-(Z_1Z_2)^{\frac{n}{2}-2}(Z_1^2+Z_2^2)\\ & = & (Z_1Z_2)^2(Z_1^{\frac{n}{2}-2}-Z_2^{\frac{n}{2}-2})(Z_1^{\frac{n}{2}-4}-Z_2^{\frac{n}{2}-4}),\\ F_6(n,x) & = & 2Z_2Z_1^{n-4}+2Z_1Z_2^{n-4}-(Z_1+Z_2)(Z_1Z_2)^{\frac{n}{2}-3}(Z_1^2+Z_2^2)\\ & = & Z_1Z_2[(Z_1^{\frac{n}{2}-1}-Z_2^{\frac{n}{2}-1})(Z_1^{\frac{n}{2}-4}-Z_2^{\frac{n}{2}-4})+(Z_1^{\frac{n}{2}-3}-Z_2^{\frac{n}{2}-3})(Z_1^{\frac{n}{2}-2}-Z_2^{\frac{n}{2}-2})].\end{aligned}$$ Notice that $Z_1>0$, $Z_2<0$, we have $Z_1^{k}-Z_2^{k}>0$ when $k$ is odd. On the other hand, for $k\geq 1$, we have $$\begin{aligned} Z_1^{2k}-Z_2^{2k}=Z_1^{2k}\left[1-\left(\frac{x}{Z_1}\right)^{4k}\right]<0.\end{aligned}$$ Therefore, we have $F_4(n,x)\geq 0$, $F_5(n,x)\geq 0$, $F_6(n,x)>0$, and so $F_3(n,x)>0$ and $F_2(n,x)<0$. Hence, $$E(A_n)-E(D_n)\leq E(A_6)-E(D_6)\doteq 6.60272-7.20775<0.$$ The proof is thus complete. Main results ============ Let $G$ be a graph in $\mathbb{U}_{n}$ with vertex set $V$ and the unique cycle $C_g$. We use $M_G$ to denote one arbitrary selected prefect matching of $G$. Let $x, y\in V$. Denote by $d_G(x,y)$ ($d_G(x,C_g)$, respectively) the distance between vertex $x$ and $y$ ($C_g$, respectively). Define $d=d(G)=\max_{x\in V(G)}\{d_G(x,C_g)\}$, $V_1=V_1(G)=\{x\in V|d_G(x,C_g)=d(G)\}$, and $t=t(G)=|V_1(G)|$. Clearly, the vertices in $V_1$ are pendent vertices when $d(G)\geq 1$. \[lem girth at least 8-d equals 0\] Let $n\equiv 0$ [(mod $4$)]{.nodecor}, $n\geq 8$. Then $C_n\succ B_n$. By Lemma \[lem delete-edge\], $$b_{2i}(C_n) = \left\{\begin{array}{ll} b_{2i}(P_n) + b_{2i-2}(P_{n-2}), & \mbox{if}\ 2i\neq n \\ b_{2i}(P_n) + b_{2i-2}(P_{n-2})-2, & \mbox{if}\ 2i=n \end{array}\right.,$$ and $$b_{2i}(B_n)=b_{2i}(G_1) + b_{2i-2}(K_2\cup F_{n-4})-2b_{2i-4}(K_2\cup F_{n-6}),$$ where $G_1$ is the tree of order $n$ obtained by attaching a path with $4$ edges to one of the two vertices of degree $2$ of $F_{n-4}$. By Lemmas \[lem tree maximal-minimal\] and \[lem delete-edge-2\], $P_n\succ G_1$, $P_{n-2}\succ F_{n-2}\succ K_2\cup F_{n-4}$. On the other hand, we have $b_{n}(C_n)=b_{n}(B_n)$ and $b_{4}(C_n)>b_{4}(B_n)$. Hence $C_n\succ B_n$. \[lem girth at least 8-d equals 1\] Let $G\in \mathbb{U}_{n}$, $g(G)\equiv 0$ [(mod $4$)]{.nodecor}, $g(G)\geq 8$, and $d(G)=1$. Then $G\succ B_n$. It is easy to see that $n\geq 10$ and $t(G)$ is even. If $t=g$, then $G=S_n^{\frac{n}{2}}$, and so $G\succ B_n$ by Lemma \[lem Sn and Bn\]. So in the following, we suppose $2\leq t\leq g-2$. Let $C_g=y_1y_2\ldots y_g$, and $x_{k_1}y_{k_1}$, $x_{k_2}y_{k_2}$, …, $x_{k_t}y_{k_t}$ ($1=k_1< k_2<\cdots < k_t$) be all the edges outside $C_G$. Then there must exist an index $k_i$ such that $k_{i}+1, k_{i}+2$ (mod $g$) are not in the set $\{k_1, k_2, \ldots, k_t\}$, that is $y_{k_{i}+1}y_{k_{i}+2}\in M_G$. Without loss of generality, we assume that $k_i=1$. Since $G$ has a perfect matching, we have that $4\leq k_2 < k_3 < \cdots < k_{t}\leq g$, and $k_2-3$ is odd, $k_{i+1}-k_i$ is odd for $2\leq i\leq t-1$, and $g-k_{t}$ is even. By Lemma \[lem delete-edge\], we have $ b_{2i}(G) = b_{2i}(G-x_1) + b_{2i-2}(G-x_1-y_1)$, and $$b_{2i}(G-x_1) = \left\{\begin{array}{ll}b_{2i}(G-x_1-y_1y_2)+ b_{2i-2}(G-x_1-y_1-y_2), & \mbox{if}\ 2i\neq g \\ b_{2i}(G-x_1-y_1y_2)+ b_{2i-2}(G-x_1-y_1-y_2)-2, & \mbox{if}\ 2i=g \end{array}\right..$$ Note that $G-x_1-y_1$ is a conjugated tree of order $n-2$ with $\Delta\leq 3$, by Lemma \[lem minimal-tree-degree-3\], $G-x_1-y_1\succeq F_{n-2}$. Denote $T_1=G-x_1-y_1y_2$ and $T_2=G-x_1-y_1-y_2$. Notice that if $k_{i+1}-k_i=2k+1>1$, then from $T_1$, we can obtain a different tree $T_2$ with $\Delta \leq 3$ by carrying out $k$ steps of e.g.t. Therefore we can finally get $H_{n-1}$ from $T_1$ by carrying out e.g.t repeatedly, if necessary. By Lemma \[lem edge-growing transformation\], we have $T_1\succeq H_{n-1}$. Similarly, we can obtain that $T_2\succeq H_{n-3}$. By Lemma \[lem formular for Bn and An\], we have $$b_{2i}(B_n)= b_{2i}(H_{n-1}) + 2b_{2i-2}(H_{n-3})+2 b_{2i-6}(F_{n-8}).$$ Hence for $2i\neq g$, we have $$\begin{array}{lll} b_{2i}(G) & \geq & b_{2i}(H_{n-1}) + b_{2i-2}(F_{n-2})+b_{2i-2}(H_{n-3})\\ & = & b_{2i}(H_{n-1}) + 2b_{2i-2}(H_{n-3})+b_{2i-4}(F_{n-4})\\ & = & b_{2i}(B_n)+b_{2i-4}(F_{n-4})-2b_{2i-6}(F_{n-8})\\ & = & b_{2i}(B_n)+b_{2i-4}(F_{n-6})+b_{2i-8}(F_{n-8})+b_{2i-8}(F_{n-10})\\ & \geq & b_{2i}(B_n), \end{array}$$ and $$b_{g}(G) \geq b_{g}(B_n)+b_{g-4}(F_{n-6})+b_{g-8}(F_{n-8})+b_{g-8}(F_{n-10})-2\geq b_{g}(B_n),$$ since $g\leq n-2$, $b_{g-4}(F_{n-6})\geq 1$ and $b_{g-8}(F_{n-8})\geq 1$. On the other hand, it is obvious that $b_{4}(G) > b_{4}(B_n)$. Thus $G\succ B_n$. \[lem girth at least 8-d equals 2\] Let $G\in \mathbb{U}_{n}$, $g(G)\equiv 0$ [(mod $4$)]{.nodecor}, $g(G)\geq 8$, and $d(G)=2$. Then $G\succ B_n$. Let $C_g=z_1z_2\ldots z_g$. We apply induction on $t$. Suppose $t=1$. Assume that $d_G(x_1,z_1)=2$ and $x_1y_1z_1$ be a path with length $2$ in $G$. Then $d_G(y_1)=2$ and $x_1y_1\in M_G$. Since $G$ has a perfect matching, either $z_1z_2\in M_G$ or $z_1z_g\in M_G$. We may assume that $z_1z_2\in M_G$. Let $E_1=\{z_{i_1}y_{i_1}, z_{i_2}y_{i_2}, \ldots, z_{i_k}y_{i_k}\}$ ($i_1<i_2<\cdots <i_k$) be the set of all edges in $M_G\setminus x_1y_1$ outside $C_G$. If $E_1$ is not empty, then we have that $3\leq i_1 < i_2 < \cdots < i_{k}\leq g$, and $i_1-3$ is even, $i_{j+1}-i_j$ is odd for $1\leq j\leq k-1$, and $g-i_{k}$ is even. By Lemma \[lem delete-edge\], we have $$\begin{array}{lll} b_{2i}(G) & = & b_{2i}(G-x_1) + b_{2i-2}(G-x_1-y_1)\\ & = & b_{2i}(G-x_1-y_1) + b_{2i-2}(G-x_1-y_1-z_1) +b_{2i-2}(G-x_1-y_1). \end{array}$$ Denote $G_1=G-x_1-y_1$ and $G_2=G-x_1-y_1-z_1$. Then $G_1\in\mathbb{U}_{n-2}$ with $d(G_1)\leq 1$. By Lemmas \[lem girth at least 8-d equals 0\] and \[lem girth at least 8-d equals 1\], we have $G_1\succ B_{n-2}$. By an argument similar to the proof in Lemma \[lem girth at least 8-d equals 1\], we can obtain $G_2\succeq H_{n-3}$. Therefore by Lemmas \[lem delete-edge\] and \[lem formular for Bn and An\], $$\begin{array}{lll} b_{2i}(G) & \geq & b_{2i}(B_{n-2}) + b_{2i-2}(B_{n-2})+ b_{2i-2}(H_{n-3}) \\ & = & b_{2i}(B_{n-2}) + b_{2i-2}(B_{n-2})+ b_{2i-2}(H_{n-5})+ 2b_{2i-4}(H_{n-7}) + 2b_{2i-6}(F_{n-8})\\ & = & b_{2i}(B_{n-2}) + b_{2i-2}(B_{n-2})+ b_{2i-2}(A_{n-4}) + 2b_{2i-6}(F_{n-8}) \\ & = & b_{2i}(B_{n-2}) + b_{2i-2}(B_{n-2})+ b_{2i-2}(B_{n-4}) + 2b_{2i-6}(F_{n-8})-2b_{2i-8}(F_{n-12}) \\ & = & b_{2i}(B_{n-2}) + b_{2i-2}(B_{n-2})+ b_{2i-2}(B_{n-4}) + 2b_{2i-6}(F_{n-10})+2b_{2i-8}(F_{n-10}) \\ & = & b_{2i}(B_{n}) + 2b_{2i-6}(F_{n-10})+2b_{2i-8}(F_{n-10}) \\ & \geq & b_{2i}(B_{n}). \end{array}$$ Since $G_1\succ B_{n-2}$, there exist $i$ such that $b_{2i}(G)>b_{2i}(B_{n})$. Hence $G\succ B_n$ for $t=1$. Assume now that $t\geq 2$ and the assertion holds for smaller values of $t$. Let $V_1(G)=\{x_1, x_2, \ldots, x_t\}$, and for each $i\in \{1, 2, \ldots,t \}$, $x_iy_iz_{k_i}$ ($1=k_1<k_2<\cdots<k_t\leq g$) be a path with length $2$ in $G$. Then $d_G(y_i)=2$ and $x_iy_i\in M_G$. For convenience, let $k_{t+1}=k_1$. We consider the following two cases: [**Case 1.**]{} There exist two indices $k_i$ and $k_{i+1}$ such that $z_{k_i}z_{k_{i+1}}$ is an edge on $C_g$. Without loss of generality, we assume that $k_i=1, k_{i+1}=2$. Now let $G'$ be the graph obtained from $G$ by deleting the edge $z_2y_2$ and adding one new edge $y_2y_1$. Then $G'\in \mathbb{U}_{n}$, and $d(G')=3$. [**Claim 1.**]{} $G\succ G'$. By Lemma \[lem delete-edge\], we have $$\begin{array}{lll} b_{2i}(G) & = & b_{2i}(G-x_2) + b_{2i-2}(G-\{x_2,y_2\}),\\ b_{2i}(G') & = & b_{2i}(G'-x_2) + b_{2i-2}(G'-\{x_2,y_2\}), \end{array}$$ and $$\begin{array}{lll} b_{2i}(G-x_2) & = & b_{2i}(G-\{x_2,x_1\})+ b_{2i-2}(G-\{x_2,x_1,y_1\})\\ & = & b_{2i}(G-\{x_2,x_1,y_2\})+b_{2i-2}(G-\{x_2,x_1,y_2,z_2\})\\ & & +b_{2i-2}(G-\{x_2,x_1,y_1,y_2\})+b_{2i-4}(G-\{x_2,x_1,y_1,y_2,z_2\}),\\ b_{2i}(G'-x_2) & = & b_{2i}(G'-\{x_2,x_1\})+ b_{2i-2}(G'-\{x_2,x_1,y_1\})\\ & = & b_{2i}(G'-\{x_2,x_1,y_2\})+ 2b_{2i-2}(G'-\{x_2,x_1,y_2,y_1\}). \end{array}$$ Denote $G_3=G-\{x_2,x_1,y_2,z_2\}$, $G_4=G-\{x_2,x_1,y_1,y_2,z_2\}$ and $G_5=G-\{x_2,x_1,y_2,y_1\}$. Since $G-\{x_2,y_2\}=G'-\{x_2,y_2\}$, $G-\{x_2,x_1,y_2\}=G'-\{x_2,x_1,y_2\}$, and $G-\{x_2,x_1,y_1,y_2\}=G'-\{x_2,x_1,y_2,y_1\}$, we have $$b_{2i}(G)- b_{2i}(G') = b_{2i-2}(G_3)+b_{2i-4}(G_4) -b_{2i-2}(G_5).$$ Furthermore, $$\begin{array}{lll} b_{2i-2}(G_5) & = & b_{2i-2}(G_5-\{z_1z_2\})+b_{2i-4}(G_5-\{z_1,z_2\})-2b_{2i-g}(G_5-C_g)\\ & = & b_{2i-2}(G_5-\{z_1z_2,z_2z_3\})+b_{2i-4}(G_5-\{z_1z_2\}-\{z_2,z_3\})\\ & & +b_{2i-4}(G_5-\{z_1,z_2\})-2b_{2i-g}(G_5-C_g)\\ & = & b_{2i-2}(G_5-\{z_2\})+b_{2i-4}(G_5-\{z_2,z_3\})\\ & & +b_{2i-4}(G_5-\{z_1,z_2\})-2b_{2i-g}(G_5-C_g),\\ b_{2i-2}(G_3) & = & b_{2i-2}(G_3-\{y_1\})+b_{2i-4}(G_3-\{y_1,z_1\})\\ & = & b_{2i-2}(G_5-\{z_2\})+b_{2i-4}(G_5-\{z_1,z_2\}). \end{array}$$ Note that $G_5-\{z_2,z_3\}=G_4-z_3$. Hence $G_4\succ [G_5-\{z_2,z_3\}]\cup K_1$ by Lemma \[lem delete-vertex\]. Therefore $$b_{2i}(G)- b_{2i}(G') = b_{2i-4}(G_4) - b_{2i-4}(G_5-\{z_2,z_3\})+2b_{2i-g}(G_5-C_g)\geq 0,$$ and $$b_{6}(G)- b_{6}(G') = b_{2}(G_4)-b_{2}(G_5-\{z_2,z_3\})>0.$$ Thus the result $G\succ G'$ holds. Now it suffices to prove that $G'\succeq B_n$. By Lemma \[lem delete-edge\], we have $$\begin{array}{lll} b_{2i}(G') & = & b_{2i}(G'-x_2) + b_{2i-2}(G'-\{x_2,y_2\})\\ & = & b_{2i}(G'-\{x_2,y_2\}) + b_{2i-2}(G'-\{x_2,y_2,y_1\})+ b_{2i-2}(G'-\{x_2,y_2\})\\ & = & b_{2i}(G'-\{x_2,y_2\}) + b_{2i-2}(G'-\{x_2,y_2,y_1,x_1\})+ b_{2i-2}(G'-\{x_2,y_2\}). \end{array}$$ Denote $G_6=G'-\{x_2,y_2\}$ and $G_7=G'-\{x_2,y_2,y_1,x_1\}$. Then it is easy to see that $G_6\in\mathbb{U}_{n-2}$, $G_7\in\mathbb{U}_{n-4}$, and for $i=6,7$, we have $d(G_i)\leq 1$ or $d(G_i)=2$ and $t(G_i)<t(G)$. Therefore by Lemmas \[lem girth at least 8-d equals 0\], \[lem girth at least 8-d equals 1\] and the induction hypothesis, $G_6\succ B_{n-2}$, $G_7\succ B_{n-4}$ and so we have $G'\succ B_n$. [**Case 2.**]{} For the indices $1=k_1<k_2<\ldots<k_t\leq g$, we have $k_{i+1}-k_i\geq 2$ for $1\leq i\leq t-1$ and $k_t\leq g-2$. Since $G$ has a perfect matching, we may assume that $z_1z_2\in M_G$, and $z_{k_i}w_{k_i}\in M_G$ for $2\leq i\leq t$, where $w_{k_i}\in\{z_{k_i-1},z_{k_i+1}\}$. Similarly, we have $$b_{2i}(G) = b_{2i}(G-x_1-y_1) + b_{2i-2}(G-x_1-y_1-z_1) + b_{2i-2}(G-x_1-y_1),$$ and $$G-x_1-y_1\succ B_{n-2}.$$ Denote $T=G-x_1-y_1-z_1$. Let $T_1$ be the tree obtained from $T$ by deleting the edge $x_{i}y_{i}$ and adding one new edge $x_iw_{{k_i}}$ ($2\leq i\leq t$), we say that $T_1$ is obtained from $T$ by Operation I. [**Claim 2.**]{} $T\succ T_1$. By Lemma \[lem delete-edge\], we have $$\begin{array}{lll} b_{2i}(T) & = & b_{2i}(T-x_i)+ b_{2i-2}(T-\{x_i,y_i\}), \\ b_{2i}(T_1) & = & b_{2i}(T_1-x_i)+b_{2i-2}(T_1-\{x_i,w_{{k_i}}\}). \end{array}$$ Note that $T-x_i=T_1-x_i$, and $T_1-\{x_i,w_{{k_i}}\}$ is isomorphic to a proper subgraph of $T-\{x_i,y_i\}$, then $T-\{x_i,y_i\}\succ T_1-\{x_i,w_{{k_i}}\}$ by Lemma \[lem delete-edge-2\]. Therefore $T\succ T_1$. Let $T'$ be the tree obtained from $T$ by deleting $t-1$ edges $x_{2}y_{2}, x_{3}y_{3},\ldots, x_ty_t$ and adding $t$ new edges $x_{2}w_{{k_2}}, x_{3}w_{{k_3}},\ldots, x_tw_{{k_t}}$. Then from $T$ we can obtain $T'$ by applying Operation I $t-1$ times. By Claim 2, we have $T\succ T'$. Clearly, $T'$ is a tree with $\Delta\leq 3$. Now we can assume that $z_{j_1}, z_{j_2},\ldots, z_{j_{l}}$ ($2<j_1 < j_2 < \cdots < j_{l}\leq g$) are all vertices with degree $3$ in $T'$. Then we have $j_1-2$ is odd, $j_{i+1}-j_i$ is odd for $1\leq i\leq l-1$, and $g-j_{l}$ is even. By an argument similar to the proof in Lemma \[lem girth at least 8-d equals 1\], we can obtain $T'\succeq H_{n-3}$. Therefore we have $T\succ H_{n-3}$ and so similar to the above case $t=1$, we can finally obtain $G\succ B_n$. The proof is thus complete. Let $G$ be a graph in $\mathbb{U}_{n}$ with $g(G)\equiv 0$ [(mod $4$)]{.nodecor}, $d(G)\geq 3$. Suppose $C_g=z_1z_2\ldots z_g$, $x_1,y_1\in V_1(G)$, and $x_1x_2 x_3\ldots x_dz_1$, $y_1y_2x_3 \ldots x_dz_1$ be two paths with length $d$ in $G$. For convenience, denote $x_{d+1}=z_1$. If $G'$ is the graph obtained from $G$ by deleting two edges $x_3y_2$, $y_2y_1$ and adding two new edges $y_1x_1$ and $y_2x_2$, then we say that $G'$ is obtained from $G$ by Operation II. Clearly, $G'\in\mathbb{U}_{n}$. \[lem Operation 2\] Let $G$ be defined as above. If $G'$ is obtained from $G$ by Operation II, then $G\succ G'$. By Lemma \[lem delete-edge\], we have $$\begin{array}{lll} b_{2i}(G) & = & b_{2i}(G-y_1) + b_{2i-2}(G-\{y_1,y_2\})\\ & = & b_{2i}(G-\{y_1,y_2\})+ b_{2i-2}(G-\{y_1,y_2,x_3\}) + b_{2i-2}(G-\{y_1,y_2\}),\\ b_{2i}(G') & = & b_{2i}(G'-y_1) + b_{2i-2}(G'-\{y_1,x_1\})\\ & = & b_{2i}(G'-\{y_1,y_2\})+ b_{2i-2}(G'-\{y_1,y_2,x_2) + b_{2i-2}(G'-\{y_1,x_1\}). \end{array}$$ It is easy to see that $G'-\{y_1,y_2\}=G-\{y_1,y_2\}\cong G'-\{y_1,x_1\}$. Let $G_1=G-\{x_1,x_2,y_1,y_2\}$. Then $$\begin{array}{lll} b_{2i-2}(G-\{y_1,y_2,x_3\}) & = & b_{2i-2}((G_1-x_3)\cup K_2)\\ & = & b_{2i-2}(G_1-x_3)+b_{2i-4}(G_1-x_3),\\ b_{2i-2}(G'-\{y_1,y_2,x_2\}) & = & b_{2i-2}(G_1)\\ & = & b_{2i-2}(G_1-x_3)+b_{2i-4}(G_1-\{x_3,x_4\}).\\ \end{array}$$ Since $G_1-x_3\succ (G_1-\{x_3,x_4\})\cup K_1$ by Lemma \[lem delete-vertex\], we have $b_{2i}(G)\geq b_{2i}(G')$ and $b_{6}(G)> b_{6}(G')$. Thus the result $G\succ G'$ holds. Let $G$ be a graph in $\mathbb{U}_{n}$ with $g(G)\equiv 0$ [(mod $4$)]{.nodecor}, $d(G)=3$. Suppose $C_g=z_1z_2\ldots z_g$, $x_1,y_1\in V_1(G)$, and $x_1x_2 x_3z_1$, $y_1y_2y_3z_i$ ($2\leq i\leq g$) be two paths with length $3$ in $G$, where $d_G(x_3)=d_G(y_3)=2$. If $G'$ is the graph obtained from $G$ by deleting two edges $y_1y_2$, $y_2y_3$ and adding two new edges $y_1x_1$ and $y_2x_2$, then we say that $G'$ is obtained from $G$ by Operation III. Clearly, $G'\in\mathbb{U}_{n}$. \[lem Operation 3\] Let $G$ be defined as above. If $G'$ is obtained from $G$ by Operation III, then $G\succ G'$. The proof is similar to that of Lemma \[lem Operation 2\]. \[thm girth at least 8\] Let $G\in \mathbb{U}_{n}$, $g(G)\equiv 0$ [(mod $4$)]{.nodecor}, $g(G)\geq 8$. Then $G\succ B_n$. We apply induction on $d$. As the case $d\leq 2$ was proved by Lemmas \[lem girth at least 8-d equals 0\], \[lem girth at least 8-d equals 1\], and \[lem girth at least 8-d equals 2\], we now suppose that $d\geq 3$ and the assertion holds for smaller values of $d$. Let $C_g=z_1z_2\ldots z_g$. Assume that $d_G(x_1,z_1)=d$ and $x_1x_2 \ldots x_dz_1$ be a path with length $d$ in $G$. Then $d_G(x_2)=2$ and $x_1x_2\in M_G$. For convenience, denote $x_{d+1}=z_1$. Let $G_1=G-\{x_1,x_2\}$ and $G_2=G-\{x_1,x_2,x_3\}$. By Lemma \[lem delete-edge\], we have $$\begin{aligned} b_{2i}(G) = b_{2i}(G_1) + b_{2i-2}(G_1)+ b_{2i-2}(G_2).\label{coefficient formular} \end{aligned}$$ Note that $$\begin{aligned} b_{2i}(B_n) = b_{2i}(B_{n-2})+b_{2i-2}(B_{n-2})+b_{2i-2}(B_{n-4}) \ (n\geq 10).\label{coefficient formular-Bn} \end{aligned}$$ Now we prove the result for the given $d$ by induction on $t$. [**Case 1.**]{} Suppose $t=1$. [**Subcase 1.1.**]{} $d_G(x_3)=d_G(x_4)=2$. Note that $x_3x_4\in M_G$. It is easy to see that $G_1\in \mathbb{U}_{n-2}$, $G_2-x_4\in \mathbb{U}_{n-4}$, $d(G_1)<d$ and $d(G_2-x_4)<d$. So by the induction hypothesis, $G_1\succeq B_{n-2}$, $G_2-x_4\succeq B_{n-4}$. By Lemma \[lem delete-vertex\], we have $G_2\succ (G_2-x_4)\cup K_1$, and so $G_2\succ B_{n-4}\cup K_1$. It follows from Eqs. and that $G\succ B_n$. [**Subcase 1.2.**]{} $d_G(x_3)=2, d_G(x_4)=3$ and $d>3$. Note that $x_3x_4\in M_G$. Suppose $y_2\not\in\{x_3,x_5\}$ is a neighbor of $x_4$, and $y_1y_2\in M_G$. Since $t=1$, we have $d_G(y_1)=1$ and $d_G(y_2)=2$. Then we have $G_2-y_1\in \mathbb{U}_{n-4}$ since $(M_G\setminus(\{x_1x_2,x_3x_4,y_1y_2\})\cup\{y_2x_4\}$ is a perfect matching of $G_2-y_{1}$. Therefore similar to Subcase 1.1, we have $G_1\succeq B_{n-2}$ and $G_2\succ B_{n-4}\cup K_1$, and so $G\succ B_n$. [**Subcase 1.3.**]{} $d_G(x_3)=2, d_G(x_4)=3$ and $d=3$, i.e., $x_{4}=z_1$. Note that $x_3x_4\in M_G$. Since $G$ has a perfect matching, $g(G)\equiv 0$ [(mod $4$)]{.nodecor}, there exist $k\geq 1$ pendent edges $z_{i_1}y_{i_1},\ldots,z_{i_k}y_{i_k}$ ($2\leq i_1<\ldots<i_k\leq g$) such that $k$ is odd and $z_{i_k}y_{i_k}\in M_G$. Then we have $G_2-y_{i_1}\in \mathbb{U}_{n-4}$, since $$(M_G\setminus\{x_1x_2,x_3x_4,z_{i_1}y_{i_1},z_2z_3,z_4z_5,\ldots,z_{i_1-2}z_{i_1-1}\})\cup\{z_1z_2,z_3z_4,\ldots,z_{i_1-1}z_{i_1}\}$$ is a perfect matching of $G_2-y_{i_1}$. Therefore similarly, we have $G_1\succeq B_{n-2}$ and $G_2\succ B_{n-4}\cup K_1$, and so $G\succ B_n$. [**Subcase 1.4.**]{} $d_G(x_3)=3$. Since $t=1$, $x_3x_4\not\in M_G$, we assume that $x_3y_1\in M_G$. Then $d_G(y_1)=1$, $d_{G_2}(y_1)=0$ and $G_2-y_1\in \mathbb{U}_{n-4}$. Therefore similarly, we have $G_1\succ B_{n-2}$ and $G_2\succeq B_{n-4}\cup K_1$, and so $G\succ B_n$. [**Case 2.**]{} Assume now that $t\geq 2$ and the assertion holds for smaller values of $t$. Note that $G_1\in \mathbb{U}_{n-2}$ with $d(G_1)=d$ and $t(G_1)<t$. By the induction hypothesis, $G_1\succeq B_{n-2}$. [**Subcase 2.1.**]{} $d_G(x_3)=d_G(x_4)=2$. The proof is similar to that of Subcases 1.1. [**Subcase 2.2.**]{} $d_G(x_3)=3$ and $x_3x_4\not\in M_G$. The proof is similar to that of Subcases 1.4. [**Subcase 2.3.**]{} $d_G(x_3)=2$, $d_G(x_4)=3$, and $d>3$. Suppose $y_2\not\in\{x_3,x_5\}$ is a neighbor of $x_4$, and $y_1y_2\in M_G$. If $d_G(y_2)=3$, let $y_3\not\in\{x_4,y_1\}$ be a neighbor of $y_2$, and $y_4y_3\in M_G$. Then we have $d_G(y_1)=d_G(y_4)=1$, $d_G(y_3)=2$, and $G_2-y_1\in \mathbb{U}_{n-4}$, since $(M_G\setminus(\{x_1x_2,x_3x_4,y_1y_2\})\cup\{y_2x_4\}$ is a perfect matching of $G_2-y_{1}$. Therefore similarly, we have $G_2\succ B_{n-4}\cup K_1$, and so $G\succ B_n$. [**Subcase 2.4.**]{} $d_G(x_3)=3$ and $x_3x_4\in M_G$. Suppose $y_2\not\in\{x_2,x_4\}$ is a neighbor of $x_3$, and $y_1y_2\in M_G$. Then we have $d_G(y_1)=1$ and $d_G(y_2)=2$. Let $G'$ be the graph obtained from $G$ by Operation II. It follows from Lemma \[lem Operation 2\] that $G\succ G'$. Similarly, we have $$b_{2i}(G') = b_{2i}(G'-\{y_1,x_1\}) + b_{2i-2}(G'-\{y_1,x_1\})+ b_{2i-2}(G'-\{y_1,x_1,x_2,y_2\}),$$ and $G'-\{y_1,x_1\}\succ B_{n-2}$, and $G'-\{y_1,x_1,x_2,y_2\}\succeq B_{n-4}$. Therefore $G'\succ B_n$. [**Subcase 2.5.**]{} $d_G(x_3)=2$ and $d=3$. Now $x_4=z_1$. Since $t\geq 2$, suppose that $y_1\in V_1(G)$ and $y_1y_2y_3z_i$($i\neq 2$) be a path with length $3$ in $G$. Then by the above subcases, we may assume that $d_G(y_3)=2$. Let $G'$ be the graph obtained from $G$ by Operation III, then we have $G\succ G'$ by Lemma \[lem Operation 3\]. And similar to the Subcase 2.4, we have $G'\succ B_n$. The proof is thus complete. (1831.4, 291.0)(0,0) (0,0)[![The graphs considered in Theorem \[thm girth 4-A\_n\].[]{data-label="fig-lemma-An"}](fig-lemma-An "fig:")]{} (135.76,38.20) (150.0, 60.0)\[l\] $I_1$ (808.17,41.93) (150.0, 60.0)\[l\] $I_2$ (1507.92,50.63) (150.0, 60.0)\[l\] $I_3$ \[thm girth 4-A\_n\] Let $G\in \mathbb{U}_{n}$, $g(G)=4$, $G\not\cong A_n$. If there are just two edges of $M_G$ in $C_4$, then $G\succ A_n$. We apply induction on $d$. Suppose $G\not\cong A_n$ is a graph in $\mathbb{U}_{n}$ with $g=4$, and there are just two edges of $M_G$ in $C_4$. For $d\leq 1$, there is nothing to prove. Suppose $d=2$. Then $G$ is isomorphic to one of the following graphs $B_8$, $I_1$, $I_2$ and $I_3$, as shown in Figure \[fig-lemma-An\]. By Lemma \[lem Bn and An\], we have $B_8\succ A_8$. It is easy to obtain that $$\begin{array}{lll} \phi(I_{1},x) & = & x^{8}-8x^6+16x^4-9x^2,\\ \phi(I_{2},x) & = & x^{10}-10x^8+30x^6-34x^4+12x^2,\\ \phi(I_{3},x) & = & x^{12}-12x^{10}+48x^8-84x^6+64x^4-16x^2,\\ \phi(A_8,x) & = & x^8-8x^6+16x^4-6x^2,\\ \phi(A_{10},x) & = & x^{10}-10x^8+30x^6-28x^4+6x^2,\\ \phi(A_{12},x) & = & x^{12}-12x^{10}+48x^8-74x^6+40x^4-6x^2. \end{array}$$ Hence $I_{1}\succ A_{8}$, $I_{2}\succ A_{10}$, and $I_{3}\succ A_{12}$. Now suppose that $d\geq 3$ and the assertion holds for smaller values of $d$. By an argument similar to the proof of Theorem \[thm girth at least 8\], we can obtain $G\succ A_n$. (2482.2, 296.0)(0,0) (0,0)[![The graphs considered in Theorem \[thm girth 4-D\_n\].[]{data-label="fig-lemma-Dn"}](fig-lemma-Dn "fig:")]{} (171.40,54.93) (150.0, 60.0)\[l\] $I_4$ (780.67,48.28) (150.0, 60.0)\[l\] $I_5$ (1409.27,38.20) (150.0, 60.0)\[l\] $I_6$ (2040.66,39.44) (150.0, 60.0)\[l\] $I_7$ \[thm girth 4-D\_n\] Let $G\in \mathbb{U}_{n}$, $g(G)=4$, $G\not\cong D_n$. If there is just one edge of $M_G$ in $C_4$, then $G\succ D_n$. We apply induction on $d$. For $d\leq 1$, there is nothing to prove. Suppose $d=2$. Then $G$ is isomorphic to $I_4$, as shown in Figure \[fig-lemma-Dn\]. It is easy to obtain that $$\begin{array}{lll} \phi(I_{4},x) & = & x^{10}-10x^8+29x^6-32x^4+12x^2-1,\\ \phi(D_{10},x) & = & x^{10}-10x^8+29x^6-28x^4+10x^2-1. \end{array}$$ Hence $I_4\succ D_{10}$. Now suppose that $d\geq 3$ and the assertion holds for smaller values of $d$. We use the same notations as in Theorem \[thm girth at least 8\]. Then the proof is similar to that of Theorem \[thm girth at least 8\]. We can divide two cases $x_dz_1\in M_g$ and $x_dz_1\not\in M_g$ to proceed. The difference is that we need to prove the result $G\succ D_n$ for the case: $d=3$, $t=1$, $d_G(x_3)=2$ and $z_2z_3\in M_4$. Since $G\not\cong D_n$, $G$ is isomorphic to one of the following graphs $I_5$, $I_6$ and $I_7$, as shown in Figure \[fig-lemma-Dn\]. It is easy to obtain that $$\begin{array}{lll} \phi(I_{5},x) & = & x^{10}-10x^8+30x^6-33x^4+11x^2-1,\\ \phi(I_{6},x) & = & x^{10}-10x^8+30x^6-33x^4+12x^2-1,\\ \phi(I_{7},x) & = & x^{12}-12x^{10}+48x^8-83x^6+62x^4-16x^2+1,\\ \phi(D_{12},x) & = & x^{12}-12x^{10}+47x^8-72x^6+46x^4-12x^2+1. \end{array}$$ Hence $I_5\succ D_{10}$, $I_6\succ D_{10}$ and $I_7\succ D_{12}$. \[thm girth 4-E\_n\] Let $G\in \mathbb{U}_{n}$, $g(G)=4$, $G\not\cong E_n$. If there are no edges of $M_G$ in $C_4$, then $G\succ E_n$. 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--- abstract: 'Ultra-cold bosons in zig-zag optical lattices present a rich physics due to the interplay between frustration, induced by lattice geometry, two-body interaction and three-body constraint. Unconstrained bosons may develop chiral superfluidity and a Mott-insulator even at vanishingly small interactions. Bosons with a three-body constraint allow for a Haldane-insulator phase in non-polar gases, as well as pair-superfluidity and density wave phases for attractive interactions. These phases may be created and detected within the current state of the art techniques.' author: - 'S. Greschner' - 'L. Santos' - 'T. Vekua' title: 'Ultra-cold bosons in zig-zag optical lattices' --- #### Introduction Atoms in optical lattices offer extraordinary possibilities for the controlled emulation and analysis of lattice models and quantum magnetism [@Lewenstein2007]. Various lattice geometries are attainable by means of proper laser arrangements, including triangular [@Becker2010] and Kagome [@Jo2012] lattices, opening fascinating possibilities for the study of geometric frustration, which may result in flat bands in which the constrained mobility may largely enhance the role of interactions [@Huber2010]. Moreover, the value and sign of inter-site hopping may be modified by means of shaking techniques [@Eckardt2005; @Zenesini2009], allowing for the study of frustrated antiferromagnets with bosonic lattice gases [@Struck2011]. Interatomic interactions may be controlled basically at will by means of Feshbach resonances [@Chin2010]. In particular, large on-site repulsion may allow for the suppression of double occupancy in bosonic gases at low fillings (hard-core regime). Interestingly, it has been recently suggested that, due to a Zeno-like effect, large three-body loss rates may result in an effective three-body constraint, in which no more than two bosons may occupy a given lattice site [@Daley2009]. This constraint opens exciting novel scenarios, especially in what concerns stable Bose gases with attractive on-site interactions, including color superfluids in spinor Bose gases [@Titvinidze2011] and pair-superfluid phases [@Daley2009; @Bonnes2011; @Chen2011]. The suppression of three-body occupation has been hinted in recent experiments [@Mark2012]. Under proper conditions, lattice gases may resemble to a large extent effective spin models, e.g. hard-core bosons may be mapped into a spin-$1/2$ XY Heisenberg model [@Lewenstein2007]. Lattice bosons at unit filling resemble to a large extent spin-$1$ chains [@DallaTorre2006], and in the presence of inter-site interactions, as it is the case of polar gases [@Lahaye2009], have been shown to present a gapped Haldane-like phase [@Haldane1983] (dubbed Haldane-insulator (HI) [@DallaTorre2006; @Berg2008]) characterized by a non-local string-order [@DenNijs1989]. In this Letter we analyze the physics of ultra-cold bosons in zig-zag optical lattices. We show that the interplay of frustration and interactions lead to a different physics for unconstrained and constrained (with up to two particles per site) bosons. For unconstrained bosons, geometric frustration induces chiral superfluidity, and allows for a Mott-insulator phase even at vanishingly small interactions. For constrained bosons, we show that a Haldane-insulator phase becomes possible even for non-polar gases. Moreover, pair-superfluid [@Daley2009; @Bonnes2011; @Chen2011] and density-wave phases may occur for attractive on-site interactions. A direct first-order phase transition from Haldane-insulator to pair-superfluid is observed and explained. These phases may be realized and detected with existing state of the art techniques. ![Zig-zag chains formed by an incoherent superposition between a triangular lattice [@Becker2010] $V_1({\vec r}\equiv(x,y))=V_{10}\left [ \sin^2\left ({\vec b}_1\cdot{\vec r}/2\right )+\sin^2\left ({\vec b}_2\cdot {\vec r}/2\right ) +\sin^2\left (({\vec b}_1-{\vec b}_2)\cdot{\vec r}/2\right )\right ]$, with $k$ the laser wavenumber, ${\vec b}_1=\sqrt{3}k {\vec e}_y$ and ${\vec b}_2=\sqrt{3}k(\sqrt{3}{\vec e}_x/2-{\vec e}_y/2)$, and an additional lattice $V_2({\vec r})=V_{20}\sin^2(\sqrt{3}ky/4 - \pi/4)$. In the figure, in which $V_{20}/V_{10}=2$, darker regions mean lower potential. The hopping rate between nearest (next-nearest) neighbors is $t>$ ($t'$).](fig1.eps){width="5.0cm"} \[fig:1\] #### Zig-zag lattices. In the following we consider bosons in zig-zag optical lattices. As shown in Fig. \[fig:1\], this particular geometry may result from the incoherent superposition of a triangular lattice with elementary cell vectors $\vec{a}_1=a {\vec e}_x$ and $\vec{a}_2=a(\frac{1}{2}{\vec e_x}+\frac{\sqrt{3}}{2}{\vec e}_y)$ (formed by three laser beams of wavenumber $k=4\pi/3a$ oriented at $120$ degrees from each other, as discussed in Ref. [@Becker2010]) and a superlattice with lattice spacing $\sqrt{3}a$ oriented along $y$. For a sufficiently strong superlattice, zig-zag ladders are formed, and the hopping between ladders may be neglected. We will hence concentrate in the following on the physics of bosons in a single zig-zag ladder, which is to a large extent given by the rates $t$ and $t'$ characterizing the hopping along the two directions ${\vec a}_{1,2}$ (Fig. \[fig:1\]). As shown in Ref. [@Struck2011], a periodic lattice shaking may be employed to control the value of $t$ and $t'$ independently. Interestingly, their sign may be controlled as well. In the following we consider an inverted sign for both hoppings, which result in an anti-ferromagnetic coupling between sites [@Struck2011]. #### Model. Ordering the sites as indicated in Fig. \[fig:1\], the physics of the system is given by a Bose-Hubbard Hamiltonian with on-site interactions characterized by the coupling constant $U$, nearest-neighbor hopping $t<0$ and next-nearest-neighbor hopping $t'<0$: $$\begin{aligned} \label{eq:H-BH} H&=&\sum_i \left [ -\frac{t}{2} b_i^\dag b_{i+1} - \frac{t'}{2} b_i^\dag b_{i+2} + {\rm H.c} \right ]\\ &+&\frac{U}{2}\sum_i n_i(n_i-1)+U_3\sum_i n_i(n_i-1)(n_i-2),\nonumber\end{aligned}$$ where $b_i^\dag,b_i$ are the bosonic creation/annihilation operators of particles at site $i$, $n_i=b_i^\dag b_i$, and we have added the possibility of three-body interactions, characterized by the coupling constant $U_3$. We assume below an average unit filling ${\bar n}=1$. #### Unconstrained bosons. We discuss first the ground-state properties of unconstrained bosons ($U_3=0$). At $U=0$, the Hamiltonian  is diagonalized in quasi-momentum space $H=\sum_k \epsilon(k) b_k^\dag b_k$, with the dispersion $\epsilon(k)= |t| (\cos k +j\cos 2k)$, with $j\equiv t'/t$. Depending on the frustration $j$ we may distinguish two regimes. If $j<1/4$, the dispersion $\epsilon(k)$ presents a single minimum at $k=\pi$, and hence small $U$ will introduce a superfluid (SF) phase, with quasi-condensate at $k=\pi$. If $j>1/4$, $\epsilon(k)$ presents two non-equivalent minima at $k=k0\equiv\pm \arccos[-1/4j]$. As shown below, interactions favor the predominant population of one of these minima, and the system enters a chiral superfluid (CSF) phase with a non-zero local boson current characterized by a finite chirality $\langle \kappa_i\rangle$, with $\kappa_i=\frac{i}{2}(b_{i}^\dag b_{i+1}-{\rm H.c.})$. At $j=1/4$, the Lifshitz point, the dispersion becomes quartic at the $k=\pi$ minimum, $\epsilon(k)\sim (k-\pi)^4$, the effective mass $m=(\partial^2 \epsilon(k)/\partial k^2)_{k=\pi}^{-1}=1/t(1-4j)$ diverges, and even vanishingly small interactions become relevant. To study the effect of interactions we combine numerical calculations based on the density matrix renormalization group (DMRG) method [@White] (with up to $N=300$ sites keeping per block on average $400$ states for gapped phases and $600$ states for gapless ones), and bosonization techniques to unveil the low-energy behavior of model . For $j<1/4$, we employ standard bosonization transformations [@Giamarchi], with an additional oscillating factor $b_i\to (-1)^ie^{i\sqrt{\pi}\theta(x)}$, to obtain the low-energy effective theory, which is given by the sine-Gordon model $$\label{sine-Gordon} {\mathcal H}\!=\! \frac{v_s}{2}\left[ \frac{(\partial_x \phi)^2}{K}+ K(\partial_x \theta)^2 \right]\!-\!{\mathcal M} \cos[2\pi \bar n x-\sqrt{4\pi} \phi ],$$ where $\theta$ and $\partial_x\phi$ describe phase and density fluctuations of bosons respectively, $[\theta(x),\partial_y\phi]=i\delta(x-y)$, $v_s$ is the sound velocity and $K$ the Luttinger parameter. In the weak-coupling, $Um\ll 1$, hydrodynamic relations are expected to hold: $v_s(j)\sim \sqrt{\bar n U/m\pi^2}=v_s(0)\sqrt{1-4j}$ and $K(j)\sim \sqrt{\bar n\pi^2/ Um}=K(0) \sqrt{1-4j}$, clearly showing that $j$ enhances correlations. At $j=1/4$, $m$ diverges and the system enters a Mott-insulator (MI) even for vanishingly small $U$  (Fig. \[fig:2\](a)). The SF-MI transition takes place however in the strong-coupling regime in which $v_s$ and $K$ must be determined numerically. We obtain $K$ from the single-particle correlations $G_{ij} =\langle b_i b^{\dagger}_j\rangle$ which in the SF decay as $\sim (-1)^{i-j}{|i-j|^{-1/2K}}$. The value $K=2$ marks the boundary between SF ($U<U_c$, $K>2$) and MI ($U>U_c$, $K<2$, and ${\mathcal M}>0 $). The MI phase is characterized by a hidden parity order [@Berg2008], ${\cal O}^2_P=\lim_{|i-j|\rightarrow\infty}\langle (-1)^{\sum_{i<l<j}\delta n_l}\rangle\sim\langle \cos \sqrt{\pi} \phi \rangle^2$, which has been recently measured in site-resolved experiments [@Endres2011]. ![Phase diagram for unconstrained bosons as a function of the frustration parameter $j$ and (a) the on-site interaction $U$ (with $U_3=0$) and (b) the three-body repulsion $U_3$ (and $U=0$). In the figures, $\bigcirc$ indicate the boundary of the chiral phases characterized by long-range ordered chirality-chirality correlations $\left<\kappa_i\kappa_j\right>$; $\square$ indicate the boundary of the SF-phases indicated by the critical Luttinger parameter $K=2$. Note that narrow CMI and CHI phases may occur as well. For $U,\, U_3 \geq 0.5$ keeping $n_{max}=4$ bosons per site in the DMRG-simulation can be shown to be sufficient.[]{data-label="fig:2"}](bh_u2_u3.eps){width="8.5cm"} The $j>1/4$ case is best understood from bosonization in the $j\gg 1$ regime. We may then introduce two pairs of bosonic fields ($\theta_{1},\phi_{1}$) and ($\theta_{2},\phi_{2}$), describing, respectively, the subchains of even and odd sites. The effective model is governed by the Hamiltonian density $$\begin{aligned} \label{effectivetwocomponent} {\mathcal H}&=&\sum_{\alpha=\pm}\frac{ v_{\alpha}}{2}\left[ \frac{(\partial_x \phi_{\alpha})^2}{ K_{\alpha}}+ K_{\alpha}(\partial_x \theta_{\alpha})^2 \right] \\ &+& \lambda\partial_x \theta_+\sin \sqrt{2\pi}\theta_-- 2{\mathcal M} \cos\sqrt{2\pi} \phi_{+} \cos\sqrt{2\pi} \phi_{-}\nonumber\end{aligned}$$ where $\theta_\pm=(\theta_1\pm \theta_2)/\sqrt{2}$, $\phi_\pm=(\phi_1\pm \phi_2)/\sqrt{2}$, $v_{\pm}$, $K_{\pm}$, and ${\mathcal M}$ are phenomenological parameters (in the regimes displayed on Figure 2 (a)), $\lambda\sim j^{-1}$. Note that the chirality is given by $\kappa_i\to \sin \sqrt{2\pi}\theta_-(x)$. In weak-coupling, $Um'\ll1$, with $m'=(\partial^2 \epsilon(k)/\partial k^2)_{k_0}^{-1}=4j/t(16j^2-1)$, $v_{\pm}\sim \sqrt{\bar n U/m'\pi^2}$ and $K_{\pm}\sim \sqrt{\bar n\pi^2/ Um'}$. In this case only the term $\partial_x \theta_+\sin \sqrt{2\pi}\theta_-$ is relevant, resulting in $\langle \sin \sqrt{2\pi}\theta_-\rangle\neq 0$ [@Nersesyan]. Hence, a small $U$ is expected to favor a CSF for $j>1/4$, as our numerical results confirm (Fig. \[fig:2\](a)). The CSF phase is characterized by $G_{ij}\sim (-1)^{i-j}e^{-i\kappa(i-j)}{|i-j|^{-1/4 K_+}}$, where $\kappa\sim \langle \kappa_i\rangle$. Moreover, depending on the values of $K_{\pm}$ bosonization opens the possibility of two consecutive phase transitions with increasing $U$ starting from the CSF phase [@Supplementary], which we have confirmed with our DMRG calculations (Fig. \[fig:2\](a)). First a KT transition occurs from CSF to chiral-Mott (CMI), a narrow Mott phase with finite chirality. Then an Ising transition is produced from CMI to non-chiral MI. At both KT transition lines in Fig. \[fig:2\](a) (SF-MI and CSF-CMI), up to a logarithmic prefactor, $G_{ij}\sim (-1)^{i-j}e^{-i \kappa(i-j)}{|i-j|^{-1/4}}$, where in CSF $ \kappa\neq 0$. #### Constrained bosons. As mentioned above, sufficiently large three-body losses may result in a three-body constraint $(b_i^\dag)^3=0$ ($U_3=\infty$) [@Daley2009]. In that case, Model  may be mapped to a large extent onto a frustrated spin-$1$ chain model [@commentSpin1], which, presents the possibility of a gapped Haldane phase, characterized by a non-local string order. Hence, interestingly, constrained bosons in a zig-zag lattice may be expected to allow for the observation of the HI phase in the absence of polar interactions. Indeed, a model with $U=0$ and finite $U_3$ shows that at the Lifshitz point, $j=1/4$, a HI phase is stabilized for arbitrarily weak $U_3$ (Fig. \[fig:2\](b)). The effective theory describing the HI is again the sine-Gordon model (\[sine-Gordon\]) with $K<2$. However, now ${\cal M}<0$, which selects a hidden string order ${\cal O}^2_S\equiv\lim_{|i-j|\rightarrow\infty}\langle \delta n_i \exp [i\pi\sum_{i<l<j}\delta n_l]\delta n_j\rangle\sim \langle \sin \sqrt{\pi}\theta\rangle^2$ [@Berg2008]. Resembling the case of Fig. \[fig:2\](a), SF, HI, chiral-HI (CHI) and CSF phases occur (Fig. \[fig:2\](b)). These phases are expected for $U_3=\infty$ from known results in frustrated spin-$1$ chains [@Kolezhuk; @Hikihara2000; @Hikihara2002]. Our DMRG simulations suggest that all these phases meet at $j=1/4$ for $U_3\to 0$. Figure \[fig:3\] shows the phase diagram for constrained bosons ($U_3=\infty$). Starting from the HI phase, increasing $U>0$ can induce a Gaussian HI-MI phase transition, characterized by a vanishing ${\cal M}=0$ in , resembling the phase transition between Haldane and large-D phases induced by single-ion anisotropy in spin-1 chains [@Schulz]. The SF phase is separated from the MI and HI by KT transitions, whereas at the CSF boundary with the MI (HI) a CMI (CHI) occurs as mentioned above (these very narrow regions are not resolved in Fig. \[fig:3\]). Interestingly, constrained bosons allow as well for the exploration of attractive two-body interactions, $U<0$, without collapse. The $U<0$ phases are also depicted in Fig. \[fig:3\]. For sufficiently large $|U|$, bosons tend to cluster in pairs and, as already discussed in Ref. [@Daley2009], for $j=0$ an Ising transition between a SF and a pair superfluid (PSF) occurs[@commentIsing], analogous to the XY1 to XY2 phase transition in spin-$1$ chains induced by single-ion anisotropy [@Schulz]  (this transition has been recently studied for 2D lattices as well [@Bonnes2011; @Chen2011]). The PSF phase is characterized by an exponentially decaying $G_{ij}$ but algebraically decaying pair-correlation function $G_{ij}^{(2)}=\langle (b_i^\dag)^2(b_j)^2\rangle$. Indeed a PSF occurs for sufficiently large $|U|$, also for $j>1/4$ which is characterized in bosonization in Eq.  by a gapped antisymmetric sector, with pinned $\phi_-$, and a gapless symmetric sector [@Vekua]. Though one may anticipate an Ising phase transition between the CSF (with broken discrete parity symmetry) and the PSF (with restored symmetry), the behavior of ${\cal O}_P^2$ and $\kappa$ (not shown) hints to a weakly first-order nature. ![Phase diagram for constrained bosons as a function of the on-site interaction $U$ and the frustration parameter $j$. Narrow CMI and CHI phases may occur along the phase transition lines from CSF to MI and CSF to HI, respectively, but their extension would be negligible in the figure. For the precise location of the PSF-DW and HI-MI transition we have additionally analyzed the energy-level-crossings with, respectively, periodic and twisted-boundary conditions. The KT transitions from SF to MI and HI have been determined by the extraction of the Luttinger parameter $K$ [@commentLuttinger].](bh3_pd.eps){width="8.0cm"} \[fig:3\] Small $U<0$ disfavors singly-occupied sites and thus enhances ${\cal O}_S^2$ and the bulk excitation gap of the HI phase (see Figs. \[fig:4\] and \[fig:5\]). However, since large $U<0$ removes singly occupied sites completely, just like strong nearest neighbour repulsion, it is expected that the HI phase eventually will transform for growing $|U|$ into a gapped density-wave (DW) phase via Ising phase transition [@DallaTorre2006], and string order will evolve into DW order (Fig. \[fig:4\] shows how ${\cal O}_s$ merges with ${\cal O}_{DW}\equiv \lim_{j\rightarrow\infty}(-1)^j \langle n_in_{i+j}\rangle$ for $U/t<-3$). The DW phase is characterized by an exponential decay of both $G_{ij}$ and $G_{ij}^{(2)}$ though a finite ${\cal O}_{DW}$. Our DMRG results confirm this scenario (see Fig. \[fig:4\]), showing that a DW phase is located between the above mentioned PSF regions (Fig. \[fig:3\]). Interestingly the DW phase remains in between both PSF regions all the way into $U\rightarrow-\infty$. In that regime, we may project out singly-occupied sites, and introduce a pseudo-spin-$1/2$, identifying $|0 \rangle\to|\!\!\downarrow \rangle , |2\rangle\to|\!\!\uparrow \rangle $, and defining the spin operators $\tau_i^-\to (-1)^i b_i^2/\sqrt{2}$, $2\tau_i^z\to b_i^{\dagger}b_i-1$. The effective model to leading order in $1/|U|$ is a spin-$1/2$ chain: $$\label{effectivespinhalf} H_{\frac{1}{2}}\!=\!J \!\sum_i \!\!\left[{{\boldsymbol}\tau}_i{{\boldsymbol}\tau}_{i+1}\!+\! j^2(\tau_i^z \tau_{i+2}^z\!-\!\tau_i^x \tau_{i+2}^x\!-\!\tau_i^y \tau_{i+2}^y) \right],$$ where $J={t^2}/{|U|}$. For $j=0$, this is a $SU(2)$ symmetric chain, whereas the $j^2J$ terms break the symmetry down to $U(1)$, moving the effective theory obtained after bosonization of $H_{\frac{1}{2}}$ towards the irrelevant direction (in the renormalization group sense). As a result of this, a gapless XY phase of the spin-1/2 chain is expected, i.e. a PSF phase. Higher order terms in $1/|U|$ (not shown explicitely) break, even for $j=0$, the $SU(2)$ symmetry to $U(1)$ in the irrelevant direction. However, interestingly, the ring exchange along the elementary triangle of the zig-zag chain, with amplitude $jt^3 /U^2$, forces the effective theory towards the relevant direction, leading to a gapped Néel phase of the spin-1/2 chain, i.e. the DW phase. The competition between exchange along the lattice bonds and ring-exchange leads hence to two consecutive KT phase transitions induced by $j$, for $j\ll 1$ first from PSF to DW, followed by DW back to PSF. The width of the DW phase is $\sim t/|U|$, and it extends all the way into the $U\to -\infty$ limit. Finally, our DMRG simulations show a narrow region where a direct, apparently first-order, HI-PSF transition occurs, characterized by discontinuous jumps of ${\cal O}_{S,P}^2$ (Fig. \[fig:5\]). This first-order nature is explained because on one hand increasing $|U|$ within the HI phase increases ${\cal O}_S^2$ due to the suppression of singly-occupied sites, and on the other hand, for $0.6 \lesssim j \lesssim 0.75$ (Fig. \[fig:3\]), a growing $|U|$ destroys the insulating state in favour of a PSF phase, where string order cannot exist. On the contrary, ${\cal O}_S^2$ diminishes for decreasing $|U|$ when approaching the HI-CSF boundary (Fig. \[fig:5\]) {width="8.0cm"} \[fig:4\] {width="8.0cm"} \[fig:5\] #### Conclusions. In summary, the interplay between geometrical frustration and interactions leads to rich physics for ultra-cold bosons in zig-zag optical lattices. Unconstrained bosons may present chiral superfluidity, and Mott insulator for vanishingly small interactions. Constrained bosons may allow for the observation of Haldane-insulator without the necessity of polar interactions, as well as pair-superfluid and density wave phases at attractive interactions. All the predicted phases may be detected using state of the art techniques. The SF and CSF phases may be distinguished by means of time-of-flight (TOF) techniques, in a similar way as recently done for condensates in triangular optical lattices [@Struck2011]. The DW and PSF phases are characterized by double or zero occupancy, which could be detected using parity measurements as those introduced in Refs [@Bakr2010; @Sherson2010], and could be discerned from each other by the absence/presence of interference fringes in TOF [@Greiner2002]. 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--- abstract: | In this paper we provide a complete algebraic characterization of elementary equivalence of rings with a finitely generated additive group in the language of pure rings. The rings considered are arbitrary otherwise.\ [**2010 MSC:**]{} 03C60\ [**Keywords:**]{} Ring, Elementary Equivalence, Largest Ring of a Bilinear Map author: - 'Alexei Miasnikov, Mahmood Sohrabi[^1]' title: On elementary equivalence of rings with a finitely generated additive group --- Introduction ============ This paper continues the authors’ efforts [@MS; @MS2010; @MS2016], in providing a comprehensive and uniform approach to various model-theoretical questions on algebras and nilpotent groups. By a *scalar ring* we mean a commutative associative unitary ring. Assume $A$ is a scalar ring. We say that $R$ is an *$A$-algebra* if $R$ is abelian group equipped with an $A$-bilinear binary operation. We use the term *ring* for a ${{\mathbb}{Z}}$-algebra where ${{\mathbb}{Z}}$ is the ring of rational integers, reserving the term scalar ring for commutative associative unitary rings. The ring $R$ is said to be a *finite dimensional ${{\mathbb}{Z}}$-algebra* or an *FDZ-algebra* for short if the additive group $R^+$ of the ring $R$ is finitely generated as an abelian group. The main problem we tackle here is to characterize the elementary equivalence of FDZ-algebras via a complete set of elementary invariants. The invariants will be purely algebraic. Statements of the main results {#approach:sec} ------------------------------ In this paper the language $L$ denotes the language of pure rings without a constant for multiplicative identity. That is because an arbitrary ring may not have a unit. By $L_1$ we mean the usual language of rings with identity. For us an $A$-module $M$ is a two-sorted structure $\langle M, A, s \rangle$, where $M$ is an abelian group, $A$ is a scalar ring and $s$ is the predicate describing the action of $A$ on $M$. Denote the language by $L_2$. We often drop $s$ from our notation. Since the scalar ring are always assumed to be commutative we do not specify whether the modules are left or right modules. Here is our first main result. \[elemmod:thm\]Let $A$ be an FDZ-scalar ring and let $M$ be a finitely generated $A$-module. Then there exists a sentence $\psi_{M,A}$ of the language $L_2$ such that $\langle M,A\rangle\models \psi_{M,A}$ and for any FDZ-scalar ring $B$ any finitely generated $B$-module $N$, we have $$\langle N,B\rangle\models \psi_{M,A} \Leftrightarrow \langle N,B\rangle \cong \langle M,A\rangle.$$ The proof of the theorem appears at the end of Section \[Z-interpret:sec\]. Indeed Theorem \[elemmod:thm\] implies the next three statements. The first two state the same result. \[scalarrings:cor\] For any FDZ-scalar ring $A$ there exists a formula $\psi_A$ of $L_1$ such that $A\models \psi_A$ and for FDZ-scalar ring $B$ we have $$B\models \psi_A \Leftrightarrow A\cong B.$$ \[scalarrings:cor2\]Let $\mathcal{K}$ be the class of all FDZ-scalar rings. Then any $A$ from $\mathcal{K}$ is finitely axiomatizable inside $\mathcal{K}$. Let us denote by $L_3$ the first-order language of two-sorted algebras. An algebra $\langle C, A\rangle$ consists of an arbitrary ring $C$, and the scalar ring $A$ (and a predicate describing the scalar multiplication which is dropped from the notation). As mentioned it is actually a corollary of Theorem \[elemmod:thm\]. We provide a brief of it at the beginning of Section \[main:sec\]. \[elem-iso-alg:thm\] Let $\mathcal{A}$ be the class of all two-sorted algebras $\langle C, A\rangle$ where $C$ is finitely generated as an $A$-module and $A^+$ is finitely generated as an abelian group. For each $\langle C, A\rangle\in \mathcal{A}$ there exists a formula $\phi_{C,A}$ of $L_3$ such that $\langle C, A\rangle\models \phi_{C,A}$ and for any $\langle D, B\rangle \in \mathcal{A}$, $$\langle D, B\rangle\models \phi_{C,A} \Leftrightarrow \langle C,A\rangle \cong \langle D,B\rangle$$ as two-sorted algebras. To state the main result of the paper we need to introduce some more definitions and notation. Consider an arbitrary ring $R$. Define the *two-sided annihilator ideal* of $R$ by $$Ann(R)=\{x\in R: xy=yx=0, \forall y\in R\}.$$ By $R^2$ we denote the ideal of $R$ generated by all products $x\cdot y$ (or $xy$ for short) of elements of $R$. Consider a scalar ring $A$ and let $R$ be an $A$-algebra. Assume $I$ is an ideal of $R$. Let $$Is_A(I){\stackrel{\text{def}}{=}}\{ x\in R: ax\in I,\text{ for some }a\in A\setminus\{0\}\}.$$ It is easy to show that $Is_A(I)$ is an ideal in $R$. We simply denote $Is_{{\mathbb}{Z}}(I)$ by $Is(I)$. Now assume $R$ is an FDZ-algebra. An *addition* $R_0$ of $R$ is a direct complement of the ideal $\Delta(R){\stackrel{\text{def}}{=}}Is(R^2)\cap Ann(R)$ in $Ann(R)$. Such a complement exists in this situation since $Ann(R)$ is a finitely generated abelian group and $Ann(R)/\Delta(R)$ is free abelian. It is clear that $R_0$ is actually an ideal of $R$. The quotient $R_F{\stackrel{\text{def}}{=}}R/R_0$ is called the *foundation* of $R$ associated to the addition $R_0$. Finally for an FDZ-algebra $R$ set $M(R){\stackrel{\text{def}}{=}}Is(R^2+Ann(R))$ and $N(R){\stackrel{\text{def}}{=}}Is(R^2)+Ann(R)$. Note that $M(R)/N(R)$ is a finite abelian group. We are now ready to state the main result of this paper. \[mainnice:thm\] Assume $R$ and $S$ are FDZ-algebras. Then the following are equivalent. 1. $R\equiv S$ as arbitrary rings. 2. Either $M(R)=N(R)$ and $R\cong S$, or there exists a monomorphism $\phi:R\to S$ of rings and additions $R_0$ and $S_0$ of $R$ and $S$ respectively such that 1. $\phi$ induces an isomorphism $R/R_0\cong S/S_0$, 2. $\phi$ induces an isomorphism $\dfrac{M(R)}{N(R)}\cong \dfrac{M(S)}{N(S)}$, 3. $\phi$ restricts to a monomorphism from $R_0$ into $S_0$, where the index $[S_0:\phi(R_0)]$ is finite and prime to the index $[M(R):N(R)]\neq 1$. The direction $(1.)\Rightarrow (2.)$ is called the *Characterization Theorem* and will be proved in Section \[main:sec\]. The direction $(2.)\Rightarrow (1.)$ called naturally the *converse of the characterization theorem*, stated in somewhat different terms, is given by Theorem \[converse\]. An FDZ-algebra is called *regular* if for some addition (and therefore for any addition) $R_0$ there exists a subring $R_F$ of $R$ containing $R^2$ such that $R\cong R_F \times R_0$. In Lemma \[regular-M=N:lem\] we shall prove that $R$ is a regular FDZ-algebra if and only if $M(R)=N(R)$. So the following statement was actually embedded in Theorem \[mainnice:thm\]. \[regular:cor\] Let $R$ be a regular FDZ-algebra. Then for an FDZ-algebra $S$, $$R\equiv S \Leftrightarrow R\cong S.$$ Finally we call an FDZ-algebra $R$ *tame* if $Ann(R)\leq Is(R^2)$. The following theorem is the generalization of Corollary \[scalarrings:cor2\] to the class ${\mathcal}{T}$ of all tame FDZ-algebras. \[tame:thm\] Let ${\mathcal}{T}$ be the class of all tame FDZ-algebras. Then any $R$ from ${\mathcal}{T}$ is finitely axiomatizable inside the class of all FDZ-algebras. Our approach ------------ Let us give an informal account of our methods in proving Theorem \[mainnice:thm\]. Recall that an arbitrary ring $R$ is an abelian group together with a bilinear map: $$f_R:R\times R \to R, \quad (x,y)\mapsto xy.$$ The bilinear map $f$ induces a full non-degenerate map $$f_{RF}: \frac{R}{Ann(R)}\times \frac{R}{Ann(R)} \to R^2.$$ By Theorem \[ringinter\], from Section \[bilin\] there exists a canonical scalar ring $P(f_R)$ of $f_{RF}$ and its actions on $R/Ann(R)$ and $R^2$ are interpretable in $f_R$. Moreover the largest subring $A(R)$ of $P(f_R)$ consisting of those $\alpha$ making the canonical homomorphism: $$\eta: R^2 \to \frac{R}{Ann(R)}$$ $A(R)$-linear is a definable subring of $R$. So indeed the two-sorted algebras $\langle R/Ann(R), A(R)\rangle$ and $\langle R^2, A(R)\rangle$ are both interpretable in the pure ring $R$. Then the main theorem will follow from Theorem \[elem-iso-alg:thm\] and a few other technical results. Bilinear maps and the relevant terminology will be discussed in Section \[bilin\]. Theorem \[mainnice:thm\] will be proved in Section \[main:sec\]. The converse Theorem \[converse\] of Theorem \[mainnice:thm\] will appear in Section \[converse:sec\], thereby providing a complete algebraic characterization of elementary equivalence of FDZ-algebras. Organization of the paper ------------------------- We finish the introduction by describing the organization of the paper. In Section \[pre:sec\] we provide background and describe our notation. In particular we shall review logical notation and background, bilinear maps and their model theory and finally a little bit of algebras. In Section \[Z-interpret:sec\] we shall discuss FDZ-scalar rings (associative commutative and unitary), resulting in proofs of Theorem \[elemmod:thm\] and Corollary \[scalarrings:cor\]. In Section \[main:sec\] we obtain a necessary condition for elementary equivalence of arbitrary FDZ-algebras and provide a proof of the characterization direction of Theorem \[mainnice:thm\] as well as a proof of Theorem \[tame:thm\]. In Section \[converse:sec\] we prove Theorem \[converse\] which indeed proves the converse of the characterization theorem. Preliminaries {#pre:sec} ============= Bilinear maps ------------- Assume $M_1$, $M_2$ and $N$ are $A$-modules, where $A$ is a commutative associative ring with unit. The map $$f:M_1\times M_2\rightarrow N$$ is called $A$-bilinear if $$f(ax,y)=f(x,ay)=af(x,y)$$ for all $x\in M_1$, $y\in M_2$ and $a \in A$. An $A$-bilinear mapping $f:M_1\times M_2\rightarrow N$ is called *non-degenerate in the first variable* if $f(x,y)=0$ for all $y$ in $M_2$ implies $x=0$. Non-degeneracy with respect to the second variable is defined similarly. The mapping $f$ is called *non-degenerate* if it is non-degenerate with respect to first and second variables. We call the bilinear map $f$, a *full bilinear mapping* if $N$ is generated as an $A$-module by elements $f(x,y)$, $x\in M_1$ and $y\in M_2$. Preliminaries on logic ---------------------- For the most part we follow standard model theory texts such as [@hodges] regarding notation and model theory. An arbitrary ring $R$ is a structure with signature $\langle +, \cdot, 0 \rangle$ and with the corresponding language is called $L$. A scalar ring $A$ is a structure with signature $\langle +, \cdot, 0,1 \rangle$ and the corresponding language is called $L_1$. ### Interpretations {#ss1} \[interpret1\]Let $\mathfrak{B}$ and $\mathfrak{U}$ be structures of signatures $\Delta$ and $\Sigma$ respectively. We may assume that $\Sigma$ and $\Delta$ do not contain any function symbols replacing them if necessary with predicates (i.e. replacing operations with their graphs). The structure $\mathfrak{U}$ is said to be *interpretable* in $\mathfrak{B}$ with parameters $\bar{b}\in |\mathfrak{B}|^m$ or *relatively interpretable* in $\mathfrak{B}$ if there is a set of first-order formulas $$\Psi=\{A(\bar{x},\bar{y}), E(\bar{x},\bar{y}_1,\bar{y}_2),\Psi_{\sigma}(\bar{x}, \bar{y}_1, \ldots , \bar{y}_{t_{\sigma}}): \sigma \textrm{ a predicate of signature } \Sigma \}$$ of signature $\Delta$ such that 1. $A(\bar{b})=\{\bar{a}\in|\mathfrak{B}|^n:\mathfrak{B}\models A(\bar{b},\bar{a})\}$ is not empty, 2. $E(\bar{x},{\overline}{y}_1,{\overline}{y}_2)$ defines an equivalence relation $\epsilon_{\bar{b}}$ on $A(\bar{b})$, 3. if the equivalence class of a tuple of elements $\bar{a}$ from $A(\bar{b})$ modulo the equivalence relation $\epsilon_{\bar{b}}$ is denoted by $[\bar{a}]$, for every $n$-ary predicate $\sigma$ of signature $\Sigma$, the predicate $P_{\sigma}$ is defined on $A(\bar{b})/\epsilon_{\bar{b}}$ by $$P_{\sigma}([\bar{b}],[{\overline}{a}_1], \ldots, [{\overline}{a}_n])\Leftrightarrow_{\text{def}}\mathfrak{B}\models \Psi_{\sigma}(\bar{b}, {\overline}{a}_1, \ldots, {\overline}{a}_n),$$ 4. There exists a map $f:A(\bar{b})\rightarrow |{\mathfrak}{U}|$ such that the structures $\mathfrak{U}$ and $\Psi(\mathfrak{B},\bar{b})=\langle A(\bar{b})/\epsilon_{\bar{b}},P_{\sigma}:\sigma\in \Sigma \rangle$ are isomorphic via the map $\tilde{f}:A(\bar{b})/\epsilon_{\bar{b}}\rightarrow |{\mathfrak}{U}|$ induced by $f$. Let $\Phi(x_1,\ldots, x_n)$ be a first-order formula of signature $\Delta$. If $\mathfrak{U}$ is interpretable in $\mathfrak{B}$ for any parameters $\bar{b}$ such that $\mathfrak{B}\models \Phi(\bar{b})$ then $\mathfrak{U}$ is said to be *regularly interpretable* in $\mathfrak{B}$ with the help of the formula $\Phi$. If the tuple $\bar{b}$ is empty, $\mathfrak{U}$ is said be *absolutely interpretable* in $\mathfrak{B}$. Now let $T$ be a theory of signature $\Delta$. Suppose that $S: Mod (T) \rightarrow K$ is a functor defined on the class $Mod (T)$ of all models of the theory $T$ (a category with isomorphisms) into a certain category $K$ of structures of signature $\Sigma$. If there exists a system of first-order formulas $\Psi$ of signature $\Delta$, which absolutely interprets the system $S(\frak{B})$ in any model $\frak{B}$ of the theory $T$ we say that $S(\frak{B})$ is *absolutely interpretable in $\frak{B}$ uniformly with respect to $T$*. For example, the annihilator $Ann(R)$ of a ring $R$ is interpretable (or in this case definable) in $R$ uniformly with respect to the theory of groups. On the other hand, the ideal $R^2$, generally speaking, is not interpretable in $G$ uniformly with respect to the theory of groups. However, it is so if $R^2$ is of *finite width* i.e. there is an $s\in {\mathbb{N}}$ such that $$R^2=\left\{\sum_{i=1}^sx_iy_i:x_i,y_i\in R\right\}.$$ For example in an FDZ-algebra $R^2$ is absolutely definable in $R$ uniformly with respect to $Th(R)$. Note that the ideal $R^2$ will have width less than or equal to $s$ if $R$ satisfies the first-order sentence $$\label{R^2Sen:eqn} \begin{split}\phi_w:\forall x &\left((\exists x_1, \ldots, x_{s+1},y_1,\ldots ,y_{s+1}~ x=\sum_{i=1}^{s+1}x_iy_i)\right. \\ &\left.\to (\exists z_1, \ldots, z_{s},t_1,\ldots ,t_{s}~ x=\sum_{i=1}^{s}t_is_i)\right).\end{split}$$ ### $A$-Modules as two-sorted structures Assume $A$ is a scalar ring and $M$ is an $A$-module. For us the $A$-module $M$ is a two-sorted structure $M_A=\langle M, A, s\rangle$ where $A$ is a ring, $M$ is an abelian group and $s=s(x,y,z)$, where $x$ and $z$ range over $M$ and $y$ ranges over $A$ is the predicate describing the action of $A$ on $M$, that is $\langle M,A,s\rangle \models s(m,a,n)$ if and only if $a\cdot m =n$. Sometimes we drop the predicate $s$ from our notation and write $M_A=\langle M, A\rangle$. When we say that *the ring $A$ and its action on $M$ are interpretable in a structure $\mathfrak{U}$* we mean that the one-sorted structure naturally associated to $M_A$ is interpretable in $\mathfrak{U}$. Note that if a multi-sorted structure has signature without any function symbols then there is a natural way to associate a one-sorted structure to it. We always assume that our signatures do not contain any function symbols, since functions can be interpreted as relations. Therefore when we talk about interpretability of multi-sorted structures in each other or interpretability of a multi-sorted structure into a one-sorted one we mean the interpretability of the associated one-sorted structures. Recall that a *homomorphism $\theta : \langle M,A, s\rangle \to \langle N,B, t\rangle$ of two-sorted modules* is a pair $(\theta_1, \theta_2)$ where $\theta_1: M\to N$ is a homomorphism of abelian groups and $\theta_2:A \to B$ is a homomorphism of rings satisfying $$s(m_1,a,m_2)\Leftrightarrow t(\theta_1(m_1), \theta_2(a), \theta_1(m_2)), \quad \forall a\in A, \forall m_1,m_2\in M.$$ A homomorphism $\theta$ as above is said be *an isomorphism of two-sorted modules* if $\theta_1$ and $\theta_2$ are isomorphisms of the corresponding structures. Largest ring of a bilinear map {#bilin} ------------------------------ In this section all the modules are considered to be faithful and scalar rings are always commutative associative with a unit. An $A$-module $M$ is said to be *faithful* if $am=0$ for $a\in A$ and all $m\in M$ implies $a=0$. Let $f:M_1\times M_2\rightarrow N$ be a non-degenerate full $A$-bilinear mapping for some ring $A$. Let $M$ be an $A$-module and let $\mu:A\rightarrow P$ be an inclusion of rings. Then the $P$-module $M$ is an *$P$-enrichment* of the $A$-module $M$ with respect to $\mu$ if for every $a\in A$ and $m \in M$, $am=\mu(a)m$. Let us denote the set of all $A$ endomorphisms of the $A$-module $M$ by $End_A(M)$. Suppose the $A$-module $M$ admits a $P$-enrichment with respect to the inclusion of rings $\mu:A\rightarrow P$. Then every $\alpha\in P$ induces an $A$-endomorphism, $\phi_{\alpha}:M\rightarrow M$ of modules defined by $\phi_{\alpha}(m)=\alpha m$ for $m \in M$. This in turn induces an injection $\phi_P:P\rightarrow End_A(M)$ of rings. Thus we associate a subring of the ring $End_A(M)$ to every ring $P$ with respect to which there is an enrichment of the $A$-module $M$. Let $f:M_1\times M_2\rightarrow N$ be a full $A$-bilinear mapping and $\mu:A\rightarrow P$ be an inclusion of rings. The mapping $f$ admits $P$-enrichment with respect to $\mu$ if the $A$-modules $M_1$, $M_2$ and $N$ admit $P$ enrichments with respect to $\mu$ and $f$ remains bilinear with respect to $P$. We denote such an enrichment by $E(f,P)$. We define an ordering $\leq$ on the set of enrichments of $f$ by allowing $E(f,P_1)\leq E(f,P_2)$ if and only if $f$ as an $P_1$ bilinear mapping admits a $P_2$ enrichment with respect to inclusion of rings $P_1\rightarrow P_2$. The largest enrichment $E_H(f,P(f))$ is defined in the obvious way. We shall prove existence of such an enrichment for a large class of bilinear mappings. The following proposition taken from [@alexei86] is essential for our work. \[P(f)\] If $f:M_1\times M_2 \rightarrow N$ is a non-degenerate full $A$-bilinear mapping over a commutative associative ring $A$ with unit, then $f$ admits the largest enrichment. An $A$-endomorphism $\alpha$ of the $A$-module $M$ is called *symmetric* if $$f(\alpha x,y)=f(x,\alpha y)$$ for every $x,y \in M$. Let us denote the set of all such endomorphisms by $Sym_f(M)$, i.e. $$Sym_f(M)=\{\alpha\in End_A(M):f(\alpha x,y)=(x,\alpha y),\quad \forall x,y \in M \}.$$ Set $$Z=\{\beta \in Sym_f(M):\alpha \circ \beta= \beta\circ\alpha,\quad \forall \alpha \in Sym_f(M)\}.$$ Then $Z$ is non-empty since the unit 1 belongs to $Z$ and it is actually an $A$-subalgebra of $End_A(M)$. For each $n$, let $Z_n$ be the set of all endomorphisms $\alpha$ in $Z$ that satisfy the formula $$\begin{split} S_n(\alpha)\Leftrightarrow & \forall \bar{x},\bar{y},\bar{u},\bar{v} \left(\sum_{i=1}^nf(x_i,y_i)=\sum_{i=1}^{n}f(u_i,v_i) \rightarrow\right. \\ & \left.\sum_{i=1}^nf(\alpha x_i,y_i)=\sum_{i=1}^{n}f(\alpha u_i,v_i) \right). \end{split}$$ i.e. $$Z_n=\{\alpha\in Z: S_n(\alpha)\}.\label{zn}$$ Each $Z_n$ is also an $R$-subalgebra of $Z$. Now set $$P(f)=\cap_{i=1}^\infty Z_n.$$ The identity mapping is in every $Z_n$ so $P(f)$ is not empty. Since the mapping $f$ is full, for every $x\in N$ there are $x_i$ and $y_i$, in $M$ such that $x=\sum_{i=1}^nf(x_i,y_i)$ for some $n$. The $P(f)$-module $M$ is faithful by construction. Now we can define the action of $P(f)$ on $N$ by setting $\alpha x=\sum_{i=1 }^nf(\alpha x_i,y_i)$. The action is clearly well-defined since $\alpha$ satisfies all the $S_n(\alpha)$ and makes $N$ into a $P(f)$-module. Moreover for any $x,y\in M$ and $A\in P(f)$ we have $$f(\alpha x,y)=f(x,\alpha y)=\alpha f(x,y),$$ that is, $f$ is $P(f)$-bilinear. In order to prove that the ring $P(f)$ is the largest ring of scalars, we prove that for any ring $P$ with respect to which $f$ is bilinear, $\phi_P(P) \subseteq P(f)$. Since $f$ is $P$-bilinear $\phi_P(P) \subseteq Sym_f(M)$. Let $p \in P$ then for $\alpha\in Sym_f(M)$ and $x,y \in M$, $$\label{myrem}\begin{split} f(\alpha \circ\phi_{p}(x),y)&=f(\phi_{p}(x),\alpha y)\\ &=\alpha f(x,\alpha y)=\alpha f(\alpha x,y)\\ &= f(\phi_{p}\circ \alpha (x),y). \end{split}$$ Non-degeneracy of $f$ implies that $\phi_{p}\circ \alpha=\alpha\circ \phi_{p}$. Therefore $\phi_P(P)\subseteq Z$. It is clear that $\phi_{p}$ belongs to every $Z_n$ by bilinearity of $f$ with respect to $P$. Therefore $\phi_P(P)\subseteq P(f)$, hence $E(f,P)\leq E(f,P(f))$. Largest ring of scalars as a logical invariant {#largest-ring:sec} ---------------------------------------------- Indeed the ring $P(f)$ is interpretable in the bilinear map $f$ providing that $f$ satisfies certain conditions in addition to the ones in Proposition \[P(f)\]. The mapping $f$ is said to have *finite width* if there is a natural number $s$ such that for every $u\in N$ there are $x_i\in M_1$ and $y_i\in M_2$ such that $$u=\sum_{i=1}^sf(x_i,y_i).$$ The least such number, $w(f)$, is the *width* of $f$. A set $E_1=\{e_1,\ldots e_n\}$ is a *left complete system* for a non-degenerate mapping $f$ if $f(E_1,y)=0$ implies $y=0$. The cardinality of a minimal left complete system for $f$ is denoted by $c_1(f)$. A right complete system and the number $c_2(f)$ are defined correspondingly. The *type* of a bilinear mapping $f$, denoted by $\tau(f)$ , is the triple $$(w(f),c_1(f), c_2(f)).$$ The mapping $f$ is said to be of finite type if $w(f)$, $c_1(f)$ and $c_2(f)$ all exist. If $f,g:M_1\times M_2 \rightarrow N$ are bilinear maps of finite type we say that the type of $g$ is less than the type of $f$ and write $\tau(g)\leq \tau(f)$ if $w(g)\leq w(f)$, $c_1(g)\leq c_1(f)$ and $c_2(g)\leq c_2(f)$. Let $A$ be a scalar ring. Assume $M_1$, $M_2$ and $N$ are faithful $A$-modules. Let $f:M_1\times M_2\rightarrow N$ be a $A$-bilinear map. We associate two structures to $f$. The first one is $$\mathfrak{U}(f)=\langle M_1,M_2,N, \delta\rangle.$$ where $M_1$, $M_2$ and $N$ are abelian groups and $\delta$ describes the bilinear map. The other one is $$\mathfrak{U}_A(f)=\langle A,M_1,M_2,N,\delta,s_{M_1},s_{M_2}, s_N\rangle,$$ where $A$ is a scalar ring and $s_{M_1}$, $s_{M_2}$ and $s_N$ describe the actions of $A$ on the modules $M_1$, $M_2$ and $N$ respectively. We state the following theorem without proof. Readers may refer to the cited reference for a proof. \[ringinter\]Let $f:M_1\times M_2 \to N$ be a non-degenerate full bilinear mapping of finite type and let $P(f)$ be the largest ring of scalars of $f$. Then $\mathfrak{U}_{P(f)}(f)$ is absolutely interpretable in $\mathfrak{U}(f)$. Moreover the same formulas interpret $\mathfrak{U}_{P(g)}(g)$ in $\mathfrak{U}(g)$ if $g$ is a full non-degenerate bilinear map with $\tau(g)\leq \tau(f)$. \[M1notM2\] Proposition \[P(f)\] and Theorem \[ringinter\] are stated for bilinear maps $f:M_1\times M_2\to N$ where $M_1=M_2$. In the following we reduce the general case $M_1\neq M_2$ to the case in these statements. So assume $f$ is full and non-degenerate. Now define $$f^p: (M_1\oplus M_2) \times (M_1\oplus M_2) \to N\oplus N$$ by $$f^p(x_1\oplus y_1, x_2 \oplus y_2)=f(x_1,y_2) \oplus f(x_2,y_1),$$ for all $x_i \in M_1$ and $y_i\in M_2$, $i=1,2$. The map is clearly non-degenerate and full. So by Proposition \[P(f)\] the largest ring $P=P(f^p)$ of scalars exists. Define $P_1$ as the subset of all elements of $P$ stabilizing both $M_1$ and $M_2$. Indeed $P_1$ is a non-empty subring of $P$ and makes $f$ a $P_1$-bilinear map. On the other hand any ring $R$ making $f$ bilinear has to embed into $P_1$ otherwise $P(f^p)$ would not define the largest enrichment. So indeed $P(f)=P_1$. Moreover $f^p$ is absolutely interpretable in $f$. Indeed in this case $M_1$ and $M_2$ are absolutely definable subgroups of $M_1\oplus M_2$ and so the ring $P_1$ is absolutely definable in $P(f^p)$. Therefore $P_1$ is interpretable in $f$. If one only cares about algebraic properties of $P(f)$ there is a simpler definition of $P(f)$. Given a full nondegenerate $R$-bilinear map $$f:M_1\times M_2\to N,$$ one can identify $P(f)$ with the subring $S\leq End_R(M_1)\times End_R(M_2)\times End_R(N)$ consisting of all triples $A=(\phi_1,\phi_2,\phi_0)$ such that for all $x\in M_1$ and $y\in M_2$ $$\label{myrem1}f(\phi_1(x), y)=f(x,\phi_2(y))=\phi_0(f(x,y)).$$ Moreover one could drop the reference to $R$ and simply take $P(f)$ to be the subring $S'$ of $End(M_1)\times End(M_2)\times End(N)$ whose elements satisfy . This is simply because $E(f,P(f))\leq E(f,S')$, while $E(f,P(f))$ is the largest enrichment. Some preliminary facts on algebras ---------------------------------- Assume $R$ is an FDZ-algebra. Since $R^+$ is a finitely generated group then $R^+$ is generated by a finite ordered set of its elements say $u_1, \ldots , u_M$ such that $U_i/U_{i+1}$ is a cyclic group generated by $u_i+U_{i+1}$ where $U_{i}$ is the subgroup generated $u_{i}, \ldots u_M$ or in notation $U_i=\langle u_{i}, \ldots u_M\rangle$. The order $e_i$ of $u_i+U_{i+1}$ is called the *period* of $u_i$. If $u_i$ has infinite order then we write $e_i=\infty$. We say that $\bar{u}=(u_1,\ldots, u_M)$ is a *pseudo-basis* of period $(e_1, \ldots , e_M)$. If $e_i<\infty$ there are fixed integers $t_{ik}$ such that $e_iu_i=\sum_{i+1}^M t_{ik}u_k.$ These $t_{ik}$ are called *torsion structure constants* associated to $\bar{u}$. We assume an arbitrary but fixed order on the $t_{ik}$. It’s an easy corollary of the structure theorem for finitely generated groups that the number $M$, period $\bar{e}$ and the structure constants $t_{ik}$ uniquely determine $R^+$ up to isomorphism. Now consider the ring structure of an FDZ-algebra $R$ and consider a pseudo-basis $\bar{u}$ as above. Then there are fixed integer constants $t_{ijk}$ such that $$u_iu_j=\sum_{k=1}^M t_{ijk}u_k.$$ The numbers $t_{ijk}$ are called the *multiplicative structure constants associated to $\bar{u}$*. Again we assume a fixed order on the set of all $t_{ijk}$ obtained as above. Now it is an elementary exercise to check that the number $M$, periods $\bar{e}$, the constants $t_{ik}$ and the $t_{ijk}$ fix the ring $R$ up to isomorphism of rings. ### Largest ring of scalars $A(R)$ Let $R$ be an $A$-algebra where $A$ is a scalar ring. Here we only consider those algebras which are faithful with respect to the action of their rings of scalars. Let $\mu: A\to A_1$ be an inclusion of rings. We say that an $A$-algebra $R$ has an $A_1$-enrichment with respect to $\mu$ if $R$ is an $A_1$-algebra and $ \alpha r =\mu(\alpha)r$, $r\in R$, $\alpha \in A$. Denote by $A(R)$ the largest, in the sense defined just above, commutative subring of $End_A(R/Ann(R))$ that satisfies the following conditions: 1. $R/Ann(R)$ and $R^2$ are faithful $A(R)$-modules. 2. The full non-degenerate bilinear mapping $$f_F: R/Ann(R)\times R/Ann(R)\to R^2$$ induced by the product in $R$ is $A(R)$-bilinear. 3. The canonical homomorphism $\eta:R^2 \to R/Ann(R)$ is $A(R)$-linear. \[A(R)exists:prop\] For any algebra $R$ the ring $A(R)$ is definable, it is unique, and does not depend on the choice of the initial ring of scalars. Elementary equivalence of FDZ-scalar rings {#Z-interpret:sec} ========================================== In this section we describe by first-order formulas some algebraic invariants of any scalar ring $A$ with finitely generated additive group $A^+$. In particular we provide a proof of Theorem \[elemmod:thm\]. Interpretability of decomposition of zero into the product of prime ideals with fixed characteristic {#Prime-decom:sec} ---------------------------------------------------------------------------------------------------- Let $A$ be a scalar ring. Suppose that we have a decomposition of zero into the product of finitely generated prime ideals: $$0={{\mathfrak}{p}}_1\cdot {{\mathfrak}{p}}_2 \cdots {{\mathfrak}{p}}_m,\qquad (\mathfrak{P})$$ Let $Char({{\mathfrak}{p}}_i)=\lambda_i$ be the characteristic of the integral domain $A/{{\mathfrak}{p}}_i$ and $$Char(\mathfrak{P})=(\lambda_1,\ldots , \lambda_m).$$ The purpose of this subsection is to obtain a formula interpreting the decomposition of type $(\mathfrak{P})$ in $A$ with the fixed characteristic $Char(\mathfrak{P})$, where the interpretation is uniform with respect to $Th(A)$. A sequence of lemmas will follow. We omit some proofs as they are obvious. \[P1\] Consider the formula $$Id(x,\bar{y})=\exists z_1,\ldots, \exists z_n (x=y_1z_1+\ldots + y_nz_n).$$ For any tuple $\bar{a}=(a_1,\ldots, a_n)\in A^n$ the formula $Id(x,\bar{a})$ defines in $A$ the ideal $id(\bar{a})$, generated by the elements $a_1,\ldots, a_n$. \[P2\] The formula $$P(\bar{y})=\forall x_1,\forall x_2(Id(x_1x_2,\bar{y})\rightarrow (Id(x_1,\bar{y})\vee Id(x_2,\bar{y})))$$ is true for the tuple $\bar{a}$ of elements of the ring $A$ if and only if the ideal $id(\bar{a})$ is prime. \[P7\] There exists a formula $Id_i(x,\bar{y}_1,\ldots , \bar{y}_i)$, such that for any tuples $\bar{a}_1$, …, $\bar{a}_i$, $Id_i(x,\bar{a}_1,\ldots , \bar{a}_i)$ defines the ideal ${{\mathfrak}{p}}_1\cdots {{\mathfrak}{p}}_i$ in $A$ where ${{\mathfrak}{p}}_k=id(\bar{a}_k)$. Indeed the ideal ${{\mathfrak}{p}}_1\cdots {{\mathfrak}{p}}_i$ is generated by all the products of the form $y_1\cdots y_i$ where $y_k$ is an element of the tuple $\bar{a}_k$ and the number of such products is finite. So an application of Lemma \[P1\] will imply the above statement. \[P3\] The formula: $$D(\bar{y_1},\ldots, \bar{y_m})=\forall x( \bigwedge_{i=1}^mP(\bar{y}_i) \wedge Id_m(x,\bar{y}_1,\ldots , \bar{y}_m)\rightarrow x=0)$$ is true for tuples $\bar{a}_1,\ldots, \bar{a}_m$ if and only if the ideals ${{\mathfrak}{p}}_i=Id(\bar{a}_i)$ satisfy the decomposition $(\mathfrak{P})$. \[P4\] The formula $$D_{\Lambda}(\bar{y}_1,\ldots , \bar{y}_m)=D(\bar{y}_1, \ldots \bar{y}_m) \wedge \bigwedge_{i=1}^m \forall x Id(\lambda_ix,\bar{y}_i)\wedge\bigwedge_{i=1}^m \exists z \neg Id(z,\bar{y}_i)$$ where $\Lambda=(\lambda_1,\ldots \lambda_m)=Char(\mathfrak{P})$ is true for tuples $\bar{a}_1,\ldots ,\bar{a}_m$ of elements of $A$ if and only if all the following statements hold: - the ideals ${{\mathfrak}{p}}_i=id(\bar{a}_i)$ satisfy the decomposition $(\mathfrak{P})$, - if $\lambda_i>0$ then $Char(A/{{\mathfrak}{p}}_i)=\lambda_i$, - the integral domains $A/{{\mathfrak}{p}}_i$ are all non-zero. Denote by $0(\mathfrak{P})$ the number of zeros in the tuple $(\lambda_1,\ldots , \lambda_m)$. \[decom\]Let $\mathfrak{P}=({{\mathfrak}{p}}_1,\ldots, {{\mathfrak}{p}}_m)$ be a collection of finitely generated prime ideals of the scalar ring $A$, satisfying the decomposition $(\mathfrak{P})$ and possessing the least number $0(\mathfrak{P})$ among all such decompositions. Then for any scalar ring $B$ such that $A\equiv B$ in $L_1$, then the formula $D_{\Lambda}(\bar{y}_1,\ldots , \bar{y}_m)$ is true in $B$ on tuples $\bar{b}_1$, …, $\bar{b}_m$, if and only if: 1. $Id(x,\bar{b}_i)$ defines the prime ideal ${{\mathfrak}{q}}_i=id(\bar{b}_i)$, 2. $0={{\mathfrak}{q}}_1\cdot {{\mathfrak}{q}}_2\cdots {{\mathfrak}{q}}_m$, 3. $Char(B/{{\mathfrak}{q}}_i)=\lambda_i$, $i=1, \ldots ,m$. Items 1 and 2 follow from Lemma \[P3\]. If $\lambda_i>0$ then Char$(B/q_i)=\lambda_i$ according to Lemma \[P4\]. Consequently, $0(\mathfrak{P})\geq 0(\mathfrak{Q})$ where $\mathfrak{Q}=({{\mathfrak}{q}}_1,\ldots , {{\mathfrak}{q}}_m)$. If $0(\mathfrak{P})>0(\mathfrak{Q})$ then starting from $\mathfrak{Q}$ we construct the formula $D_{\mu}$, $\mu=char(\mathfrak{Q})$. From $A\equiv B$ and Lemma \[P4\] we obtain that there exists a tuple $\mathfrak{P}'=({{\mathfrak}{p}}'_1,\ldots, {{\mathfrak}{p}}'_m)$ such that $0(\mathfrak{P}')\leq 0(\mathfrak{Q})<0(\mathfrak{P})$ which contradicts the choice of $\mathfrak{P}$. Consequently $0(\mathfrak{P})=0(\mathfrak{Q})$ and hence $Char(\mathfrak{P})=Char(\mathfrak{Q})$. The proposition is proved. \[decom:rem\] Any Noetherian commutative associative ring with a unit possesses a decomposition of zero $0={{\mathfrak}{p}}_1\ldots {{\mathfrak}{p}}_m$, satisfying the assumptions of Proposition \[decom\]. \[primary:thm\] For any Noetherian associative commutative ring $A$ with a unit, there exists an interpretable decomposition of zero into a product of prime ideals, where the interpretation is uniform with respect to $Th(A)$. The proposition is a direct corollary of Proposition \[decom\] and Remark \[decom:rem\]. The case of FDZ-scalar rings {#elem-eq-rings:sec} ---------------------------- Now let $A$ be an FDZ-scalar ring. We shall denote by $r(A)$ the minimal number of generators of $A^+$ as an abelian group, say, the number of cyclic factors in the invariant decomposition of $A^+$. In case that $M$ is a finitely generated $A$-module where $A$ is as above the minimal number of generators of $M$ as an abelian group is denoted by $r(M)$, while the minimal number of generators for $M$ as an $A$-module is denoted by $r_A(M)$. \[P9\] There exists a sentence $ch_{\lambda}$ of $L_1$ such that for any integral domain $A$ with finitely generated additive group $A^+$: $$char(A)=\lambda \Leftrightarrow A\models ch_{\lambda}.$$ To prove the claim notice that if $\lambda$ is a prime then we can set $ch_{\lambda}= \forall x (\lambda x=0)$. For $\lambda =0$ it is enough to note that for the integral domain $A$, $char(A)=0$ if and only if $2\neq 0$ and $1/2\notin A$. In fact if $char(A)=0$ then $2\neq 0$ and if $1/2\in A$ then $A\geq \mathbb{Z}[1/2]$ but $\mathbb{Z}[1/2]$ is not finitely generated. Contradicting with the assumption that $A^+$ is finitely generated. Conversely if $char(A)=p\neq 2$, then $pA=0$. So $A$ contains the finite field ${{\mathbb}{Z}}/p{{\mathbb}{Z}}$ and so $1/2\in A$. \[P10\] Let $A$ be an FDZ-scalar ring with $r(A)=n$. Then there exists a sentence $\varphi_{n}$ of $L_1$ such that $ A \models \varphi_{n}$ and for FDZ-scalar ring $B$, $$B \models \varphi_n \Leftrightarrow r(B)\leq n.$$ Let us first assume that $A$ is an integral domain. By Lemma \[P9\] there is a sentence $ch_{\lambda}$ that defines the characteristic $char(A)=\lambda$ of $A$ in the language of rings and hence $char(B)=\lambda$ if $B\models ch_{\lambda}$. If $char(A)=\lambda\neq 0$ then $A$ is finite and $\varphi_n$ will say that $ch_\lambda$ and $A$ does not have more than $\lambda^n$ elements. If $char(A)=0$ then $r(A)=n$ if and only if $|A^+/2A^+|=2^n$. So in this case $\varphi_n$ will say that $ch_0$ and there are precisely $2^n$ distinct elements in $A$ modulo $2A$. Now assume $A$ is not necessarily an integral domain. Then $A$ is Noetherian and by Remark \[decom:rem\] it admits a decomposition of zero $$0={{\mathfrak}{p}}_1\cdot {{\mathfrak}{p}}_2 \cdots {{\mathfrak}{p}}_m,\qquad (\mathfrak{P})$$ with $\Lambda=char(\mathfrak{P})$ where the prime ideals ${{\mathfrak}{p}}_i$ are finitely generated. Set $O_i={{\mathfrak}{p}}_0\cdots{{\mathfrak}{p}}_i$, where ${{\mathfrak}{p}}_0=A$. Set also $\bar{O}_i=O_i/O_{i+1}$. Note that $r(A)$ is bounded by $$\sum_{i=0}^{m-1}r(\bar{O}_i).$$ So it is enough to come up with sentences $\varphi_i$ each expressing a bound for $r(\bar{O}_i)$. By lemma \[P4\] there are tuples of elements $\bar{a}_i$ ,$i=1, \ldots, m$ satisfying $D_{\Lambda}(\bar{a}_1, \ldots \bar{a}_m)$. Moreover if $B$ is any ring similar to $A$ with tuples of elements $\bar{b}_1, \ldots, \bar{b}_m$ which satisfy $D_\Lambda(b_1,\ldots ,b_m)$ then by Lemma \[P5\], $B$ has a decomposition of zero $\mathfrak{Q}$ with same exact properties of $\mathfrak{P}$. Moreover the formula $id_i(x,\bar{a}_1, \ldots \bar{a}_i)$ from Lemma \[P1\] defines $O_i$ in $A$ and $id_i(x,\bar{b}_1, \ldots \bar{b}_i)$ defines similar term in $B$. The quotients $\bar{O}_i$ are finitely generated $A$-modules over the integral domains $A/{{\mathfrak}{p}}_i$. Assume $r(A/{{\mathfrak}{p}}_{i+1})=n_i$ and $r_{A/{{\mathfrak}{p}}_{i+1}}(\bar{O})=s_i$. Note that $r(\bar{O}_i)\leq n_is_i$. So it is enough to define $n_i$ and $s_i$ in the language of rings. By definability of the ${{\mathfrak}{p}}_i$ and the $O_i$ it is easy to write a sentence in the language of rings saying that $r_{A/{{\mathfrak}{p}}_{i+1}}(\bar{O}_i)\leq s_i$. By the first paragraph of this proof and definability of ${{\mathfrak}{p}}_{i+1}$ there is also a sentence in the language of rings saying that $r(A/{{\mathfrak}{p}}_{i+1})\leq n_i$. Note that the same formulas work for a ring $B$ as above. \[10b\] Assume $\mathcal{K}$ is the class of all FDZ-scalar rings. Assume $\mathcal{I}_n$ is the subclass of $\mathcal{K}$ consisting of all integral domains $A$ of characteristic zero with $r(A)\leq n$, for some natural number $n>0$. Then there exists a sentence $\Phi_n$ of the language of rings such that for any $A\in \mathcal{K}$ $$A\models \Phi_n \Leftrightarrow A\in \mathcal{I}_n.$$ A ring $A$ being an integral domain is axiomatizable by one ring theory sentence. The formula $ch_0$ from Lemma \[P9\] is true in any $A\in \mathcal{I}_n$ and conversely implies that $A\in \mathcal{K}$ has characteristic 0 once $A$ satisfies it. The formula $\varphi_n$ from Lemma \[P10\] is satisfied by any $A\in \mathcal{I}_n$ and conversely will force $r(A)\leq n$ for any $A\in \mathcal{K}$ satisfying it. The conjunction of these sentences is the desired one. \[10a\] Let $A\in \mathcal{I}_n$. Then, there exists a formula $\phi_{{{\mathbb}{Z}}}$ of the language of rings such that $$A \models \phi_{{\mathbb}{Z}}\Leftrightarrow A\cong {{\mathbb}{Z}}.$$ By Corollary \[10b\] the formula $\Phi_1$ characterizes members of $\mathcal{I}_1$ among those of $\mathcal{K}$. But $\mathcal{I}_1$ has only one member up to isomorphism, namely ${{\mathbb}{Z}}$. So we may set $\phi_{{\mathbb}{Z}}=\Phi_1$. \[julia:lem\] Consider the class $\mathcal{I}_n$ introduced in Corollary \[10b\]. Then there exists a formula $R_n(x)$ defining the subring $\mathbb{Z}\cdot 1_A$ in any member $A$ of $\mathcal{I}_n$. We need to note that the field of fractions $F$ of $A$ is an extension of field of rationals ${{\mathbb}{Q}}$ with dimension $n$ over ${{\mathbb}{Q}}$. So $F$ is a field of algebraic numbers of finite degree over ${{\mathbb}{Q}}$. Now by Theorem on page 956 of [@julia] the ring of integers ${{\mathbb}{Z}}$ is definable in $F$ by a formula $\Phi_{{{\mathbb}{Z}}(F)}(x)$. An inspection of the proof shows that the formula defines ${{\mathbb}{Z}}$ in any algebraic extension $K$ of ${{\mathbb}{Q}}$ with $[K:{{\mathbb}{Q}}]\leq [F:{{\mathbb}{Q}}]=r(A)=n$ (See the formula on line 15 of page 952 as well as the one in lines 20-21 of page 956 in [@julia]). Moreover $F$ is uniformly interpretable in $A$. Though elementary, let us elaborate on this claim here a bit. Recall that $F$ is realized as $X/\sim$ where $$X=\{(x,y):x\in A, y\in A\setminus \{0\}\},$$ and $\sim$ is the equivalence relation on $X$ defined by $$(x,y)\sim (z,w) \Leftrightarrow xw=yz.$$ Addition and multiplication are defined on $X/\sim$ in the obvious manner using addition and multiplication on $A$. The same formulas interpret the field of fractions $K$ of any integral domain of characteristic zero $B$ in $B$. So combining the results here we have an interpretation of ${{\mathbb}{Z}}$ in $A$. But the above interpretation of ${{\mathbb}{Z}}$ in $A$ also provides a formula defining ${{\mathbb}{Z}}$ (as a subset of $A$) in $A$ in the following way. Note that there is an interpretable monomorphism $\mu: A \to F$ defined by $\mu (a)=[(a,1)]$ where $|F|=X/\sim$ is considered as the set of equivalence classes $[(x,y)]$ described above. Now the copy of ${{\mathbb}{Z}}$ sitting in $F$ is included in the image of $\mu$ so the copy of ${{\mathbb}{Z}}$ in $A$ is a definable subset of $A$ as $\mu^{-1}(\Phi_{{{\mathbb}{Z}}(F)}(\mu(A))$. Since by Corollary \[10a\] ${{\mathbb}{Z}}$ is axiomatizable in $\mathcal{I}_n$ by one formula, there exists a formula defining ${{\mathbb}{Z}}$ in any member of $\mathcal{I}_n$. \[P5\] There exists a formula $R_{n,\Lambda}(x,\bar{y})$ such that for any scalar ring $A$ with unit and $r(A)\leq n$ and for any prime ideal ${{\mathfrak}{p}}=id(\bar{a})$ of $A$ if $char(A/{{\mathfrak}{p}})=\lambda$ then the formula $R_{n,\Lambda}(x,\bar{a})$ defines the subring $$\mathbb{Z}\cdot 1+{{\mathfrak}{p}}=\{z\cdot 1+x:z\in \mathbb{Z}, x\in {{\mathfrak}{p}}\},$$ in $A$. Indeed the ideal ${{\mathfrak}{p}}=id(\bar{a})$ is defined in $A$ by the formula $Id(x,\bar{a})$. Consequently the ring $A/{{\mathfrak}{p}}$ and the canonical epimorphism $A \rightarrow A/{{\mathfrak}{p}}$ are interpretable in $A$. Therefore to obtain $R_{n,\Lambda}(x,\bar{y})$ it is sufficient to define the subring $\mathbb{Z}\cdot 1$ in $A/{{\mathfrak}{p}}$. In the case of $Char (A/{{\mathfrak}{p}})=0$ we use the formula $R_n(x)$ from Lemma \[julia:lem\]. As for the case of $char(A/{{\mathfrak}{p}})>0$ the set $\mathbb{Z}\cdot 1 +{{\mathfrak}{p}}$ is finite in $A/{{\mathfrak}{p}}$ and hence definable in $A/{{\mathfrak}{p}}$. \[P6\] Assume $A$ is a scalar ring and $r(A)\leq n$, admitting a decomposition $$0={{\mathfrak}{p}}_1\ldots {{\mathfrak}{p}}_m, \quad (\mathfrak{P})$$ into prime ideals ${{\mathfrak}{p}}_i=id(\bar{a}_i)$ with $char(A/{{\mathfrak}{p}}_i)=\lambda_i$, $i=1, \ldots , n$. Let $\Lambda=(\lambda_1, \ldots , \lambda_m)$. Then there exists a first-order formula $R_{n,\Lambda}(x, \bar{y}_1, \ldots , \bar{y}_m)$ of $L_1$ such that the formula $R_{n,\Lambda}(x,\bar{a}_1, \ldots, \bar{a}_m)$ defines in $A$ the subring $$A_{\mathfrak{P}}=\bigcap_{i=1}^m(\mathbb{Z}\cdot 1 + {{\mathfrak}{p}}_i),$$ For each $1\leq i \leq m$ consider the formula $R_{n,\lambda_i}(x,\bar{y}_i) $ introduced in Lemma \[P5\]. So we can set $$R_{n,\Lambda}(x,\bar{a}_1, \ldots, \bar{a}_m)=\bigwedge_{i=1}^mR_{n,\lambda_i}(x,\bar{y}_i).$$ To a decomposition of $0$ in $A$ as above we associate the series of ideals $$A> {{\mathfrak}{p}}_1>{{\mathfrak}{p}}_1{{\mathfrak}{p}}_2>\ldots> {{\mathfrak}{p}}_1\cdots {{\mathfrak}{p}}_m =0$$ of the ring $A$ which will be called a $\mathfrak{P}$-series. The ring $A_{\mathfrak{P}}$ from Lemma \[P6\] acts on all the quotients ${{\mathfrak}{p}}_1\cdots {{\mathfrak}{p}}_i/{{\mathfrak}{p}}_1\cdots {{\mathfrak}{p}}_{i+1}$ as each subring $\mathbb{Z}\cdot 1 +{{\mathfrak}{p}}_{i+1}$ of $A$ acts on the corresponding quotient ${{\mathfrak}{p}}_1\cdots {{\mathfrak}{p}}_i/{{\mathfrak}{p}}_1\cdots {{\mathfrak}{p}}_{i+1}$ for each $i=1, \ldots, m$. Recall that $L_2$ is the language of two-sorted modules. Take a module $\langle M, A\rangle$. If $A$ is a scalar ring admitting a decomposition of zero $\mathfrak{P}$, then $\mathfrak{P}$-series of the ring $A$ induces a series of $A$-modules $$M\geq {{\mathfrak}{p}}_1M\geq {{\mathfrak}{p}}_1{{\mathfrak}{p}}_2M\geq \ldots \geq {{\mathfrak}{p}}_1\cdots {{\mathfrak}{p}}_mM=0,$$ which will also be called a *$\mathfrak{P}$-series for the $A$-module $M$* or a *special series* for $M$. The following lemma is a direct corollary of Proposition \[decom\]. \[P8\] There exists a formula $\phi_i(x,\bar{y}_1, \ldots ,\bar{y}_i)$ of $L_2$ such that if ${{\mathfrak}{p}}_1\cdots {{\mathfrak}{p}}_m=0$ is a decomposition of zero in the scalar ring $A$ and ${{\mathfrak}{p}}_k=id(\bar{a}_k)$, then $\phi_i(x,\bar{a}_1, \ldots , \bar{a}_i)$ defines the submodule $M_i={{\mathfrak}{p}}_1\cdots {{\mathfrak}{p}}_iM$ in the two-sorted model $\langle M,A \rangle$. The following proposition collects the main results of this section so far. \[big\] Let ${\mathcal}{M}$ be the class of all two-sorted modules $\langle M,A\rangle$ where $M$ is a finitely generated module over an FDZ-scalar ring $A$. Pick $\langle M, A\rangle$ in ${\mathcal}{M}$ with $r(M)\leq n$. Assume there are tuples $\bar{a}_1$,$ \ldots $, $\bar{a}_m$ of elements of $A$ which satisfy $D_{\Lambda}(\bar{x}_1, \ldots , \bar{x}_m)$. Then the following hold uniformly with respect to all models $\langle N, B\rangle$ of $Th(\langle M, A\rangle)$ from ${\mathcal}{M}$ which contain tuples $\bar{b}_{i}$ which satisfy $D_{\Lambda}(\bar{b}_1, \ldots , \bar{b}_m)$. 1. $Id(x,\bar{b}_i)$ defines in $B$ the prime ideal ${{\mathfrak}{q}}_i=id(\bar{b}_i)$; 2. If $\mathfrak{Q}=({{\mathfrak}{q}}_1, \ldots , {{\mathfrak}{q}}_m)$ then $char(\mathfrak{q})=\Lambda$; 3. $0={{\mathfrak}{q}}_1\cdots {{\mathfrak}{q}}_m$; 4. $\phi_i(x,\bar{q}_1, \ldots , \bar{q}_i)$ defines the $i$-th term $N_i={{\mathfrak}{q}}_1\cdots {{\mathfrak}{q}}_iN$ of the special $\mathfrak{Q}$-series for $N$ in $\langle N,B \rangle$; 5. $r(B)\leq n$. 6. The formula $R_{n,\Lambda}(x,\bar{b}_1, \ldots , \bar{b}_m)$, defines the subring $$B_{\mathfrak{Q}}=\bigcap_{i=1}^m(\mathbb{Z}\cdot 1+{{\mathfrak}{q}}_i),$$ in $B$. Items (1)-(4) follow from Lemma \[decom\], \[P6\] and \[P9\]. Part (5) follows directly from Lemma \[P9\]. To prove (6.) we note that by (5), $r(B)\leq n$. So the statement follows from Lemma \[P8\]. Finally we are ready to finish the proof of the main technical result of this section. Recall that by a ${{\mathbb}{Z}}$-pseudo-basis for finitely generated abelian group $M$ we simply mean a minimal generating set for $M$ as an abelian group. Assume $\bar{u}=(u_1, \ldots , u_s)$ is an ordered ${{\mathbb}{Z}}$-pseudo-basis for $M$ and let $M_i$ be the subgroup of $M$ generated by $u_i, \ldots, u_s$. Again, recall that the period $e_i$ of $u_i$ is the order of the cyclic group $M_i/M_{i+1}$ if $M_i/M_{i+1}$ is finite, and we set $e_i=\infty$ if the corresponding quotient is infinite. \[basedef\] Let $\langle M, A\rangle$ and $\langle N, B\rangle$ be finitely generated modules over FDZ-scalar rings $A$ and $B$ respectively. Let the collection of prime ideals $\mathfrak{P}=({{\mathfrak}{p}}_1,\ldots , {{\mathfrak}{p}}_m)$, ${{\mathfrak}{p}}_i=id(\bar{a}_i)$, satisfy the usual conditions, say as in Proposition \[big\]. Assume $\bar{c}=(c_1, \ldots , c_n)$ is a ${{\mathbb}{Z}}$-pseudo-basis of period $\bar{f}=(f_1, \ldots f_n)$ associated to the $\mathfrak{P}$-series of $A$, and $\bar{u}=(u_1, \ldots, u_s)$ is a pseudo-basis of period $\bar{e}=(e_1, \ldots , e_s)$ associated with the $\mathfrak{P}$-series for $M$. Then there exists a formula $$\phi_{\mathfrak{P},n}(x_1, \ldots , x_n,y_1, \ldots , y_s, \bar{y}_1, \ldots , \bar{y}_m)$$ defining in the two-sorted structure $M_A^*=\langle M, A, \bar{a}_1,\ldots , \bar{a}_m\rangle$ the set of all ${{\mathbb}{Z}}$-pseudo-bases of periods $\bar{f}$ and $\bar{e}$ associated with the $\mathfrak{P}$-series for $A$ and $M$, respectively. Moreover the formula $\phi_{\mathfrak{P},n}(\bar{x}, \bar{y}, \bar{b}_1, \ldots, \bar{b}_m)$ defines in the model $N_B^*=\langle N,B, \bar{b}_1, \ldots \bar{b}_m\rangle$ the set of all ${{\mathbb}{Z}}$-pseudo-bases $\bar{d}$ and $\bar{v}$ of $N$ associated to the corresponding special $\mathfrak{Q}$-series of $B$ and $N$ if $\bar{b}_1$, …, $\bar{b}_m$ satisfy the formula $D_{\Lambda}$. By Proposition \[big\] the models $\langle M_i, A_{\mathfrak{P}}, \bar{a}_i\rangle$ are definable in $M_A^*$ uniformly with respect to all models $N_B^*$, with $N$ and $B$ satisfying the hypotheses, of $Th(M_A^*)$. So it suffices to find a formula $\phi_i$ for each $i$, defining a basis for $M_i/M_{i+1}$ of fixed period $\bar{e}_i$. the model $\langle M_i, A_{\mathfrak{P}}\rangle$ is interpretable in $\langle M,A\rangle$ with the help of any tuples of generating elements $\bar{a}_1$, …, $\bar{a}_m$ satisfying the formula $D_{\Lambda}$. Since $M_{i+1}={{\mathfrak}{p}}_{i+1}M_i$, the model $\langle {\overline}{M}_i,A_i \rangle$ where ${\overline}{M}_i=M_i/M_{i+1}$ and $A_i=A_{\mathfrak{P}}/({{\mathfrak}{p}}_{i+1}\cap A_{\mathfrak{P}})$ is obviously interpretable in $\langle M_i, A_{\mathfrak{P}}\rangle$ with the help of $\bar{a}_{i+1}$. In the view of the fact that $A_i$ is either $\mathbb{Z}$ or the finite field $\mathbb{Z}/p\mathbb{Z}$ and the action of $A_i$ on ${\overline}{M}_i$ is interpretable in $\langle M,A\rangle$ it is easy to write out a formula defining all bases of ${\overline}{M}_i$ of given period $\bar{e_i}$ and thus to construct the desired formula $\phi_i$. Again $\phi_i$ depends on the tuples $a_i$ as far as they satisfy $D_\Lambda$. So again by Proposition \[big\] the formulas $\phi_i$ define all ${{\mathbb}{Z}}$-pseudo-bases for the $N_i/N_{i+1}$ in a model $N_B^*$ of $Th(M_A^*)$ where the $N_i$ are defined in $N^*_B$ with the same formulas that define the $M_i$ in $M^*_A$. Assume $A$ and $M$ satisfy the usual conditions, $\bar{c}=(c_1, \ldots , c_n)$ is a ${{\mathbb}{Z}}$-pseudo-basis of $A$ of period $\bar{f}=(f_1, \ldots f_n)$ and $\bar{u}=(u_1, \ldots , u_s)$ is a ${{\mathbb}{Z}}$-pseudo-basis of $M$ of period $\bar{e}=(e_1, \ldots , e_s)$. Then 1. for any $1\leq i,j\leq n$ there exist integers $s_k(c_i,c_j)$ such that $c_ic_j=\sum_{k=1}^ns_k(c_i,c_j)c_k$, 2. for any $1\leq i \leq n$ and $1\leq j \leq s$ there exist integers $s'_k(c_i,u_j)$ such that $c_iu_j=\sum_{k=1}^ss'_k(c_i,u_j)u_k$, 3. for any $1 \leq i \leq n$ if $f_i< \infty $ then there exist integers $t_k(f_ic_i)$ such that $f_ia_i=\sum_{k=1}^n t_k(f_ic_i)c_i$, 4. for any $1\leq i \leq s$ if $e_i< \infty$ then there exist integers $t'_k(e_iu_i)$ such that $e_iu_i=\sum_{k=1}^s t'_k(e_iu_i)u_i$. The integers introduced above are called *the structural constants* associated to the pseudo-bases $\bar{a}$ and $\bar{u}$. We assume an arbitrary but fixed ordering on the set of structure constants. It is easy to verify that $\langle M, A \rangle$ is determined up to isomorphism, as a two-sorted module, by the periods $\bar{e}$, $\bar{f}$ and the associated structure constants. Finally we are ready to complete the proof of Theorem \[elemmod:thm\]. *Proof of Theorem \[elemmod:thm\].* From Proposition \[basedef\] we have the formula $\varphi_{\mathfrak{P},n}$ which defines all $\mathbb{Z}$-pseudo-bases $\bar{c}$ and $\bar{u}$ of periods $\bar{f}$ and $\bar{e}$ for $A$ and $M$, respectively, in $\langle M,A\rangle$. Again by Proposition \[basedef\] the same formula defines in $\langle N,B\rangle$ similar ${{\mathbb}{Z}}$-pseudo-bases $\bar{d}$ and $\bar{v}$ of $B$ and $N$. We need only to describe the structural constants associated with the pseudo-bases $\bar{c}$ and $\bar{u}$ for $A$ and $M$ respectively. This can be done by a formula, say $\psi_{A,M}$, of the language $L_2$ because all these constants are integers and there are only finitely many of them. Obviously this implies that the ${{\mathbb}{Z}}$-pseudo-bases $(\bar{u},\bar{c})$ and $(\bar{v},\bar{d})$ are ${{\mathbb}{Z}}$-pseudo-bases of $\langle M,A\rangle$ and $\langle N, B\rangle$ respectively of the same periods and structure constants. So the theorem follows. Elementary equivalence of FDZ-algebras {#main:sec} ====================================== Finally here we prove the main theorem of this paper. Let us first put together a proof of Theorem \[elem-iso-alg:thm\]. *Proof of Theorem \[elem-iso-alg:thm\].* The proof is entirely similar to that of Theorem \[elemmod:thm\]. In addition to the structure constants listed in items (1)-(4) above for a two-sorted module we need to describe the structure constants defining the multiplication for the ring $C$. Keeping the same notation as the proof of mentioned theorem and replacing $M$ by $C$ we need to describe the integers $t''_k(u_iu_j)$ for every $1\leq i,j\leq s$, where $u_iu_j= \sum_{k=1}^s t''_k(u_iu_j)u_k$. Again these new structure constants are also integers and could be captured in the first-order theory of $C$. The new structure constants together with the ones from items (1)-(4) above describe $\langle C, A\rangle$ up to isomorphism by a single first-order formula $\phi_{C,A}$ of $Th(\langle C,A\rangle)$. Clearly if an algebra $\langle D, B\rangle$ from $\mathcal{A}$ satisfies $\phi_{C,A}$ then $\langle C, A \rangle\cong \langle D, B\rangle.$ We recall some notation and introduce some new ones. For an FDZ-algebra $R$ define $M(R){\stackrel{\text{def}}{=}}Is(R^2+Ann(R))$ and $N(R){\stackrel{\text{def}}{=}}Is(R^2)+Ann(R)$. Note that that $M(R)/N(R)$ is a finite abelian group. The ideals $M(R)$ and $N(R)$ are uniformly definable in $R$ if $R$ is an FDZ-algebra. The ideal $Ann(R)$ is clearly uniformly definable. The ideal $R^2$ is uniformly definable among all FDZ-models of $Th(R)$, since $R^2$ has finite width. Assume $I$ is uniformly definable in $R$. first $Is(I)/I$ is an abelian group of finite order, say $m$. Then the formula expressing $mx\in I$ uniformly defines $Is(I)$ among all FDZ-models of $Th(R)$. That is because they need to satisfy the following sentences for all $n\in {\mathbb{N}}\setminus \{0\}$. $$\label{TypeIso:eqn} \Psi_n: \forall x (nx \in I \to mx\in I).$$ \[A(R)-inter:lem\] For an FDZ-algebra $R$ the maximal ring of scalars $A(R)$ and its actions on $R/Ann(R)$ and $R^2$ are absolutely interpretable in $R$. The full non-degenerate bilinear map $f_{RF}:R/Ann(R)\times R/Ann(R) \to R^2$ induced by the product in $R$ is absolutely interpretable in $R$. So $P=P(f_{RF})$ an its actions on $R/Ann(R)$ and $R^2$ are interpretable in $R$ by Theorem \[ringinter\] since $f_{RF}$ is full, non-degenerate and of finite type. Note that $$A(R)=\{\alpha \in P: (\alpha x) + Ann(R)=\alpha(x+Ann(R)), \forall x\in R^2 \}.$$ Indeed $A(R)$ is clearly a definable unitary subring of $P$. This finishes the proof. \[zinter:cor\] If $R\equiv S$ are FDZ-algebras, then 1. $A(R)\cong A(S)$ 2. $R/Ann(R)\cong S/Ann(S)$ 3. $R^2 \cong S^2$ To prove (1) note that since $A(R)$ and $A(S)$ are interpreted in $R$ and $S$ with the same formulas $A(R)\equiv A(S)$. Since $R$ and $S$ are FDZ-algebras $A(R)$ and $A(S)$ are FDZ-scalar rings. So by Corollary \[scalarrings:cor\] $A(R)\cong A(S)$. To prove (2) we note that by Lemma \[A(R)-inter:lem\], the two sorted algebra $\langle R/Ann(R), A(R) \rangle$ is absolutely interpretable in $R/Ann(R)$. So indeed $\langle R/Ann(R), A(R) \rangle\equiv \langle S/Ann(S), A(S)\rangle$. By Theorem \[elem-iso-alg:thm\] we have $\langle R/Ann(R), A(R) \rangle\cong \langle S/Ann(S), A(S)$. In particular this implies $$R/Ann(R)\cong S/Ann(S)$$ as rings. (3) is similar to (2). \[def-series:lem\] Let $R$ be an FDZ-algebra and consider the series of the first-order definable ideals $$\label{def-series:eq} R \geq M(R) \geq N(R) \geq Is(R^2) \geq 0.$$ Then the action of the ring of integers ${{\mathbb}{Z}}$ is interpretable on all the infinite quotients of the consecutive terms of the series except possibly $N(R)/Is(R^2)$. However the rank of $N(R)/Is(R^2)$ as a free-abelian group is a first-order invariant. Note that $Ann(R)\leq M(R)$ and consider the canonical epimorphism $\theta: R/Ann(R)\to R/M(R)$. Then $\theta$ is interpretable in $R$ and so by proof of Corollary \[zinter:cor\] $\langle R/M(R), {{\mathbb}{Z}}\rangle$ is interpretable in $R$. The same holds for $Is(R^2)$, i.e. $\langle Is(R^2), {{\mathbb}{Z}}\rangle$ is interpretable in $R$. The quotient $M(R)/N(R)$ is finite. So there only remains one gap $N(R)\leq Is(R^2)$ on the quotient of which the action of the ring ${{\mathbb}{Z}}$ is not necessarily interpretable. Set $P(R){\stackrel{\text{def}}{=}}N(R)/Is(R^2)$. Let $\bar{u}=u_1, \ldots, u_o$ be pseudoNote that for any integer $e\geq 2$ still the fact that $P/eP$ has a basis consisting of the images of some elements of $u_{m}, \ldots , u_{n-1}$ in $P/eP$ is expressible by first-order formulas. Consequently, the fact that $u_m+ Is(R^2), \ldots, u_{n-1}+Is(R^2)$ generate a subgroup of index, say $d$, relatively prime to $e$ in $N(R)+Is(R^2)$ is a first-order property. In the above lemma we introduced positive integers $0\leq l\leq m \leq n$. In the following lemma we will refer to these number and for the sake of brevity and keeping the already complicated notation at bay we shall assume that the inequalities are strict, i.e. $l<m<n$. It will be clear that if any of the equalities hold the case becomes easier. \[mainalgR:lem\] Let $R$ be a FDZ-algebra and assume $\bar{u}$ is a pseudo-basis of $R$ adapted to the series $$\label{def-series:eq} R \geq M(R) \geq N(R) \geq Is(R^2) \geq 0.$$ Then there exists a formula $\Phi(\bar{x})$ of the language rings and some fixed integers $0<l<m<n<r$, such that $R\models \Phi(\bar{u})$ and $\Phi(\bar{u})$ expresses that 1. 1. $u_1+M(R), \ldots , u_{l-1}+M(R)$ is a basis of the free abelian group $R/M(R)$, 2. $u_{l}+Is(R^2), \ldots , u_{m-1}+Is(R^2)$ is a pseudo-basis of the finite abelian group $M(R)/N(R)$ providing the invariant factor decomposition for $M(R)/N(R)$, i.e. $$\frac{M(R)}{N(R)}\cong \frac{{{\mathbb}{Z}}}{e_l{{\mathbb}{Z}}}\oplus\cdots\oplus \frac{{{\mathbb}{Z}}}{e_{m-1}{{\mathbb}{Z}}},$$ where $e_l|e_{l+1}|\cdots |e_{m-1}$, 3. The images of $u_m=e_lu_l$,$\ldots$, $u_{m+p-1}=e_{m-1}u_{m-1}, u_{m+p}$, $\ldots$, $u_{n-1}$ in $N(R)/Is(R^2)$, where $p=m-l$, generate a subgroup of index, say $d$, prime to $e=e_{l}\cdots e_{m-1}$ of $N(R)/Is(R^2)$. 4. $u_{n}, \ldots , u_l$ is a pseudo-basis of $Is(R^2)$, 2. for all $1\leq i,j\leq r$ and some fixed integers $t_{ijk}$, ${\displaystyle}u_iu_j=\sum_{k=n}^rt_{ijk}u_k$, 3. for those $l\leq i \leq r$ such that $e_i < \infty$ and some fixed $t_{ik}$, ${\displaystyle}e_iu_i=\sum_{k=m}^rt_{ik}u_k$ In particular for $l\leq i \leq m-1$, $$e_iu_i = u_{m+i}+\sum_{k=n}^rt_{ik}u_k.$$ Pick a pseudo-basis of $R$ as in the statement. Note that $Ann(R)\leq M(R)$ and consider the canonical epimorphism $\theta: R/Ann(R)\to R/M(R)$. Then $\theta$ is interpretable in $R$ and so by proof of Corollary \[zinter:cor\] $\langle R/M(R), {{\mathbb}{Z}}\rangle$ is interpretable in $R$. The same holds for $Is(R^2)$, i.e. $\langle Is(R^2), {{\mathbb}{Z}}\rangle$ is interpretable in $R$. The quotient $M(R)/N(R)$ is finite. So there only remains one gap $N(R)\leq Is(R^2)$ on the quotient of which the action of the ring ${{\mathbb}{Z}}$ is not necessarily interpretable in $R$. Set $P(R){\stackrel{\text{def}}{=}}N(R)/Is(R^2)$. For any integer $e\geq 2$ still the fact that $P/eP$ has a basis consisting of the images of elements $u_{m}, \ldots , u_{n-1}$ in $P/eP$ is expressible by first-order formulas. Consequently the fact that $u_m+ Is(R^2), \ldots, u_{n-1}+Is(R^2)$ generate a subgroup of index, say $d$, relatively prime to $e$ in $N(R)+Is(R^2)$ is a first-order property. Here for $e$ we pick the order of the finite group $M(R)/N(R)$, i.e. $$e=e_l\cdots e_{m-1}.$$ The reason for this choice will be made clear in the next lemma. The structure constants $t_{ijk}$ are $t_{ik}$ are fixed integers and depend only on $\bar{u}$ and $R$. \[mainalgS:thm\] Assume $R\equiv S$ and $S\models \Phi(\bar{v})$ where $\Phi(\bar{x})$ is the formula obtained in Lemma \[mainalgR:lem\]. Then the following hold: 1. 1. $v_1+M(S), \ldots , v_{l-1}+M(S)$ is a basis of the free abelian group $S/M(S)$, 2. $v_l+N(S), \ldots , v_{m-1}+N(S)$ is a pseudo-basis of the finite abelian group $M(S)/N(S)$ 3. There are elements $w_{m},\ldots ,w_{n-1}$ of $S$ and integers $d_m, \ldots, d_{n-1}$ such that $w_{m}+Is(S^2),\ldots ,w_{n-1}+Is(S^2)$ is a basis of $M(S)/Is(S^2)$ and $$\begin{aligned} \langle d_{m}w_{m}+Is(S^2)&,\ldots ,d_{n-1}w_{n-1}+Is(S^2)\rangle \\ &= \langle v_m+Is(S^2),\ldots ,v_{n-1}+Is(S^2)\rangle\end{aligned}$$ and $gcd(d,e)=1$, where $d=d_m\cdots d_{n-1}$, and $e=e_l\cdots e_{m-1}$. 4. $v_n, \ldots , v_M$ is a pseudo-basis of $Is(R^2)$. 2. for all $1\leq i,j\leq r$ and some fixed integers $t_{ijk}$, ${\displaystyle}v_iv_j=\sum_{k=n}^rt_{ijk}v_k$. 3. for those $l\leq i \leq r$ such that $e_i < \infty$ and some fixed $t_{ik}$, ${\displaystyle}e_iv_i=\sum_{k=m}^rt_{ik}v_k$. In particular if $l\leq i \leq m-1$ then $$e_iv_i = v_{m+i} +\sum_{k=n}^r t_{ik}v_k.$$ The fact that the $v_i$ satisfy 1.(a),1.(b) and 1.(d) is corollary of the statements from Lemma \[mainalgR:lem\] and uniformity of the interpretations. For 1.(c) note that by 1.(c) of Lemma \[mainalgR:lem\] the $v_i+Is(S^2)$, $i=m, \ldots ,n-1$ will in general generate a subgroup $Q(S)$ of $P(S)=M(S)/Is(S^2)$ of finite index. By the structure theorem for finitely generated abelian groups there is a basis $\{w_i+Is(S^2): m \leq i\leq n-1\}$ of $P(S)$, and integers $d_i$, $i=m,\ldots n-1$, such that $d_iw_i+Is(S^2)$ is a basis of $Q(S)$. So $d=d_{m}\cdots d_{n-1}$ is the index of $Q(S)$ in $P(S)$. Recall that the images of the above $w_i$’s have to form a pseudo-basis of $P(S)/eP(S)$. So one can easily check that $gcd(e,d)=1$. For (2) and (3) everything is clear. However the constants $t_{ijk}$ and $t_{ik}$ will not necessarily determine $S$ up to isomorphisms since $\bar{v}$ in general will generate only a subring (of finite-index as an abelian group) of $S$, clear from 1.(c). \[regular-M=N:lem\] The following are equivalent for an FDZ-algebra. 1. $R$ is regular. 2. For any addition $R_0$, $R\cong R/R_0\times R_0$. 3. $M(R)=N(R)$. The equivalence of (1) and (2) is clear. Let us show that (2) implies (3). So assume $x\in M(R)$. Then for some non-zero $m\in {\mathbb{N}}$, $mx\in R^2+Ann(R)=R^2\times R_0$. Since $R=R_F \times R_0$ there exists unique $y\in R_F$ and $z\in R_0$ such that $x=y+z$. Since $mx\in R^2 \times R_0$ and $mz\in R_0$ there exist $y_1\in R^2$ and $y_2\in R_0$ such that $my=y_1+y_2$. Since $R^2\leq R_F$ and $my\in R_F$ we have $y_2\in R_F$. Therefore $y_2=0$, $my=y_1$, and $y\in Is(R^2)$. So $x\in N(R)$. It remains to show $(3)\Rightarrow (1)$. Consider the canonical map $\pi:R \to R/Is(R^2)$. Since $M(R)=N(R)$, there exists a direct complement $C$ for $N(R)/Is(R^2)$ in $R/Is(R^2)$. It is easy to see that $R=\pi^{-1}(C)\times R_0$ for any addition $R_0$. Since multiplication in $R/Is(R^2)$ is trivial $\pi^{-1}(C)$ is indeed a subring of $R$ and it clearly contains $R^2$. *Proof of $ (1) \Rightarrow (2)$ of Theorem \[mainnice:thm\].* We follow terminology and notation of Lemma \[mainalgR:lem\] and Theorem \[mainalgS:thm\]. Indeed we will show that $S$ fits the description in Theorem \[mainalgS:thm\] if and only if all the conditions in the statement of Theorem \[mainnice:thm\] are satisfied by $S$. Let us first consider the case $M(R)\neq N(R)$. Indeed if $S$ satisfies conditions of Theorem \[mainalgS:thm\] then the assignment $u_i \mapsto v_i$, $i=1, \ldots , M$ will extend to a monomorphism $\phi$ of rings since $\bar{u}$ and $\bar{v}$ are pseudo-basis of the same length, periods and structure constants. Only $im(\phi)$ which is the subring of $S$ generated by the $v_i$ may not contain all of $S$. Conversely if $\phi:R\to S$ satisfies the conditions of Theorem \[mainnice:thm\] it is clear that the $u_i\in R$ with description from Lemma \[mainalgR:lem\] will map under $\phi$ to some $v_i\in S$ as in the Theorem \[mainalgS:thm\]. Now if $M(R)=N(R)$ then $M(S)=N(S)$. By Lemma \[regular-M=N:lem\] $X\cong X/X_0\times X_0$, where $X=R,S$ for any additions $R_0$ and $S_0$ of $R$ and $S$ respectively. Now $u_i\mapsto v_i$, $i \neq m, \ldots ,n-1$ will induce an isomorphism between $R/R_0$ and $S/S_0$ while $R_0\cong S_0$ since they are both free abelian groups of the same finite rank. \[nutral:lem\] Note that the elements $v_{m+p}, \ldots, v_n$ and the corresponding $u_i$’s, aside the rank of the subrings they generate which are just abelian groups with zero multiplication, will play no structural role in either of the rings $R$ and $S$ and split from them. Indeed all the structure constants $t_{ik}=0$ and $t_{ijk}=0$ if any of $i$,$j$ or $k$ is between $m+p$ and $n-1$. *Proof of Theorem \[tame:thm\].* Since By assumption $Ann(R)\leq Is(R^2)$ then for any addition $R_0$ we have $R_0=0$. Now we get a series: $$R \geq Is(R^2) \geq Ann(R)+R^2 \geq R^2 \geq 0.$$ The only problem is that even though $Is(R^2)$ is definable in $R$ in order for the corresponding formula to define $Is(S^2)$ in an FDZ-algebra $S$, $S$ has to satisfy the infinite type $\{\Psi_n:n\in {\mathbb{N}}^+\}$ from Equation . Note that $Ann(X)$ is definable in any algebra $X$ by the same sentence, while $R^2$ is definable in $R$ and the same formula defines $S^2$ is any algebra satisfying the sentence $\phi_w$ in Equation . Now assume the order of $Is(R^2)/R^2$ is $q$. Then there exists a first-order sentence $\phi_{Is(R^2)}$ true in $R$ which will imply $\forall x ( x \in Ann(S) \to qx \in S^2)$ in any FDZ-algebra $S$ satisfying it and therefore implying that $Ann(S)\leq Is(S^2)$ and therefore that $(Ann(S)+S^2)/S^2$ is finite. So indeed to prove the theorem we need to deal with the gaps in the following series: $$R \geq Ann(R)+R^2 \geq R^2 \geq 0.$$ Recall from Corollary \[zinter:cor\] that $\langle R/Ann(R), {{\mathbb}{Z}}\rangle$ and $\langle R^2, {{\mathbb}{Z}}\rangle$ are interpretable in $R$, while by Theorem \[elem-iso-alg:thm\] each of them is axiomatized in the class of two-sorted FDZ-algebras by one sentence. It follows now that $\langle R/(Ann(R)+R^2), {{\mathbb}{Z}}\rangle$ is interpretable in $R$ while the former was axiomatizable by one sentence. Therefore the isomorphism type of each quotient coming from the sequence $$R \geq Ann(R)+R^2 \geq R^2 \geq 0$$ can be captured by one sentence of $L$. This implies the statement. The converse of the characterization theorem {#converse:sec} ============================================ In this section we prove the converse of Theorem \[mainalgS:thm\] and therefore we shall provide a proof for $(2) \Rightarrow (1)$ direction of Theorem \[mainnice:thm\]. Let us fix some notation first. Let ${\mathcal{D}}$ be a non-principal ultrafilter on an index set $I$. By $R^*$ for a ring $R$ we mean the ultrapower $R^I/{\mathcal{D}}$ of $R$. The equivalence class of $x \in R^I$ in $R^*$ is denoted by $x^*$. \[equality:lem\] Let $R$ be an FDZ-algebra and let ${\mathcal{D}}$ be a non-principal ultrafilter on $I$. Then 1. $(R^2)^*= (R^*)^2$, 2. $(Is(R^2))^*=Is((R^*)^2)=Is((R^2)^*),$ 3. $Ann(R^*)=(Ann(R))^*$, 4. If $R_0$ is an addition of $R$ then $(R_0)^*$ is an addition of $R^*$. For (1) the inclusion $\geq$ follows from the fact that $(R^I)^2$ is generated by $xy=z$, $x, y\in R^I$, where $z(i)=x(i)y(i)\in R^2$ for ${\mathcal{D}}$-almost every $i\in I$. But then the equivalence class of $xy$ is in $(R^2)^*$. The other inclusion follows from the fact that $R^2$ is of finite width. Indeed, pick $z\in R^I$ such that the equivalence class $z^*\in (R^2)^*$, and assume that width of $R^2$ is $s$. Then for ${\mathcal{D}}$-almost every $i\in I$, $z(i)=\sum_{j=1}^s x_j(i)y_j(i)$. Define $z_j\in R^I$, $j=1,\ldots s$, by $z_j(i)=x_j(i)y_j(i)$. Obviously $z^*_j\in (R^*)^2$, $j=1, \ldots ,s$, and $z^*= z^*_1+\cdots +z^*_s$. This implies the result. For (2) the equality of the last two terms follows from (1.). Moreover $$x^*\in Is((R^2)^*)\Leftrightarrow m(x^*)\in (R^2)^* \Leftrightarrow (mx)^*\in (R^2)^* \Leftrightarrow x^*\in (Is(R^2))^*.$$ (3) is clear. (4) is implied by (2) and (3) and the definition of an addition. \[satabelian:lem\] Assume $A$ is a free abelian group of rank $n$ with basis $v_1, \ldots ,v_n$. Let $A^*$ the ultrapower of $A$ over an $\aleph_1$-incomplete ultrafilter ${\mathcal{D}}$ and let - $(b_{ij})$ be an $n\times n$ matrix with integer entries and the determinant $det((b_{ij}))=\pm 1$ - $\alpha_i$, $i=1, \ldots, n$, be elements of the ultrapower ${{\mathbb}{Z}}^*$ of the ring of integers ${{\mathbb}{Z}}$ over ${\mathcal{D}}$, such that $p\nmid \alpha_i$ for any prime number $p$, where $|$ denotes division in the ring ${{\mathbb}{Z}}^*$. Then there is an automorphism $\psi:A^* \to A^*$ extending $v_i\mapsto \sum_{k=1}^n{\alpha_kb_{ik}}v_k$. Recall that $A^*$ is an $\aleph_1$-saturated abelian group. By the structure theory of saturated abelian groups (see either of [@Szmielew] or [@eklof]) there is an automorphism $\eta$ of $A^*$ such that $\eta(v_k)=\alpha_k v_k$, for each $k=1, \ldots n$. Note that the automorphism $\eta$ is not necessarily a ${{\mathbb}{Z}}^*$-module automorphism. However since $det(b_{ik})=\pm 1$ there is ${{\mathbb}{Z}}^*$-module automorphism of $A^*$ extending $v_i\mapsto \sum_{k=1}^nb_{ik}v_k$. This proves the statement. \[converse\] Assume $R$ is a FDZ-algebra with a pseudo-basis $\bar{u}$ as in Lemma \[mainalgR:lem\] and $S$ an FDZ-algebra as in Theorem \[mainalgS:thm\]. Then $$R\equiv S.$$ In order to prove the statement we prove that ultrapowers $R^*=R^\mathbb{N}/{\mathcal{D}}$ and $S^*=S^{\mathbb{N}}/{\mathcal{D}}$ of $R$ and $S$ over any $\omega_1$-incomplete ultrafilter $(\mathbb{N},{\mathcal{D}})$ are isomorphic. By Remark \[nutral:lem\], $u_k$ and $w_k$, $k=m+p-1,\ldots, n-1$ generate zero multiplication subrings of $R$ and $S$ which split from the respective rings. So just to make notation simpler we assume that $$n=m+p,$$ i.e. $n-m=m-l=p$. Recall the definition of $Q(S)$ from Theorem \[mainalgS:thm\]. Let $B=(b_{ik})$, $l\leq i\leq m-1$, $m\leq k\leq n-1$ be the $(m-l)\times (m-l)$ change of basis matrix between the bases $v_k+Is(S^2)$ and $d_kw_k+Is(S^2)$, i.e. $$v_i=\sum_{k=m}^{n-1}b_{ik}d_kw_k.$$ Now recall that $e=e_{l}\cdots e_{m-1}$, $d=d_{m}\cdots d_{n-1}$ and gcd$(d,e)=1$. Assume $\pi$ denotes the set of all prime numbers and that, $\pi_k$ is the set of all prime numbers $p$, such that $p|d_k$, $l\leq k \leq m-1$. Let us denote the $j$’th prime number in $\pi\setminus \pi_k$ by $p_{kj}$ and the product of the first $j$ primes in $\pi\setminus \pi_k$ by $q_{kj}$. For each $j\in {\mathbb{N}}$ and for all $l\leq i \leq m-1$ define $$w_{ij}=\sum_{k=m}^{n-1}b_{ik}(d_k+q_{kj}e)w_k.$$ Now let, $w_i^*\in H^*$ and $q_k^*\in {{\mathbb}{Z}}^*$ denote the classes of $(w_{ij})_{j\in {\mathbb{N}}}$ and $(q_{kj})_{j\in {\mathbb{N}}}$ respectively. Indeed $$\label{wi*:eqn}w^*_i=\sum_{k=m}^{n-1}b_{ik}(d_k+q_k^*e)w_k.$$ Let us set $\alpha_k=d_k+q_k^*e$ for each relevant $k$. Next we claim that the $\alpha_k$ satisfy hypothesis (b) of Lemma \[satabelian:lem\], that is, no prime $p$ divides $\alpha_k=d_k+q_k^*e$ for each $k$, $k=m, \ldots, n-1$. To prove this we recall that $q_{kj}=p_{k1}\cdots p_{kj}$ where the $p_{k1}, \ldots ,p_{kj}$ are the first $j$ primes that do not divide $d_k$. Pick a prime $p$. If $p\in \pi_d$, i.e. $p|d_k$ and $p|(d_k+q_{kj}e)$, then $p|q_{kj}e$ which contradicts the choice of $q_{kj}$ and the fact that gcd$(d_k,e)=1$. So for such $p$, $p\nmid (d_k+q_{kj}e)$. Now pick a prime $p\in \pi\setminus \pi_k$, i.e $p\nmid d_{k}$. Then $p=p_{kt}$ for some $t\in {\mathbb{N}}$, meaning that $p$ is a factor of $q_{kj}$ for every $j\geq t$. So $p|q_{kj}e$ for every $j\geq t$. Therefore, for every such $j$ if $p|(d_k+q_{kj}e)$ then $p|d_k$, which is impossible. So for every $j\geq t$, $p\nmid d_k+q_{kj}e$. So indeed for any prime $p$, $p\nmid (d_k+q_{k}^*e)$. Let $R_0$ ($S_0$) be the addition of $R$ (resp. $S$) generated by $u_i$ (resp. $v_i$), $i=m, \ldots , n-1$. By Lemma \[equality:lem\] the ${{\mathbb}{Z}}^*$-submodule $R^*_0$ ($S^*_0$) of $Ann(R^*)$ ($Ann(S^*)$) generated be the $u_i$ ($v_i$), $i=m, \ldots , n-1$ is an addition of $R^*$ (resp. $S^*$) and $R^*_0=(R^*)_0=(R_0)^*$ (the same in $S$). By Lemma \[satabelian:lem\] there exists an isomorphism $\psi:R^*_0\to S^*_0$ extending $u_i \to w_i^*$. By construction there exists a monomorphism $\phi:R \to S$ of groups such that: $$\phi(u_i)=\left\{ \begin{array}{ll} v_i & \text{if } i\neq m,\ldots, n-1\\ \\ \sum_{k=m}^{n-1}b_{ik}d_kw_k & \text{if }i= m,\ldots, n-1 \end{array}\right.$$ Actually $\phi$ defined above is the same $\phi$ as in Theorem \[mainalgS:thm\] only the $v_i$, $m\leq i \leq n-1$, are written with respect to the new basis of $S_0$ consisting of the $w_i$. For each $j\in {\mathbb{N}}$ one could twist the monomorphism $\phi:R\to S$ to get a new one denoted by $\phi_j:R\to S$ and defined by: $$\phi_j(u_i)=\left\{ \begin{array}{ll} v_i & \text{if }i\neq l, \ldots , n-1\\ \\ v_i+\sum_{k=m}^{n-1}q_{kj}\hat{e}_ib_{ik}w_k& \text{if }i= l,\ldots ,m-1\\ \\ \sum_{k=m}^{n-1}(d_kc_{ik}+q_{kj}eb_{ik})w_k & \text{if }i= i_1+1,\ldots, i_1+n \end{array}\right.$$ where $\hat{e}_i=e/e_i$. Again note that $R$ and $im(\phi_j)\leq S$ are generated by the pseudo-bases of the same lengths, periods and structure constants. Let $\phi^*:R^*\to S^*$ be the monomorphism induced $(\phi_j)_{j\in {\mathbb{N}}}$. Next consider the subring $R^*_f$ of $R^*$ generated by $$\{\alpha u_i, u_j: i\neq l,\ldots,m-1, \alpha\in {{\mathbb}{Z}}^*, j=l,\ldots, m-1\}.$$ We assume the same definitions in $S^*$ too. We claim that $R^*=R^*_f + R^*_0$. Firstly $R^*$ is generated by all the $\alpha u_i$, $\alpha\in {{\mathbb}{Z}}^*$ in the obvious manner. All these generators belong to $R^*_f+ R^*_0$ with the possible exceptions when $i=l, \ldots m-1$. However by Lemma \[equality:lem\] $$\frac{Is((R^*)^2)+ Ann(R^*))}{Is((R^*)^2+ Ann(R^*))}\cong \frac{Is(R^2)+Ann(R)}{Is(R^2+ Ann(R))}$$ is a finite abelian group and so we only need integer multiples of the $u_i$, $i=l, \ldots m-1$ in the generating set. This proves the claim. Now given $x\in R^*$ there are $y\in R^*_f$ and $z\in R^*_0$ such that $x=y+z$. Now define a map $\eta:R^* \to S^*$ by $$\eta(x)= \phi^*(y)+\psi(z).$$ To show that $\eta$ is well-defined we need to check if $\phi^*$ and $\psi$ agree on $R^*_f\cap R^*_0$. We note that $$R^*_f\cap R^*_0=\langle e_iu_i: l\leq i \leq m-1\rangle,$$ i.e. the subgroup generated by the $e_iu_i$ as above. Now $$\begin{aligned} \phi^*(e_iu_i)&=e_i( v_i+\sum_{k=m}^{n-1}q^*_{k}\hat{e}_ib_{ik}w_k)\\ &= e_iv_i+ \sum_{k=m}^{n-1}q^*_{k}eb_{ik}w_k\\ &= \sum_{k=m}^{n-1}d_kb_{ik}w_k+ \sum_{k=m}^{n-1}q^*_{k}eb_{ik}w_k\\ &= w_{m+i}^*\\ &=\psi (u_{m+i})\\ &=\psi(e_iu_i).\end{aligned}$$ $\eta$ is a homomorphism since $\phi^*$ and $\psi$ are so and $\psi$ maps a subring of $Ann(R^*)$ into a subring of $Ann(S^*)$. It is injective since both $\phi^*$ and $\psi$ are injective and they agree on $R^*_f\cap R^*_0$. Finally $$S^*=S^*_f + S^*_0= \phi^*(R^*_f) + S^*_0$$ and by construction $H^*_0=im(\psi)$. Therefore $\eta$ is surjective. We have proved that $\eta: R^* \to S^*$ is an isomorphism of rings and by the Keisler-Shelah’s theorem, we have proved that $$R \equiv S.$$ Let $A_1^*$ be the ${{\mathbb}{Z}}^*$-submodule of $G^*_0$ generated by $u_{}$, $k=i_1+1, \ldots, i_1+n$ and $B^*$ be the direct complement of $A^*$ in $H^*_0$ generated by the $v_k$, $i_1+n< k\leq i_2$, as a ${{\mathbb}{Z}}^*$-submodule. We need only prove the surjectivity. Let us identify the $u_i$ and the $v_i$ with their images under the respective diagonal mappings of $G$ and $H$ into $G^*$ and $H^*$. The homomorphism $\phi^*$ induces $\bar{\phi^*}: Ab(G^*)\to Ab(H^*)$. Since $H^*$ is nilpotent, to prove the surjectivity of $\phi^*$ it is enough to prove that $\bar{\phi^*}$ is surjective. Now the equalities from Lemma \[equality:lem\] imply $$\frac{G^*}{(G^*)'}\cong \frac{G^*}{Is((G^*)')}\times \frac{Is((G^*)')}{(G^*)'},$$ where $Is((G^*)')/(G^*)'\cong (Is(G')/G')^*\cong Is(G')/G'$ is a finite abelian group, and $G^*/Is((G^*)')$ is torsion-free. The same facts hold for $H$. By construction all $\bar{\phi}_j:Ab(G)\to Ab(H)$ induce the same isomorphism $\bar{\phi}_j|_{Is(G')/G'}:Is(G')/G'\to Is(H')/H'$. So checking whether $\bar{\phi}^*$ is surjective reduces to checking whether the homomorphism $\psi^*:G^*/Is((G^*)')\to H^*/Is((H^*)')$ induced by $\bar{\phi^*}$ is surjective. Let ${{\mathbb}{Z}}^*={{\mathbb}{Z}}^{\mathbb{N}}/{\mathcal{D}}$. Since both $G^*/Is((G^*)')$ and $H^*/Is((H^*)')$ can be considered as free ${{\mathbb}{Z}}^*$-modules of finite rank and $\psi^*$ is also clearly ${{\mathbb}{Z}}^*$-linear it is enough to prove that all $v_{i}Is((H^*)')$, $i=i_1+1, \ldots , i_1+n,$ are in the image of this map. Let us denote by $\widetilde{H_1}$ the ${{\mathbb}{Z}}^*$-submodule generated by these elements and agree that $$\bar{v}_i=v_iIs((H^*)').$$ Obviously $\widetilde{H_1}$ can be considered as a free ${{\mathbb}{Z}}^*$-module with basis the $\bar{v}_{i}$, $i=i_1+1, \ldots , i_1+n$. We claim that the $\gamma_i$ obtained above form a ${{\mathbb}{Z}}^*$-basis for $H^*/Is((H^*)')$. Let us now set $\alpha_k=d_k+q_k^*e$. Assume $A_k$ is the ${{\mathbb}{Z}}^*$-module generated by $\bar{v}_k$. In particular these elements are units in the ring ${{\mathbb}{Z}}^*$. This implies that $det(\cal{M})$ is a unit in ${{\mathbb}{Z}}^*$ so $\cal{M}$ is an invertible matrix as a matrix with entries in ${{\mathbb}{Z}}^*$. Zilber’s example {#Zilber:sec} ================ Here we present an example due to B. Zilber [@Z71] which was the first example of two finitely generated nilpotent groups $G$ and $H$ where $G\equiv H$ but $G\ncong H$. We shall show that $G$ and $H$ are best described as abelian deformations of one another. Consider the groups $G$ and $H$ presented (in the category of 2-nilpotent groups) as follows. $$G=\langle a_1,b_1,c_1,d_1|a^5_1 \text{ is central}, [a_1,b_1][c_1,d_1]=1\rangle,$$ $$H=\langle a_2,b_2,c_2,d_2|a^5_2 \text{ is central}, [a_2,b_2]^2[c_2,d_2]=1\rangle.$$ B. Zilber [@Z71] proved that $G\equiv H$ but $G\ncong H$. Let us first apply a Titze transformation to both $G$ and $H$ to get $$G\cong \langle a_1,b_1,c_1,d_1,f_1|f_1 \text{ is central}, a_1^5f^{-1}_1=1, [a_1,b_1][c_1,d_1]=1\rangle$$ $$H\cong \langle a_2,b_2,c_2,d_2,f_2|f_2 \text{ is central}, a_2^5f_2^{-1}=1, [a_2,b_2]^2[c_2,d_2]=1\rangle.$$ Now we are going to show that $H$ is an abelian deformation of $G$. So define a group $K$ by $$K= \langle a_3,b_3,c_3,d_3,f_3|f_3 \text{ is central}, a_3^5f^{-2}_3=1, [a_3,b_3][c_3,d_3]=1\rangle.$$ Note that we can choose $G_0=\langle f_1=a_1^5\rangle$ and $K_0=\langle f_3\rangle$. We also have that $Is(G'\cdot G_0)=G'\cdot Is(G_0),$ and $Is(K'\cdot K_0)=K'\cdot Is(K_0)$, and so $$Is(G'\cdot G_0)/Is(G')\cdot G_0=Is(G_0)/G_0=\langle a_1|a_1^5=1\rangle$$ and $$Is(K'\cdot K_0)/Is(K')\cdot K_0=Is(K_0)/K_0=\langle a_3|a_3^5=1\rangle.$$ Indeed using notation of Definition \[abdefcohom:defn\], $K=\text{Abdef}(G,d,c)$ where $d=2$ and $(c)$ is $1\times 1$ matrix (1). Indeed we only changed the structure constant $t_{f_1}(a_1^5)$ to deform $G$ to $K$. However in $K$ $$(a_3^3f^{-1}_3)^2=a_3^6f_3^{-2}=a_3(a_3^5f_3^{-2})=a_3.$$ Now apply the corresponding Titze transformation to the presentation of $K$ to get $$K= \langle a^3_3f_3^{-1},b_3,c_3,d_3,f_3|f_3 \text{ is central}, (a_3^3f_3^{-1})^5f_3^{-1}=1, [a_3^3f_3^{-1},b_3]^2[c_3,d_3]=1\rangle.$$ Obviously $H\cong K.$ [99]{} P. C. Eklof, and R. F. Fischer, The elementary theory of abelian groups, Ann. Math. Logic, 4(2) (1972) 115-171. W. Hodges, Model Theory, Encyclopedia of mathematics and its applications: V. 42, Cambridge University Press, 1993. A. G. Myasnikov, Definable invariants of bilinear mappings, (Russian) Sibirsk. Mat. Zh. 31(1) (1990) 104-115, English trans. in: Siberian Math. J. 31(1) (1990) 89-99. A. G. Myasnikov, The structure of models and a criterion for the decidability of complete theories of finite-dimensional algebras, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 53(2) (1989) 379-397; English translation in Math. USSR-Izv. 34(2) (1990) 389-407. A. G. Myasnikov, V. N. Remeslennikov, Definability of the set of Mal’cev bases and elementary theories of finite-dimensional algebras I, (Russian) Sibirsk. Math. Zh., 23(5) (1982) 152-167. English transl., Siberian Math. J. 23 (1983) 711-724. A. G. Myasnikov, V. N. Remeslennikov, Definability of the set of Mal’cev bases and elementary theories of finite-dimensional algebras II, (Russian) Sibirsk. Math. Zh. 24(2) (1983) 97-113. English transl., Siberian Math. J. 24 (1983) 231-246. A. G. Myasnikov and M. Sohrabi, Groups elementarily equivalent to a free 2-nilpotent group of finite rank, Alg. and Logic, 48(2) (2009), 115-139 A. G. Myasnikov, M. Sohrabi, Groups elementarily equivalent to a free nilpotent group of finite rank, Ann. Pure Appl. Logic 162(11) (2011), 916-933 A. G. Myasnikov and M. Sohrabi, $\omega$-stability and Morley rank of bilinear maps, rings and nilpotent groups, arXiv:1410.2280v2. groups, J. London Math. Society 44(2) (1991) 173-183. J. Robinson, The undecidability of algebraic rings and fields, Proc. Amer. Math. Soc. 10 (1959) 950-957. W. Szmielew, Elementary properties of abelian groups, Fund. Math. 41 (1955) 203-271. [^1]: Address: Stevens Institute of Technology, Department of Mathematical Sciences, Hoboken, NJ 07087, USA. Email: msohrab1@stevens.edu

--- abstract: 'Let $A$ be a $K$-subalgebra of the polynomial ring $S=K[x_1,\ldots,x_d]$ of dimension $d$, generated by finitely many monomials of degree $r$. Then the Gauss algebra $\mathbb{G}(A)$ of $A$ is generated by monomials of degree $(r-1)d$ in $S$. We describe the generators and the structure of $\mathbb{G}(A)$, when $A$ is a Borel fixed algebra, a squarefree Veronese algebra, generated in degree $2$, or the edge ring of a bipartite graph with at least one loop. For a bipartite graph $G$ with one loop, the embedding dimension of $\mathbb{G}(A)$ is bounded by the complexity of the graph $G$.' address: - 'Jürgen Herzog, Fachbereich Mathematik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany' - 'Raheleh Jafari, Mosaheb Institute of Mathematics, Kharazmi University, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.' - 'Abbas Nasrollah Nejad, Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran' author: - Jürgen Herzog - Raheleh Jafari - Abbas Nasrollah Nejad title: On the Gauss algebra of toric algebras --- [^1] Introduction {#introduction .unnumbered} ============ Let $V\subseteq \PP_K^{n-1}$ be a projective variety of dimension [$d-1$]{} over an algebraically closed field $K$ of characteristic zero. Denote by $V_{\mathrm{sm}}$ the set of non-singular points of $V$ and by $\GG( {d-1},n-1)$ the Grassmannian of [$d-1$]{}-planes in $ {\PP_K^{n-1}}$. The *Gauss map* of $V$ is the morphism $$\gamma: V_{\mathrm{sm}}\longrightarrow \GG( {d-1},n-1),$$ which sends each point $p\in V_{\mathrm{sm}}$ to the embedded tangent space $\mathrm{T}_{p}V $ of $V$ at the point $p$. The closure of the image of $\gamma$ in $\GG( {d-1},n-1)$ is called the *Gauss image* of $V$, or *the variety of tangent planes* and is denoted by $\gamma(V)$. The homogeneous coordinate ring of $\gamma(V)$ in the Plücker embedding of the Grassmannian $\mathbb{G}( {d-1},n-1)$ of $ {(d-1)}$-planes is called the *Gauss algebra* of $V$. The Gauss map is a classical subject in algebraic geometry and has been studied by many authors. For example, it is known that the Gauss map of a smooth projective variety is finite [@GH; @Zak]; in particular, a smooth variety and its Gauss image have the same dimension with the obvious exception of a linear space. Zak [@Zak Corollary 2.8] showed that, provided $V$ is not a linear subvariety of $\PP_K^n$, the dimension of the Gauss image satisfies the inequality $\dim V-\dim \Sing(V)-1\leq \dim \gamma(V)\leq \dim V$, where $\Sing(V)$ denotes the singular locus of $V$. For an algebraic proof of Zak’s inequality, see [@AKB]. We take up the situation where $V\subset \PP_K^{n-1}$ is a unirational variety. To elaborate on the algebraic side of the picture, consider the polynomial ring $S=K[x_1,\ldots,x_d]$. Let $\gg=g_1,\ldots,g_n$ be a sequence of non-constant homogeneous polynomials of the same degree in $S$ generating the $K$-subalgebra $A=K[\gg]\subseteq S$ of dimension $d$. Then the Jacobian matrix $\Theta(\gg)$ of $\gg$ has rank $d$ [@Aron1 Proposition 1.1]. In this situation we define the *Gauss algebra* associated with $\gg$ as the $K$-subalgebra generated by the set of $d\times d$ minors of $\Theta(\gg)$ [@BGS Definition 2.1]. Since the definition does not depend on the choice of the homogeneous generators of $A$, we simply denote the Gauss algebra associated with $\gg$, by $\GG(A)$, and call it the Gauss algebra of $A$. The Gauss algebra $\mathbb{G}(A)$ is isomorphic to the coordinate ring of the Gauss image of the projective variety defined parametrically by $\gg$ in the Plücker embedding of the Grassmannian $\mathbb{G}( {d-1},n-1)$ of [$d-1$]{}-planes. Moreover, there is an injective homomorphism of $K$-algebras $\GG(A){\hookrightarrow}A$ inducing the rational map from $\proj{A}$ to its Gauss image [@BGS Lemma 2.3]. In this paper, we study the Gauss algebra of toric algebras. If $A\subset S$ is a toric algebra with monomial generators $\gg=g_1,\ldots,g_n$ of the same degree, then all minors of $\Theta(\gg)$ are monomials. In particular, the Gauss algebra is a toric algebra. For example, it has been shown that the Gauss algebra of a Veronese algebra is again Veronese [@BGS Proposition 3.2]. Veronese algebras are special cases of a more general class of algebras, namely the class of Borel fixed algebras. As a generalization of the above mentioned result, we show that the Gauss algebra of any Borel fixed algebra is again Borel fixed, see Theorem \[Borel\_fix\]. This approach provides a simple proof for [@BGS Proposition 3.2]. Veronese algebras are actually principal Borel fixed algebras, that is, the Borel set defining the algebra admits precisely one Borel generator. In general the number of Borel generators of the Borel fixed algebra $A$ and that of $\GG(A)$ may be different. However, in Theorem \[principal\] we show that the Gauss algebra of a principal Borel fixed algebra is again principal. This has the nice consequence that the Gauss algebra of a principal Borel fixed algebra is a normal Cohen-Macaulay domain, and its defining ideal is generated by quadrics. Note that in general the property of $A$ being normal does not imply that $\GG(A)$ is normal, and vice versa (Example \[non-nomral\] and Theorem \[2-Veronese\](d)). The Gauss algebra of a squarefree Veronese algebra is much harder to understand. We can give a full description of $\GG(A)$, when $A$ is a squarefree Veronese algebra generated in degree $2$. In Theorem \[2-Veronese\] we show that $\GG(A)$ is defined by all monomials $u$ of degree $d$ and $|\supp(u)|\geq 3$, provided $d\geq 5$. Algebras of this type may be viewed as the base ring of a polymatroid. In particular, $\GG(A)$ is normal and Cohen–Macaulay. However, $\GG(A)$ is not normal for $d=4$. Yet for any $d$, the Gauss map $\gamma: \proj{A}\dashrightarrow \proj{\GG(A)}$ is birational. In the last section of this paper we study the Gauss algebra of the edge ring of a finite graph. Let $G$ be a loop-less connected graph with $d$ vertices. It is well-known that the dimension of the edge ring $A=K[G]$ of $G$ is $d$, if $G$ is not bipartite, and is $d-1$ if $G$ is bipartite. In our setting, $\GG(A)$ is defined under the assumption that $\dim A=d$. By using a well-known theorem [@GKS] of graph theory, the generators of $\GG(A)$, when $G$ is not bipartite, correspond to $d$-sets $E$ of edges of $G$, satisfying the property that the subgraph with edges $E$ has an odd cycle in each of its connected components. In the bipartite case we form the graph $G^L$, where $L$ is a non-empty subset of the vertex set of $G$, by adding a loop to $G$ for each vertex in $L$. Then $A=K[G^L]$ has dimension $d$, and there is bijective map from the set of pairs $(V,T)$ to the set of monomial generators of $\GG(A)$, where $V$ is a non-empty subset of $L$ and $T$ is a set of edges which form a spanning forest $G(T)$ of $G$ with the property that each connected component of $G(T)$ contains exactly one vertex of $V$. From this description it follows that if $|L|=1$, then the embedding dimension of the Gauss algebra is bounded by the complexity of the graph, which by definition, is the number of spanning trees of the graph. This is an important graph invariant. The number of spanning trees provides a measure for the global reliability of a network. For a complete bipartite graph $K_{m;n}$ the embedding dimension of the Gauss algebra is $\binom{n+m-2}{n-1}\binom{n+m-2}{m-1}$, see Example \[bipartite\], while the number of spanning tress is $m^{n-1}n^{m-1}$, see for instance [@HW Theorem 1]. In general the defining ideal of the Gauss algebra admits many binomial generators. Thus it is not surprising that the Gauss algebra is rarely a hypersurface ring. This is for example the case, when $G$ is a cycle with one loop or a path graph with two loops attached. The Gauss algebra of an odd (resp.  even) cycle of length $d$ with one loop attached is a hypersurface ring of dimension $d$ (resp. $d-1$). More generally, we expect that if $G$ is a bipartite graph on $[d]$, $L=\{i\}$ and $A$ be the edge ring of $G^L$, then $\GG(A)$ is a hypersurface ring of dimension $d-1$, if and only if $G$ is an even cycle. Toric algebras ============== In this section, we collect some basic fact [s]{} about the Gauss algebra of a toric algebra. Let $S=K[x_1,\ldots,x_d]$ be a polynomial ring over $K$, where $K$ is a field of characteristic zero. Let $\gg=g_1,\ldots,g_n$ be a sequence of monomials with $g_i= {x_1^{a_{1i}}\cdots x_d^{a_{di}}}$ for $i=1,\ldots,n$. We associate to the sequence $\gg$ two matrices, namely $\Theta(\gg)$ and $\mathrm{Log}(\gg)$, where $\Theta(\gg)$ is the Jacobian matrix of $\gg$ and $\mathrm{Log}(\gg)=(a_{ij})$ is the exponent matrix (or log-matrix ) of $\gg$, whose columns are the exponent vectors of the monomials in $\gg$. We denote the $r$-minor $$\det \begin{bmatrix} \frac{\partial g_{i_1}}{\partial x_{j_1}} & \cdots & \frac{\partial g_{i_1}}{\partial x_{j_r}} \\ \vdots & \ddots & \vdots \\ \frac{\partial g_{i_r}}{\partial x_{j_1}} & \cdots & \frac{\partial g_{i_r}}{\partial x_{j_r}}\\ \end{bmatrix}= \det \begin{bmatrix} a_{i_1j_1}\frac{g_{i_1}}{x_{j_1}} &\cdots & a_{i_1j_r}\frac{g_{i_1}}{x_{j_r}} \\ \vdots & \ddots & \vdots \\ a_{i_rj_1}\frac{g_{i_r}}{x_{j_1}} &\cdots & a_{i_rj_r}\frac{g_{i_r}}{x_{j_r}} \\ \end{bmatrix}$$ by $[i_1,\ldots,i_r\ | \ j_1,\ldots,j_r ]_{\Theta(\gg)}$. The multi-linearity property of the determinant implies that $$x_{j_1}\ldots x_{j_r} [i_1,\ldots,i_r\ | \ j_1,\ldots,j_r ]_{\Theta(\gg)}=g_{i_1}\ldots g_{i_r}[i_1,\ldots,i_r\ | \ j_1,\ldots,j_r ],$$ where [$[i_1,\ldots,i_r\ | \ j_1,\ldots,j_r ]$ is the $r$-minor corresponding to the rows $i_1,\ldots,i_r$ and columns $j_1,\ldots,j_r$ of the transpose]{} of $\mathrm{Log}(\gg)$. Therefore, $r$-minors of $\Theta(\gg)$ are monomials of the form $$\label{minor-monomail} [i_1,\ldots,i_r\ | \ j_1,\ldots,j_r ]\cdotp\frac{g_{i_1}\ldots g_{i_r}}{x_{j_1}\ldots x_{j_r}}.$$ By the relation (\[minor-monomail\]), the Jacobian matrix and the log-matrix of $\gg$ have the same rank (see also [@Simis-1998 Proposition 1.2]). Let $A=K[\gg]$ be the toric $K$-algebra with generators $\gg=g_1,\ldots,g_n$. It is well know [n]{} that the dimension of $A$ is the rank of the matrix $\log(\gg)$. Thus, if all monomials of $\gg$ are of degree $r$ and the rank of $\log({\gg})$ is $d$, then the Gauss algebra $\GG(A)$ of $A$ is a toric algebra generated by monomials of degree $(r-1)d$. Then (\[minor-monomail\]) implies that $$\GG(A)=K[(g_{i_1}\cdots g_{i_d})/(x_1\cdots x_d) :\ \ \det(\mathrm{Log}(g_{i_1},\ldots, g_{i_d}))\neq 0].$$ The injective $K$-algebra homomorphism $ \GG(A)\hookrightarrow A$ is defined by multiplying each generator of $\GG(A)$ by $x_1\cdots x_d$. Therefore, $$\GG(A)\simeq K[g_{i_1}\cdots g_{i_d} :\ \det(\mathrm{Log}(g_{i_1},\ldots, g_{i_d}))\neq 0]\subseteq A.$$ The morphism $\GG(A)\hookrightarrow A$ induces the rational Gauss map $$\gamma: \proj{A}\dashrightarrow \proj{\GG(A)} .$$ \[birational\] Let $A=K[g_1,\ldots, g_n]$ be a standard graded $K$-subalgebra of $K[x_1,\ldots,x_d]$, up to degree renormalization, and $X=\proj{A}$. Since $$X=\bigcup_{i=1}^n \mathrm{Spec}(K[g_1/g_i,\ldots,g_n/g_i]),$$ it follows that the field $ K(X)$ of rational functions of $X$ is equal to the field of fractions of any of the algebras $K[g_1/g_i,\ldots,g_n/g_i]$. Let $B\subset A$ be an extension of homogeneous standard graded algebras, and $X=\proj{A}$ and $Y=\proj{B}$. Let $A$ be a domain. Then the corresponding dominant rational map $X \dashrightarrow Y$ is birational if and only if $K(X)=K(Y)$. Therefore if $A=K[g_1,\ldots,g_n]$ is the toric algebra as above, then the morphism $\gamma:\proj{A} \dashrightarrow\proj{\GG(A)}$ is birational if and only if for all $i<j$, the fractions $g_i/g_j$ can be expressed as a product of fractions of the form $(g_{i_1}\cdots g_{i_d})/(g_{j_1}\cdots g_{j_d})$ with $\det(\mathrm{Log}(g_{i_1},\ldots, g_{i_d}))\neq 0$ and $\det(\mathrm{Log}(g_{j_1},\ldots, g_{j_d}))\neq 0$. For example, $\gamma: \proj{A} \dashrightarrow\proj{\GG(A)}$ is birational, when $A\subseteq k[s,t]$ is the coordinate ring of the projective monomial curve parametrized by the generators of $A$ [@BGS Proposition 3.8]. In general, normality, Cohen-Macaulayness or other homological or algebraic properties are not preserved when passing from $A$ to $\GG(A)$. For example, the squarefree $r$-Veronese algebra $A=K[V_{r,d}]$ is normal Cohen-Macaulay, while for $r=2$ the Gauss algebra $\GG(A)$ is normal and Cohen–Macaulay if and only if $d\geq 5$, see Theorem \[Veronese\]. The following example shows that the Gauss algebra of a non-normal toric algebra may be normal. \[non-nomral\] Let $A=K[s^6, s^5t, s^4t^2,s^3t^3, t^6]\subset K[s,t]$ be the homogeneous coordinate ring of the projective monomial curve embedded in $\PP_K^4$. By [@BGS Lemma 3.7], the $K$-algebra $A$ is not an isolated singularity and hence is not normal. However, the Gauss algebra $\GG(A)$ is the $8$-Veronese algebra in $k[t,s]$, which is normal, Cohen-Macaulay and an isolated singularity. Borel-fixed algebras ==================== We start with the following lemma which is crucial for the kind of algebras studied in this section. \[Tran-det\] Let $g_1,\ldots,g_d\in S = K[x_1,\ldots,x_d]$ be homogeneous polynomials, and let $\varphi: S\to S$ be a linear automorphism. Then $$\det(\Theta(\varphi(g_1),\ldots, \varphi(g_d)) {)}=\det (\varphi)\cdot\varphi(\det(\Theta(g_1,\ldots,g_d)) {)}.$$ Consider the linear transformation $\varphi(x_i)=\sum_{j=1}^{d}a_{ji}x_j$, $i=1,\ldots,d$. For polynomial $g\in K[x_1,\ldots,x_d] $ a direct computation with derivatives shows that $$\dfrac{\partial \varphi (g)}{\partial x_i}=a_{i1} \varphi(\dfrac{\partial g}{\partial x_1})+\cdots+ a_{id} \varphi(\dfrac{\partial g}{\partial x_d}).$$ We have $$\begin{aligned} \nonumber \det(\Theta(\varphi(g_1), \ldots, \varphi(g_{ {d}}))) &=& \det\left( \begin{bmatrix} a_{11}& \cdots & a_{1d}\\ \vdots & \ddots & \vdots\\ a_{d1}& \cdots & a_{dd} \end{bmatrix} \begin{bmatrix} \varphi(\dfrac{\partial g_1}{\partial x_1})& \cdots & \varphi(\dfrac{\partial g_d}{\partial x_1})\\ \vdots & \ddots & \vdots\\ \varphi(\dfrac{\partial g_1}{\partial x_d})& \cdots & \varphi(\dfrac{\partial g_d}{\partial x_d}) \end{bmatrix}\right)\\ \nonumber \\ \nonumber&=& \det(\varphi).\det(\varphi(\Theta(g_1,\ldots,g_d)))\\ \nonumber &=& \det(\varphi).\varphi(\det((\Theta(g_1,\ldots,g_d)))).\end{aligned}$$ Recall that a set $G=\{g_1,\ldots,g_n\}$ of monomials of the same degree in $K[x_1,\ldots,x_d]$ is called [*Borel set*]{}, if the monomial ideal generated by $G$ is fixed under the action of all linear automorphisms $\varphi: S\to S$ defined by nonsingular upper triangular matrices. The ideal generated by a Borel set, is called a Borel-fixed ideal. If $\chara(K)=0$, as we always assume in this paper, the Borel-fixed ideals are just the strongly stable monomial ideals, that is, the monomial ideals $I$ with the property that $x_i(u/x_j)\in I$ for the all monomial generators $u$ of $I$, and all integers $i<j$ such that $x_j$ divides $u$. Let $B\subseteq G$. Then the elements of $B$ are called [*Borel generators*]{} of $G$, if $G$ is the smallest Borel set containing $B$. In this case if $B=\{u_1,\ldots,u_t\}$, we write $G=\langle u_1,\ldots,u_t\rangle$. For instance, the Borel set generated by $\{x_1x_3, x_2x_4\}$ is $\langle x_1x_3, x_2x_4\rangle=\{x_1^2,x_1x_2,x_1x_3,x_1x_4,x_2^2,x_2x_3,x_2x_4\}$. A Borel set $G$ is called [*principal*]{} if there exists $u\in G$ such that $G=\langle u\rangle$. Let $G$ be a Borel set of monomials of degree $r$. The Borel generators of $G$ are characterized by the property that they are maximal among the monomials of $G$ with respect to the following partial order on the monomials: let $u= x_{i_1}x_{i_2}\ldots x_{i_r}$ and $v=x_{j_1}x_{j_2}\ldots x_{j_r}$ with $i_1\leq i_2\leq \ldots \leq i_r$ and $j_1\leq j_2\leq \ldots \leq j_r$. Then we set $u\prec v$, if $i_k\leq j_k$ for $k=1,\ldots,r$. In particular, if $v=x_{i_1}^{c_1}\cdots x_{i_r}^{c_r}$ with $c_i>0$, and $u=x_1^{a_1}\cdots x_d^{a_d}$. Then $u\npreceq v$ if and only if there exists $j$, such that $$\begin{aligned} \label{borel-order} a_{i_j+1}+\cdots+a_d\geq c_{j+1}+\cdots+c_r+1.\end{aligned}$$ Let $G=\{g_1,\ldots,g_n\}\subset K[x_1,\ldots,x_d]$ be a Borel set. Then we call $A=K[g_1,\ldots,g_n]$, a [*Borel fixed algebra*]{}, if $\dim(A)=d$. Note that $\dim(A)=d$, if and only if there exists $j$ such that $x_d|g_j$. Indeed, since $G$ is a Borel set, the condition implies that $\{x_1^r,x_1^{r-1}x_2,\ldots,x_1^{r-1}x_d\}\subseteq G$, where $r$ is the degree of the monomials in $G$. The log-matrix of these elements is upper triangular, and so has rank $d$. This shows that $\dim(A)=d$. Indeed, $A$ is isomorphic to the polynomial ring $K[x_1,\ldots,x_d]$ by multiplication by $1/x^{r-1}$. \[Borel\_fix\] The Gauss algebra of a Borel-fixed algebra is a Borel-fixed algebra. Let $A$ be a Borel-fixed algebra with monomial generators $G=\{g_1,\ldots,g_n\}$. Let $G'$ be the set of the corresponding monomial generators of $\GG(A)$. We want to show that $G'$ is a Borel set. For this, it is enough to show that the ideal $I'$ generated by $G'$ is a Borel-fixed ideal. Let $g$ be a monomial generator in $I'$. Then $g=\det(\Theta(g_{i_1},\ldots,g_{i_d}))$. Let $I$ be the monomial ideal generated by $G$. By Lemma \[Tran-det\], for any upper triangular automorphism $\varphi:\; S\to S$, one has $$\varphi(g)=\varphi(\det(\Theta(g_{i_1},\ldots,g_{i_d})))=\det(\varphi)^{-1}(\det\Theta(\varphi(g_{i_1}),\ldots,\varphi(g_{i_d}))).$$ Since $I$ is a Borel-fixed ideal, each $\varphi(g_i)$ is a $K$-linear combination of elements of $G$. By using the fact that $\Theta(-)$ is a multilinear function, we get $\varphi(g)\in I'$. This shows that $\GG(A)$ is Borel-fixed. \[Veronese\] The Gauss algebra of an $r$-Veronese algebra is an $(r-1)d$- Veronese algebra. Consider the monomials $g_1= x_1x_d^{r-1},\ldots, g_{d-1}=x_{d-1}x_d^{r-1}, g_d=x_d^r$. As the log-matrix of $\gg$ is non-singular, the monomial $g_1\cdots g_d/x_1\cdots x_d= x_d^{(r-1)d}$ belongs to the Gauss algebra. Since the $r$-Veronese is a Borel-fixed ideal, the assertion follows from Theorem \[Borel\_fix\]. In general the number of Borel generators of the Borel fixed algebra $A$ and that of $\GG(A)$ may be different. In fact, let $\{x_2x_3, x_1x_4\}$ be the set of Borel generators of $A$. Then $A=K[x_1^2,x_1x_2,x_2^2,x_1x_3,x_2x_3,x_1x_4]$ and the log-matrix of the generators of $A$ is $$\begin{bmatrix} 2&1&0&1&0&1\\ 0&1&2&0&1&0\\ 0&0&0&1&1&0\\ 0&0&0&0&0&1 \end{bmatrix}.$$ Therefore $\GG(A)=K[x_1^4,x_1^3x_2,x_1^2x_2^2,x_1x_2^3,x_1^3x_3,x_1^2x_2x_3,x_1x_2^2x_3]$, and $x_1x_2^2x_3$ is the single Borel generator of $\GG(A)$. However if $A$ is principal Borel, then $\GG(A)$ is principal Borel as well. More precisely we have the following \[principal\] Let $A$ be a principal Borel-fixed algebra with Borel generator $m=x_{i_1}^{a_{i_1}}\cdots x_{i_r}^{a_{i_r}}$ with $a_{i_j}>0$, for $j=1,\ldots,r$. Then $\mathbb{G}(A)$ is a principal Borel-fixed algebra with Borel generator $$m'=\frac{m^{i_r}}{x_{i_1}^{i_1-1}x_{i_2}^{i_2-i_1}\cdots x_{i_{r-1}}^{i_{r-1}-i_{r-2}}x_{i_r}^{i_r-i_{r-1}+1}}.$$ We first show that $m'\in \GG(A)$. Let $g_{k,l}=x_l(m/x_{i_k})$, $k=1,\ldots,r$, $l=i_{k-1},\ldots,i_k-1$ for all $k$, where $i_0=1$. Then the $g_{k,l}$ belong to $\langle m\rangle$, and $$m'=(\prod_{k=1}^{r}\prod_{l=i_{k-1}}^{i_k-1}g_{k,l})/x_1x_2\cdots x_d.$$ We order the monomials $g_{k,l}$ lexicographically and consider the corresponding log-matrix $A$. The $i$th row with $i\notin\{i_1,\ldots,i_r\}$ has only one non-zero entry which is $1$. So in order to compute the determinant of the log-matrix, we reduce to the computation of the cofactor of that nonzero entry, indeed we skip the $i$th row and the column corresponding to the nonzero entry. This can be done for all $i\notin\{i_1,\ldots,i_r\}$. Then we obtain the log-matrix $M$ of the following sequence of monomials $$m, x_{i_1}\frac{m}{x_{i_2}},\ldots, x_{i_{r-1}}\frac{m}{x_{i_r}}$$ with respect to $x_{i_1},\ldots, x_{i_r}$. Subtracting the first column of $M$ from the other columns of $M$, we obtain the following matrix $$\begin{bmatrix} a_{i_1}& 1&0&\cdots&\cdots & \cdots& 0\\ a_{i_2}&-1&1&0&\cdots & \cdots& 0\\ a_{i_3} & 0 & -1& 1 & 0 &\cdots&0\\ \vdots & \vdots &\ddots& \ddots&\ddots& {\ddots}&\vdots\\ \vdots & \vdots && \ddots&\ddots& 1 & 0\\ a_{i_{r-1}}& 0 & \cdots&\cdots& 0 & -1 & 1\\ a_{i_r}& 0 & \cdots&\cdots&\cdots& 0 & -1\\ \end{bmatrix}$$ Now for each $i>1$, we add the $i$th row to the first row. The result is a lower triangular matrix with non-zero entries on the diagonal. This shows that $A$ is non-singular, and proves that $m'$ is a generator of the Borel-fixed algebra $\GG(A)$. Since $\GG(A)$ is a Borel-fixed ideal, by Theorem \[Borel\_fix\], it is enough to prove that for any monomial $g$ in $\GG(A)$, one has $g\preceq m'$. Let $m=x^{a_{i_1}}_{i_1}\cdots x^{a_{i_r}}_{i_r}$, $m'=x^{a'_{i_1}}_{i_1}\cdots x^{a'_{i_r}}_{i_r}$. By definition of $m'$, we have $a'_{i_j}=i_ra_{i_{j+1}}-i_{j+1}+i_j$ for $j=2,\ldots,r-1$, and $a'_{i_r}=i_r(a_{i_r}-1)+i_{r-1}-1$. Let $$g=(\prod^{i_r}_{i=1}g_i)/x_1x_2\cdots x_{i_r},$$ where $g_1,\ldots,g_{i_r}$ belong to the minimal monomial generating set of $A$, the latter having a non-singular log-matrix. If $g\npreceq m'$, then by Borel order property (\[borel-order\]), there exists $1\leq j\leq r-1$, such that $\prod^{i_r}_{j=1}g_j$ is divisible by $w=x_{i_j}^{b_{i_j}}\cdots x_{i_r}^{b_{i_r}}$, and $$\begin{aligned} \sum^{i_r}_{l=i_j+1}b_l-(i_r-i_j)&\geq &1+\sum^{i_r}_{l=i_{j+1}}a'_{i_l}\\ &=&\sum_{k=j}^{r-2}(i_ra_{i_{k+1}}-i_{k+1}+i_k)+i_r(a_{i_{r}}-1)+i_{r-1}\\ &=&(\sum^r_{l=j+1}a_{i_l}-1)i_r+i_j.\end{aligned}$$ Therefore, $$\begin{aligned} \label{clv} \sum^r_{l=i_j+1}b_{i_l}\geq (\sum^r_{l=j+1}a_{i_l})i_r. \end{aligned}$$ We may write $g_s$ as a product of monomials $g_s=f_{s}h_{s}$ with $\supp(f_{s})\subseteq\{i_1,\ldots,i_j\}$ and $\supp(h_{s})\subseteq\{i_{j+1},\ldots,i_r\}$. As $g_s\preceq x^{a_{i_1}}_{i_1}\cdots x^{a_{i_r}}_{i_r}$, we have $\deg(h_{s})\leq\sum^r_{l=j+1}a_{i_j}$ and, since $w$ divides $g_1\cdots g_{i_r}$, we get $$\sum^{i_r}_{l=i_j+1}b_{i_l}\leq \sum^d_{s=1}\deg(h_s)\leq d\sum^r_{l=j+1}a_{i_l}.$$ Together with (\[clv\]), it follows that $\sum^{i_r}_{s=1}\deg(h_s)=i_r\sum^r_{l=j+1}a_{i_l}$ and, this implies $\deg(h_s)=\sum^r_{l=j+1}a_{i_l}$. Let $L$ be the log-matrix of $g_1,\ldots,g_{i_r}$. Then the summation of the last $i_r-i_j$ entries of each column of $L$ is equal to $\sum^r_{l=j+1}a_{i_l}$, and so the summation of the first $j$ entries of each column is equal to $i_r-\sum^r_{l=j+1}a_{i_l}$. This implies that $L$ is singular, a contradiction. \[Emma\] Let $A$ be a principal Borel-fixed algebra. Then $\GG(A)$ is normal and for suitable monomial order its defining ideal has a quadratic Gröbner basis. By the above theorem, $\GG(A)$ is a principal Borel fixed algebra. A principal Borel set is a polymatroid. Therefore $\GG(A)$ is normal, see [@HH Corollary 6.2]. In [@dN] it is shown that the principal Borel-fixed sets are sortable, and so $\GG(A)$ has quadratic Gröbner basis. Let $A$ be a Borel-fixed algebra such that $\dim A=\dim\GG(A)=d$. Then the Guass map $\gamma: \proj{A}\dashrightarrow \proj{\GG(A)}$ is birational. By the hypothesis on the dimension of $\GG(A)$, there exists a generator $u$ of $\GG(A)$ such that $x_d|u$. For $1\leq i<j\leq d$ we have $$\frac{x_i}{x_j}=\frac{x_i(u/x_d)}{x_j(u/x_d)},$$ which implies that $\gamma$ is birational, since any quotient of monomials in $A$ is the product of some of the $x_i/x_j$, see Remark \[birational\]. Squarefree Veronese algebras ============================= Let $V_{r,d}$ be the set of all squarefree monomials of degree $r$ in $S=K[x_1,\ldots,x_d]$. The $K$-subalgebra $A=K[V_{r,d}]$ of $S$ is called the *squarefree $r$-Veronese* algebra. By Proposition \[Veronese\], the Gauss algebra associated to a Veronese algebra is again a Veronese algebra. The situation for squarefree Veronese algebra is more complicated. Denote by $\mathrm{Mon}_S(t,r)$ the set of all monomials $u$ of degree $r$ in $S$, such that $|\supp(u)|\geq t$, where $\supp(u)=\{i \ : \ x_i|u\}$. \[kirschtorte\] The monomial ideal generated by $\mathrm{Mon}_S(t,r)$ is polymatroidal. In particular, the $K$-algebra $K[\mathrm{Mon}_S(t,r)]$ is normal and Cohen–Macaulay. The normality of the $K$-algebra $K[\mathrm{Mon}_S(t,r)]$ follows from [@HH Theorem 6.1], once we have shown that the ideal generated by $\mathrm{Mon}_S(t,r)$ is polymatroidal. Let $u=x_1^{a_1}\cdots x_d^{a_d},v=x_1^{b_1}\cdots x_d^{b_d}\in\mathrm{Mon}_S(t,r)$. By symmetry we may assume that $a_1>b_1$. Suppose $a_1>1$, then $x_iu/x_1\in \mathrm{Mon}_S(t,r)$ for any $i\neq 1$, and so the exchange property holds. Next suppose that $a_1=1$, then $b_1=0$. If $\supp(u)$ has more than $t$ elements, we may replace $x_1$ by any variable $x_i\in\supp(v)$. Finally, suppose that $\supp(u)$ has exactly $t$ elements. Since $x_1\notin\supp(v)$, there exists $x_j\in\supp(v)\setminus\supp(u)$. Replacing $x_1$ by $x_j$, the exchange property is satisfied. In the following result we describe the structure of the Gauss algebra of the squarefree $2$-Veronese algebra $K[V_{2,d}]$. Note that for $d\leq 3$, the Gauss algebra is isomorphic to a polynomial ring. \[2-Veronese\] Let $A=K[V_{2,d}]$, with $d\geq4$. Then 1. $\GG(A)=K[\mathrm{Mon}_S(3,4)\setminus\{x_1x_2x_3x_4\}]$, if $d=4$; 2. $\GG(A)=K[\mathrm{Mon}_S(3,d)]$, if $d\geq 5$; 3. the embedding dimension of $\GG(A)$ is $$\mathrm{edim} \GG(k[A])=\left\{ \begin{array}{ll} e-1, & if\, \hbox{d=4,} \\ e, & if \, \hbox{d=5,} \end{array} \right.$$ where $e=\binom{2d-1}{d}-(d-1)\binom{d}{2}-d$; 4. the Gauss algebra is a normal Cohen–Macaulay domain, if and only if $d\geq 5$; 5. the Gauss map $\gamma: \proj{A}\dashrightarrow \proj{\GG(A))}$ is birational. For the proof of the theorem, we need the following \[minor\] Let $G$ [be]{} a loop-less connected graph with the same number of vertices and edges. Then the log-matrix of the edge ideal of $G$ is non-singular if and only if $G$ contains an odd cycle. First we show that any monomial of the form $m=g/x_1\cdots x_d$ belongs to $\mathrm{Mon}_S(3,d)$, where $g=g_{i_1}\cdots g_{i_d}$ is a product of pairwise distinct elements of $V_{2,d}$. This then yields the inclusion $\GG(A)\subseteq K[\mathrm{Mon}_S(3,d)]$. Suppose that the number of elements in the support of $m$ is less than $3$. Then at least $d-2$ variables have degree $1$ in $g$. Hence $g$ can be written as a product of at most $d-1$ monomials in $A$, which is a contradiction. Now, to prove (a) and (b), let $m$ be an element of $\mathrm{Mon}_S(3,d)$. For $d=4$ and $d=5$ the assertions can be shown by direct computations. Let $d>5$, and first assume that $m=x_1\cdots x_d$. If $d$ is odd, then let $$g_1=x_1x_2, g_2=x_2x_3,\ldots, g_{d-1}=(x_{d-1}x_d), g_d=(x_dx_1).$$ Then the log-matrix of $g_1,\ldots,g_d$ is non-singular by Lemma \[minor\]. If $d$ is even, then let $$g_1=x_1x_2, g_2=x_2x_3,g_3=x_3x_1,g_4=x_4x_5,g_5=x_5x_6,\ldots, g_{d-1}=(x_{d-1}x_d), g_d=(x_dx_4).$$ Now, the log-matrix is $$\begin{bmatrix} A &0\\ 0&B\\ \end{bmatrix},$$ where $A$ and $B$ are incidence matrices of odd cycles, and so it is non-singular. Next assume that $m\neq x_1\cdots x_d$. Without loss of generality we may assume that $m=x_1^{r_1}\cdots x_{d-1}^{r_{d-1}}$. Since $\deg(m)=d$, there exists $i$ such that $r_i>1$. Let $u=m/x_i$. Then $u\in \Mon_{S'}(3,d-1)$, where $S'=K[x_1,\ldots,x_{d-1}]$. By induction, $(x_1\cdots x_{d-1})u=g_1\cdots g_{d-1}$ with $g_i\in A$ and $L(g_1,\ldots, g_{d-1})$ non-singular. Let $g_d=x_ix_d$. Then $(x_1\dots x_d)m=g_1\cdots g_{d-1}g_d$. Since all the entries of the last row of $\mathrm{Log}(g_1,\ldots,g_d)$ are zero, except the last one, which is equal to $1$, we see that $\mathrm{Log}(g_1,\ldots,g_d)$ is non-singular. \(c) follows from (b) by a simple counting argument. (d): If $d\geq 5$, it follows from (b) and Proposition \[kirschtorte\] that $\GG(A)$ is normal, and Cohen–Macaulay by Hochster [@Ho]. On the other hand a calculation with Singular [@DGPS] shows that for $d=4$, the $h$-vector of $\GG(A)$ has a negative component. Therefore, in this case $\GG(A)$ is not Cohen–Macaulay. (e): By Remark \[birational\], it is suffices to show that for every $1\leq i<j\leq n$, $$K[\dfrac{x_ix_j}{x_rx_s} \ | \ 1\leq r<s\leq n] \subset K[\frac{u}{v}\ | \ u,v\in \mathrm{Mon}(3,d) ].$$ For $1\leq i<j\leq d$ one has $$\begin{aligned} \label{quotient} \dfrac{x_i}{x_j}=\dfrac{x_k x_l^{d-2}x_i}{x_k x_l^{d-2}x_j},\end{aligned}$$ with $i,j,k,l$ pairwise distinct. Hence $(x_ix_j)/(x_rx_s)$ has an expression as in (\[quotient\]), if $\{i,j\}\sect \{r,s\}\neq \emptyset$. Otherwise, $$\frac{x_ix_j}{x_rx_s}=\frac{(x_jx_s^{d-2}x_i)(x_ix_r^{d-2}x_j)}{(x_jx_s^{d-2}x_r)(x_ix_r^{d-2}x_s)}.$$ \(a) Let $A=K[V_{r,d}]$. We may assume that $d\geq r+2$, otherwise $\GG(A)$ is a polynomial ring. Then $$\GG(A)\subseteq\{x_1^{a_1}\cdots x_d^{a_d}\in\Mon_{S}(r+1,(r-1)d)\ : \ a_i\leq d-2 \quad \text{for}\quad 1\leq i\leq d\}.$$ For $r=2$, the equality holds if and only if $d\geq 5$. It would be interesting to know for which $r>2$ and $d$ the equality holds. \(b) According to White’s conjecture [@W], the base ring of a polymatroid is generated by the so-called exchange relations, which are quadratic binomials. Since $\Mon_{S}(3,d)$ is polymatroidal, we expect that the Gauss algebra of $K[V_{2,d}]$ has quadratic relations. Edge rings ========== Let $G$ be a simple graph on the vertex set $V(G)=[d]$ and edge set $E(G)=\{e_1,\ldots,e_m\}$. For given subset $V\subseteq [d]$, we set $x_V=\prod_{i\in V}x_i$. In the case that $V$ is an edge $e=\{i,j\}$, we simply write $e$ instead of $x_V=x_ix_j$. The edge ideal $I(G)$, of $G$, is the ideal generated by the monomials $e\in E(G)$. Note that the log-matrix of $E(G)$ is the incidence matrix of $G$. Let $V\subseteq V(G)$ and $E\subseteq E(G)$ with $| V|=| E|$. We denote by $\Delta_{V,E}$ the minor of the log-matrix $\log(E(G))$, with rows $V$ and columns $E$. \[Lemma\] Let $V\subseteq V(G)$ and $E\subseteq E(G)$ with $|V|=|E|=r$, and let $\Delta_{V,E}$ be the minor of the log-matrix $\log(E(G))$, with rows $V$ and columns $E$. Suppose the edges in $E$ can be labeled as $e_1,\ldots,e_r$, such that $$\begin{aligned} \label{maybe} | V\cap(e_1\cup\cdots\cup e_i)| =i \quad \text{for} \quad i=1,\ldots,r. \end{aligned}$$ Then $\Delta_{V,E}\neq 0$. The converse holds if $G$ is a bipartite graph. Suppose condition (\[maybe\]) holds. Let $M$ be the matrix with rows $V$ and columns $E$. Let $V\cap e_1=\{v\}$. Then the first column of $M$ has only one non-zero entry, corresponding to vertex $v$. Let $V'=V\setminus\{v\}$, then $$|V'\cap\{e_2,\ldots,e_i\}|=i-1 \quad \text{for}\quad i=2,\ldots,r.$$ Now, by the induction hypothesis the matrix $M'$ whose rows are $V'$ and whose columns are $e_2,\ldots,e_r$, is non-singular. It follows that $M$ is non-singular. Conversely, assume that $\Delta_{V,E}\neq 0$. Then we claim that there exists a column $e_1$ in $E$ such that $| V\cap e_1|=1$. Indeed, if $| V\cap e_i|>1$, for $i=1,\ldots,r$, then $M$ is the incidence matrix of a bipartite graph. Now, Lemma \[minor\] implies $\Delta_{V,E}=0$, contradiction. Let $V'=V\setminus\{v_1\}$, where $V\cap e_1=\{v_1\}$. Then the matrix $M'$ whose rows are $V'$ and whose columns are $e_2,\ldots,e_r$, is non-singular. Now, $|V'\cap\{e_2,\ldots,e_i\}|=i-1$ for $i=2,\ldots,r$, by induction. This implies that $| V'\cap\{e_1,\ldots,e_i\}|=i$ for $i=2,\ldots,r$. \[schwarzwaelder\] Let $G$ be a graph with $c$ connected components, and $V\subset V(G)$, with $| V|\leq d-c$. Then there exists $E\subseteq E(G)$ with $| E|=|V|$ such that $\Delta_{V,E}\neq 0$. Let $| V|=r$. Since $r\leq d-c$, we can choose a set $E$ of $r$ edges such that $e\cap V\neq\emptyset$ for each $e\in E$. Now, the matrix $M$ with rows $V$ and columns $E$, is not the incidence matrix of a forest, since for a forest the number of vertices is strictly bigger than the number of edges. Hence there exists an edge $e_1$ in $E$ such that $e_1\cap V=\{v\}$. Removing the edge $e_1$ from $G$, the number of connected components $c'$ of $G\setminus\{e_1\}$ is at most $c+1$. Let $V'=V\setminus\{v\}$. Then $|V'|\leq d-c'$. By induction there exist edges $e_2,\ldots,e_r$ such that $|V'\cap e_2\cup\cdots\cup e_i|=i-1$ for $i=2,\ldots,r$. It follows that $e_1,\ldots,e_r$ satisfies the condition (\[maybe\]). Therefore the desired result follows from Lemma \[Lemma\]. Let $G$ be a simple bipartite graph. Let $L$ [be]{} a non-empty subset of $[d]$, and let $G^L$ be the graph which is obtained from $G$ by attaching a loop to $G$ at each vertex belonging to $L$. For given set $T\subseteq E(G)$, let $G(T)$ denote the graph with $V(G(T))=V(G)$ and $E(G(T))=T$. \[notneeded\] Let $G$ be a bipartite graph with $r$ components and $L$ be a subset of $[d]$. Let $A$ be the edge ring of $G^L$. Then the following statements hold. 1. $A$ has dimension $d$ if and only if $L$ contains at least one vertex of each component of $G$. 2. If the condition (a) [is]{} satisfied, then the Gauss algebra $\GG(A)$ is generated by the monomials $$g_{V,T}= x_{V}\frac{e_{T}}{x_{V^c}},$$ where $V$ is a non-empty subset of $L$, $V^c=[d]\setminus V$, and $e_T=\prod_{e\in T}e$ where $T\subseteq E(G)$ satisfies 1. $G(T)$ is a forest [, which may have isolated vertices as some of its connected components]{}; 2. each connected component of $G(T)$ contains exactly one vertex of $V$. In particular, when $|V|=1$, then the cardinality of the minimal set of generators of $\GG(A)$ is bounded by the number of spanning trees of $G$. Moreover, $g_{V,T}=g_{V,T'}$ if and only if each vertex of $G$ has the same degree in $T$ and $T'$. (a). [As $G$ is a bipartite graph, the log-matrix of $G$ is singular by Lemma \[minor\]. ]{} We show that the log-matrix of $G^L$ has a non-singular maximal minor, if and only if $L$ contains at least one vertex of each component of $G$. Let $L_i=G_i\cap L$, then $G_1^{L_1},\ldots,G_r^{L_r}$ are the connected components of $G^L$, and the log-matrix of $G^L$ has maximal rank if and only if the log-matrix of each $G_i^{L_i}$ has maximal rank. [Therefore, it is enough to show that the log-matrix of a connected graph $G$, with at least one loop, is non-singular. Assume that there is a loop at vertex $1$. Then the $1$st column has only one non-zero entry at $1$st row. Let $A$ denote the log-matrix of $G$ and $|V(G)|=n$. In order to compute the rank of $A$, we may skip the $1$st row and the $1$st column, obtaining a new matrix $A_1$, which has maximal rank $n-1$, by Corollary \[schwarzwaelder\]. Therefore the rank of $A$ is equal to $n$. ]{} (b). We first show that the conditions (i),(ii) are equivalent to 1. $|T|=|V^c|$; 2. the elements of $T$ can be labeled as $e_1,\ldots,e_m$ such that $$|V^c\cap(e_1\cup\cdots\cup e_i)|=i \quad \text{for} \quad i=1,\ldots,m.$$ Suppose that (i) and (ii) are satisfied. If $T=\emptyset$, then the equivalence of (i),(ii) with $\alpha,\beta$ is trivial. Now, assume that $T\neq\emptyset$, and let $G(T)_1,\ldots,G(T)_t$ be the connected components of $G(T)$ with $|V(G(T)_i)|\geq 2$ and $v_i$ be the vertex of $V$ belonging to $G(T)_i$. Since $G(T)_i$ is a tree, we may label the edges of $G(T)_i$ as $e_{i_1},\ldots,e_{i_{s_i}}$ such that $v_i\in e_{i_1}$ and $|e_{i_j}\cap(e_{i_1}\cup\cdots\cup e_{i_{j-1}})|=1$ for all $j=1,\ldots,s_i$. Then the sequence of edges $$e_{1_1},\ldots,e_{1_{s_1}},e_{2_1},\ldots,e_{2_{s_2}},e_{3_1},\ldots$$ satisfies conditions ($\alpha$),($\beta$). Conversely, condition $(\beta)$ guarantees that $G(T)$ does not contain any cycle, and so it is a forest, which by $(\alpha)$ has $d-|V|$ edges. Therefore $|V|$ is equal to the number of connected components of $G(T)$. let $G(T)_1,\ldots,G(T)_t$ be the connected components of $G(T)$ with $|V(G(T)_i)|\geq 2$, and let $e_{i_j}$ be the first edge, with respect to the labeling in ($\beta$), such that $e_{i_j}\cap G(T)_j\neq\emptyset$. Then $|e_{i_j}\cap V|=1$, so each connected component of $G(T)$ contains at least one vertex in $V$. Since $G(T)$ has $|V|$ number of components, each component should contain exactly one element of $V$. Let $g$ belong to the minimal set of generators of $\GG(A)$. Then $g=\frac{g_1\cdots g_d}{x_1\cdots x_d}$, where $g_i$ is a monomial generator of $A$ and the log-matrix of $g_1,\ldots,g_d$ is non-singular. Since the incidence matrix of a bipartite graph is singular by Lemma \[minor\], at least one $g_i$ corresponds to a loop. After relabeling, we may assume that $\{g_1,\ldots,g_d\}=\{x_1^2,\ldots,x_s^2,e_1,\ldots,e_{d-s}\}$ and $$\ \log(x_1^2\cdots x_s^2e_{1}\cdots e_{{d-s}})= \begin{bmatrix} 2&0&\cdots&0&a_{1,1}&\cdots&a_{1,d-s}\\ 0&2&&0&a_{2,1}&\cdots&a_{2,d-s}\\ \vdots&&\ddots&\vdots&\vdots&\vdots\\ 0&0&\cdots&2&a_{s,1}&\cdots&a_{s,d-s}\\ 0&0&\cdots&0&a_{s+1,1}&\cdots&a_{s+1,d-s}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&0&a_{d,1}&\cdots&a_{d,d-s} \end{bmatrix},$$ where $A=[a_{r,t}]$ is the log-matrix of $e_{1}\cdots e_{{d-s}}$. Let $V=\{1,\ldots,s\}$ and $T=\{e_1,\ldots,e_{d-s}\}$. Then $$g=g_{V,T}=\frac{x_1^2\cdots x_s^2e_{1}\cdots e_{{d-s}}}{x_1\cdots x_d}=x_1\cdots x_s\frac{e_{1}\cdots e_{{d-s}}}{x_{s+1}\cdots x_{d}},$$ and the log-matrix is non-singular if and only if the submatrix $A'$ with rows $s+1,\ldots,d$ and columns $s+1,\ldots, d$, is non-singular, and by Lemma \[Lemma\], $A'$ is non-singular if and only if the condition ($\beta$) is satisfied. Let $G$ be a path graph with $d$ vertices, and edges $\{1,2\},\{2,3\},\ldots,\{d-1,d\}$. Let $L\subseteq [d]$ and $A$ be the edge ring of $G^L$. [The induced subgraph by any set $T\subset E(G)$, can be considered as a disjoint union of intervals. Since $G$ is a path graph, $G(T)$ is a forest. By Theorem \[notneeded\], the product of edges in $T$ should be divisible by all vertices in $[d]\setminus V$. Therefore $G(T)$, covers $[d]$. In other words, the generators of $\GG(A)$ correspond to interval partitions of $[d]$ with the property that each interval contains exactly one element of $V$. Let $[d]=\cup^r_{i=1}[a_i,b_i]$ with $[a_i,b_i]\sect[a_j,b_j]=\emptyset$ for all $i\neq j$, and with $|V\cap[a_i,b_i]|=1$ for $i=1,\ldots,r$. The corresponding generator of $\GG(A)$, is $$\begin{aligned} \label{easy} (\prod^r_{i=1}\prod_{j\in ]a_i,b_i[ }x_j)\prod_{j\in L}x_j^{c_j},\end{aligned}$$ where $c_j=2$, if $j$ belongs to a proper interval, and is $c_j=1$, otherwise. Here $[a_i,b_i]$ is said to be proper if $b_i-a_i>0$. ]{} The above discussions show that if $L=[d]$, then the number of generators of $\GG(A)$ is the number [ $$\lambda_d=\sum^{d}_{r=1}\sum_{a\in P_r}\prod_{i=1}^r(a_{i+1}-a_i),$$ where $P_r=\{a=(a_0,\ldots,a_r) : 0=a_0<a_1< a_2<\cdots <a_r=d)\}$, for $r=1,\ldots,d$. ]{} The sequence $(\lambda_d)_{d\geq 1}$ begins as follows $$1,3,8,21,55,144,377,\ldots.$$ The recursive formula $\lambda_d=3\lambda_{d-1}-\lambda_{d-2}$, describes the beginning of the sequence. This seems to be the rule for the whole sequence $(\lambda_d)_{d\geq 1}$. In the case that $L=\{i<j\}$, $\GG(A)$ is generated by $j-i+2$ monomials $$x^2_ix_2\cdots x_{d-1} \ ,\ x_j^2x_2\cdots x_{d-1} \ , \ (x_i^2x_j^2x_2\cdots x_{d-1})/x_{k}x_{k+1} \quad \text{for} \quad i\leq k\leq j-1.$$ An easy calculation shows that the log-matrix of $\GG(A)$ has rank $j-i+1$. In particular $\GG(A)$ is a hypersurface ring. It can be shown that the multiplicity of $\GG(A)$ is $j-i$. When $i=1,j=d$, the defining equation of $\GG(A)$ is $$f=\left\{\begin{array}{ll} \prod^{d/2}_{i=1}y^2_{2i}-y_1y_{d+1}\prod_{i=1}^{d/2-1}y^2_{2i+1}, & \text{ if } d \text{ is even; }\\ \\ y_1\prod^{(d-1)/2}_{i=1}y^2_{2i+1}-y_{d+1}\prod_{i=1}^{(d-1)/2}y^2_{2i+1}, & \text{ if } d \text{ is odd, }\end{array} \right.$$ and if $j=i+1$, then the defining equation is quadratic. By computing the singular locus, we see that $\GG(A)$ is normal if and only if $d=2$. Let $G$ be a cycle with $d$ vertices, and edges $\{1,2\},\{2,3\},\ldots,\{d-1,d\},\{1,d\}$. Let $L=\{1\}$ and $A$ be the edge ring of $G^L$. When $d$ is even, the spanning trees of $G$ correspond to the generators of $\GG(A)$. Each spanning tree of $G$ is obtained by removing one edge from $G$, and so the generators of $\GG(A)$ are $$\begin{aligned} \label{easy-cycle} x_1^2\prod^d_{i=3}x_i \ , \ x_1^2\prod^{d-1}_{i=2}x_i \ , \ x_1^3\frac{\prod^d_{i=2}x_i}{x_jx_{j+1}} \quad \text{for} \quad j=2,\ldots,d-1. \end{aligned}$$ When $d$ is odd, in addition to the monomials in (\[easy-cycle\]), $\GG(A)$ has one more generator, namely $x_1\cdots x_d$. For even $d$, $\dim(\GG(A))=d-1$, and for odd $d$, $\dim(\GG(A))=d$. Hence in both cases $\GG(A)$ is a hypersurface ring with defining equation $$f=\left\{\begin{array}{ll} y_{d-1}\prod^{d/2-1}_{i=1}y_{2i}-y_{d}\prod_{i=1}^{d/2-1}y_{2i-1}, & \text{ if } d \text{ is even; }\\ \\ y_d\prod^{(d-1)/2}_{i=1}y_{2i}-y_{d+1}\prod_{i=1}^{(d-1)/2}y_{2i-1}, & \text{ if } d \text{ is odd. }\end{array} \right.$$ The initial monomial of $f$ (with respect to any monomial order) is squarefree. Therefore, $\GG(A)$ is normal. Let $G$ be a bipartite graph on $[d]$, $L=\{i\}$ and $A$ be the edge ring of $G^L$. The above examples and computational evidence indicate that $\GG(A)$ is a hypersurface ring of dimension $d-1$ if and only if $G$ is an even cycle. \[bipartite\] Let $G=K_{n,m}$ be a complete bipartite graph with partition sets $X=\{x_1,\ldots,x_n\}$ and $Y=\{y_1,\ldots,y_m\}$. Let $A$ be the edge ring of $G$ with one loop at vertex $x_1$. Then $$\GG(A)=K[x_1^2(x_1,\ldots,x_n)^{m-1}(y_1,\ldots,y_m)^{n-1}].$$ Indeed, any generator of $\GG(A)$ can be written as $x_1^2(e_1\cdots e_{n+m})/x_1\cdots x_ny_1\cdots y_m$, where $e_j$ is an edge of $G$, which follows $\GG(A)\subseteq K[x_1^2(x_1,\ldots,x_n)^{m-1}(y_1,\ldots,y_m)^{n-1}]$. Let $f$ be a monomial in the generating set of $(x_1,\ldots,x_n)^{m-1}(y_1,\ldots,y_m)^{n-1}$. Then $f=x_{i_1}\cdots x_{i_{m-1}}y_{j_1}\cdots y_{j_{n-1}}$, for some $1\leq i_1\leq \cdots\leq i_{m-1}\leq n$ and $1\leq j_1\leq \cdots \leq j_{n-1}\leq m$. Now, let $T$ be the subgraph of $G$ with $V(T)=V(G)$ and $E(T)$ equal to $$\{e_1=x_1y_{j_1},\ldots, e_{n-1}=x_{n-1}y_{j_{n-1}},e_{n}=y_1x_{i_1},\ldots,e_{n+m-2}=y_{m-1}x_{i_{m-1}}, e_{n+m-1}=x_ny_n\}.$$ Then $T$ is a spanning tree of $G$, which implies that $$x_1^2f=x_1^2e_1\cdots e_{n+m-1}/x_1\cdots x_ny_1\cdots y_m$$ is a generator of $\GG(A)$. As a consequence, the embedding dimension of $\GG(A)$ is $\binom{m+n-2}{n-1}\binom{m+n-2}{m-1}$. However, the number of spanning tress of $G$ is $n^{m-1}m^{n-1}$, see [@HW Theorem 1]. Therefore, among the spanning trees of $G$, many of them correspond to the same generator in $\GG(A)$. [99]{} P. Brumatti, P. Gimenez and A. Simis, On the Gauss algebra associated to a rational map $\PP^d\to \PP^n$, J. Algebra [**207**]{} (1998), no. 2, 557–571. Decker, W.; Greuel, G.-M.; Pfister, G.; Sch[ö]{}nemann, H.: — [A]{} computer algebra system for polynomial computations. (2018). E. De Negri, Toric rings generated by special stable sets of monomials, Math. Nachr. [**203**]{} (1999), 31–45. P. Griffiths and J. Harris, Algebraic geometry and local differential geometry, Ann. Sci. École Norm. Sup. (4) [**12**]{} (1979), no. 3, 355–452. J. W. Grossman, D. M. Kulkarni and I. E. Schochetman, On the minors of an incidence matrix and its Smith normal form, Linear Algebra Appl. [**218**]{} (1995), 213–224. N. Hartsfield and J. S. 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--- abstract: | [*Abstract:*]{} We consider a degenerate stochastic differential equation that has a sticky point in the Markov process sense. We prove that weak existence and weak uniqueness hold, but that pathwise uniqueness does not hold nor does a strong solution exist. .2cm *Subject Classification: Primary 60H10; Secondary 60J60, 60J65* author: - 'Richard F. Bass' title: A stochastic differential equation with a sticky point --- Introduction {#S:intro} ============ The one-dimensional stochastic differential equation \[intro-E0\] dX\_t=(X\_t) dW\_t has been the subject of intensive study for well over half a century. What can one say about pathwise uniqueness when $\sigma$ is allowed to be zero at certain points? Of course, a large amount is known, but there are many unanswered questions remaining. Consider the case where $\sigma(x)=|x|^\al$ for $\al\in (0,1)$. When $\al \ge 1/2$, it is known there is pathwise uniqueness by the Yamada-Watanabe criterion (see, e.g., [@stoch Theorem 24.4]) while if $\al<1/2$, it is known there are at least two solutions, the zero solution and one that can be constructed by a non-trivial time change of Brownian motion. However, that is not the end of the story. In [@xtoal], it was shown that there is in fact pathwise uniqueness when $\al<1/2$ provided one restricts attention to the class of solutions that spend zero time at 0. This can be better understood by using ideas from Markov process theory. The continuous strong Markov processes on the real line that are on natural scale can be characterized by their speed measure. For the example in the preceding paragraph, the speed measure $m$ is given by $$m(dy)=1_{(y\ne 0)} |y|^{-2\al}\, dy+\gamma\delta_0(dy),$$ where $\gamma\in [0,\infty]$ and $\delta_0$ is point mass at 0. When $\gamma=\infty$, we get the 0 solution, or more precisely, the solution that stays at 0 once it hits 0. If we set $\gamma=0$, we get the situation considered in [@xtoal] where the amount of time spent at 0 has Lebesgue measure zero, and pathwise uniqueness holds among such processes. In this paper we study an even simpler equation: \[intro-E1\] dX\_t=1\_[(X\_t0)]{} dW\_t,X\_0=0, where $W$ is a one-dimensional Brownian motion. One solution is $X_t=W_t$, since Brownian motion spends zero time at 0. Another is the identically 0 solution. We take $\gamma\in (0,\infty)$ and consider the class of solutions to which spend a positive amount of time at 0, with the amount of time parameterized by $\gamma$. We give a precise description of what we mean by this in Section \[S:SMM\]. Representing diffusions on the line as the solutions to stochastic differential equations has a long history, going back to Itô in the 1940’s, and this paper is a small step in that program. For this reason we characterize our solutions in terms of occupation times determined by a speed measure. Other formulations that are purely in terms of stochastic calculus are possible; see the system – below. We start by proving weak existence of solutions to for each $\gamma\in (0,\infty)$. We in fact consider a much more general situation. We let $m$ be any measure that gives finite positive mass to each open interval and define the notion of continuous local martingales with speed measure $m$. We prove weak uniqueness, or equivalently, uniqueness in law, among continuous local martingales with speed measure $m$. The fact that we have uniqueness in law not only within the class of strong Markov processes but also within the class of continuous local martingales with a given speed measure may be of independent interest. We then restrict our attention to and look at the class of continuous martingales that solve and at the same time have speed measure $m$, where now \[intro-E3\] m(dy)=1\_[(y0)]{} dy+\_0(dy) with $\gamma\in (0,\infty)$. Even when we fix $\gamma$ and restrict attention to solutions to that have speed measure $m$ given by , pathwise uniqueness does not hold. The proof of this fact is the main result of this paper. The reader familiar with excursions will recognize some ideas from that theory in the proof. Finally, we prove that for each $\gamma\in (0,\infty)$, no strong solution to among the class of continuous martingales with speed measure $m$ given by exists. Thus, given $W$, one cannot find a continuous martingale $X$ with speed measure $m$ satisfying such that $X$ is adapted to the filtration of $W$. A consequence of this is that certain natural approximations to the solution of do not converge in probability, although they do converge weakly. Besides increasing the versatility of , one can easily imagine a practical application of sticky points. Suppose a corporation has a takeover offer at \$10. The stock price is then likely to spend a great deal of time precisely at \$10 but is not constrained to stay at \$10. Thus \$10 would be a sticky point for the solution of the stochastic differential equation that describes the stock price. Regular continuous strong Markov processes on the line which are on natural scale and have speed measure given by are known as sticky Brownian motions. These were first studied by Feller in the 1950’s and Itô and McKean in the 1960’s. A posthumously published paper by Chitashvili ([@Chitashvili]) in 1997, based on a technical report produced in 1988, considered processes on the non-negative real line that satisfied the stochastic differential equation \[one-sided\] dX\_t=1\_[(X\_t0)]{} dW\_t+1\_[(X\_t=0)]{} dt, X\_t0, X\_0=x\_0, with $\theta\in (0,\infty)$. Chitashvii proved weak uniqueness for the pair $(X,W)$ and showed that no strong solution exists. Warren (see [@Warren1] and also [@Warren2]) further investigated solutions to . The process $X$ is not adapted to the filtration generated by $W$ and has some “extra randomness,” which Warren characterized. While this paper was under review, we learned of a preprint by Engelbert and Peskir [@Engelbert-Peskir] on the subject of sticky Brownian motions. They considered the system of equations $$\begin{aligned} dX_t&=1_{(X_t\ne 0)}\, dW_t, \label{EPeq1}\\ 1_{(X_t=0)}\, dt&=\frac{1}{\mu}\, d\ell^0_t(X),\label{EPeq2}\end{aligned}$$ where $\mu\in (0,\infty)$ and $\ell^0_t$ is the local time in the semimartingale sense at 0 of $X$. (Local times in the Markov process sense can be different in general.) Engelbert and Peskir proved weak uniqueness of the joint law of $(X,W)$ and proved that no strong solution exists. They also considered a one-sided version of this equation, where $X\ge 0$, and showed that it is equivalent to . Their results thus provide a new proof of those of Chitashvili. It is interesting to compare the system – investigated by [@Engelbert-Peskir] with the SDE considered in this paper. Both include the equation . In this paper, however, in place of we use a side condition whose origins come from Markov process theory, namely: $$\begin{aligned} X &\mbox{\rm is a continuous martingale with speed measure }\label{RBeq2}\\ &~~~~~ m(dx)= dx+\gamma \delta_0(dx),\nn\end{aligned}$$ where $\delta_0$ is point mass at 0 and “continuous martingale with speed measure $m$” is defined in . One can show that a solution to the system studied by [@Engelbert-Peskir] is a solution to the formulation considered in this paper and vice versa, and we sketch the argument in Remark \[comparison\]. However, we did not see a way of proving this without first proving the uniqueness results of this paper and using the uniqueness results of [@Engelbert-Peskir]. Other papers that show no strong solution exists for stochastic differential equations that are closely related include [@Barlow-skew], [@Barlow-LMS], and [@Karatzasetal]. After a short section of preliminaries, Section \[S:prelim\], we define speed measures for local martingales in Section \[S:SMM\] and consider the existence of such local martingales. Section \[S:WU\] proves weak uniqueness, while in Section \[S:SDE\] we prove that continuous martingales with speed measure $m$ given by satisfy . Sections \[S:approx\], \[S:est\], and \[S:PU\] prove that pathwise uniqueness and strong existence fail. The first of these sections considers some approximations to a solution to , the second proves some needed estimates, and the proof is completed in the third. We would like to thank Prof. H. Farnsworth for suggesting a mathematical finance interpretation of a sticky point. Preliminaries {#S:prelim} ============= For information on martingales and stochastic calculus, see [@stoch], [@KaratzasShreve] or [@RevuzYor]. For background on continuous Markov processes on the line, see the above references and also [@Ptpde], [@ItoMcKean], or [@Knight]. We start with an easy lemma concerning continuous local martingales. \[prelim-L1\] Suppose $X$ is a continuous local martingale which exits a finite non-empty interval $I$ a.s. If the endpoints of the interval are $a$ and $b$, $a<x<b$, and $X_0=x$ a.s., then $$\E \angel{X}_{\tau_I}=(x-a)(b-x),$$ where $\tau_I$ is the first exit time of $I$ and $\angel{X}_t$ is the quadratic variation process of $X$. Any such local martingale is a time change of a Brownian motion, at least up until the time of exiting the interval $I$. The result follows by performing a change of variables in the corresponding result for Brownian motion; see, e.g., [@stoch Proposition 3.16]. Let $I$ be a finite non-empty interval with endpoints $a<b$. Each of the endpoints may be in $I$ or in $I^c$. Define $g_I(x,y)$ by $$g_I(x,y)=\begin{cases} 2(x-a)(b-y)/(b-a),\phantom{\Big]}& a\le x<y\le b;\\ 2(y-a)(b-x)/(b-a),& a\le y\le x\le b. \end{cases}$$ Let $m$ be a measure such that $m$ gives finite strictly positive measure to every finite open interval. Let $$G_I(x)=\int_I g_I(x,y)\, m(dy).$$ If $X$ is a real-valued process adapted to a filtration $\{\sF_t\}$ satisfying the usual conditions, we let \[prelim-E301\] \_[I]{}={t&gt;0: X\_tI}. When we want to have exit times for more than one process at once, we write $\tau_{I}(X)$, $\tau_{I}(Y)$, etc. Define \[prelim-E302\] T\_x={t&gt;0: X\_t=x}. A continuous strong Markov process $(X,\P^x)$ on the real line is regular if $\P^x(T_y<\infty)>0$ for each $x$ and $y$. Thus, starting at $x$, there is positive probability of hitting $y$ for each $x$ and $y$. A regular continuous strong Markov process $X$ is on natural scale if whenever $I$ is a finite non-empty interval with endpoints $a<b$, then $$\P^x(X_{\tau_I}=a)=\frac{b-x}{b-a}, \qq \P^x(X_{\tau_I}=b)=\frac{x-a}{b-a}$$ provided $a<x<b$. A continuous regular strong Markov process on the line on natural scale has speed measure $m$ if for each finite non-empty interval $I$ we have $$\E^x \tau_I=G_I(x)$$ whenever $x$ is in the interior of $I$. It is well known that if $(X,\P^x)$ and $(Y,\bQ^x)$ are continuous regular strong Markov processes on the line on natural scale with the same speed measure $m$, then the law of $X$ under $\P^x$ is equal to the law of $Y$ under $\bQ^x$ for each $x$. In addition, $X$ will be a local martingale under $\P^x$ for each $x$. Let $W_t$ be a one-dimensional Brownian motion and let $\{L^x_t\}$ be the jointly continuous local times. If we define \[cE21\] \_t=L\_t\^y m(dy), then $\al_t$ will be continuous and strictly increasing. If we let $\beta_t$ be the inverse of $\al_t$ and set \[prelim-E1\] X\^M\_t=x\_0+W\_[\_t]{}, then $X^M$ will be a continuous regular strong Markov process on natural scale with speed measure $m$ starting at $x_0$. See the references listed above for a proof, e.g., [@stoch Theorem 41.9]. We denote the law of $X^M$ started at $x_0$ by $\P^{x_0}_M$. If $(\Omega, \sF, \P)$ is a probability space and $\sG$ a $\sigma$-field contained in $\sF$, a regular conditional probability $\bQ$ for $\P(\cdot\mid \sG)$ is a map from $\Omega\times \sF$ to $[0,1]$ such that\ (1) for each $A\in \sF$, $\bQ(\cdot, A)$ is measurable with respect to $\sF$;\ (2) for each $\omega\in \Omega$, $\bQ(\omega,\cdot)$ is a probability measure on $\sF$;\ (3) for each $A\in \sF$, $\P(A\mid \sG)(\omega)=\Q(\omega,A)$ for almost every $\omega$. Regular conditional probabilities do not always exist, but will if $\Omega$ has sufficient structure; see [@stoch Appendix C]. The filtration $\{\sF_t\}$ generated by a process $Z$ is the smallest filtration to which $Z$ is adapted and which satisfies the usual conditions. We use the letter $c$ with or without subscripts to denote finite positive constants whose value may change from place to place. Speed measures for local martingales {#S:SMM} ==================================== Let $a:\R\to \R$ and $b:\R\to \R$ be Borel measurable functions with $a(x)\le b(x)$ for all $x$. If $S$ is a finite stopping time, let $$\tau^S_{[a,b]}=\inf\{t>S: X_t\notin [a(X_S),b(X_S)]\}.$$ We say a continuous local martingale $X$ started at $x_0$ has speed measure $m$ if $X_0=x_0$ and \[SMM-E1\] =G\_[\[a(X\_S),b(X\_S)\]]{}(X\_S), whenever $S$ is a finite stopping time and $a$ and $b$ are as above. \[R-speed\][ We remark that if $X$ were a strong Markov process, then the left hand side of would be equal to $\E^{X_S} \tau^0_{[a,b]}$, where $\tau^0_{[a,b]}=\inf\{t\ge 0: X_t\notin [a,b]\}$. Thus the above definition of speed measure for a martingale is a generalization of the one for one-dimensional diffusions on natural scale. ]{} \[SMM-T1\] Let $m$ be a measure that is finite and positive on every finite open interval. There exists a continuous local martingale $X$ with $m$ as its speed measure. Set $X$ equal to $X^M$ as defined in . We only need show that holds. Since $X$ is a Markov process and has associated with it probabilities $\P^x$ and shift operators $\theta_t$, then $$\tau^S_{[a,b]}-S=\sigma_{[a(X_0),b(X_0)]}\circ \theta_S,$$ where $\sigma_{[a(X_0),b(X_0)]}=\inf\{t>0: X_t\notin [a(X_0),b(X_0)]\}$. By the strong Markov property, \[SMM-E2\] =\^[X\_S]{} \_[\[a(X\_0), b(X\_0)\]]{} For each $y$, $\sigma_{[a(X_0),b(X_0)]}=\tau_{[a(y),b(y)]} $ under $\P^y$, and therefore $$\E^y \sigma_{[a(X_0),b(X_0)]}=G_{[a(y),b(y)]}(y).$$ Replacing $y$ by $X_S(\omega)$ and substituting in yields . \[SMM-P1\] Let $X$ be any continuous local martingale that has speed measure $m$ and let $f$ be a non-negative Borel measurable function. Suppose $X_0=x_0$, a.s. Let $I=[a,b]$ be a finite interval with $a<b$ such that $m$ does not give positive mass to either end point. Then \[SMM-E3m\] \_0\^[\_I]{} f(X\_s) ds=\_I g\_I(x,y) f(y) m(dy). It suffices to suppose that $f$ is continuous and equal to 0 at the boundaries of $I$ and then to approximate an arbitrary non-negative Borel measurable function by continuous functions that are 0 on the boundaries of $I$. The main step is to prove \[SMM-E4\] \_0\^[\_I(X)]{} f(X\_s) ds= \_0\^[\_I(X\^M)]{} f(X\^M\_s) ds. Let $\eps>0$. Choose $\delta$ such that $|f(x)-f(y)|<\eps$ if $|x-y|<\delta$ with $x,y\in I$. Set $S_0=0$ and $$S_{i+1}=\inf\{t>S_i: |X_t-X_{S_i}|\ge \delta\}.$$ Then $$\E\int_0^{\tau_I} f(X_s)\, ds=\E\sum_{i=0}^\infty \int_{S_i\land \tau_I}^{S_{i+1}\land \tau_I} f(X_s)\, ds$$ differs by at most $\eps \E\tau_I$ from $$\begin{aligned} \E\sum_{i=0}^\infty f(X_{S_i\land \tau_I})& (S_{i+1}\land \tau_I-S_i\land \tau_I)\label{SMM-E2a}\\ &=\E\Big[\sum_{i=0}^\infty f(X_{S_i\land \tau_I})\E[S_{i+1}\land \tau_I -S_i\land \tau_I\mid \sF_{S_i\land \tau_I}]\,\Big].\nn\end{aligned}$$ Let $a(x)=a\lor (x-\delta)$ and $b(x)=b\land (x+\delta)$. Since $X$ is a continuous local martingale with speed measure $m$, the last line in is equal to \[SMM-E2b\] \_[i=0]{}\^f(X\_[S\_i\_I]{}) G\_[\[a(X\_[S\_i\_I]{}),b( X\_[S\_i\_I]{})\]]{}(X\_[S\_i\_I]{}). Because $\E \tau_{[-N,N]}<\infty$ for all $N$, then $X$ is a time change of a Brownian motion. It follows that the distribution of $\{X_{S_i\land \tau_I(X)}, i\ge 0\}$ is that of a simple random walk on the lattice $\{x+k\delta\}$ stopped the first time it exits $I$, and thus is the same as the distribution of $\{X^M_{S_i\land \tau_I(X^M)}, i\ge 0\}$. Therefore the expression is is equal to the corresponding expression with $X$ replaced by $X^M$. This in turns differs by at most $\E \eps \tau_I(X^M)$ from $$\E\int_0^{\tau_I(X^M)} f(X^M_s)\, ds.$$ Since $\eps$ is arbitrary, we have . Finally, the right hand side of is equal to the right hand side of by [@Ptpde Corollary IV.2.4]. Uniqueness in law {#S:WU} ================= In this section we show that if $X$ is a continuous local martingale under $\P$ with speed measure $m$, then $X$ has the same law as $X^M$. Note that we do not suppose *a priori* that $X$ is a strong Markov process. We remark that the results of [@ES3] do not apply, since in that paper a generalization of the system – is studied rather than the formulation given by together with . \[WU-T1\] Suppose $\P$ is a probability measure and $X$ is a continuous local martingale with respect to $\P$. Suppose that $X$ has speed measure $m$ and $X_0=x_0$ a.s. Then the law of $X$ under $\P$ is equal to the law of $X^M$ under $\P^{x_0}_M$. Let $R>0$ be such that $m(\{-R\})=m(\{R\})=0$ and set $I=[-R,R]$. Let $\ol X_t=X_{t\land \tau_I(X)}$ and $\ol X^M_t=X^M_{t\land \tau_I(X^M)}$, the processes $X$ and $X^M$ stopped on exiting $I$. For $f$ bounded and measurable let $$H_\lam f=\E \int_0^{\tau_I({\ol X})} e^{-\lam t} f({\ol X}_t)\, dt$$ and $$H_\lam^M f(x)=\E^{x} \int_0^{\tau_I({\ol X}^M)} e^{-\lam t} f({\ol X}^M_t)\, dt$$ for $\lam\ge 0$. Since ${\ol X}$ and ${\ol X}^M$ are stopped at times $\tau_I({\ol X})$ and $\tau_I({\ol X}^M)$, resp., we can replace $\tau_I({\ol X})$ and $\tau_I({\ol X}^M)$ by $\infty$ in both of the above integrals without affecting $H_\lam$ or $H_\lam^M$ as long as $f$ is 0 on the boundary of $I$. Suppose $f(-R)=f(R)=0$. Then $H_\lam^M f(-R)$ and $H_\lam^M f(R)$ are also 0, since we are working with the stopped process. We want to show \[WU-E2\] H\_f=H\_\^M f(x\_0), 0. By Theorem \[SMM-P1\] we know holds for $\lam=0$. Let $K=\E \tau_I(X)$. We have $\E^{x_0} \tau_I(X^M)=K$ as well since both $X$ and $X^M$ have speed measure $m$. Let $\lam=0$ and $\mu\le 1/2K$. Let $t>0$ and let $Y_s={\ol X}_{s+t}$. Let $\bQ_t$ be a regular conditional probability for $\P(Y\in \cdot\mid \sF_t)$. It is easy to see that for almost every $\omega$, $Y$ is a continuous local martingale under $\bQ_t(\omega, \cdot)$ started at ${\ol X}_t$ and $Y$ has speed measure $m$. Cf. [@Ptpde Section I.5] or [@xtoal]. Therefore by Theorem \[SMM-P1\] $$\E_{\bQ_t}\int_0^\infty f(Y_s)\, ds=H^M_0 f({\ol X}_t).$$ This can be rewritten as \[WU-E31\] =H\^M\_0f([X]{}\_t), as long as $f$ is 0 on the endpoints of $I$. Therefore, recalling that $\lam=0$, $$\begin{aligned} H_\mu H_\lam^M f&=\E\int_0^\infty e^{-\mu t} H_\lam^M f({\ol X}_t)\, dt\label{WU-E3}\\ &=\E\int_0^\infty e^{-\mu t} \E\Big[\int_0^\infty e^{-\lam s}f({\ol X}_{s+t})\, ds\mid \sF_t\Big]\, dt\nn\\ &=\E\int_0^\infty e^{-\mu t}e^{\lam t}\int_t^\infty e^{-\lam s}f({\ol X}_s)\, ds\, dt\nn\\ &=\E\int_0^\infty \int_0^s e^{-(\mu-\lam)t}\, dt\, e^{-\lam s}f({\ol X}_s)\, ds\nn\\ &=\E\int_0^\infty \frac{1-e^{-(\mu-\lam)s}}{\mu-\lam}e^{-\lam s}f({\ol X}_s)\, ds\nn\\ &=\frac{1}{\mu-\lam}\E\int_0^\infty e^{-\lam s}f({\ol X}_s)\, ds -\frac{1}{\mu-\lam}\E\int_0^\infty e^{-\mu s} f({\ol X}_s)\, ds.\nn\\ &=\frac{1}{\mu-\lam}H_\lam^M f(x_0) -\frac{1}{\mu-\lam}\E\int_0^\infty e^{-\mu s} f({\ol X}_s)\, ds.\nn\end{aligned}$$ We used in the second equality. Rearranging, \[SMM-E32\] H\_f=H\_\^M f(x\_0)+(-)H\_(H\_\^Mf). Since ${\ol X}$ and ${\ol X}^M$ are stopped upon exiting $I$, then $H^M_\lam f=0$ at the endpoints of $I$. We now take with $f$ replaced by $H_\lam^M f$, use this to evaluate the last term in , and obtain $$H_\mu f=H_\lam^Mf({ x}_0)+(\lam-\mu)H^M_\lam(H^M_\lam f)(x_0)+(\lam-\mu)^2 H_\mu(H^M_\lam(H^M_\lam f)).$$ We continue. Since $$|H_\mu g|\le \norm{g} \E\tau_I(X)=\norm{g} K$$ and $$\norm{H^M_\lam g} \le \norm{g} \E \tau_I(X^M)= \norm{g} K$$ for each bounded $g$, where $\norm{g}$ is the supremum norm of $g$, we can iterate and get convergence as long as $\mu\le 1/2K$ and obtain $$H_\mu f=H^M_\lam f(x_0)+\sum_{i=1}^\infty ((\lam-\mu) H^M_\lam)^i H_\lam^M f(x_0).$$ The above also holds when ${\ol X}$ is replaced by ${\ol X}^M$, so that $$H_\mu^M f(x_0)=H^M_\lam f(x_0)+\sum_{i=1}^\infty ((\lam-\mu) H^M_\lam)^iH_\lam^M f(x_0).$$ We conclude $H_\mu f=H^M_\mu f(x_0)$ as long as $\mu\le 1/2K$ and $f$ is 0 on the endpoints of $I$. This holds for every starting point. If $Y_s={\ol X}_{s+t}$ and $\bQ_t$ is a regular conditional probability for the law of $Y_s$ under $\P^x$ given $\sF_t$, then we asserted above that $Y$ is a continuous local martingale started at ${\ol X}_t$ with speed measure $m$ under $\bQ_t(\omega, \cdot)$ for almost every $\omega$. We replace $x_0$ by ${\ol X}_t(\omega)$ in the preceding paragraph and derive $$\E\Big[\int_0^\infty e^{-\mu s}f({\ol X}_{s+t})\, ds\mid \sF_t\Big]=H^M_\mu f({\ol X}_t), \qq \mbox{\rm a.s.}$$ if $\mu\le 1/2K$ and $f$ is 0 on the endpoints of $I$. We now take $\lam=1/2K$ and $\mu\in(1/2K, 2/2K]$. The same argument as above shows that $H_\mu f=H^M_\mu f(x_0)$ as long as $f$ is 0 on the endpoints of $I$. This is true for every starting point. We continue, letting $\lam=n/2K$ and using induction, and obtain $$H_\mu f=H_\mu^M f(x_0)$$ for every $\mu\ge 0$. Now suppose $f$ is continuous with compact support and $R$ is large enough so that $(-R,R)$ contains the support of $f$. We have that $$\E\int_0^{\tau_{[-R,R]}({\ol X})} e^{-\mu t} f({\ol X}_t)\, dt =\E^{x_0}\int_0^{\tau_{[-R,R]}({\ol X}^M)} e^{-\mu t} f({\ol X}^M_t)\, dt$$ for all $\mu> 0$. This can be rewritten as \[WU-E501\] \_0\^e\^[-t]{} f([ X]{}\_[t\_[\[-R,R\]]{}(X)]{}) dt =\^[x\_0]{}\_0\^e\^[-t]{} f([X]{}\^M\_[t\_[\[R,R\]]{}(X\^M)]{}) dt. If we hold $\mu$ fixed and let $R\to \infty$ in , we obtain $$\E\int_0^\infty e^{-\mu t} f(X_t)\, dt =\E^{x_0}\int_0^\infty e^{-\mu t} f(X^M_t)\, dt$$ for all $\mu>0$. By the uniqueness of the Laplace transform and the continuity of $f, X,$ and $X^M$, $$\E f(X_t)=\E^{x_0} f(X^M_t)$$ for all $t$. By a limit argument, this holds whenever $f$ is a bounded Borel measurable function. The starting point $x_0$ was arbitrary. Using regular conditional probabilities as above, $$\E[f(X_{t+s})\mid \sF_t]=\E^{x_0} [f(X_{t+s}^M)\mid \sF_t].$$ By the Markov property, the right hand side is equal to $$\E^{X^M_t} f(X_s)=P_s f(X^M_t),$$ where $P_s$ is the transition probability kernel for $X^M$. To prove that the finite dimensional distributions of $X$ and $X^M$ agree, we use induction. We have $$\begin{aligned} \E\prod_{j=1}^{n+1} f_j(X_{t_j}) &=\E_i \prod_{j=1}^{n} f_j(X_{t_j})\E_i[f_{n+1}(X_{t_{n+1}})\mid \sF_{t_n}]\\ &=\E_i \prod_{j=1}^{n} f_j(X_{t_j})P_{t_{n+1}-t_n}f_{n+1}(X_{t_n}).\end{aligned}$$ We use the induction hypothesis to see that this is equal to $$\E^{x_0} \prod_{j=1}^{n} f_j(X^M_{t_j})P_{t_{n+1}-t_n}f_{n+1}(X^M_{t_n}).$$ We then use the Markov property to see that this in turn is equal to $$\E^{x_0}\prod_{j=1}^{n+1} f_j(X^M_{t_j}).$$ Since $X$ and $X^M$ have continuous paths and the same finite dimensional distributions, they have the same law. The stochastic differential equation {#S:SDE} ==================================== We now discuss the particular stochastic differential equation we want our martingales to solve. We specialize to the following speed measure. Let $\gamma\in (0,\infty)$ and let \[SDE-E31\] m(dx)= dx+\_0(dx), where $\delta_0$ is point mass at 0. We consider the stochastic differential equation \[SMM-E200\] X\_t=x\_0+\_0\^t 1\_[(X\_s0)]{} dW\_s. A triple $(X,W,\P)$ is a weak solution to with $X$ starting at $x_0$ if $\P$ is a probability measure, there exists a filtration $\{\sF_t\}$ satisfying the usual conditions, $W$ is a Brownian motion under $\P$ with respect to $\{\sF_t\}$, and $X$ is a continuous martingale adapted to $\{\sF_t\}$ with $X_0=x_0$ and satisfying . We now show that any martingale with $X_0=x_0$ a.s. that has speed measure $m$ is the first element of a triple that is a weak solution to . Although $X$ has the same law as $X^M$ started at $x_0$, here we only have one probability measure and we cannot assert that $X$ is a strong Markov process. We point out that [@ES3 Theorem 5.18] does not apply here, since they study a generalization of the system –, and we do not know at this stage that this formulation is equivalent to the one used here. \[SMM-T3\] Let $\P$ be a probability measure on a space that supports a Brownian motion and let $X$ be a continuous martingale which has speed measure $m$ with $X_0=x_0$ a.s. Then there exists a Brownian motion $W$ such that $(X,W,\P)$ is a weak solution to with $X$ starting at $x_0$. Moreover \[sde-E323\] X\_t=x\_0+\_0\^t 1\_[(X\_s0)]{} dX\_s. Let $$W'_t=\int_0^t 1_{(X_s\ne 0)}\, dX_s.$$ Hence $$d\angel{W'}_t=1_{(X_t\ne 0)}\, d\angel{X}_t.$$ Let $0<\eta<\delta$. Let $S_0=\inf\{t: |X_t|\ge \delta\}$, $T_i=\inf\{t>S_i: |X_t|\le \eta\}$, and $S_{i+1}=\inf\{t>T_i: |X_t|\ge \delta\}$ for $i=0,1,\ldots$. The speed measure of $X$ is equal to $m$, which in turn is equal to Lebesgue measure on $\R\setminus\{0\}$, hence $X$ has the same law as $X^M$ by Theorem \[WU-T1\]. Since $X^M$ behaves like a Brownian motion when it is away from zero, we conclude $1_{[S_i,T_i]}\,d\angel{X}_t=1_{[S_i,T_i]}\, dt$. Thus for each $N$, $$\int_0^t 1_{\cup_{i=0}^N [S_i,T_i]}(s)\, d\angel{X}_s =\int_0^t 1_{\cup_{i=0}^N [S_i,T_i]}(s)\, ds.$$ Letting $N\to \infty$, then $\eta\to 0$, and finally $\delta\to \infty$, we obtain $$\int_0^t 1_{(X_s\ne 0)}\, d\angel{X}_s=\int_0^t 1_{(X_s\ne 0)}\, ds.$$ Let $V_t$ be an independent Brownian motion and let $$W''_t=\int_0^t 1_{(X_s=0)}\, dV_s.$$ Let $W_t=W'_t+W''_t$. Clearly $W'$ and $W''$ are orthogonal martingales, so $$d\angel{W}_t=d\angel{W'}_t+d\angel{W''}_t =1_{(X_t\ne 0)}\, dt+1_{(X_t=0)}\, dt=dt.$$ By Lévy’s theorem (see [@stoch Theorem 12.1]), $W$ is a Brownian motion. If $$M_t=\int_0^t 1_{(X_s=0)}\, dX_s,$$ by the occupation times formula ([@RevuzYor Corollary VI.1.6]), $$\angel{M}_t=\int_0^t 1_{(X_s=0)}\, d\angel{X}_s =\int 1_{\{0\}}(x) \ell^x_t(X)\, dx=0$$ for all $t$, where $\{\ell^x_t(X)\}$ are the local times of $X$ in the semimartingale sense. This implies that $M_t$ is identically zero, and hence $X_t=W'_t$. Using the definition of $W$, we deduce \[sde-E5.95\] 1\_[(X\_t0)]{} dW\_t=1\_[(X\_t0)]{} dX\_t=dW’\_t=dX\_t, as required. We now show weak uniqueness, that is, if $(X,W,\P)$ and $(\wt X,\wt W,\wt \P)$ are two weak solutions to with $X$ and $\wt X$ starting at $x_0$ and in addition $X$ and $\wt X$ have speed measure $m$, then the joint law of $(X,W)$ under $\P$ equals the joint law of $(\wt X,\wt W)$ under $\wt \P$. This holds even though $W$ will not in general be adapted to the filtration of $X$. We know that the law of $X$ under $\P$ equals the law of $\wt X$ under $\wt \P$ and also that the law of $W$ under $\P$ equals the law of $\wt W$ under $\wt \P$, but the issue here is the joint law. Cf. [@Cherny]. See also [@Engelbert-Peskir]. \[WU-T21\] Suppose $(X,W,\P)$ and $(\wt X,\wt W,\wt \P)$ are two weak solutions to with $X_0=\wt X_0=x_0$ and that $X$ and $\wt X$ are both continuous martingales with speed measure $m$. Then the joint law of $(X,W)$ under $\P$ equals the joint law of $(\wt X,\wt W)$ under $\wt \P$. Recall the construction of $X^M$ from Section \[S:prelim\]. With $U_t$ a Brownian motion with jointly continuous local times $\{L^x_t\}$ and $m$ given by , we define $\al_t$ by , let $\beta_t$ be the right continuous inverse of $\al_t$, and let $X^M_t=x_0+U_{\beta_t}$. Since $m$ is greater than or equal to Lebesgue measure but is finite on every finite interval, we see that $\al_t$ is strictly increasing, continuous, and $\lim_{t \to \infty} \al_t=\infty$. It follows that $\beta_t$ is continuous and tends to infinity almost surely as $t\to \infty$. Given any stochastic process $\{N_t, t\ge 0\}$, let $\sF^N_\infty$ be the $\sigma$-field generated by the collection of random variables $\{N_t, t\ge 0\}$ together with the null sets. We have $\beta_t=\angel{X^M}_t$ and $U_t=X^M_{\al_t}-x_0$. Since $\beta_t$ is measurable with respect to $\sF^{X^M}_\infty$ for each $t$, then $\al_t$ is also, and hence so is $U_t$. In fact, we can give a recipe to construct a Borel measurable map $F:C[0,\infty)\to C[0,\infty)$ such that $U=F(X^M)$. Note also that $X^M_t$ is measurable with respect to $\sF^U_\infty$ for each $t$ and there exists a Borel measurable map $G:C[0,\infty)\to C[0,\infty)$ such that $X^M=G(U)$. In addition observe that $\angel{X^M}_\infty=\infty$ a.s. Since $X$ and $X^M$ have the same law, then $\angel{X}_\infty=\infty$ a.s. If $Z_t$ is a Brownian motion with $X_t=x_0+Z(\zeta_t)$ for a continuous increasing process $\zeta$, then $\zeta_t=\angel{X}_t$ is measurable with respect to $\sF^X_\infty$, its inverse $\rho_t$ is also, and therefore $Z_t=X_{\rho_t}-x_0$ is as well. Moreover the recipe for constructing $Z$ from $X$ is exactly the same as the one for constructing $U$ from $X^M$, that is, $Z=F(X)$. Since $X$ and $X^M$ have the same law, then the joint law of $(X,Z)$ is equal to the joint law of $(X^M,U)$. We can therefore conclude that $X$ is measurable with respect to $\sF^Z_\infty$ and $X=G(Z)$. Let $$Y_t=\int_0^t 1_{(X_s=0)}\, dW_s.$$ Then $Y$ is a martingale with $$\angel{Y}_t=\int_0^t 1_{(X_s=0)}\, ds =t-\angel{X}_t.$$ Observe that $\angel{X,Y}_t=\int_0^t 1_{(X_s\ne 0)}1_{(X_s=0)}\, ds=0$. By a theorem of Knight (see [@Knight2] or [@RevuzYor]), there exists a two-dimensional process $V=(V_1,V_2)$ such that $V$ is a two-dimensional Brownian motion under $\P$ and $$(X_t,Y_t)=(x_0+V_1(\angel{X}_t),V_2(\angel{Y}_t), \qq \mbox{\rm a.s.}$$ (It turns out that $\angel{Y}_\infty=\infty$, but that is not needed in Knight’s theorem.) By the third paragraph of this proof, $X_t=x_0+V_1(\angel{X}_t)$ implies that $X_t$ is measurable with respect to $\sF^{V_1}_\infty$, and in fact $X=G(V_1)$. Since $\angel{Y}_t=t-\angel{X}_t$, then $(X_t,Y_t)$ is measurable with respect to $\sF^V_\infty$ for each $t$ and there exists a Borel measurable map $H: C([0,\infty), \R^2)\to C([0,\infty), \R^2)$, where $C([0,\infty), \R^2)$ is the space of continuous functions from $[0,\infty)$ to $\R^2$, and $(X,Y)=H(V)$. Thus $(X,Y)$ is the image under $H$ of a two-dimensional Brownian motion. If $(\wt X, \wt W, \wt \P)$ is another weak solution, then we can define $\wt Y$ analogously and find a two-dimensional Brownian motion $\wt V$ such that $(\wt X, \wt Y)=H(\wt V)$. The key point is that the same $H$ can be used. We conclude that the law of $(X,Y)$ is uniquely determined. Since $$(X,W)=(X,X+Y-x_0),$$ this proves that the joint law of $(X,W)$ is uniquely determined. \[comparison\] In Section \[S:prelim\] we constructed the continuous strong Markov process $(X^M,\P_M^x)$ and we now know that $X$ started at $x_0$ is equal in law to $X^M$ under $\P^{x_0}_M$. We pointed out in Remark \[R-speed\] that in the strong Markov case the notion of speed measure for a martingale reduces to that of speed measure for a one dimensional diffusion. In [@Engelbert-Peskir] it is shown that the solution to the system – is unique in law and thus the solution started at $x_0$ is equal in law to that of a diffusion on $\R$ started at $x_0$; let $\wt m$ be the speed measure for this strong Markov process. Thus to show the equivalence of the system – to the one given by and , it suffices to show that $\wt m=m$ if and only if holds, where $m$ is given by and $\gamma=1/\mu$. Clearly both $\wt m$ and $m$ are equal to Lebesgue measure on $\R\setminus \{0\}$, so it suffices to compare the atoms of $\wt m$ and $m$ at 0. Suppose holds and $\gamma=1/\mu$. Let $A_t=\int_0^t 1_{\{0\}}(X_s)\, ds$. Thus asserts that $A_t=\frac{1}{\mu}\ell_t^0$. Let $I=[a,b]=[-1,1]$, $x_0=0$, and $\tau_I$ the first time that $X$ leaves the interval $I$. Setting $t=\tau_I$ and taking expectations starting from 0, we have $$\E^0 A_{\tau_I}=\frac1{\mu} \E^0 \ell^0_{\tau_I}.$$ Since $\ell^0_t$ is the increasing part of the submartingale $|X_t-x_0|-|x_0|$ and $X_{\tau_I}$ is equal to either 1 or $-1$, the right hand side is equal to $$\frac{1}{\mu} \E^0|X_{\tau_I}|=\frac1{\mu}.$$ On the other hand, by [@Ptpde (IV.2.11)], $$\E^0 A_{\tau_I}=\int_{-1}^1 g_I(0,y) 1_{\{0\}}(y)\, \wt m(dy) =\wt m(\{0\}).$$ Thus $\wt m=m$ if $\gamma=1/\mu$. Now suppose we have a solution to the pair and and $\gamma=1/\mu$; we will show holds. Let $R>0$, $I=[-R,R]$, and $\tau_I$ the first exit time from $I$. Set $B_t=\frac{1}{\mu} \ell^0_t$. For any $x\in I$, we have by [@Ptpde (IV.2.11)] that \[Rc1\] \^x A\_[\_I]{}=\_[-1]{}\^1 g\_I(x,y)1\_[{0}]{}(y) m(dy) =g\_I(x,0). Taking expectations, \[Rc2\] \^x B\_[\_I]{}=\^x\[|X\_[\_I]{}-x|-|x|\]. Since $X$ is a time change of a Brownian motion that exits $I$ a.s., the distribution of $X_{\tau_I}$ started at $x$ is the same as that of a Brownian motion started at $x$ upon exiting $I$. A simple computation shows that the right hand side of agrees with the right hand side of . By the strong Markov property, $$\E^0[A_{\tau_I}-A_{\tau_I\land t}\mid \sF_t]= \E^{X_t} A_{\tau_I}=\E^{X_t}B_{\tau_I} =\E^0[B_{\tau_I}-B_{\tau_I\land t}\mid \sF_t]$$ almost surely on the set $(t\le \tau_I)$. Observe that if $U_t=\E^0[A_{\tau_I}-A_{\tau_I\land t}\mid \sF_t]$, then we can write $$U_t =\E^0[A_{\tau_I}-A_{\tau_I\land t}\mid \sF_t] =\E^0[A_{\tau_I}\mid \sF_t] -A_{\tau_I\land t}$$ and $$U_t=\E^0[B_{\tau_I}-B_{\tau_I\land t}\mid \sF_t] =\E^0[B_{\tau_I}\mid \sF_t] -B_{\tau_I\land t}$$ for $t\le \tau_I$. This expresses the supermartingale $U$ as a martingale minus an increasing process in two different ways. By the uniqueness of the Doob decomposition for supermartingales, we conclude $A_{\tau_I\land t}=B_{\tau_I\land t}$ for $t\le \tau_I$. Since $R$ is arbitrary, this establishes . (The argument that the potential of an increasing process determines the process is well known.) \[eitherpaper\][In the remainder of the paper we prove that there does not exist a strong solution to the pair and nor does pathwise uniqueness hold. In [@Engelbert-Peskir], the authors prove that there is no strong solution to the pair and and that pathwise uniqueness does not hold. Since we now know there is an equivalence between the pair and and the pair and , one could at this point use the argument of [@Engelbert-Peskir] in place of the argument of this paper. Alternatively, in the paper of [@Engelbert-Peskir] one could use our argument in place of theirs to establish the non-existence of a strong solution and that pathwise uniqueness does not hold. ]{} Approximating processes {#S:approx} ======================= Let $\wt W$ be a Brownian motion adapted to a filtration $\{\sF_t, t\ge 0\}$, let $\eps\le \gamma$, and let $X_t^\eps$ be the solution to \[approx-E671\] dX\^\_t=\_(X\_t\^) dW\_t, X\^\_0=x\_0, where $$\sigma_\eps(x)=\begin{cases} 1,& |x|>\eps;\\ \sqrt{\eps/\gamma},& |x|\le \eps.\end{cases}$$ For each $x_0$ the solution to the stochastic differential equation is pathwise unique by [@LeGall] or [@Nakao]. We also know that if $\P^x_\eps$ is the law of $X^\eps$ starting from $x$, then $(X^\eps, \P^x_\eps)$ is a continuous regular strong Markov process on natural scale. The speed measure of $X^\eps$ will be $$m_\eps(dy)=dy+\frac{\gamma}{\eps} 1_{[-\eps,\eps]}(y)\, dy.$$ Let $Y^\eps$ be the solution to \[approx-E672\] dY\^\_t=\_[2]{}(Y\^\_t) dW\_t, Y\^\_0=x\_0. Since $\sigma_\eps\le 1$, then $d\angel{X^\eps}_t\le dt$. By the Burkholder-Davis-Gundy inequalities (see, e.g., [@stoch Section 12.5]), \[c6.2A\] |X\^\_t-X\^\_s|\^[2p]{}c|t-s|\^p for each $p\ge 1$, where the constant $c$ depends on $p$. It follows (for example, by Theorems 8.1 and 32.1 of [@stoch]) that the law of $X^\eps$ is tight in $C[0,t_0]$ for each $t_0$. The same is of course true for $Y^\eps$ and $\wt W$, and so the triple $(X^\eps,Y^\eps,\wt W)$ is tight in $(C[0,t_0])^3$ for each $t_0>0$. Let $P_t^\eps$ be the transition probabilities for the Markov process $X^\eps$. Let $C_0$ be the set of continuous functions on $\R$ that vanish at infinity and let $$L=\{f\in C_0: |f(x)-f(y)|\le |x-y|, x,y\in \R\},$$ the set of Lipschitz functions with Lipschitz constant 1 that vanish at infinity. One of the main results of [@Lsg] (see Theorem 4.2) is that $P_t^\eps$ maps $L$ into $L$ for each $t$ and each $\eps<1$. \[approx-T101\] If $f\in C_0$, then $P_t^\eps f$ converges uniformly for each $t\ge 0$. If we denote the limit by $P_tf$, then $\{P_t\}$ is a family of transition probabilities for a continuous regular strong Markov process $(X, \P^x)$ on natural scale with speed measure given by . For each $x$, $\P^x_{\eps}$ converges weakly to $\P^x$ with respect to $C[0,N]$ for each $N$. *Step 1.* Let $\{g_j\}$ be a countable collection of $C^2$ functions in $L$ with compact support such that the set of finite linear combinations of elements of $\{g_j\}$ is dense in $C_0$ with respect to the supremum norm. Let $\eps_n$ be a sequence converging to 0. Suppose $g_j$ has support contained in $[-K,K]$ with $K>1$. Since $X_t^\eps$ is a Brownian motion outside $[-1,1]$, if $|x|>2K$, then $$|P_t^\eps g_j(x)|=|\E^x g_j(X_t^\eps)|\le \norm{g_j}\, \P^x(|X^\eps|\mbox{ hits $|x|/2$ before time } t),$$ which tends to 0 uniformly over $\eps<1$ as $|x|\to \infty$. Here $\norm{g_j}$ is the supremum norm of $g_j$. By the equicontinuity of the $P_t^{\eps} g_j$, using the diagonalization method there exists a subsequence, which we continue to denote by $\eps_n$, such that $P_t^{\eps_n} g_j$ converges uniformly on $\R$ for every rational $t\ge 0$ and every $j$. We denote the limit by $P_tg_j$. Since $g_j\in C^2$, $$\begin{aligned} P_t ^\eps g_j(x)-P_s^\eps g_j(x)&=\E^x g_j(X_t^\eps)-\E^x g_j(X_s^\eps)\\ &=\E^x \int_s^t \sigma_\eps(X_r^\eps)g'_j(X_r^\eps)\, d\wt W_r+\tfrac12 \E^x \int_s^t \sigma_\eps(X_r^\eps)^2g''_j(X_r^\eps)\, dr\\ &=\tfrac12 \E^x \int_s^t \sigma_\eps(X_r^\eps)^2g''_j(X_r^\eps)\, dr,\end{aligned}$$ where we used Ito’s formula. Since $\sigma_\eps$ is bounded by 1, we obtain $$|P_t^\eps g_j(x)-P_s^\eps g_j(x)|\le c_j |t-s|,$$ where the constant $c_j$ depends on $g_j$. With this fact, we can deduce that $P_t^{\eps_n} g_j$ converges uniformly in $C_0$ for every $t\ge 0$. We again call the limit $P_t g_j$. Since linear combinations of the $g_j$’s are dense in $C_0$, we conclude that $P_t^{\eps_n} g$ converges uniformly to a limit, which we call $P_tg$, whenever $g\in C_0$. We note that $P_t$ maps $C_0$ into $C_0$. *Step 2.* Each $X_t^\eps$ is a Markov process, so $P_s^\eps(P_t^\eps g)=P_{s+t}^\eps g$. By the uniform convergence and equicontinuity and the fact that $P_s^\eps$ is a contraction, we see that $P_s(P_tg)=P_{s+t}g$ whenever $g\in C_0$. Let $s_1<s_2<\cdots s_j$ and let $f_1, \ldots f_j$ be elements of $L$. Define inductively $g_j=f_j$, $g_{j-1}=f_{j-1}(P_{s_j-s_{j-1}}g_j)$, $g_{j-2}=f_{j-2}(P_{s_{j-1}-s_{j-2}}g_{j-1})$, and so on. Define $g_j^\eps$ analogously where we replace $P_t$ by $P_t^\eps$. By the Markov property applied repeatedly, $$\E^x[f_1(X^\eps_{s_1})\cdots f_j(X^\eps_{s_j})] =P^\eps_{s_1} g_1^\eps(x).$$ Suppose $x$ is fixed for the moment and let $f_1, \cdots, f_j\in L$. Suppose there is a subsequence $\eps_{n'}$ of $\eps_n$ such that $X^{\eps_{n'}}$ converges weakly, say to $X$, and let $\P'$ be the limit law with corresponding expectation $\E'$. Using the uniform convergence, the equicontinuity, and the fact that $P_t^\eps$ maps $L$ into $L$, we obtain \[Approx-E31\] ’\[f\_1(X\_[s\_1]{})f\_j(X\_[s\_j]{})\] =P\_[s\_1]{} g\_1(x). We can conclude several things from this. First, since the limit is the same no matter what subsequence $\{\eps_{n'}\}$ we use, then the full sequence $\P^x_{\eps_n}$ converges weakly. This holds for each starting point $x$. Secondly, if we denote the weak limit of the $\P^x_{\eps_n}$ by $\P^x$, then holds with $\E'$ replaced by $\E^x$. From this we deduce that $(X,\P^x)$ is a Markov process with transition semigroup given by $P_t$. Thirdly, since $\P^x$ is the weak limit of probabilities on $C[0,\infty)$, we conclude that $X$ under $\P^x$ has continuous paths for each $x$. *Step 3.* Since $P_t$ maps $C_0$ into $C_0$ and $P_tf(x)=\E^x f(X_t)\to f(x)$ by the continuity of paths if $f\in C_0$, we conclude by [@stoch Theorem 20.9] that $(X,\P^x)$ is in fact a strong Markov process. Suppose $f_1, \ldots, f_j$ are in $L$ and $s_1<s_2<\cdots < s_j<t<u$. Since $X_t^\eps$ is a martingale, $$\E^x_\eps\Big[X_u^\eps\prod_{i=1}^j f_i(X_{s_i}^\eps) \Big]= \E^x\Big[X_t^\eps\prod_{i=1}^j f_i(X_{s_i}^\eps) \Big].$$ Moreover, $X_t^\eps$ and $X_u^\eps$ are uniformly integrable due to . Passing to the limit along the sequence $\eps_n$, we have the equality with $X^\eps$ replaced by $X$ and $\E^x_\eps$ replaced by $\E^x$. Since the collection of random variables of the form $\prod_i f_i(X_{s_i})$ generate $\sigma(X_r; r\le t)$, it follows that $X$ is a martingale under $\P^x$ for each $x$. *Step 4.* Let $\delta, \eta>0$. Let $I=[q,r]$ and $I^*=[q-\delta,r+\delta]$. In this step we show that \[approx-E102\] \_I(X)= \_I g\_I(0,y) m(dy). First we obtain a uniform bound on $\tau_{I^*}(X^\eps)$. If $A^\eps_t=t\land \tau_{I^*}(X^\eps)$, then $$\E[A^\eps_\infty-A^\eps_t\mid \sF_t]=\E^{X_t^\eps} A^\eps_\infty \le \sup_x \E^x \tau_{I^*}(X^\eps).$$ The last term is equal to $$\sup_x \int_{I^*} g_{I^*}(x,y) \Big(1+\frac{\gamma}{\eps}1_{I^*}(y)\Big)\, dy.$$ A simple calculation shows that this is bounded by $$c(r-q+2\delta)^2+c\gamma (r-q+2\delta),$$ where $c$ does not depend on $r, q, \delta$, or $\eps$. By Theorem I.6.10 of [@pta], we then deduce that $$\E \tau_{I^*}(X^\eps)^2 =\E (A^\eps_\infty)^2<c<\infty,$$ where $c$ does not depend on $\eps$. By Chebyshev’s inequality, for each $t$, $$\P(\tau_{I^*}(X^\eps)\ge t)\le c/t^2.$$ Next we obtain an upper bound on $\E \tau_I(X)$ in terms of $g_{I^*}$. We have $$\begin{aligned} \P(\tau_I(X)>t)&=\P(\sup_{s\le t} |X_s|\le r,\inf_{s\le t}|X_s|\ge q)\\ &\le \limsup_{\eps_n\to 0} \P(\sup_{s\le t}|X_s^{\eps_n}|\le r,\inf_{s\le t}|X_s|^\eps\ge q)\\ &\le \limsup_{\eps_n\to 0} \P(\tau_{I^*}(X^{\eps_n})>t)\le c/t^2.\end{aligned}$$ Choose $u_0$ such that $$\int_{u_0}^\infty \P(\tau_I(X)>t)\, dt<\eta, \qq \int_{u_0}^\infty \P(\tau_{I^*}(X^{\eps_n})>t)\, dt<\eta$$ for each $\eps_n$. Let $f$ and $g$ be continuous functions taking values in $[0,1]$ such that $f$ is equal to $1$ on $(-\infty,r]$ and $0$ on $[r+\delta, \infty)$ and $g$ is equal to 1 on $[q,\infty)$ and 0 on $(-\infty,q-\delta]$. We have $$\begin{aligned} \P(\sup_{s\le t}|X_s|\le r,\inf_{s\le t}|X_s\ge q) &\le \E[ f(\sup_{s\le t}|X_s|)g(\inf_{s\le t}|X_s|)]\\ &=\lim_{\eps_n\to 0} \E[ f(\sup_{s\le t}|X_s^{\eps_n}|)g(\inf_{s\le t}|X_s^{\eps_n}|)].\end{aligned}$$ Then [$$\begin{aligned} \int_0^{u_0} \P(\tau_I(X)>t)\, dt&=\int_0^{u_0} \P(\sup_{s\le t}|X_s|\le r,\inf_{s\le t}|X_s|\ge q)\, dt\\ &\le \int_0^{u_0}\E[ f(\sup_{s\le t}|X_s|)g(\inf_{s\le t}|X_s|)]\, dt\\ &=\int_0^{u_0} \lim_{\eps_n\to 0} \E [f(\sup_{s\le t}|X_s^{\eps_n}|)g(\inf_{s\le t}|X^{\eps_n}_s|)]\, dt\\ &=\lim_{\eps_n\to 0} \int_0^{u_0} \E [f(\sup_{s\le t}|X_s^{\eps_n}|)g(\inf_{s\le t}|X^{\eps_n}_s|)]\, dt\\ &\le \limsup_{\eps_n\to 0}\int_0^{u_0}\P(\sup_{s\le t}|X_s^{\eps_n}|\le r+\delta,\inf_{s\le t}|X_s|\ge q-\delta)\, dt\\ &\le \limsup_{\eps_n\to 0} \int_0^{u_0}\P(\tau_{I^*}(X^{\eps_n})\ge t)\, dt\\ &\le \limsup_{\eps_n\to 0} \E \tau_{I^*}(X^{\eps_n}).\end{aligned}$$ ]{} Hence $$\E \tau_I(X)\le \int_0^{u_0}\P(\tau_I(X)>t)\, dt+\eta \le \limsup_{\eps_n\to 0} \E \tau_{I^*}(X^{\eps_n})+\eta.$$ We now use the fact that $\eta$ is arbitrary and let $\eta\to 0$. Then $$\begin{aligned} \E \tau_I(X)&\le \limsup_{\eps_n\to 0} \E \tau_{I^*}(X^{\eps_n})\\ &=\limsup_{\eps_n\to 0} \int_{I^*} g_{I^*}(0,y)\Big(1+\frac{\gamma}{\eps}1_{[-\eps,\eps]}(y)\Big)\, dy\\ &=\int_{I^*} g_{I^*}(0,y)m(dy).\end{aligned}$$ We next use the joint continuity of $g_{[-a,a]}(x,y)$ in the variables $a,x$ and $y$. Letting $\delta\to 0$, we obtain $$\E \tau_I(X)\le \int_I g_I(0,y)\, m(dy).$$ The lower bound for $\E \tau_I(X)$ is done similarly, and we obtain . *Step 5.* Next we show that $X$ is a regular strong Markov process. This means that if $x\ne y$, $\P^x(X_t=y\mbox{ for some }t)>0$. To show this, assume without loss of generality that $y<x$. Suppose $X$ starting from $x$ does not hit $y$ with positive probability. Let $z=x+4|x-y|$. Since $\E^x\tau_{[y,z]}<\infty$, then with probability one $X$ hits $z$ and does so before hitting $y$. Hence $T_z=\tau_{[y,z]}<\infty$ a.s. Choose $t$ large so that $\P^x(\tau_{[y,z]}>t)<1/16$. By the optional stopping theorem, $$\E^x X_{T_z\land t}\ge z\P^x(T_z\le t)+y\P^x(T_z> t) =z-(z-y)\P^x(T_z>t).$$ By our choice of $z$, this is greater than $x$, which contradicts that $X$ is a martingale. Hence $X$ must hit $y$ with positive probability. Therefore $X$ is a regular continuous strong Markov process on the real line. Since it is a martingale, it is on natural scale. Since its speed measure is the same as that of $X^M$ by , we conclude from [@Ptpde Theorem IV.2.5] that $X$ and $X^M$ have the same law. In particular, $X$ is a martingale with speed measure $m$. *Step 6.* Since we obtain the same limit law no matter what sequence $\eps_n$ we started with, the full sequence $P_t^\eps$ converges to $P_t$ and $\P^x_\eps$ converges weakly to $\P^x$ for each $x$. All of the above applies equally well to $Y$ and its transition probabilities and laws. Recall that the sequence $(X^\eps, Y^\eps, \wt W)$ is tight with respect to $(C[0,N])^3$ for each $N$. Take a subsequence $(X^{\eps_n}, Y^{\eps_n}, \wt W)$ that converges weakly, say to the triple $(X,Y,W)$, with respect to $(C[0,N])^3$ for each $N$. The last task of this section is to prove that $X$ and $Y$ satisfy . \[approx-T55\] $(X,W)$ and $(Y,W)$ each satisfy . We prove this for $X$ as the proof for $Y$ is exactly the same. Clearly $W$ is a Brownian motion. Fix $N$. We will first show \[Approx-E64\] \_0\^t 1\_[(X\_s0)]{} dX\_s=\_0\^t 1\_[(X\_s0)]{} dW\_s if $t\le N.$ Let $\delta>0$ and let $g$ be a continuous function taking values in $[0,1]$ such that $g(x)=0$ if $|x|<\delta$ and $g(x)=1$ if $|x|\ge 2\delta$. Since $g$ is bounded and continuous and $(X^{\eps_n},\wt W)$ converges weakly to $(X,W)$, then $(X^{\eps_n}, \wt W, g(X^{\eps_n}))$ converges weakly to $(X,W, g(X))$. Moreover, since $g$ is 0 on $(-\delta, \delta)$, then \[CE1\] \_0\^t g(X\^[\_n]{}\_s) dW\_s= \_0\^t g(X\^[\_n]{}\_s) dX\^[\_n]{}\_s for ${\eps_n}$ small enough. By Theorem 2.2 of [@Kurtz-Protter], we have $$\Big(\int_0^t g(X^{\eps_n}_s)\, d\wt W_s, \int_0^t g(X^{\eps_n}_s)\, dX^{\eps_n}_s\Big)$$ converges weakly to $$\Big(\int_0^t g(X_s)\, dW_s, \int_0^t g(X_s)\, dX_s\Big).$$ Then $$\begin{aligned} \E&\arctan\Big(\Big|\int_0^t g(X_s)\, dW_s- \int_0^t g(X_s)\, dX_s\Big|\Big)\\ &=\lim_{n\to \infty} \E\arctan\Big(\Big|\int_0^t g(X^{\eps_n}_s)\, d\wt W_s- \int_0^t g(X^{\eps_n}_s)\, dX^{\eps_n}_s\Big|\Big)=0,\end{aligned}$$ or $$\int_0^t g(X_s)\, dW_s= \int_0^t g(X_s)\, dX_s, \qq\mbox{\rm a.s.}$$ Letting $\delta\to 0$ proves . We know $$X^M_t=\int_0^t 1_{(X^M_s\ne 0)}\, dX^M_s.$$ Since $X^M$ and $X$ have the same law, the same is true if we replace $X^M$ by $X$. Combining with proves . Some estimates {#S:est} ============== Let $$j^\eps(s)=\begin{cases} 1,&|X^\eps_s|\in [-\eps,\eps] \mbox{ or } |Y^\eps_s|\in [-2\eps,2\eps] \mbox{ or both};\\ 0,& \mbox{otherwise}.\end{cases}$$ Let $$J^\eps_t=\int_0^t j^\eps_s\, ds.$$ Set $$Z_t^\eps=X_t^\eps-Y^\eps_t,$$ suppose $Z_0^\eps=0$, and define $\psi_\eps(x,y)=\sigma_\eps(x)-\sigma_{2\eps}(y)$. Then $$dZ^\eps_t=\psi_\eps(X^\eps_t, Y^\eps_t)\, d\wt W_t.$$ Let $$\begin{aligned} S_1&=\inf\{t: |Z_t^\eps|\ge 6\eps\},\label{est-E21}\\ T_i&=\inf\{t\ge S_i: |Z_t^\eps|\notin [4\eps,b]\},\nn\\ S_{i+1}&=\inf\{t\ge T_i: |Z_t^\eps|\ge 6\eps\}, \qq\mbox{\rm and}\nn\\ U_b&=\inf\{t: |Z_t^\eps|=b\}.\nn\end{aligned}$$ \[est-P1\] For each $n$, $$\P(S_n< U_b)\le \Big(1-\frac{2\eps}{b}\Big)^n.$$ Since $X^\eps$ is a recurrent diffusion, $\int_0^t 1_{[-\eps,\eps]}(X_s^\eps) \, ds$ tends to infinity a.s. as $t\to \infty$. When $x\in [-\eps,\eps]$, then $|\psi_{\eps}(x,y)|\ge c\eps$, and we conclude that $\angel{Z^\eps}_t\to \infty$ as $t\to \infty$. Let $\{\sF_t\}$ be the filtration generated by $\wt W$. $Z^\eps_{t+S_n}-Z^\eps_{S_n}$ is a martingale started at 0 with respect to the regular conditional probability for the law of $(X^\eps_{t+S_n}, Y^\eps_{t+S_n})$ given $\sF_{S_n}$. The conditional probability that it hits $4\eps$ before $b$ if $Z^\eps_{S_n}=6\eps$ is the same as the conditional probability it hits $-4\eps$ before $-b$ if $Z_{S_n}^\eps=-6\eps$ and is equal to $$\frac{b-6\eps}{b-4\eps}\le 1-\frac{2\eps}{b}.$$ Since this is independent of $\omega$, we have $$\P\Big(|Z^\eps_{t+S_n}-Z^\eps_{S_n}|\mbox{ hits }4\eps \mbox{ before hitting } b \mid \sF_{S_n}\Big)\le 1-\frac{2\eps}{b}.$$ Let $V_n=\inf\{t> S_n: |Z_t^\eps|=b\}$. Then $$\begin{aligned} \P(S_{n+1}<U_b)&\le \P(S_n<U_b, T_{n+1}<V_n)\\ &=\E[\P(T_{n+1}<V_n\mid \sF_{S_n}); S_n<U_b]\\ &\le \Big(1-\frac{2\eps}{b}\Big)\P(S_n<U_b).\end{aligned}$$ Our result follows by induction. \[est-P2\] There exists a constant $c_1$ such that $$\E J^\eps_{T_n}\le c_1n\eps$$ for each $n$. For $t$ between times $S_n$ and $T_n$ we know that $|Z_t^\eps|$ lies between $4\eps$ and $b$. Then at least one of $X^\eps_t\notin [-\eps,\eps]$ and $Y^\eps_t\notin [-2\eps,2\eps]$ holds. If exactly one holds, then $|\psi_\eps(X_t^\eps,Y_t^\eps)|\ge 1-\sqrt{2\eps/\gamma}\ge 1/2$ if $\eps$ is small enough. If both hold, we can only say that $d\angel{Z^\eps}_t\ge 0$. In any case, $$d\angel{Z^\eps}_t\ge \tfrac14\,dJ^\eps_t$$ for $S_n\le t\le T_n$. $Z_t^\eps$ is a martingale, and by Lemma \[prelim-L1\] and an argument using regular conditional probabilities similar to those we have done earlier, \[est-E301\] 44(b-6)(2)=c. Between times $T_n$ and $S_{n+1}$ it is possible that $\psi_\eps(X_t^\eps,Y_t^\eps)$ can be 0 or it can be larger than $c\sqrt{\eps/\gamma}$. However if either $X_t^\eps\in [-\eps,\eps]$ or $Y_t^\eps\in [-2\eps,2\eps]$, then $\psi_\eps(X_t^\eps,Y_t^\eps)\ge c\sqrt{\eps/\gamma}$. Thus $$d\angel{Z^\eps}_t\ge c\eps\,dJ^\eps_t$$ for $T_n\le t\le S_{n+1}$. By Lemma \[prelim-L1\] \[est-E302\] c\^[-1]{}c\^[-1]{}(2)(10)=c. Summing each of and over $j$ from 1 to $n$ and combining yields the proposition. \[est-P3\] Let $K>0$ and $\eta>0$. There exists $R$ depending on $K$ and $\eta$ such that $$\P(J^\eps_{\tau_{[-R,R]}(X^\eps)}<K)\le \eta, \qq \eps\le 1/2.$$ Fix $\eps\le 1/2$. We will see that our estimates are independent of $\eps$. Note $$J^\eps_t\ge H_t=\int_0^t 1_{[-\eps,\eps]}(X_s^\eps)\,ds.$$ Therefore to prove the proposition it is enough to prove that $$\P^0_\eps(H_{\tau_{[-R,R]}(X^\eps)}<K)\le \eta$$ if $R$ is large enough. Let $I=[-1,1]$. We have $$\E^0_\eps H_{\tau_I(X^\eps)}\ge \int_{-1}^1 g_{I} (0,y)\frac{\gamma}{\eps} 1_{[-\eps,\eps]}(y) \, dy\ge c_1.$$ On the other hand, for any $x\in I$, $$\E^x_0 H_{\tau_{I}(X^\eps)}=\int_I g_I(x,y) \frac{\gamma}{\eps}1_{[-\eps,\eps]}(y) \, dy\le c_2.$$ Combining this with $$\E^0_\eps[H_{\tau_I(X^\eps)}-H_t\mid \sF_t]\le \E^{X_t^\eps}_\eps H_{\tau_{I}(X^\eps)}$$ and Theorem I.6.10 of [@pta] (with $B=c_2$ there), we see that $$\E H_{\tau_I(X^\eps)}^2\le c_3.$$ Let $\al_0=0$, $\beta_i=\inf\{t>\al_i: |X_t^\eps|=1\}$ and $\al_{i+1}=\inf\{t>\beta_i: X_t^\eps=0\}.$ Since $X_t^\eps$ is a recurrent diffusion, each $\al_i$ is finite a.s. and $\beta_i\to \infty$ as $i\to \infty$. Let $V_i=H_{\beta_i}-H_{\al_i}$. By the strong Markov property, under $\P^0_\eps$ the $V_i$ are i.i.d. random variables with mean larger than $c_1$ and variance bounded by $c_4$, where $c_1$ and $c_4$ do not depend on $\eps$ as long as $\eps<1/2$. Then $$\begin{aligned} \P^0_\eps\Big(\sum_{i=1}^k V_i\le c_1k/2\Big)&\le \P^0_\eps\Big(\sum_{i=1}^k (V_i-\E V_i)\ge c_1k/2\Big)\\ &\le \frac{\Var(\sum_{i=1}^k V_i)}{(c_1k/2)^2}\\ &\le 4c_4/c_1^2k.\end{aligned}$$ Taking $k$ large enough, we see that $$\P^0_\eps\Big(\sum_{i=1}^k V_i\le K\Big)\le \eta/2.$$ Using the fact that $X_t^\eps$ is a martingale, starting at 1, the probability of hitting $R$ before hitting 0 is $1/R$. Using the strong Markov property, the probability of $|X|$ having no more than $k$ downcrossings of $[0,1]$ before exiting $[-R,R]$ is bounded by $$1-\Big(1-\frac{1}{R}\Big)^k.$$ If we choose $R$ large enough, this last quantity will be less than $\eta/2$. Thus, except for an event of probability at most $\eta$, $X_t^\eps$ will exit $[-1,1]$ and return to 0 at least $k$ times before exiting $[-R,R]$ and the total amount of time spent in $[-\eps,\eps]$ before exiting $[-R,R]$ will be at least $K$. \[est-P4\] Let $\eta>0, R>0,$ and $I=[-R,R]$. There exists $t_0$ depending on $R$ and $\eta$ such that $$\P^0_\eps(\tau_I(X^\eps)>t_0)\le \eta, \qq \eps\le 1/2.$$ If $\eps\le 1$, $$\E^0_\eps \tau_R(X^\eps)=\int_I g_I(x,y)\, m_\eps(dy).$$ A calculation shows this is bounded by $cR^2+cR$, where $c$ does not depend on $\eps$ or $R$. Applying Chebyshev’s inequality, $$\P^0_\eps(\tau_I(X^\eps)>t_0)\le \frac{\E^0_\eps\tau_I(X^\eps)}{t_0},$$ which is bounded by $\eta$ if $t_0\ge c(R^2+R)/\eta$. Pathwise uniqueness fails {#S:PU} ========================= We continue the notation of Section \[S:est\]. The strategy of proving that pathwise uniqueness does not hold owes a great deal to [@Barlow-LMS]. \[PU-T1\] There exist three processes $X,Y$, and $W$ and a probability measure $\P$ such that $W$ is a Brownian motion under $\P$, $X$ and $Y$ are continuous martingales under $\P$ with speed measure $m$ starting at 0, holds for $X$, holds when $X$ is replaced by $Y$, and $\P(X_t\ne Y_t\mbox{ for some }t>0)>0$. Let $(X^\eps, Y^\eps, \wt W)$ be defined as in and and choose a sequence $\eps_n$ decreasing to 0 such that the triple converges weakly on $C[0,N]\times C[0,N]\times C[0,N]$ for each $N$. By Theorems \[approx-T101\] and \[approx-T55\], the weak limit, $(X,Y,W)$ is such that $X$ and $Y$ are continuous martingales with speed measure $m$, $W$ is a Brownian motion, and holds for $X$ and also when $X$ is replaced by $Y$. Let $b=1$ and let $S_n$, $T_n$, and $U_b$ be defined by . Let $A_1(\eps,n)$ be the event where $T_n< U_b$. By Proposition \[est-P1\] $$\P(A_1(\eps,n))= \P(S_n< U_b)\le \Big(1-\frac{2\eps}{b}\Big)^n.$$ Choose $n\ge \beta/\eps$, where $\beta$ is large enough so that the right hand side is less than $1/5$ for all $\eps$ sufficiently small. By Proposition \[est-P2\], $$\E J_{T_n}^\eps\le c_1n \eps=c_1\beta.$$ By Chebyshev’s inequality, $$\P(J^\eps_{T_n}\ge 5c_1\beta)\le \P(J_{T_n}^\eps\ge 5\E J_{T_n}^\eps) \le 1/5.$$ Let $A_2(\eps, n)$ be the event where $J^\eps_{T_n}\ge 5c_1\beta$. Take $K=10c_1\beta$. By Proposition \[est-P3\], there exists $R$ such that $$\P(J^\eps_{\tau_{[-R,R]}(X^\eps)}<K)\le 1/5.$$ Let $A_3(\eps,R,K)$ be the event where $J^\eps_{\tau_{[R,R]}(X^\eps)}< K$. Choose $t_0$ using Proposition \[est-P4\], so that except for an event of probability $1/5$ we have $\tau_{[-R,R]}(X^\eps)\le t_0$. Let $A_4(\eps,R, t_0)$ be the event where $\tau_{[-R,R]}(X^\eps)\le t_0$. Let $$B(\eps)=(A_1(\eps,n)\cup A_2(\eps,n)\cup A_3(\eps,R,K)\cup A_4(\eps, R, t_0))^c.$$ Note $\P(B(\eps))\ge 1/5$. Suppose we are on the event $B(\eps)$. We have $$J^\eps_{T_n}\le 5c_1\beta< K\le J^\eps_{\tau_{[-R,R]}(X^\eps)}.$$ We conclude that $T_n< \tau_{[-R,R]}(X^\eps)$. Therefore, on the event $B(\eps)$, we see that $T_n$ has occurred before time $t_0$. We also know that $U_b$ has occurred before time $t_0$. Hence, on $B(\eps)$, $$\P(\sup_{s\le t_0} |Z_s^\eps|\ge b)\ge 1/5.$$ Since $Z^\eps=X^\eps-Y^\eps$ converges weakly to $X-Y$, then with probability at least $1/5$, we have that $\sup_{s\le t_0} |Z_s|\ge b/2$. This implies that $X_t\ne Y_t$ for some $t$, or pathwise uniqueness does not hold. We also can conclude that strong existence does not hold. The argument we use is similar to ones given in [@Cherny], [@Engelbert], and [@Kurtz]. \[PU-T2\] Let $W$ be a Brownian motion. There does not exist a continuous martingale $X$ starting at 0 with speed measure $m$ such that holds and such that $X$ is measurable with respect to the filtration of $W$. Let $W$ be a Brownian motion and suppose there did exist such a process $X$. Then there is a measurable map $F:C[0,\infty)\to C[0,\infty)$ such that $X=F(W)$. Suppose $Y$ is any other continuous martingale with speed measure $m$ satisfying . Then by Theorem \[WU-T1\], the law of $Y$ equals the law of $X$, and by Theorem \[WU-T21\], the joint law of $(Y,W)$ is equal to the joint law of $(X,W)$. Therefore $Y$ also satisfies $Y=F(W)$, and we get pathwise uniqueness since $X=F(W)=Y$. However, we know pathwise uniqueness does not hold. We conclude that no such $X$ can exist, that is, strong existence does not hold. [99]{} M.T. Barlow, Skew Brownian motion and a one-dimensional stochastic differential equation. *Stochastics **25*** (1988) 1–2. M.T. Barlow, One-dimensional stochastic differential equation with no strong solution. *J. London Math. Soc. **26*** (1982) 335–345. R.F. Bass, Markov processes with Lipschitz semigroups, *Trans. Amer. Math. Soc.* [**267**]{} (1981), 307–320. R.F. Bass, *Probabilistic Techniques in Analysis*, New York, Springer, 1995. R.F. Bass, *Diffusions and Elliptic Operators*, New York, Springer, 1997. R.F. 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McKean, Jr., *Diffusion Processes and their Sample Paths*. Springer-Verlag, Berlin, 1974. I. Karatzas and S.E. Shreve, *Brownian Motion and Stochastic Calculus, 2nd ed.* Springer-Verlag, New York, 1991. I. Karatzas, A.N. Shiryaev, and M. Shkolnikov, On the one-sided Tanaka equation with drift. *Electron. Commun. Probab. **16*** (2011), 664–677. F.B. Knight, *Essentials of Brownian Motion and Diffusion*. American Mathematical Society, Providence, R.I., 1981. F.B. Knight, A reduction of continuous square-integrable martingales to Brownian motion. *Martingales*, 19–31, Springer, Berlin, 1970 T.G. Kurtz, The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities. *Electron. J. Prob. **12*** (2007) 951–965. T.G. Kurtz and P. Protter, Weak limit theorems for stochastic integrals and stochastic differential equations. *Ann. Probab. **19*** (1991) 1035–1070. J.-F. 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[**General aspects of heterotic string compactifications**]{}\ [**on stacks and gerbes**]{} Lara B. Anderson$^1$, Bei Jia$^2$, Ryan Manion$^3$, Burt Ovrut$^4$, Eric Sharpe$^2$ [cc]{} ------------------------------------ $^1$ Center for the $\: \:$ Fundamental Laws of Nature Jefferson Laboratory Harvard University 17 Oxford Street Cambridge, MA 02138 ------------------------------------ & ---------------------------- $^2$ Department of Physics Robeson Hall, 0435 Virginia Tech Blacksburg, VA 24061 ---------------------------- \ -------------------------------- $^3$ Department of Mathematics David Rittenhouse Laboratory 209 South 33rd Street University of Pennsylvania Philadelphia, PA 19104-6395 -------------------------------- & ------------------------------ $^4$ Department of Physics David Rittenhouse Laboratory 209 South 33rd Street University of Pennsylvania Philadelphia, PA 19104-6395 ------------------------------ [lara@physics.harvard.edu]{}, [beijia@vt.edu]{}, [rymanion@gmail.com]{}, [ovrut@elcapitan.hep.upenn.edu]{}, [ersharpe@vt.edu]{} $\,$ In this paper we work out some basic results concerning heterotic string compactifications on stacks and, in particular, gerbes. A heterotic string compactification on a gerbe can be understood as, simultaneously, both a compactification on a space with a restriction on nonperturbative sectors, and also, a gauge theory in which a subgroup of the gauge group acts trivially on the massless matter. Gerbes admit more bundles than corresponding spaces, which suggests they are potentially a rich playground for heterotic string compactifications. After we give a general characterization of heterotic strings on stacks, we specialize to gerbes, and consider three different classes of ‘building blocks’ of gerbe compactifications. We argue that heterotic string compactifications on one class is equivalent to compactification of the same heterotic string on a disjoint union of spaces, compactification on another class is dual to compactifications of other heterotic strings on spaces, and compactification on the third class is not perturbatively consistent, so that we do not in fact recover a broad array of new heterotic compactifications, just combinations of existing ones. In appendices we explain how to compute massless spectra of heterotic string compactifications on stacks, derive some new necessary conditions for a heterotic string on a stack or orbifold to be well-defined, and also review some basic properties of bundles on gerbes. July 2013 Introduction ============ The compactification of heterotic superstrings on smooth Calabi-Yau threefolds has led to realistic $N=1$ supersymmetric particle physics in four-dimensions. For the $E_{8} \times E_{8}$ heterotic string, the generic structure of such vacua was presented in [@Donagi:2004qk; @Donagi:2004ia; @Donagi:2004su; @Donagi:2004ub]. Building upon these results, many phenomenologically relevant low-energy theories with MSSM-like matter spectra have been constructed, see for example [@Bouchard:2005ag; @Braun:2005ux; @Braun:2005bw; @Braun:2005zv; @Anderson:2009mh; @Anderson:2011ns; @Anderson:2012yf; @Braun:2011ni] for constructions and related work. However, the limitation of these vacua to equivariant vector bundles over smooth Calabi-Yau manifolds seems overly restrictive, and it is of considerable interest to try to construct heterotic vacua over more general backgrounds. The purpose of this paper is to outline basic results and general issues in making sense of heterotic string compactifications on stacks, generalized spaces admitting metrics, spinors, and all the other items needed to make sense of a string compactification. This essentially completes a program started many years ago to understand the basics of string compactifications on stacks, see [*e.g.*]{} [@kps; @nr; @msx; @glsm; @summ; @cdhps; @karp1; @karp2; @ps5; @me-tex; @me-qts]. The original hope of this program was to find new SCFT’s, new string compactifications, arising from these generalized spaces. Although that has not proven to be the case, much has been learned about the structure of string compactifications, as we shall review. One of the most physically interesting kinds of stacks are known as gerbes. The worldsheet theory of a string compactification on a gerbe can be understood in two more or less[^1] equivalent ways: - as a sigma model on a space, but with a (combinatorial[^2]) restriction on allowed nonperturbative sectors, or - as a gauge theory in which a (finite) subgroup of the gauge group acts trivially on the massless matter. Viewed from the first perspective, it is clear that there is a potential problem with cluster decomposition in these theories. For (2,2) SCFT’s, this issue was addressed in [@summ], where it was argued that the SCFT is equivalent to that on a disjoint union of spaces with variable $B$ fields, a result listed there as the ‘decomposition conjecture.’ A sigma model on a disjoint union also violates cluster decomposition, but in an extremely mild fashion, easily understood. This duality has since proven crucial for understanding physics issues in many GLSM’s, see [*e.g.*]{} [@cdhps; @hori2; @ed-nick-me; @ncgw; @hkm; @enstx], and also has been used to make predictions for Gromov-Witten invariants of gerbes, predictions which have been checked in [*e.g.*]{} [@ajt1; @ajt2; @ajt3; @t1; @gt1; @xt1]. Viewed from the second perspective, there are analogous issues concerning whether and how physics can see a trivially-acting finite group. This was addressed in [@nr; @msx; @glsm], and will be reviewed later in this paper. Massless spectra of (2,2) SCFT’s are computed[^3] to contain multiple dimension zero operators, another sign of cluster decomposition issues. These multiple dimension zero operators are (discrete Fourier transforms of) identity operators counting the number of components in the corresponding disjoint union of spaces [@summ]. These ideas have also been recently been applied to four-dimensional supergravity theories[^4] [@nati0; @git-sugrav; @banks-seib; @sugrav-g]. For example, gerbes admit line bundles with fractional Chern classes, so the Bagger-Witten [@bw1] quantization condition on cohomology classes of Kähler forms is modified when the supergravity moduli space admits a gerbe structure. More generally, a general introduction to four-dimensional supergravities whose moduli spaces are stacks (generic in Calabi-Yau compactification) is in [@sugrav-g]. Furthermore, it was shown in [@js1]\[appendix B\] that four-dimensional supergravity anomalies have a natural description in terms of stacks. See for example [@bgcmru; @bgcmu] for other applications. This paper is concerned with heterotic string compactifications on stacks and, in particular, gerbes. As the introduction above alludes, there are many more bundles on gerbes than on corresponding spaces, which naively suggests that there could be a rich new landscape of (0,2) SCFT’s and heterotic string compactifications obtainable from heterotic compactifications on gerbes. Our results break into three fundamental building blocks or classes: - For heterotic compactifications on gerbes in which the gauge bundle is a pullback from the base (equivalently, when the group that acts trivially on the base, also acts trivially on the bundle), the heterotic string compactification is consistent, and is equivalent to a compactification on a disjoint union of spaces. Compactifications of this form are discussed in section \[sect:het-decomp\]. - For heterotic compactifications on ${\mathbb Z}_2$ gerbes in which the ${\mathbb Z}_2$ acts nontrivially on a rank 8 bundle, these compactifications do not decompose, and (we conjecture) are T-dual to ordinary heterotic compactifications (on spaces) with a different left-moving GSO. In other words, a Spin$(32)/{\mathbb Z}_2$ compactification on such a gerbe is equivalent to an $E_8 \times E_8$ compactification on a space. Compactifications of this form are described in section \[sect:het-gsomods\]. - We conjecture when the bundle is nontrivial over the gerbe, but not rank 8 or the gerbe is not ${\mathbb Z}_2$, a perturbative heterotic string compactification is not consistent. That said, we do provide some seemingly consistent (0,2) SCFT’s defined by gerbes and bundles of this form, but unfortunately they do not seem to be useful for heterotic string compactification. Compactifications of this form are discussed in section \[sect:type3:twisted\]. In addition, it is also possible to build examples displaying combinations of these classes, which are discussed in section \[sect:combos\]. In appendix \[app:spectra\] we describe how to compute massless spectra in heterotic string compactifications on general stacks. Along the way, we derive some new necessary conditions for well-definedness of a SCFT associated to a heterotic string on a stack, generalizing old statements that “$c_1 \equiv 0$ mod 2” for a consistent heterotic compactification. Appendix \[app:linebundles\] describes in some depth line bundles on gerbes over projective spaces, as a good prototype for other bundles on more general gerbes. Appendix \[app:chern-reps\] discusses how Chern classes and characters are defined for stacks, and in particular, discusses $c^{\rm rep}$ and ${\rm ch}^{\rm rep}$, versions of Chern classes and characters which encode information about twisted sectors and which play a vital role in index theory. Appendix \[app:canonical-roots\] contains a short discussion of roots of canonical bundles on gerbes, a technical matter that sometimes arises in computations. One of the original motivations of this work was the hope that the third class above would yield new consistent heterotic string compactifications and new consistent (0,2) SCFT’s. Although it seems there are new consistent (0,2) SCFT’s, we will argue that they do not seem to define new consistent supersymmetric heterotic string compactifications. In hindsight, we can understand that result as follows[^5]. In an ineffective orbifold (one in which part of the orbifold group acts trivially on the space), the twisted sectors contain massless states whose wavefunctions have support over the entire space. This would seem to imply that there are ‘extra’ ten-dimensional massless states, but this would be a contradiction, since the ten-dimensional supergravity theory is known and fixed. Furthermore, so long as we work at low energies and close to large-radius limits, a ten-dimensional supergravity analysis should be applicable. In type II strings, this conundrum was implicity solved by the decomposition conjecture [@summ]: strings on gerbes are the same as strings on disjoint unions of spaces. The ‘extra’ states are there, but simply fill out copies of the supergravity theory. In heterotic strings, we will see a mix of several solutions: in some cases, an analogue of the decomposition conjecture exists; in other cases, the theory is dual to a compactification on a manifold; in yet other cases, the compactification does not seem to be consistent. Generalities {#sect:generalreview} ============ Strings on stacks ----------------- Stacks are a form of ‘generalized spaces,’ admitting smooth structures, metrics, bundles, and other structures needed to define sigma models. In particular, stacks are defined by the incoming maps from other spaces, making them a natural setting for defining sigma models. Stacks have been discussed as target ‘spaces’ for nonlinear sigma models in a number of references, including[^6] [@kps; @nr; @msx; @glsm; @summ; @cdhps; @karp1; @karp2; @ps5; @me-tex; @me-qts] for two-dimensional (2,2) supersymmetric and [@git-sugrav; @sugrav-g] for four-dimensional ${\cal N}=1$ supersymmetric sigma models. References for physicists on the mathematics of stacks are, unfortunately, somewhat harder to locate. In the mathematics literature, standard references on algebraic stacks include [@vistoli; @gomez; @lmb] and good references on topological stacks include [@bx; @heinloth1; @metzler1; @noohi1; @noohi2; @noohi3; @heinloth2; @bss1]. In addition, we have striven to write our own papers to be reasonably self-contained (see for example [@msx] for more information, oriented towards physicsts). We can make more concrete sense of strings on stacks as follows. Every[^7] smooth (Deligne-Mumford) stack $\mathfrak{X}$ has a presentation of the form of a global quotient $[X/G]$, where $X$ is a smooth manifold and $G$ is a group which need neither be finite nor act effectively. To such a presentation, we associate a $G$-gauged nonlinear sigma model on $X$. Now, such presentations are not unique: a given stack can have many presentations of this[^8] form. In two dimensional (2,2) theories, it is believed, and has been extensively checked, that renormalization group flow ‘washes out’ such presentation dependence, so universality classes depend only upon the stack, not any particular presentation. Thus, one can meaningfully associate a two-dimensional CFT to a stack, not merely a presentation thereof. In four dimensions, by contrast, this is not believed to be the case. For example, although gauge couplings are dynamically generated in two dimensions, they are not dynamically generated in four dimensions, and the stack does not determine a gauge coupling. Thus, in four dimensions we can not uniquely associate physics to stacks, though we can certainly do the converse, and use stacks to understand some parts of the physics of four-dimensional gauge theories, as in [@sugrav-g]. This paper is concerned with issues around perturbative heterotic strings on stacks, [*i.e.*]{} (0,2) SCFT’s. In principle, a perturbative heterotic string will be defined by a Calabi-Yau stack $\mathfrak{X}$ together with a gauge bundle ${\cal E}$ over the stack, satisfying certain anomaly cancellation conditions. We understand (0,2) SCFT’s in the same fashion as above: we pick a presentation of the stack of the form $[X/G]$. Given such a presentation, the gauge bundle is then a $G$-equivariant bundle ${\cal E}$ over $X$. To this data, we associate a $G$-gauged heterotic sigma model on $X$ with gauge bundle ${\cal E}$. As before, there can be multiple presentations of a stack with different UV physics, so we conjecture that renormalization group flow washes out such presentation-dependence, and only associate universality classes of renormalization group flow to stacks. Not every $(X,{\cal E},G)$ will define a consistent heterotic string theory; for example, the data above must satisfy anomaly cancellation. One part of anomaly cancellation is clear: before gauging, the heterotic sigma model on $X$ with bundle ${\cal E}$ must be anomaly-free, meaning that ${\rm ch}_2(TX) = {\rm ch}_2({\cal E})$. Demanding that the gauge theory be anomaly-free can impose further constraints. One well-known example is level-matching. As discussed in [*e.g.*]{} [@freedvafa], for orbifolds, level-matching is believed to be equivalent to matching of second Chern classes in equivariant cohomology. (In particular, equivariant Chern classes can be defined intrinsically on the stack, they are independent of the choice of presentation and descend to well-defined objects on the stack.) Equivariant cohomology can be defined on stacks, and in fact forms the ‘naive’ cohomology theory of a stack. (See appendix \[app:chern-reps\] for more subtle notions.) However, level-matching (in the form described in [@freedvafa]) is not sufficient to guarantee that a given theory is consistent [@dienespriv; @klst], and we shall see explicit examples later in section \[sect:type3:twisted\]. In appendix \[app:spectra:fockconstraints\], we discuss another set of consistency conditions that arise, essentially a generalization of the statement that “$c_1 \equiv 0$ mod 2.” Specifically, these conditions state that on each component $\alpha$ of the inertia stack, the $\langle \alpha \rangle$-equivariant line bundle $$K_{\alpha} \otimes \det {\cal E}^{\alpha}_0$$ admit a square root. We defer further discussion of this condition to appendix \[app:spectra:fockconstraints\]. One of the original goals of this project was to find a suitable generalization of anomaly cancellation, a set of sufficient conditions, valid for arbitrary stacks, that would guarantee that the resulting $G$-gauged heterotic sigma model is consistent, but we have been unable to do this. Instead, we only have the necessary conditions above. We leave the problem of finding sufficient conditions for future work. The most interesting examples of heterotic strings on stacks are the special case of strings on gerbes. In previous work [@summ], it was argued that (2,2) supersymmetric strings on gerbes are equivalent to strings on disjoint unions of spaces. For the heterotic string, we shall argue that such a decomposition only exists in general if the gauge bundle is a pullback from a bundle on the base space. More general, ‘twisted,’ bundles exist, and at least sometimes can appear in heterotic compactifications. In fact, it was one of the original goals of this work to construct new (0,2) SCFT’s by using twisted bundles, though as we shall argue later, that does not seem to be the case. In any event, most of this paper will focus on the special case of heterotic strings on gerbes, so in the remainder of this section we shall review some pertinent facts. Review of gerbes {#sect:rev-gerbes} ---------------- So far we have realized heterotic strings on stacks as gauged nonlinear sigma models. The special case of gerbes is realized when a subgroup of the gauge group acts trivially on the target space. In this case, even though part of the gauge group acts trivially on the target, it need not act trivially on the gauge bundle, and this will be responsible for ‘twisted’ bundles. For purposes of disambiguation, let us distinguish our usage of the term from other appearances in the literature. In some papers, gerbes are used formally to describe characteristic classes of $B$ fields, just as principal bundles can be used to describe characteristic classes of gauge fields, and sometimes they are used in that sense to help characterize nontrivial $B$ fields. However, our usage in this paper is different. We are not using the term ‘gerbe’ to describe characteristic classes; instead, we are thinking of gerbes as analogues of spaces on which strings propagate, just as strings can propagate on the total space of a principal bundle. Let us now turn to reviewing gerbes. We review some basics here, see [@summ] for another pertinent general description. In general, to specify a $G$-gerbe over a space $X$, given an open cover $\{U_i\}$ of $X$, one specifies $g_{ijk} \in G$ on triple overlaps and $\varphi_{ij} \in \mbox{Aut}(G)$ on double overlaps, obeying the constraints $$\label{firstg} \varphi_{jk}\circ\varphi_{ij}\:=\:\mbox{Ad}(g_{ijk})\circ\varphi_{ik}$$ on triple overlaps and $$\label{secondg} g_{jk\ell}\,g_{ij\ell}\:=\:\varphi_{k\ell}(g_{ijk})\,g_{ik\ell}.$$ on quadruple overlaps. If we let Out$(G)$ denote the quotient of the group of all automorphisms of $G$ by inner automorphisms, then the $\varphi_{ij}$ above descend to define a principal Out$(G)$ bundle. If that bundle is trivializable, then we say the gerbe is banded. In this case, the gerbe is effectively specified just by the $g_{ijk}$’s, which define a characteristic class in $H^2(X,Z(G))$. (For example, these were the gerbes described in [@hitchin].) The more general case, in which the Out$(G)$ bundle is nontrivial, is known simply as non-banded. In terms of stacks, a stack $[X/G]$ will be a ($K$-)gerbe if a nontrivial subgroup (denoted $K$) of $G$ acts trivially on $X$, by which we mean $g \cdot x = x$ for all $x \in X$ and all $g \in K \subseteq G$. (This is known as an non-effective group action.) Although quotient spaces cannot detect trivial group actions, quotient stacks can, and moreover, so too can the physics[^9] of gauge theories. Although such trivial group actions are invisible perturbatively, they show up nonperturbatively, as has been discussed extensively in [*e.g.*]{} [@nr; @msx; @glsm; @summ]. As the physics of strings on gerbes will be important in this paper, let us briefly review how nonperturbative physics can detect trivial group actions. One short answer is that working with a gauge theory containing a non-effective group action is equivalent to restricting the allowed nonperturbative sectors[^10]. For example, consider the ${\mathbb P}^n$ model, described as a supersymmetric $U(1)$ gauge theory with $n+1$ chiral superfields of charge $1$, but let us instead give the fields charge $k$ instead of charge $1$. Mathematically, this means that a ${\mathbb Z}_k$ subgroup of $U(1)$ acts trivially on the chiral superfields, and describes the weighted projective stack ${\mathbb P}^n_{[k,k,\cdots,k]}$, which is a ${\mathbb Z}_k$ gerbe on ${\mathbb P}^n$. Physically, it is straightforward to see that the instantons in this GLSM are the same as the instantons of degree divisible by $k$ in the original ${\mathbb P}^n$ model. As a practical matter, this means that the $U(1)_A$ symmetry is broken to ${\mathbb Z}_{2k(n+1)}$ rather than ${\mathbb Z}_{2(n+1)}$, for example, and also changes correlation functions and quantum cohomology rings. More globally, if the worldsheet is compact, then the proper definition of the ‘charge’ of a field is in terms of what bundle it couples to. Changing the bundle changes the allowed zero modes, hence changes anomalies and correlation functions [@nr]. For a noncompact worldsheet, an analogous result can be obtained in two dimensions utilizing theta angles. We distinguish ‘gerbe’ cases from ‘non-gerbe’ cases by adding massive minimally-charged fields. The existence of such fields can be sensed, even if their masses are above the cutoff, by examining the periodicity of the theta angle. Since the theta angle acts as an electric field in two dimensions, if we build a capacitor, then by making the plate separation large, one can excite arbitarily-massive field configurations, hence theta angle periodicity measures existence of massive minimally-charged fields [@nr; @nati0; @banks-seib]. In four dimensions, there are analogous methods, involving for example Reissner-Nordstrom black holes and Hawking radiation [@sugrav-g]. A simple example in toroidal orbifolds may help clarify the discussion. Consider the orbifold $[X/D_4]$, where $D_4$ is an eight-element group with a ${\mathbb Z}_2$ center, such that $D_4/{\mathbb Z}_2 = {\mathbb Z}_2 \times {\mathbb Z}_2$. Assume the central ${\mathbb Z}_2$ acts trivially on $X$. From the general analysis above, one would expect that $[X/D_4] \neq [X/{\mathbb Z}_2 \times {\mathbb Z}_2]$, [*i.e.*]{} that physics ‘sees’ the trivially-acting ${\mathbb Z}_2$, and that is exactly what happens. Label the elements of $D_4$ by $$\{1, z, a, b, az, bz, ab, ba=abz \},$$ where $z$ generates the ${\mathbb Z}_2$ center, so that the coset $D_4/{\mathbb Z}_2$ is given by the images of $1$, $a$, $b$, $ab$, which in the (${\mathbb Z}_2 \times {\mathbb Z}_2$) coset we shall denote $\{ 1, \overline{a}, \overline{b}, \overline{ab} \}$. The (string) one-loop partition function of $[X/D_4]$ is obtained by summing over twisted sectors defined by all commuting pairs in $D_4$. For example, there are no $(a, ab)$, $(b, ab)$, $(a,b)$ twisted sectors, as those pairs do not commute in $D_4$. Now, if we compare the ${\mathbb Z}_2 \times {\mathbb Z}_2$ partition function, although individual twisted sector contributions match (as the ${\mathbb Z}_2$ acts trivially), the total number is different. For example, the ${\mathbb Z}_2 \times {\mathbb Z}_2$ contains contributions from $(\overline{a}, \overline{ab})$, $(\overline{b},\overline{ab})$ and $(\overline{a},\overline{b})$ twisted sectors, but there are no corresponding $(a,ab)$, $(b,ab)$, $(a,ab)$ contributions in the $D_4$ partition function. Thus, we see the one-loop partition functions of the $D_4$ and ${\mathbb Z}_2 \times {\mathbb Z}_2$ partition functions are very different, despite the fact that the theories differ by a trivially-acting gauged ${\mathbb Z}_2$. In fact, in the example above, one can show that the partition function of the $D_4$ orbifold is the same as the partition function of a disjoint union of two ${\mathbb Z}_2 \times {\mathbb Z}_2$ orbifolds, one with and the other without discrete torsion. The one-loop partition function of a disjoint union is the sum of the partition functions of the components, and discrete torsion adds a sign to the $(\overline{a}, \overline{ab})$, $(\overline{b},\overline{ab})$ and $(\overline{a},\overline{b})$ sectors, so they cancel out of the partition function for the disjoint union. This is a simple example of the ‘decomposition conjecture’ we review in section \[sect:decomp-22review\]. Notions of twisting {#sect:twisting} ------------------- Now that we have outlined gerbes and demonstrated their physical meaningfulness, let us turn to possible bundles over gerbes. A gerbe was defined by a trivial group action on the base space; however, that same group action can be nontrivial on the bundle. The resulting bundle is then interpreted as some sort of twisted bundle, in some sense, as we shall review here. There are various notions of twisted bundles in the literature. One notion, discussed for example in [@cks], is of a twisted bundle in which the twisting refers to the fact that the transition functions do not quite close on triple overlaps: instead of $$g_{\alpha \beta } g_{\beta \gamma} g_{\gamma \alpha} \: = \: 1$$ the transition functions obey $$\label{cocyc1} g_{\alpha \beta } g_{\beta \gamma} g_{\gamma \alpha} \: = \: h_{\alpha \beta \gamma} I$$ for some cocycle $h_{\alpha \beta \gamma}$. At the level of the gauge field, such a twisting means that across coordinate patches, the gauge field receives an affine translation in addition to a gauge transformation. Such twisted bundles appear physically on D-branes. After all, under a gauge transformation of the $B$ field, of the form $$B \: \mapsto \: B \: + \: d \Lambda,$$ the Chan-Paton gauge field must necessarily transform as $$A \: \mapsto \: A \: - \: \Lambda$$ in order to preserve gauge-invariance on the open string worldsheet, and such affine translations correspond, in terms of transition functions, to the modified overlap condition equation (\[cocyc1\]). However, although such twistings are possible for D-branes, no such twisting is ordinarily possible in heterotic strings, because the heterotic gauge field never picks up affine translations across coordinate patches – the heterotic gauge field and the heterotic $B$ field are related in a very different fashion than in D-branes. A second notion of twisting appears when discussing gerbes. Consider the weighted projective stack ${\mathbb P}^N_{[k,\cdots,k]}$, a ${\mathbb Z}_k$ gerbe on ${\mathbb P}^N$, described physically by an analogue of the supersymmetric ${\mathbb P}^N$ model in which chiral superfields have charge $k$ instead of $1$, as discussed earlier. Now, the total space of a line bundle ${\cal O}(-n) \rightarrow {\mathbb P}^N$ can be described as a quotient of $N+1$ fields $\phi_i$ and one field $p$ of charges $1$, $-n$, respectively. Consider instead a quotient of the fields above in which the $\phi_i$ have charge $k$ (and so describe ${\mathbb P}^N_{[k,\cdots,k]}$), and the field $p$ has charge $-1$. This quotient is the total space of a line bundle on the gerbe sometimes denoted ${\cal O}(-1/k)$. (We will discuss line bundles on gerbes in more detail in appendix \[app:linebundles\].) We can understand this second notion of twisting in much greater generality, as follows. First, for any stack $\mathfrak{X}$ presented as $\mathfrak{X} = [X/G]$ for some space $X$ and group $G$, a vector bundle (sheaf) on $\mathfrak{X}$ is the same as a $G$-equivariant vector bundle (sheaf) on $X$. Now, suppose that $G$ is an extension $$1 \: \longrightarrow \: K \: \longrightarrow \: G \: \longrightarrow \: H \: \longrightarrow \: 1,$$ where $K$ acts trivially on $X$, and $G/K \cong H$ acts effectively. In this case, $\mathfrak{X} = [X/G]$ is a $K$-gerbe. A vector bundle on $\mathfrak{X}$ is a $G$-equivariant vector bundle on $X$, and as such, the $K$ action is defined by a representation of $K$ on the fibers of that vector bundle. This is the more general picture of the second notion of twisting. Any bundle on the gerbe that is not a pullback from the base, has a nontrivial action of $K$. These two notions of twisting are not unrelated. Mathematically, it is a standard result that the category of sheaves on a gerbe decomposes into different sectors containing twisted sheaves on the underlying space, twisted by flat $B$ fields. Moreover, this decomposition is complete: there are no nonzero Ext groups between sheaves in different sectors on the same gerbe. This fact was one of the inspirations for the ‘decomposition conjecture’ presented in [@summ], which said that conformal field theories describing strings on gerbes should factorize in the same way, that the CFT’s are the same as CFT’s on disjoint unions of spaces. The resulting factorization of D-branes reflects the mathematical result above on factorization of sheaves on gerbes. For completeness, let us discuss this decomposition for the special case of ${\cal O}(1/k) \rightarrow {\mathbb P}^N_{[k,\cdots,k]}$. To be twisted in the first sense we discussed, one can show that the rank of the twisted bundle must be divisible by the order of the twisting cocycle’s cohomology class. Here, since ${\cal O}(1/k)$ has rank one, the order of the cocycle must be one. Indeed, the twistings of ${\cal O}(1/k)$ appearing involve cocycles with trivial cohomology, so there is no rank restriction. Class I: Gauge bundle a pullback from the base {#sect:het-decomp} ============================================== We have classified heterotic string compactifications on gerbes into three fundamental classes or ‘building blocks,’ from which more general compactifications can be built. In this and the next two sections, we will examine properties of those classes. The first class we consider involves the special case that the gauge bundle is a pullback from the base. This is equivalent to the statement that the subgroup $G$ of the gauge group that acts trivially on the base, also acts trivially on the fibers of the gauge bundle. In this case, we will argue that, at least for banded gerbes, the heterotic (0,2) SCFT factorizes – it is equivalent to a heterotic string on a disjoint union of spaces with bundles, following essentially the same mechanism as in (2,2) strings. Review of (2,2) decomposition conjecture {#sect:decomp-22review} ---------------------------------------- As was reviewed earlier in section \[sect:rev-gerbes\], gauge theories in which a subgroup of the gauge group acts trivially on massless matter break cluster decomposition. However, it was argued in [@summ] that such theories are equivalent to tensor products / disjoint unions of cluster-decomposition-obeying theories. For example, a gauged nonlinear sigma model of this form is equivalent to a nonlinear sigma model on a disjoint union of ordinary spaces. The latter violates cluster decomposition, but does so in an obviously trivial fashion, and so there is no essential difficulty with the quantum field theory. For (2,2) supersymmetric gauged nonlinear sigma models in two dimensions, this was encapsulated in [@summ] in the “decomposition conjecture,” which we shall generalize to heterotic strings. To make this paper self-contained, we take a moment here to review the statement of the decomposition conjecture. Suppose we have a $K$-gerbe over $[X/H]$, defined by the quotient $[X/G]$ where $$1 \: \longrightarrow \: K \: \longrightarrow \: G \: \longrightarrow \: H \: \longrightarrow \: 1.$$ Let $\hat{K}$ denote the set of irreducible representations of $K$. There is a natural action of $H$ on $\hat{K}$, defined as follows: given $h \in H$ and $\rho \in \hat{K}$, pick a lift $\tilde{h} \in G$ of $h$, and define $h \cdot \rho$ by, $$(h \cdot \rho)(g) \: \equiv \: \rho(\tilde{h}^{-1} g \tilde{h} )$$ for all $g \in K$. If $K$ is abelian, this is well-defined. If $K$ is not abelian, then it can be shown (see [@summ]\[section 4\]) that there exists an operator intertwining the representations $h \cdot \rho$ defined by any two lifts, hence $h \cdot \rho$ is well-defined in $\hat{K}$. Then, the decomposition conjecture for (2,2) theories states that a string on the gerbe $[X/G]$ is the same as a string on the disjoint union of spaces $[ (X \times \hat{K} )/H ]$, together with a flat $B$ field defined in [@summ]\[section 4\]. In the special case that the gerbe $[X/G]$ is banded, the description above simplifies. In this case, the $H$ action on $\hat{K}$ is trivial, and so the decomposition conjecture reduces to the statement that a string on the gerbe $[X/G]$ is the same as a string on a disjoint union of $| \hat{K} |$ copies of $[X/H]$, in which each copy comes with a flat $B$ field determined by acting on the characteristic class of the gerbe with the irreducible representation corresponding to that copy: $$\rho \in \hat{K}: \: H^2([X/H], Z(G)) \: \longrightarrow \: H^2([X/H], U(1) ).$$ Extensive evidence was presented in [@summ] for this conjecture, ranging from computations of orbifold spectra and partition functions to GLSM results and quantum cohomology computations. Other results have appeared since. For reasons of brevity, we only list two below: - This conjecture makes a prediction for Gromov-Witten invariants of stacks, namely that the Gromov-Witten invariants of gerbes are equivalent to Gromov-Witten invariants of disjoint unions of spaces. This was checked in the mathematics literature in [*e.g.*]{} [@ajt1; @ajt2; @ajt3; @t1; @gt1; @xt1]. - This conjecture plays an important role in understanding certain GLSM’s. Specifically, it was used in [@cdhps] to understand Landau-Ginzburg points of complete intersections of quadrics, resolving some old unanswered questions, and also providing examples of GLSM’s that realize geometry in a different way than as a critical locus of a superpotential, that contain non-birational phases, and in some cases, that RG flow to ‘noncommutative resolutions’ of singular spaces, providing the first physical realizations of those mathematical theories in CFT. The results of [@cdhps] have since been checked in [*e.g.*]{} [@hori2; @ed-nick-me] and further examples discussed in [@hori2; @hkm; @enstx]. The same methods have also been applied to make predictions for Gromov-Witten invariants of noncommutative resolutions in [@ncgw]. See also the $D_4$ orbifold discussed in section \[sect:rev-gerbes\] for another example. The result may seem obscure, but there is a simple physical reason for it, namely that in the path integral, summing over the elements of the disjoint union, together with variable $B$ fields, is equivalent to inserting a projection operator that enforces the requirement that only instantons of certain degrees contribute to the theory. Schematically, for a nonlinear sigma model, we can describe the insertion of a projection operator in the form $$\int [D \phi] e^{-S} \left( \sum_{k=0}^{n-1} e^{i k \int \phi^* \omega} \right) \: = \: \sum_{k=0}^{n-1} \int [D \phi] \exp\left( - S + i k \int \phi^* \omega \right),$$ where $\omega$ is the Kähler form on the target space. The left-hand side is the partition function with a projector onto nonperturbative states of certain degrees; the right-hand side is a partition function for a disjoint union of $n$ copies of the original target space, each with a rotated $B$ field, rotated by an amount $k \omega$. Nonbanded gerbes merely represent a more complicated variation. Heterotic decomposition conjecture ---------------------------------- In this section we will describe the heterotic analogue of the decomposition conjecture, for banded gerbes. Briefly, given a (0,2) SCFT defined by a banded gerbe $\mathfrak{X}$ over a space (or orbifold) $X$ and bundle ${\cal E} \rightarrow \mathfrak{X}$, such that ${\cal E}$ is a pullback of a bundle on $X$, then this (0,2) SCFT is the same as a (0,2) SCFT on a disjoint union of copies of $X$. Now, let us define terms more precisely. Suppose we have a $K$-gerbe over $[X/H]$, defined by the quotient $\mathfrak{X} = [X/G]$ where $$1 \: \longrightarrow \: K \: \longrightarrow \: G \: \longrightarrow \: H \: \longrightarrow \: 1.$$ Suppose we also have a holomorphic vector bundle ${\cal E}$ over $[X/G]$ ([*i.e.*]{} a $G$-equivariant bundle on $X$), defining a consistent (0,2) SCFT. We assume that ${\cal E}$ is a pullback of a bundle ${\cal E}'$ on $[X/H]$. This can be understood in several equivalent ways, for example: - $K$ acts trivially on both $X$ and ${\cal E}$, - ${\cal E}$ is in the weight-zero part of the decomposition of sheaves on $[X/G]$, which imply that the $G$-equivariant structure on ${\cal E}$ (as a bundle on $X$) descends to an $H$-equivariant structure. The heterotic decomposition conjecture for (0,2) theories is that, in these circumstances, if the gerbe is banded, a heterotic string on $([X/G], {\cal E})$ is the same as a heterotic string on the disjoint union $$\amalg_{\hat{K}} [X/H]$$ with varying $B$ fields and gauge bundle ${\cal E}'$ on each copy of $[X/H]$. As a consistency check, in the special case that ${\cal E}=T\mathfrak{X}$ ([*i.e.*]{} $TX$ with its natural $G$-equivariant structure), then ${\cal E}' = TX$ with its natural $H$-equivariant structure, and this reduces to the (2,2) decomoposition conjecture (for banded gerbes). Other examples are easy to construct. For example, if we take an anomaly-free heterotic (0,2) SCFT defined by a bundle ${\cal E}$ on a space $X$, and take a global orbifold of $X$ by a finite group that acts trivially on both $X$ and ${\cal E}$, it is trivial to see that the twisted sector states will all be copies of the untwisted sector states, in agreement with the prediction of the decomposition conjecture above that this (0,2) SCFT should be the same as that for a disjoint union of copies of $(X, {\cal E})$. Another set of examples is provided by (0,2) GLSM’s. Begin with an anomaly-free (0,2) GLSM describing a bundle ${\cal E}'$, say, $$0 \: \longrightarrow \: {\cal E}' \: \longrightarrow \: \oplus_a {\cal O}(n_a) \: \stackrel{F}{\longrightarrow} \: \oplus_i {\cal O}(m_i) \: \longrightarrow \: 0,$$ over a hypersurface in a weighted projective space ${\mathbb P}^d_{w_0, \cdots, w_d}[w_0 + \cdots + w_d]$. Now, build a new (0,2) GLSM constructed from the one above by multiplying all gauge charges by an integer $k > 0$. The result is a bundle ${\cal E}$, $$0 \: \longrightarrow \: {\cal E} \: \longrightarrow \: \oplus_a {\cal O}(k n_a) \: \stackrel{F}{\longrightarrow} \: \oplus_i {\cal O}(k m_i) \: \longrightarrow \: 0,$$ over a hypersurface in a weighted projective stack ${\mathbb P}^d_{[k w_0, \cdots, k w_d]}[k(w_0 + \cdots + w_d)]$. The bundle map $F$ and hypersurface polynomial are unchanged. If one now goes to the Landau-Ginzburg point of this theory and computes the massless spectrum, it is trivial to see that the spectrum will consist of $k$ copies of the spectrum of the original theory, in agreement with the prediction of the decomposition conjecture. The analogue of the decomposition conjecture for nonbanded gerbes is not currently known. It is tempting to speculate that it should be the statement that a heterotic string on $([X/G], {\cal E})$ is the same as a heterotic strings on $( [(X \times \hat{K})/H], {\cal E})$, where (as in the (2,2) case) $\hat{K}$ is the set of irreducible representations of $K$, and we extend ${\cal E}$ trivially from $[X/H]$ to $[(X \times \hat{K})/H]$. However, on the (2,2) locus, the special case that ${\cal E} = TX$ with its natural $G$-equivariant structure, ${\cal E}$ reinterpreted as an $H$-equivariant bundle and extended trivially over $\hat{K}$ does not in general[^11] define the tangent bundle of $[(X \times \hat{K})/H]$, and so this would not reduce correctly to (2,2) decomposition. Class II: Dualities {#sect:het-gsomods} =================== Basic proposal -------------- In the special case of a heterotic string in which a ${\mathbb Z}_2$ that acts nontrivially on the base, acts nontrivially on a rank 8 bundle, that subgroup of the gauge group is locally duplicating the effect of one of the ten-dimensional left-moving GSO projections. If one starts with a Spin$(32)/{\mathbb Z}_2$ string, then the dual looks locally like an $E_8 \times E_8$ string. In this section, we will describe[^12] a precise duality relating such Spin$(32)/{\mathbb Z}_2$ compactifications to ordinary $E_8 \times E_8$ compactifications, and discuss some examples. First, let us consider the easiest case. If the ${\mathbb Z}_2$ gerbe is trivial, the result is automatic: the worldsheet left-moving GSO projection is duplicated exactly, not just locally. When the gerbe is nontrivial, one must think a little harder to find a precise duality. We propose[^13] a duality to heterotic $E_8 \times E_8$ strings as follows. To set conventions, suppose our stack $\mathfrak{X} = [X/\tilde{G}]$, where $$1 \: \longrightarrow \: {\mathbb Z}_2 \: \longrightarrow \: \tilde{G} \: \longrightarrow \: G \: \longrightarrow \: 1$$ and ${\mathbb Z}_2$ acts trivially on $X$. Suppose furthermore that ${\cal E}$ is a bundle on $\mathfrak{X}$, [*i.e.*]{} a $\tilde{G}$-equivariant bundle on $X$, whose embedding into $E_8$ is via the standard worldsheet fermionic construction, in which left-moving fermions are in the fundamental representation of the structure group. Suppose that the ${\mathbb Z}_2$ acts nontrivially on ${\cal E}$. In general ${\cal E}$ will not admit a $G$-equivariant structure. Nevertheless, at least in the special case that the ${\mathbb Z}_2$ is central in $\tilde{G}$, the bundles ${\cal E}^* \otimes {\cal E}$ and $\wedge^{\rm even} {\cal E}$ will admit a $G$-equivariant structure, and so can be defined on $[X/G]$. Moreover, it was observed in [@dist-greene] that, for the ‘typical’ worldsheet embeddings of $SU(n)$ in $E_8$ (including the present one), massless spectra of heterotic compactifications on smooth spaces can be defined solely in terms of sheaf cohomology with coefficients in ${\cal E}^* \otimes {\cal E}$ and $\wedge^{\rm even} {\cal E}$; other sheaf cohomology groups are related by Serre duality. There is a good reason for this. In the heterotic compactifications discussed in [@dist-greene], the $SU(n)$ gauge bundle is embedded into $E_8$ by first embedding in ${\rm Spin}(2n) \subset {\rm Spin}(16)$, projecting to ${\rm Spin}(16)/{\mathbb Z}_2$ (as a result of the left GSO projection), and then ${\rm Spin}(16)/{\mathbb Z}_2$ naturally embeds into $E_8$ [@adamse8]. The only coefficient bundles that survive the left GSO projection are ${\cal E}^* \otimes {\cal E}$ and $\wedge^{\rm even} {\cal E}$; they suffice to define an $E_8$ bundle, and that is why they suffice to define massless spectra. Thus, we propose that a heterotic ${\rm Spin}(32)/{\mathbb Z}_2$ string compactified on a ${\mathbb Z}_2$ gerbe $\mathfrak{X}$ as above, with the ${\mathbb Z}_2$ central, acting by signs on a rank 8 bundle ${\cal E} \rightarrow \mathfrak{X}$, embedded in a typical fashion, defines the same SCFT as a heterotic $E_8 \times E_8$ string compactified on $[X/G]$ with $E_8$ gauge bundle determined by ${\cal E}^* \otimes {\cal E}$ and $\wedge^{\rm even} {\cal E}$ (which are defined on $[X/G]$, even if ${\cal E}$ itself is not). In the special case that the ${\mathbb Z}_2$ gerbe is trivial, the dual $E_8 \times E_8$ string on $[X/G]$ is defined by the bundle ${\cal E}$ – in this special case, the $E_8$ bundle determined by ${\cal E}^* \otimes {\cal E}$, $\wedge^{\rm even} {\cal E}$ is the same determined by the usual embedding of ${\cal E}$ into $E_8$. More generally, the $E_8$ bundle need not have a description in terms of a similarly-embedded $SU(n)$ gauge bundle; a direct construction might have to appeal to the fibered WZW methods discussed in [@anom; @gates1; @gates2; @gates3; @gates4; @gates5]. So far we have discussed Spin$(32)/{\mathbb Z}_2$ compactifications on a ${\mathbb Z}_2$ gerbe with rank 8 bundle. Now let us briefly consider an $E_8 \times E_8$ compactification on a ${\mathbb Z}_2$ gerbe with rank 8 bundle. Nearly the same analysis applies as in the Spin$(32)/{\mathbb Z}_2$ case. At the level of SCFT, before imposing the left GSO projections, the same duality argument we have just given suggests the gerbe theory should be dual to an $E_8$ bundle, as above. The left GSO for the corresponding bundle duplicates the gerbe ${\mathbb Z}_2$, and so should act trivially on the theory. The dual should be thus be interpreted as class I, and so the result should have the form of a disjoint union of two copies of an $E_8 \times E_8$ compactification. As the details are largely duplicative of the Spin$(32)/{\mathbb Z}_2$ case just discussed, and for which we will see examples below, we will not treat this case further. We have discussed bundles with structure group $SU(n)$ embedded into ${\rm Spin}(32)/{\mathbb Z}_2$ and $E_8 \times E_8$ in the form of the standard worldsheet construction, but more general embeddings exist, and admit worldsheet descriptions [@anom]. One open question we leave for future work is to generalize the duality discussed here to more general embeddings. Toroidal orbifold example {#sect:class2-ex1} ------------------------- Consider a Spin$(32)/{\mathbb Z}_2$ heterotic string compactified on a ${\mathbb Z}_2$ gerbe over $[T^4/{\mathbb Z}_2]$, with a rank eight bundle, defined as follows. The ${\mathbb Z}_2$ gerbe is $[T^4/{\mathbb Z}_4]$, where the ${\mathbb Z}_4$ acts on the $T^4$ by $$x \: \mapsto \: \exp\left( \frac{2\pi i (2k)k}{4} \right) x \: = \: (-)^k x,$$ so that there is a trivially-acting ${\mathbb Z}_2$ subgroup; only the sectors $k=1, 3$ have twisted bosons. (Mathematically, this is a nontrivial[^14] ${\mathbb Z}_2$ gerbe.) The bundle is the rank eight bundle ${\cal O}^{\oplus 8}$, on which the ${\mathbb Z}_4$ acts (effectively) by fourth roots of unity. We will compute the spectrum, and discover not only that it is consistent, but in addition that it has the same form as the spectrum of a perturbative $E_8 \times E_8$ compactification on a space, as expected from the duality proposal. We use $X^{3-4}$ to denote the bosons in the $T^4$, and $\psi^{3-4}$ their right-moving superpartners. We shall use $\lambda^{1-8}$ to denote the free left-moving fermions and $\lambda^{9-16}$ to denote the left-moving fermions in the bundle above. Let us begin the spectrum computation in the untwisted sector. First, consider (NS,NS) states. Here, the left- and right-moving vacuum energies are given by $E_{\rm left} = -1$, $E_{\rm right} = -1/2$. The ${\mathbb Z}_4$-invariant states have the form ------------------------------------------------------------------------------------------------------------------------------------- State Count ------------------------------------------------------------------------- ----------------------------------------------------------- $\left( \lambda^{1-8}_{-1/2}, \overline{\lambda}^{1-8}_{-1/2} \right)^2 spacetime vector, valued in adjoint of $so(16)$ \otimes \left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2} \right)$ $\overline{\partial} X^{1-2}_{-1} \otimes gravity, tensor multiplet contribution \left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2} \right)$ $\left( \lambda^{9-16}_{-1/2} \overline{\lambda}^{9-16}_{-1/2} \right) spacetime vector, valued in adjoint, ${\bf 1}$ of $su(8)$ \otimes \left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2} \right)$ (${\bf 1}$ from the trace) $\overline{\partial} X^{3-4}_{-1} 16 spacetime scalars (toroidal moduli), \otimes \left( \psi^{3-4}_{-1/2}, \overline{\psi}^{3-4}_{-1/2} \right)$ forming 4 hypermultiplets $\left( \left( \lambda^{9-16}_{-1/2} \right)^2, 4 spacetime scalars, \left( \overline{\lambda}^{9-16}_{-1/2} \right)^2 \right) \otimes \left( \psi^{3-4}_{-1/2}, \overline{\psi}^{3-4}_{-1/2} \right)$ valued in $\wedge^2 {\bf 8} = {\bf 28}$, $\wedge^2 {\bf \overline{8}} = {\bf \overline{28}}$ of $su(8)$, forming 1 hypermultiplet in ${\bf 28}$, another in ${\bf \overline{28}}$ ------------------------------------------------------------------------------------------------------------------------------------- There are no (R,NS) states in the untwisted sector, since $E_{\rm left} > 0$. Next, consider the twisted sector $k=1$. In the (NS,NS) sector, fields have the following boundary conditions: $$\begin{aligned} X^{1-2}(\sigma + 2 \pi) & = & + X^{1-2}(\sigma), \\ X^{3-4}(\sigma + 2 \pi) & = & - X^{3-4}(\sigma), \\ \psi^{1-2}(\sigma + 2\pi) & = & - \psi^{1-2}(\sigma), \\ \psi^{3-4}(\sigma + 2 \pi) & = & + \psi^{3-4}(\sigma), \\ \lambda^{1-8}(\sigma + 2 \pi) & = & - \lambda^{1-8}(\sigma), \\ \lambda^{9-16}(\sigma + 2 \pi) & = & - \exp\left( \frac{2\pi i}{4} \right) \lambda^{9-16}(\sigma).\end{aligned}$$ It is straightforward to compute $E_{\rm left} = -1/2$, $E_{\rm right} = 0$. The available field modes are $$X^{3-4}_{-1/2}, \: \: \: \lambda^{1-8}_{-1/2}, \overline{\lambda}^{1-8}_{-1/2}, \: \: \: \lambda^{9-16}_{-1/4}, \overline{\lambda}^{9-16}_{-3/4}.$$ There is a multiplicity of right-moving Fock vacua, arising from the periodicity of $\psi^{3-4}$. Briefly, the vacua $| \pm \mp \rangle$ are invariant, and $| \pm \pm \rangle$ get a sign under the action of the generator of ${\mathbb Z}_4$. In this sector, the ${\mathbb Z}_4$- and GSO-invariant states are --------------------------------------------------------------------------------------------------------------------------------------------- State Count -------------------------------------------------------------------- ------------------------------------------------------------------------ $\overline{\partial} X^{3-4}_{-1/2} \otimes 8 spacetime scalars | \pm \pm \rangle$ $\left( \lambda^{9-16}_{-1/4} \right)^2 \otimes | \pm \pm \rangle$ 2 spacetime scalars valued in $\wedge^2 {\bf 8} = {\bf 28}$ of $su(8)$ --------------------------------------------------------------------------------------------------------------------------------------------- There are no massless states in (R,NS) in this sector, as $E_{\rm left} = + 1/2$. Copies of the states in the $k=1$ sector occur at each of the sixteen fixed points, hence the total state count should be obtained by multiplying the totals for this sector by sixteen. Next, consider the twisted sector $k=2$. In the (NS,NS) sector, fields have the following boundary conditions: $$\begin{aligned} X^{1-4}(\sigma + 2\pi) & = & + X^{1-4}(\sigma), \\ \psi^{1-4}(\sigma + 2 \pi) & = & - \psi^{1-4}(\sigma), \\ \lambda^{1-8}(\sigma + 2 \pi) & = & - \lambda^{1-8}(\sigma), \\ \lambda^{9-16}(\sigma + 2 \pi) & = & + \lambda^{9-16}(\sigma).\end{aligned}$$ It is straightforward to compute $E_{\rm left} = 0$, $E_{\rm right} = -1/2$. The available field modes are $$\psi^{\mu}_{-1/2}, \overline{\psi}^{\mu}_{-1/2}, \: \: \: \lambda^{1-8}_{-1/2}, \overline{\lambda}^{1-8}_{-1/2}.$$ There is a multiplicity of left Fock vacua, arising from $\lambda^{9-16}$. Let $|m,n\rangle$ denote a vacuum with $m$ +’s and $n$ -’s (note $m+n=8$), then under the action of the generator of ${\mathbb Z}_4$, it is straightforward to check that $|m=0,4,8\rangle$ are invariant, $|m=2,6\rangle$ get a sign flip, and the others get other fourth roots of unity. The ${\mathbb Z}_4$- and GSO-invariant states in this sector are of the form ----------------------------------------------------------------------------------------------------------------------------------------------- State Count ---------------------------------------------------- ------------------------------------------------------------------------------------------ $|m=0,4,8\rangle \otimes \left( \psi^{1-2}_{-1/2}, spacetime vectors, in the ${\bf 1}$, ${\bf 1}$, $\wedge^4 {\bf 8} = {\bf 70}$ of $su(8)$ \overline{\psi}^{1-2}_{-1/2} \right)$ $|m=2,6\rangle \otimes \left( \psi^{3-4}_{-1/2}, 1 hypermultiplet in $\wedge^2 {\bf 8} = {\bf 28}$, $\wedge^2 {\bf \overline{8}} = \overline{\psi}^{3-4}_{-1/2} \right)$ {\bf \overline{28}}$ of $su(8)$ ----------------------------------------------------------------------------------------------------------------------------------------------- The (R,NS) sector in $k=2$ is closely related. Here, fields have the following boundary conditions: $$\begin{aligned} X^{1-4}(\sigma + 2\pi) & = & + X^{1-4}(\sigma), \\ \psi^{1-4}(\sigma + 2 \pi) & = & - \psi^{1-4}(\sigma), \\ \lambda^{1-8}(\sigma + 2 \pi) & = & + \lambda^{1-8}(\sigma), \\ \lambda^{9-16}(\sigma + 2 \pi) & = & - \lambda^{9-16}(\sigma).\end{aligned}$$ Just as in the (NS,NS) sector, $E_{\rm left} = 0$ and $E_{\rm right} = -1/2$. Here, the left Fock vacua form a spinor of the low-energy $so(16)$. The ${\mathbb Z}_4$-invariant states in this sector are of the form ------------------------------------------------------------------------------------------------------ State Count ----------------------------------------------------- ------------------------------------------------ $(\mbox{spinor}) \otimes \left( \psi^{1-2}_{-1/2}, spacetime vector, in chiral spinor of $so(16)$ \overline{\psi}^{1-2}_{-1/2} \right)$ ------------------------------------------------------------------------------------------------------ Finally, let us consider the $k=3$ sector. There are no massless states in (R,NS), so we only consider (NS,NS). Fields in this sector have the following boundary conditions: $$\begin{aligned} X^{1-2}(\sigma + 2\pi) & = & + X^{1-2}(\sigma), \\ X^{3-4}(\sigma + 2\pi) & = & - X^{3-4}(\sigma), \\ \psi^{1-2}(\sigma + 2\pi) & = & - \psi^{1-2}(\sigma), \\ \psi^{3-4}(\sigma + 2 \pi) & = & + \psi^{3-4}(\sigma), \\ \lambda^{1-8}(\sigma + 2\pi) & = & - \lambda^{1-8}(\sigma), \\ \lambda^{9-16}(\sigma + 2\pi) & = & \exp\left( \frac{\pi i}{2} \right) \lambda^{9-16}(\sigma).\end{aligned}$$ It is straightforward to compute $E_{\rm left} = -1/2$, $E_{\rm right} = 0$. The available field modes are $$\overline{\partial} X^{3-4}_{-1/2}, \: \: \: \lambda^{1-8}_{-1/2}, \overline{\lambda}^{1-8}_{-1/2}, \: \: \: \lambda^{9-16}_{-3/4}, \overline{\lambda}^{9-16}_{-1/4}.$$ Because $\psi^{3-4}$ is periodic, there is a multiplicity of right Fock vacua. The states $|+-\rangle$, $|-+\rangle$ are invariant under the generator of ${\mathbb Z}_4$, whereas the states $|++\rangle$, $|--\rangle$ get a sign flip. Putting this together, we find ${\mathbb Z}_4$- and GSO-invariant massless states of the form: ----------------------------------------------------------------------------------------------------------------- State Count --------------------------------------------------------------- ------------------------------------------------- $\left( \overline{\partial} X^{3-4}_{-1/2}, 8 scalars \overline{\partial} \overline{X}^{3-4}_{-1/2} \right) \otimes | \pm \pm \rangle$ $\left( \overline{\lambda}^{9-16}_{-1/4} \right)^2 \otimes 2 sets of scalars each in the $\wedge^2 {\bf | \pm \pm \rangle$ \overline{8}} = {\bf \overline{28}}$ of $su(8)$ ----------------------------------------------------------------------------------------------------------------- Furthermore, copies of the states above occur at each fixed point, hence the total number of states is obtained by multiplying the tally above by sixteen. Now, let us summarize our results so far. We have found the following states: 1 gravity multiplet, 1 tensor multiplet, vector multiplets transforming in the adjoint, chiral spinor of $so(16)$, vector multiplets transforming in the adjoint, ${\bf 70}$, ${\bf 1}$, ${\bf 1}$, ${\bf 1}$ of $su(8)$, 10 hypermultiplets in ${\bf 28}$ of $su(8)$ ($k=0,1,2$), 10 hypermultiplets in ${\bf \overline{28}}$ of $su(8)$ ($k=0,2,3$), 4 ($k=0$) plus 32 ($k=1$) plus 32 ($k=3$) singlet hypermultiplets. We can describe this spectrum more compactly as follows. First, the vectors transforming in the adjoint and chiral spinor of $so(16)$ clearly combine to form a vector in the adjoint of $e_8$. Second, under its $su(8)$ subalgebra, the adjoint representation of $e_7$ decomposes as [@slansky]\[table 52\] $${\bf 133} \: = \: {\bf 63} \: + \: {\bf 70},$$ so we see that the remaining non-singlet vectors combine to form the adjoint of $e_7$. In the same decomposition, $${\bf 56} \: = \: {\bf 28} \: + \: {\bf \overline{28}},$$ so we see that the hypermultiplets in the ${\bf 28}$ and ${\bf \overline{28}}$ combine to form 10 hypermultiplets in the ${\bf 56}$ of $e_7$. Putting this together, we find that the spectrum can be described as follows: 1 gravity multiplet, 1 tensor multiplet, vector multiplets transforming in the adjoint of $E_7 \times E_8 \times U(1)^3$, 10 hypermultiplets in the ${\bf 56}$ of $E_7$, 68 hypermuliplets that are singlets under $E_7 \times E_8$. The number of hypermultiplets is greater than the number of vector multiplets by 244, which is a necessary condition for anomaly cancellation. The duality proposal in this example predicts that the dual is defined by a heterotic $E_8 \times E_8$ compactification on $[T^4/{\mathbb Z}_2]$, with $E_8$ bundle defined by ${\cal E}^* \otimes {\cal E}$, $\wedge^{\rm even} {\cal E}$, for ${\cal E} = {\cal O}^8$ on $T^4$, but such that ${\cal E}^* \otimes {\cal E}$ and $\wedge^{\rm even} {\cal E}$ are odd under the action of the ${\mathbb Z}_2$ defining $[T^4/{\mathbb Z}_2]$. We do not see how such an $E_8$ bundle on $[T^4/{\mathbb Z}_2]$ could be obtained from embedding an $SU(n)$ bundle in the usual fashion, and indeed, as remarked earlier, it need not be, the duals in general may only be describable by fibered WZW models. That said, the reader should note that the spectrum computed above is nearly the same as the massless spectrum of an $E_8 \times E_8$ string compactified on a (2,2) $[T^4/{\mathbb Z}_2]$, which in general terms is consistent with the existence of a duality between the current Spin$(32)/{\mathbb Z}_2$ gerbe compactification and an $E_8 \times E_8$ compactification. So, although we cannot check the details at this time, certainly in broad brushstrokes this is consistent. Examples in Distler-Kachru models --------------------------------- In table \[table:DK-duality-exs\] we tabulate the combinatorial data for a number of anomaly-free Distler-Kachru (0,2) GLSM’s of the pertinent form. Each describes a bundle ${\cal E}$ over a Calabi-Yau hypersurface in a weighted projective stack, $${\mathbb P}^n_{[w_0, \cdots, w_n]}[w_0 + \cdots + w_n],$$ a ${\mathbb Z}_2$ gerbe over a Calabi-Yau space, where the (rank 8) bundle is given as a kernel of the form $$0 \: \longrightarrow \: {\cal E} \: \longrightarrow \: \oplus_a {\cal O}(n_a) \: \longrightarrow \: \oplus_i {\cal O}(m_i) \: \longrightarrow \: 0.$$ $w_0, \cdots, w_4$ $n_a$ $m_i$ -------------------- ------------ --------- -- $2,2,2,4$ $1^9$ $9$ $2,2,2,2,2$ $1^9, 9$ $7, 11$ $2,2,2,2,2$ $3^9, 19$ $9, 21$ $2,2,2,2,4$ $3^9, 27$ $9, 29$ $2,2,2,4,6$ $1^8, 5^2$ $9, 13$ $2,2,2,2,6$ $1^9, 9$ $3, 15$ $2,2,2,4,4$ $1^9, 15$ $5,19$ : This table lists combinatorial data for anomaly-free (0,2) GLSM’s describing rank 8 bundles over ${\mathbb Z}_2$ gerbes on Calabi-Yau’s.[]{data-label="table:DK-duality-exs"} For example, the first entry in table \[table:DK-duality-exs\] describes a rank 8 bundle given as a kernel $$0 \: \longrightarrow \: {\cal E} \: \longrightarrow \: \oplus_1^9 {\cal O}(1) \: \longrightarrow \: {\cal O}(9) \: \longrightarrow \: 0$$ over the stack ${\mathbb P}^3_{[2,2,2,4]}[10]$, a ${\mathbb Z}_2$ gerbe over ${\mathbb P}^3_{[1,1,1,2]}[5]$. We list a few rank 9 examples over ${\mathbb Z}_3$ gerbes in section \[sect:otherexs-good\]. These rank 8 examples are listed in this section because we are enumerating rank 8 bundles over ${\mathbb Z}_2$ gerbes, and the rank 9 examples are not candidates for the dualities discussed here. Curiously, we were unable to find solutions of the combinatorial consistency conditions for GLSM’s for bundles of rank less than 8. We do not know whether this reflects a fundamental limitation of GLSM’s, or merely the inadequacy of our parameter space search. Given a Distler-Kachru model with a phase describing a Landau-Ginzburg model over an orbifold of a vector space, methods exist to compute the massless spectrum in that Landau-Ginzburg phase [@kw; @dk1]. When these methods are applied to, for example, heterotic ${\rm Spin}(32)/{\mathbb Z}_2$ compactifications on typical examples from the table above, we find a large number of single vectors and matter representations which likely combine to form representations of a larger nonabelian gauge symmetry, but unfortunately the corresponding worldsheet global symmetry does not seem to be visible in the UV. We conclude that in these examples, much of the needed worldsheet global symmetry appears in the IR, where we have no direct access. This is atypical of Distler-Kachru models, where spacetime gauge symmetries typically appear as worldsheet global symmetries visible in the UV, but is not contradicted by any physics we know. In any event, spectrum computations at Landau-Ginzburg points in these theories have not proven insightful. Class III: Twisted bundles {#sect:type3:twisted} ========================== The third fundamental class of examples we shall discuss involve cases in which the trivially-acting part of the gauge group acts nontrivially on the bundle, but is not one of the special cases discussed in section \[sect:het-gsomods\] in which the effect is merely to recreate part of the ten-dimensional left-moving GSO projection. One reason for interest is that examples of this form have the potential to define new heterotic string compactifications. Other reasons also exist, revolving around making sense of heterotic orbifolds with invariant non-equivariant bundles. We review such motivations in subsection \[sect:class3-motivations\]. In subsection \[sect:otherexs-good\], we describe some (indirect) constructions of (0,2) SCFT’s of this form, via dimensional reduction of consistent four-dimensional theories, and via anomaly-free (0,2) GLSM’s. Unfortunately, although there seem to exist consistent (0,2) SCFT’s, they do not seem to yield consistent perturbative heterotic string compactifications. The essential problem is that any finite group that acts only on left-movers, locally looks like a modification of the ten-dimensional left-moving GSO projection, and as the consistent ten-dimensional GSO projections are already known, if it is not one of them, the results cannot be well-behaved. We will give several examples of six-dimensional compactifications of this form, in which the six dimensional theory has anomalies and cannot be consistent. We outline in detail some examples in which heterotic string compactifications on these (0,2) SCFT’s break down in subsections \[sect:class3-caution1\], \[sect:class3-caution2\], and \[sect:class3-caution4\]. That said, it may sometimes be possible to restore these theories by adding suitable phases to twisted sectors. For example, the ten-dimensional nonsupersymmetric $SO(8) \times SO(24)$ string seems to be obtainable by a procedure along these lines. Specifically, in the worldsheet theory, if one takes the Spin$(32)/{\mathbb Z}_2$ string and performs an additional left-moving ${\mathbb Z}_2$ orbifold on 4 complex fermions, the result satisfies level-matching but does not define a modular-invariant theory; if one then adds phases to restore modular invariance, the result is the nonsupersymmetric ten-dimensional $SO(8) \times SO(24)$ string. (See [*e.g.*]{} [@dix-harv; @klt-ns], [@polv2]\[section 11.3\] for more information on this nonsupersymmetric string.) Unfortunately, we do not have a procedure for finding such phases (or even checking whether they exist), and if they do, the previous example suggests that the results will not be supersymmetric. In addition, see [*e.g.*]{} [@afiu; @afiuv] for a different set of ideas which may be relevant, though we have not considered them carefully in this context. In subsection \[sect:possible-anomcanc\] we outline a few attempts to find a way to understand these issues in terms of some sort of anomaly cancellation. Motivations {#sect:class3-motivations} ----------- One reason for interest in this class of examples is that they potentially could describe new (0,2) SCFT’s. Another reason to be interested in them is that they may give a way of understanding heterotic compactifications on ordinary spaces but with non-equivariant bundles. In this section we will explain this motivation. Let $X$ be a Calabi-Yau manifold, with stable holomorphic vector bundle ${\cal E} \rightarrow X$ satisfying anomaly cancellation, so that the pair $(X, {\cal E})$ defines a consistent large-radius heterotic Calabi-Yau compactification. Now, suppose a finite group $G$ acts on $X$. In order to construct a $G$-orbifold of the heterotic string on $(X, {\cal E})$, we need for the bundle ${\cal E}$ to admit a $G$-equivariantizable structure, which means that for every $g \in G$, we need a lift $\tilde{g}: {\cal E} \rightarrow {\cal E}$ such that $$\xymatrix{ {\cal E} \ar[r]^{\tilde{g}} \ar[d] & {\cal E} \ar[d] \\ X \ar[r]^{g} & X }$$ and also such that the lifts obey the group law: $\tilde{g} \circ \tilde{h} = \widetilde{gh}$. We need such an equivariant structure on the bundle ${\cal E}$ for the following two reasons: - In the worldsheet theory, such an equivariant structure enables us to define a group action on the worldsheet fermions/bosons describing the bundle, such that summing over twisted sectors in the orbifold yields an honest projection operator onto $G$-invariant states. - In the low-energy supergravity, if ${\cal E}$ does not have an equivariant structure, then even if $G$ acts freely, on the quotient $X/G$ the bundle ${\cal E}$ will descend to a ‘twisted’ bundle, not an honest bundle, whose transition functions $g_{\alpha \beta}$ obey $$g_{\alpha \beta} g_{\beta \gamma} g_{\gamma \alpha} \: = \: h_{\alpha \beta \gamma} I$$ on triple overlaps, and whose gauge field $A$ obeys $$A_{\beta} \: = \: g_{\alpha \beta} A_{\alpha} g_{\alpha \beta}^{-1} \: + \: g_{\alpha \beta}^{-1} d g_{\alpha \beta} \: + \: \Lambda_{\alpha \beta} I$$ across intersections, for some affine translation $\Lambda_{\alpha \beta}$. As ten-dimensional super-Yang-Mills only describes honest bundles and ordinary gauge transformations, the structure above cannot be used to define a consistent string compactification. However, there is a workaround. If the bundle ${\cal E}$ is invariant (meaning, its characteristic classes are invariant under the group action), but not equivariant, then we can find a larger group $\tilde{G}$, an extension of $G$ by a trivially-acting subgroup, such that ${\cal E}$ does admit a $\tilde{G}$-equivariant structure, and then take a $G'$ orbifold. This is precisely an example of a heterotic string on a gerbe, in this case a gerbe over $[X/G]$. First, let us review some generalities on the construction of $G'$. There is a ‘universal’, ‘maximal’ extension $\tilde{G}_{\rm max}$, which extends $G$ by the group of all automorphisms of the total space of ${\cal E}$ that cover the action of the elements of $G$ on $X$. It fits into a short exact sequence $$1 \: \longrightarrow \: {\rm Aut}({\cal E}) \: \longrightarrow \: \tilde{G}_{\rm max} \: \longrightarrow \: G \: \longrightarrow \: 1,$$ where ${\rm Aut}({\cal E})$ is the group of global bundle automorphisms of ${\cal E}$. The group we want, $\tilde{G}$, will necessarily be a subgroup of this universal extension of $\tilde{G}_{\rm max}$. In general, the extension defining $\tilde{G}_{\rm max}$ will not be central, but if ${\cal E}$ is stable or simple then ${\rm Aut}({\cal E}) = {\mathbb C}^{\times}$, and the extension is central. The group $\tilde{G}_{\rm max}$ acts by definition on ${\cal E}$ and so defines an equivariant structure. Every other group for which one has an equivariant structure will map to $\tilde{G}_{\rm max}$ and the equivariant structure will factor through that map. Now, clearly, $\tilde{G}_{\rm max}$ is not a finite group, and we only want to consider cases in which the trivially-acting subgroup is finite. If $G$ is finite and ${\cal E}$ is stable or simple, then $\tilde{G}_{\rm max}$ is a central extension of $G$ by ${\mathbb C}^{\times}$ and, because $$H^2(G,{\mathbb C}^*) \: = \: H^2(G,{\mathbb Q}/{\mathbb Z})$$ for $G$ finite, the relevant $H^2$ is finite and so every extension is induced from some central extension $G_{min}$ of $G$ by a finite group of order bounded by the maximal order of elements in $G$. In this fashion, we can construct a $\tilde{G}$. So far, we have described how, given a bundle that is invariant but not equivariant with respect to an orbifold group $G$, one can extend $G$ to a larger finite group $\tilde{G}$, where the extension acts trivially on the base. Now, not any $\tilde{G}$ will be acceptable: the orbifold by $\tilde{G}$ must, at minimum, satisfy level-matching, and as discussed earlier, even more in order to define a consistent heterotic string compactification. For completeness, let us now consider some possible examples. One example is described in the paper [@dopr]. (See also [@Donagi:2000zf; @Donagi:2000zs; @Donagi:2000fw; @Ovrut:2002jk; @Ovrut:2003zj; @Braun:2004xv].) In that paper, the authors first construct an elliptically-fibered Calabi-Yau threefold $Z$ with fundamental group ${\mathbb Z}_2 \times {\mathbb Z}_2$, built as a freely-acting[^15] ${\mathbb Z}_2 \times {\mathbb Z}_2$ quotient of a simply-connected Calabi-Yau threefold $X$: $$Z \: = \: X / ( {\mathbb Z}_2 \times {\mathbb Z}_2 )$$ together with a bundle $V$ on $X$ that is not quite equivariant with respect to the ${\mathbb Z}_2 \times {\mathbb Z}_2$ action, and so descends to a twisted bundle on $Z$. Consider the gerbe presented as $[X/G]$, where $$1 \: \longrightarrow \: {\mathbb Z}_2 \: \longrightarrow \: G \: \longrightarrow \: {\mathbb Z}_2 \times {\mathbb Z}_2 \: \longrightarrow \: 1,$$ with the ${\mathbb Z}_2$ kernel acting trivially. (Explicitly, the extension above is the Heisenberg extension, and $G = D_4$ [@tonypriv].) The bundle $V$ above descends to a bundle on a gerbe. Furthermore, the entire bundle is an eigenbundle under the nontrivial element of the center of $D_4$, with eigenvalue $-1$ (since it must square to the identity and can not itself be the identity) [@tonypriv]. For completeness, let us now work through the example of [@dopr] in more detail. Their Calabi-Yau manifold $X$ is an elliptic fibration over a rational elliptic surface, and in fact can be described as the fiber product over ${\mathbb P}^1$ of two rational elliptic surfaces $B$, $B'$: $$X \: = \: B \times_{ {\mathbb P}^1 } B'$$ where $\pi: X \rightarrow B'$, $\pi': X \rightarrow B$, $\beta': B' \rightarrow {\mathbb P}^1$, $\beta: B \rightarrow B$: $$\xymatrix{ & X \ar[rd]^{\pi} \ar[ld]_{\pi'} & \\ B \ar[rd]_{\beta} & & B' \ar[ld]^{\beta'} \\ & {\mathbb P}^1 & }$$ $B$ and $B'$ are both chosen to admit an automorphism group containing ${\mathbb Z}_2 \times {\mathbb Z}_2$. A stable rank four vector bundle $V \rightarrow X$ is constructed as an extension $$0 \: \longrightarrow \: V_1 \: \longrightarrow \: V \: \longrightarrow \: V_2 \: \longrightarrow \: 0,$$ where $$V_i \: = \: \pi'^* W_i \otimes \pi^* L_i,$$ for $W_i$ a pair of rank 2 vector bundles on $B$ and $L_i$ a pair of line bundles on $B'$. Briefly, [@dopr] first argues that each $V_i$ is ${\mathbb Z}_2\times {\mathbb Z}_2$-equivariant. As a result, the group of extensions ${\rm Ext}^1(V_2,V_1)$ decomposes into subspaces associated with characters of ${\mathbb Z}_2 \times {\mathbb Z}_2$. By picking an extension in a subspace associated with the trivial representation, we get a bundle $V$ which is at least ${\mathbb Z}_2 \times {\mathbb Z}_2$-invariant, though not necessarily ${\mathbb Z}_2 \times {\mathbb Z}_2$-equivariant. Again, for this example to be physically meaningful, the orbifold group would have to, at minimum, satisfy level-matching. As our purpose in this section was merely to outline one of the motivations for considering heterotic compactifications on gerbes, and we will argue later that these examples are, in most cases, not physically useful, we will end our discussion here. Constructions of consistent CFT’s {#sect:otherexs-good} --------------------------------- In this section, we will describe some constructions of what seem to be consistent (0,2) SCFT’s describing heterotic strings on gerbes with fractional gauge bundles. For reasons described elsewhere, these cannot be consistently used in supersymmetric heterotic string compactifications, but nevertheless they do seem to be examples of consistent (0,2) SCFT’s. Our first example was discussed in [@anton1]. Specifically, in [@anton1]\[section 3.1\], an ${\cal N}=2$ gauge theory in four dimensions with hypermultiplets transforming in the $R$ representation of the gauge group, was reduced along a Riemann surface $C$ to a two-dimensional $(0,4)$ theory, a heterotic nonlinear sigma model whose target is the Hitchin moduli space ${\cal M}_H(G,C)$ and with a twisted gauge bundle ${\cal R}$, defined by the representation $R$ in which the hypermultiplets transform. The four-dimensional theory was partially topologically twisted along a $U(1)_R$ (and only exists for superconformal field theories for which that $U(1)_R$ is nonanomalous). In this example, the gauge bundle is twisted, in the sense that the transition functions only close to a cocycle on triple overlaps. Now, ordinarily heterotic strings cannot couple to such twisted bundles, only D-branes can couple to such twisted bundles, as described in section \[sect:twisting\]. Despite that fact, it was claimed in [@anton1]\[section 3.1\] that the $(0,4)$ theory nevertheless consistently couples to a twisted gauge bundle. In order to make that possible, the nonlinear sigma model was restricted to maps such that the pullback of the twisted bundle, is an honest bundle. Such nonlinear sigma models, with a restriction on nonperturbative sectors, are equivalent to sigma models on gerbes, as reviewed in [*e.g.*]{} section \[sect:generalreview\], and so this is an example of a heterotic string compactification on a gerbe with a non-pullback bundle. More generally, more of the analysis of [@anton1] can also be rephrased in this language, following a discussion in [@summ]\[section 12.3\], which discussed how gerbes could be used to slightly simplify the analysis of the physical realization of geometric Langlands. Briefly, Hitchin moduli spaces arising from $G$ gauge theories are defined by modding out adjoint actions, under which the center $Z(G)$ is trivial and so formally one can replace them with moduli stacks which are $Z(G)$-gerbes. After reduction to two dimensions, one obtains sigma models on gerbes, which physics sees [@summ] as a sigma model on a disjoint union of spaces, matching results of [@edanton]. In any event, after performing the dimensional reduction from a four-dimensional ${\cal N}=2$ theory to two dimensions, one gets [@anton1]\[section 3.1\] a heterotic sigma model on the Hitchin moduli space ${\cal M}_H(G,C)$, with a twisted bundle over that moduli space, twisted by an element of $H^2(Z(G))$. Since the Hitchin moduli space is defined by modding out the adjoint action of $G$, the center is trivial, and so one could naturally replace the Hitchin moduli space with a moduli stack which is a $Z(G)$ gerbe, just as in [@summ]\[section 12.3\]. A heterotic sigma model on such a stack would appear to be a sigma model on the moduli space but with a restriction on allowed maps, exactly as described in [@anton1]\[section 3.1\]. As these two-dimensional (0,2) theories are constructed by dimensional reduction of a consistent four-dimensional theory, it is difficult to believe that they are not consistent. Other naively-consistent examples can be constructed in (0,2) GLSM’s. For example, consider the two examples: - The rank 9 bundle $$0 \: \longrightarrow \: {\cal E} \: \longrightarrow \: \bigoplus_1^9 {\cal O}(1) \oplus {\cal O}(10) \: \longrightarrow \: {\cal O}(19) \: \longrightarrow \: 0$$ over ${\mathbb P}^4_{[3,3,3,3,6]}[18]$, a ${\mathbb Z}_3$ gerbe over ${\mathbb P}^4_{[1,1,1,1,2]}[6]$, - The rank 9 bundle $$0 \: \longrightarrow \: {\cal E} \: \longrightarrow \: \bigoplus_1^9 {\cal O}(1) \oplus {\cal O}(13) \: \longrightarrow \: {\cal O}(22) \: \longrightarrow \: 0$$ over ${\mathbb P}^3_{[3,3,6,9]}[21]$, a ${\mathbb Z}_3$ gerbe over ${\mathbb P}^3_{[1,1,2,3]}[7]$. It is straightforward to check, just at the level of combinatorics, that they satisfy the usual conditions for a GLSM to be anomaly-free. However, the usual danger with GLSM’s is that we do not have perfect control over the RG flow – although we have described them in terms of data associated to twisted bundles, along the RG flow they might pick up ‘phases’ (as suggested earlier), for example. In the next subsections, we shall show explicitly that examples of this form do not yield consistent supersymmetric heterotic string compactifications, unfortunately. Cautionary example {#sect:class3-caution1} ------------------ Let ${\cal E}$ be a rank 4 bundle on a Calabi-Yau $X$, defining a consistent (0,2) SCFT. Now, consider a ${\mathbb Z}_2$ orbifold in which the orbifold group acts trivially on $X$, but by a sign flip on ${\cal E}$ (so that all of ${\cal E}$ is an eigenbundle of weight $-1$). This example can be shown to satisfy level-matching in the sense of [@freedvafa], as well as the conditions in appendix \[app:spectra:fockconstraints\]. However, in principle this theory is nevertheless deeply suspicious. Since the ${\mathbb Z}_2$ acts trivially on right-moving fields, and only on left-moving fields, we could just as well think of this as a compactification of a ten-dimensional theory in which the left-moving GSO projection has been altered. Since the resulting new GSO does not coincide with either the existing Spin$(32)/{\mathbb Z}_2$ or $E_8 \times E_8$ strings, this theory must be inconsistent. (Indeed, this is the starting point for one construction of the ten-dimensional nonsupersymmetric $SO(8) \times SO(24)$ string [@dix-harv; @klt-ns; @polv2], though this orbifold must be supplemented by further phases.) Another argument for inconsistency arises from considering massless spectra. Specifically, if we take $X$ to be a K3 surface, and consider a compactification of a ten-dimensional $E_8 \times E_8$ string, in which the gauge bundle is embedded in one $E_8$, then the six-dimensional spectrum is anomalous. We summarize the details below, following the methods outlined in appendix \[app:spectra\]. (The integer $n$ is the dimension of $X$; we will quickly specialize to $n=2$, but will remain general for as long as possible.) Following the appendix, there are two components in the inertia stack, which are identical: $$I_{\mathfrak{X}} \: = \: \mathfrak{X} \amalg \mathfrak{X}.$$ Below we list results for both states and left and right $U(1)_R$ charges. First, consider the untwisted sector. In the (R,R) sector, the vacuum energy $E_{{\rm Id}} = 0$. The massless charged states are - $H^m(X, \wedge^{\rm even} {\cal E})$, charge $({\rm even}-2, m-n/2)$, giving spacetime states valued in a spinor of $so(8)$. In the (NS,R) sector, the vacuum energy $E_{{\rm Id}} = -1$. The massless charged states are - $H^m(X, {\cal E}^* \otimes {\cal E})$, charge $(0,m-n/2)$, spacetime gauge neutral, - $H^m(X, \wedge^2 {\cal E})$, charge $(2,m-n/2)$, spacetime gauge neutral, - $H^m(X, {\cal O})$, charge $(0,m-n/2)$, in the adjoint representation of $so(8)$, - $H^m(X, \wedge^2 {\cal E}^*)$, charge $(-2,m-n/2)$, spacetime gauge neutral. Now, consider the twisted sector. Here, all of ${\cal E}$ is an eigenbundle with eigenvalue $-1$. In the (R,R) sector, $E = -1/2$. There are no massless charged states in this sector. In the (NS,R) sector, $E = -1/2$. Again, there are no massless charged states in this sector. States above are listed with charges $(q_-,q_+)$. The $q_+$ charge distinguishes chiral multiplets from vector multiplets; the $q_-$ charge is the charge of the $u(1)$ that combines with $so(8)$ to build $so(10)$. For a compactification to four dimensions, ($n=3$,) states with $q_+ = -1/2$ would be spacetime fermions in chiral multiplets (and $q_+ = +1/2$ their antichiral partners); states with $q_+ = +3/2$ would be spacetime fermions in vector multiplets (and $q_+ = -3/2$ their partners). For a compactification to six dimensions, ($n=2$,) which is the pertinent case, states with $q_+ = \pm 1$ are spacetime fermions in vector multiplets; states with $q_+ = 0$ are spacetime fermions in hypermultiplets. Since we have a rank 4 bundle, in principle the $E_8$ should be broken to ${\rm Spin}(10)$, which in the worldsheet theory will be assembled from representations of $so(8) \times u(1)$ (the $so(8)$ rotating the remaining free left-moving fermions in the first $E_8$, and the $u(1)$ being an overall phase rotation on the bundle fermions, which on the (2,2) locus would become the left R symmetry). Under the $so(8)\times u(1)$ subalgebra, representations of $so(10)$ decompose as follows: $$\begin{aligned} {\bf 45} & = & {\bf 8}_{-2} \oplus {\bf 28}_0 \oplus {\bf 1}_0 \oplus {\bf 8}_2, \\ {\bf 16} & = & {\bf 8}_{-1} \oplus {\bf 8}_{+1}, \\ {\bf 10} & = & {\bf 1}_{-2} \oplus {\bf 8}_0 \oplus {\bf 1}_{2}, \\ {\bf 1} & = & {\bf 1}_0,\end{aligned}$$ where the subscript indicates the $q_-$ charge. We arrange the (untwisted sector) states into $so(10)$ representations, with the following results: - The adjoint of $so(10)$ arises from $H^*(X, {\cal O})$. Contributing terms are: - $H^*(X, {\cal O})$ in (R,R), transforming as ${\bf 8}_{-2}$, - $H^*(X, \wedge^4 {\cal E} \cong {\cal O})$ in (R,R), transforming as ${\bf 8}_{+2}$, - $H^*(X, {\rm Tr} \, {\cal E}^* \otimes {\cal E} \cong {\cal O})$ in (NS,R), transforming as ${\bf 1}_0$, - $H^*(X, {\cal O})$ in (NS,R), transforming as ${\bf 28}_0$. - Copies of ${\bf 10}$ of $so(10)$ arise from $H^*(X, \wedge^2 {\cal E})$. Contributing terms are: - $H^*(X, \wedge^2 {\cal E})$ in (R,R), transforming as ${\bf 8}_0$, - $H^*(X, \wedge^2 {\cal E})$ in (NS,R), transforming as ${\bf 1}_2$, - $H^*(X, \wedge^2 {\cal E}^* \cong \wedge^2 {\cal E})$ in (NS,R), transforming as ${\bf 1}_{-2}$. - Gauge singlets, arising as $H^*(X, {\rm End} \, {\cal E})$ (where we use End to denote the traceless endomorphisms), arising in the (NS,R) sector. In addition, there is one vector in the adjoint representation of the second $E_8$, which is always present in computations of the form of appendix \[app:spectra\]. In any event, altogether in this six-dimensional theory we have $h^0(X, {\cal O})=1$ vector multiplets in the adjoint of ${\rm Spin}(10)$, One vector multiplet in the adjoint of $E_8$, $h^1(X, \wedge^2 {\cal E}) = 36$ half-hypermultiplets[^16] in the ${\bf 10}$ of ${\rm Spin}(10)$, 20 singlet hypermultiplets for $K3$ moduli, $h^1({\rm End}\, {\cal E}) = 162$ singlet half-hypermultiplets for bundle moduli[^17], so that we find $$\begin{aligned} n_V & = & 45 \: + \: 248 \: = \: 293, \\ n_H & = & (1/2)\left( (36)(10) \: + \: 162 \right) \: + \: 20 \: = \: 281, \\ n_H - n_V & = & -12 \: \neq \: 244,\end{aligned}$$ and so we see that this cannot satisfy anomaly cancellation, mechanically verifying our previous observation that this theory cannot be consistent. More generally, any heterotic compactification on a gerbe, in which the bundle is twisted, will be of this same general type, unless the bundle has rank 8 and the trivially-acting group is ${\mathbb Z}_2$. Locally each theory will look like a compactification of a ten-dimensional theory with an altered GSO projection, and except for the case that the GSO projection switches between Spin$(32)/{\mathbb Z}_2$ and $E_8 \times E_8$, the resulting theory cannot be consistent. For purposes of comparison, and to help illuminate the methods encoded in appendix \[app:spectra\], let us also outline the results in a closely related consistent compactification. If we did not orbifold, if we took a compactification of an $E_8 \times E_8$ heterotic string on a smooth large-radius $K3$ with a rank 4 vector bundle, then from a similar computation we would find $h^0(X, {\cal O})=1$ vector multiplets in the adjoint of ${\rm Spin}(10)$, One vector multiplet in the adjoint of $E_8$, $h^1(X, {\cal E}) = 16$ half-hypermultiplets in the ${\bf 16}$ of ${\rm Spin}(10)$, $h^1(X, \Lambda^2 {\cal E}) = 36$ half-hypermultiplets in the ${\bf 10}$ of ${\rm Spin}(10)$, $h^1(X, \Lambda^3 {\cal E} = {\cal E}^*) = 16$ half-hypermultiplets in the ${\bf 16}$ of ${\rm Spin}(10)$, 20 singlet hypermultiplets for $K3$ moduli, $h^1({\rm End}\, {\cal E}) = 162$ singlet half-hypermultiplets for bundle moduli, where the representations of ${\rm Spin}(10)$ are constructed in the same fashion. Altogether, we find that $$\begin{aligned} n_V & = & 45 \: + \: 248 \: = \: 293, \\ n_H & = & (1/2)\left( (16)(16) \: + \: (36)(10) \: + \: (16)(16) \: + \: 162 \right) \: + \: 20 \: = \: 537, \\ n_H - n_V & = & 244,\end{aligned}$$ consistent with anomaly cancellation, in that standard compactification. Unfortunately, our gerbe example is not so well-behaved. Second cautionary example {#sect:class3-caution2} ------------------------- For completeness, we give here a second cautionary example, here involving a heterotic Spin$(32)/{\mathbb Z}_2$ compactification on a nontrivial toroidal orbifold. This will involve a rank 10 bundle over a ${\mathbb Z}_2$ gerbe on $[T^4/{\mathbb Z}_2]$, and although level matching holds, the spectrum is anomalous in six dimensions. The ${\mathbb Z}_2$ gerbe is defined by $[T^4/{\mathbb Z}_4]$, where the ${\mathbb Z}_4$ acts on the $T^4$ by $$x \: \mapsto \: \exp\left( \frac{2\pi i (2k)k}{4} \right) x \: = \: (-)^k x,$$ so that there is a trivially-acting ${\mathbb Z}_2$ subgroup. (This is the same ${\mathbb Z}_2$ gerbe discussed in a different context in section \[sect:class2-ex1\].) The gauge bundle is a rank 10 bundle, where the generator of ${\mathbb Z}_4$ acts on an ${\cal O}^{\oplus 2}$ factor by multiplication by $\exp(2 \pi i (2/4) ) = -1$, and on the ${\cal O}^{\oplus 8}$ factor by $\exp(2 \pi i /4)$. It is straightforward to check that this satisfies level-matching, in the sense of [@freedvafa], as well as the conditions in appendix \[app:spectra:fockconstraints\]. Let us now outline the massless spectrum. In the untwisted sector, there are massless states in the (NS,NS) sector. It is straightforward to compute $E_{\rm left} = -1$, $E_{\rm right} = -1/2$, and one has ${\mathbb Z}_4$-invariant states of the form --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- State Count ----------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------- $\left( \lambda^{1-6}_{-1/2}, \overline{\lambda}^{1-6}_{-1/2} \right)^2 spacetime vector, \otimes \left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2} \right)$ valued in adjoint of $so(12)$ $\overline{\partial} X^{1-2}_{-1} \otimes gravity, tensor multiplet contributions \left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2} \right)$ $\left( \lambda^{7-14}_{-1/2} \overline{\lambda}^{7-14}_{-1/2} \right) spacetime vector, \otimes \left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2} \right)$ valued in adjoint, ${\bf 1}$ (trace) of $su(8)$ $\left( \lambda^{15-16}_{-1/2}, \overline{\lambda}^{15-16}_{-1/2} \right)^2 spacetime vector, \otimes \left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2} \right)$ valued in adjoint of $so(4)$ $\left( \lambda^{1-6}_{-1/2}, \overline{\lambda}^{1-6}_{-1/2} \right) 4 sets of scalars, \left( \lambda^{15-16}_{-1/2}, \overline{\lambda}^{15-16}_{-1/2} \right) \otimes \left( \psi^{3-4}_{-1/2}, \overline{\psi}^{3-4}_{-1/2} \right)$ valued in $({\bf 12},{\bf 4})$ of $so(12) \times so(4)$ $\overline{\partial} X^{3-4}_{-1} 16 scalars (toroidal moduli) \otimes \left( \psi^{3-4}_{-1/2}, \overline{\psi}^{3-4}_{-1/2} \right)$ $\left( \left( \lambda^{7-14}_{-1/2} \right)^2, 4 sets of scalars, \left( \overline{\lambda}^{7-14}_{-1/2} \right)^2 \right) \otimes \left( \psi^{3-4}_{-1/2}, \overline{\psi}^{3-4}_{-1/2} \right)$ valued in $\wedge^2 {\bf 8} = {\bf 28}$, $\wedge^2 {\bf \overline{8}} = {\bf \overline{28}}$ of $su(8)$ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- There are no massless states in the untwisted (R,NS) sector, and in fact also no massless states in the $k=1$ or $k=3$ sectors. All of the remaining massless states are in the $k=2$ sector. In the (NS,NS) sector, fields have the following boundary conditions: $$\begin{aligned} X^{1-2}(\sigma + 2 \pi) & = & + X^{1-2}(\sigma), \\ X^{3-4}(\sigma + 2 \pi) & = & + X^{3-4}(\sigma), \\ \psi^{1-2}(\sigma + 2 \pi) & = & - \psi^{1-2}(\sigma), \\ \psi^{3-4}(\sigma + 2 \pi) & = & - \psi^{3-4}(\sigma), \\ \lambda^{1-6}(\sigma + 2 \pi) & = & - \lambda^{1-6}(\sigma), \\ \lambda^{7-14}(\sigma + 2 \pi) & = & - \exp\left( 2 \pi i \frac{2}{4} \right) \lambda^{7-14}(\sigma) \: = \: + \lambda^{7-14}(\sigma), \\ \lambda^{15-16}(\sigma + 2 \pi) & = & - \lambda^{15-16}(\sigma).\end{aligned}$$ It is straightforward to compute that $E_{\rm left} = 0$, $E_{\rm right} = -1/2$. There is a multiplicity of left vacua, arising from $\lambda^{7-14}$. Let $|m\rangle$ denote a vacuum with $m$ +’s and $8-m$ -’s, [*i.e.*]{} annihilated by $m$ $\lambda$’s and $8-m$ $\overline{\lambda}$’s, then under the action of the generator of ${\mathbb Z}_4$, it is straightforward to check that $|m=0,4,8\rangle$ are invariant, $|m=2,6\rangle$ get a sign flip, and the others are multiplied by various fourth roots of unity. The ${\mathbb Z}_4$-invariant states in this sector are of the form ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- State Count --------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------- $|m=0,4,8 \rangle \otimes spacetime vector, valued in ${\bf 1}$, ${\bf 1}$, $\wedge^4 {\bf 8} \left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2} = {\bf 70}$ of $su(8)$ \right)$ $|m=6,2\rangle \otimes 4 sets of scalars, in $\wedge^2 {\bf 8} = {\bf 28}$, $\wedge^2 {\bf \overline{8}} = {\bf \overline{28}}$ of $su(8)$ \left( \psi^{3-4}_{-1/2}, \overline{\psi}^{3-4}_{-1/2} \right)$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- In the $k=2$ (R,NS) sector, fields have the following boundary conditions: $$\begin{aligned} X^{1-2}(\sigma + 2 \pi) & = & + X^{1-2}(\sigma), \\ X^{3-4}(\sigma + 2 \pi) & = & + X^{3-4}(\sigma), \\ \psi^{1-2}(\sigma + 2 \pi) & = & - \psi^{1-2}(\sigma), \\ \psi^{3-4}(\sigma + 2 \pi) & = & - \psi^{3-4}(\sigma), \\ \lambda^{1-6}(\sigma + 2 \pi) & = & + \lambda^{1-6}(\sigma), \\ \lambda^{7-14}(\sigma + 2 \pi) & = & + \exp\left( 2 \pi i \frac{2}{4} \right) \lambda^{7-14}(\sigma) \: = \: - \lambda^{7-14}(\sigma), \\ \lambda^{15-16}(\sigma + 2 \pi) & = & + \lambda^{15-16}(\sigma).\end{aligned}$$ It is straightforward to compute that $E_{\rm left} = 0$, $E_{\rm right} = -1/2$. There is a multiplicity of left vacua, as $\lambda^{1-6}$ and $\lambda^{15-16}$ are periodic. In particular, $|\pm \mp \rangle_{15-16}$ are invariant under ${\mathbb Z}_4$, whereas $|\pm \pm \rangle_{15-16}$ get a sign flip. Therefore, the ${\mathbb Z}_4$-invariant massless states are of the form ------------------------------------------------------------------------------------------------------------- State Count --------------------------------------------------------- --------------------------------------------------- $| \pm \cdots \pm \rangle_{1-6} spacetime vector | \pm \mp \rangle_{15-16} \otimes \left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2} \right)$ in $({\bf 32},{\bf 2})$ of $so(12)\times so(4)$ $| \pm \cdots \pm\rangle_{1-6} |\pm \pm \rangle_{15-16} 4 sets of scalars \otimes \left( \psi^{3-4}_{-1/2}, \overline{\psi}^{3-4}_{-1/2} \right)$ in $({\bf 32}',{\bf 2}')$ of $so(12)\times so(4)$ ------------------------------------------------------------------------------------------------------------- We can rearrange the spacetime vectors more sensibly as follows. The $so(12)\times so(4) \cong so(12) \times su(2) \times su(2)$ is enhanced to an $e_7 \times su(2)$, using the fact that the adjoint representation of $e_7$ decomposes under $so(12) \times su(2)$ as [@slansky]\[table 52\] $${\bf 133} \: = \: ({\bf 66},{\bf 1}) \oplus ({\bf 32},{\bf 2}) \oplus ({\bf 1},{\bf 3}).$$ The ${\bf 66}$ is the adjoint representation of $so(12)$, which arises in $k=0$, as does the ${\bf 3}$ of $su(2)$ (half of the adjoint representation of $so(4)$), and the $({\bf 32},{\bf 2})$ arises in the sector $k=2$. Similarly, the $su(8)$ is enhanced to $e_7$. The adjoint representation of $e_7$ decomposes under $su(8)$ as [@slansky]\[table 52\] $${\bf 133} \: = \: {\bf 63} \oplus {\bf 70}.$$ The ${\bf 63}$ arises in the $k=0$ sector, and the ${\bf 70}$ in $k=2$. In addition, there are three remaining vector multiplets, in the $k=0$ and $k=2$ sectors. Therefore, the complete gauge (algebra) symmetry in this compactification is $e_7 \times e_7 \times su(2) \times u(1)^3$. The matter fields align themselves with the gauge algebra above. In the $k=0$ and $k=2$ sectors, the hypermultiplets valued in ${\bf 28}$, ${\bf \overline{28}}$ of $su(8)$ form hypermultiplets in the ${\bf 56}$ of $e_7$, using the fact that under the $su(8)$ subalgebra [@slansky]\[table 52\], $${\bf 56} \: = \: {\bf 28} \oplus {\bf \overline{28}}.$$ Similarly, since under the $so(12) \times su(2)$ subalgebra [@slansky]\[table 52\], $${\bf 56} \: = \: ({\bf 32}',{\bf 1}) \oplus ({\bf 12},{\bf 2}),$$ the $k=0$ hypermultiplet valued in $({\bf 12},{\bf 4})$ of $so(12)\times so(4)$ and the $k=2$ hypermultiplet valued in $({\bf 32}',{\bf 2}')$ form a hypermultiplet valued in $({\bf 56},{\bf 2})$ of $e_7 \times su(2)$. Let us summarize our results so far. We have found the following states: 1 gravity multiplet, 1 tensor multiplet, 1 vector multiplet in the adjoint representation of $e_7 \times e_7 \times su(2) \times u(1)^3$, 2 hypermultiplets in the $({\bf 56},{\bf 1},{\bf 1})$ of $e_7 \times e_7 \times su(2)$, 1 hypermultiplet in the $({\bf 1},{\bf 56},{\bf 2})$ of $e_7 \times e_7 \times su(2)$, 4 singlet hypermultiplets. It is straightforward to compute that there are 272 vector multiplets and 228 hypermultiplets. Since the difference is not 244, this six-dimensional theory is anomalous. Third cautionary example {#sect:class3-caution4} ------------------------ Now consider an $E_8 \times E_8$ string on a ${\mathbb Z}_3$ gerbe over a different $[T^4/{\mathbb Z}_2]$, constructed as $[T^4/{\mathbb Z}_6]$. Let the generator $g$ of ${\mathbb Z}_6$ act on the $T^4$ with coordinates $(X^3, X^4)$ as $$g: \: \left( X^3, X^4 \right) \: \mapsto \: \left( \exp(+4 \pi i/3), \exp(-4 \pi i/3) \right).$$ Define a rank 2 bundle over this stack by taking ${\cal O}^{\oplus 2}$ over $T^4$, and let $g$ act with eigenvalues $$\left( \exp(-2 \pi i/3), \exp(-4 \pi i/3) \right).$$ It is straightforward to check that this satisfies anomaly cancellation in the sense of [@freedvafa], and also the constraints in appendix \[app:spectra:fockconstraints\]. In an $E_8 \times E_8$ compactification, we can describe this as the following action on fields: $$\begin{aligned} g \cdot X^{1-2} & = & + X^{1-2}, \\ g \cdot X^3 & = & \exp(+4 \pi i/3) X^3, \\ g \cdot X^4 & = & \exp(-4 \pi i/3) X^4, \\ g \cdot \psi^{1-2} & = & + \psi^{1-2}, \\ g \cdot \psi^{3} & = & \exp(+ 4 \pi i/3) \psi^3, \\ g \cdot \psi^4 & = & \exp(-4 \pi i/3) \psi^4, \\ g \cdot \lambda^{1-6} & = & + \lambda^{1-6}, \\ g \cdot \lambda^7 & = & \exp(-2 \pi i/3) \lambda^7, \\ g \cdot \lambda^8 & = & \exp(-4 \pi i/3) \lambda^8.\end{aligned}$$ Let us now outline the massless spectrum. In the untwisted sector, there are massless states in the (NS,NS) sector. It is straightforward to compute that $E_{\rm left} = -1$, $E_{\rm right} = -1/2$, and one has ${\mathbb Z}_6$-invariant states of the form -------------------------------------------------------------------------------------------------------------------------------- State Count -------------------------------------------------------------------------------- ----------------------------------------------- $\overline{\partial} X^{1-2}_{-1} \otimes \left( \psi^{1-2}_{-1/2}, gravity, tensor multiplet contributions \overline{\psi}^{1-2}_{-1/2} \right)$ $\left( \lambda^{1-6}_{-1/2}, \overline{\lambda}^{1-6}_{-1/2} \right)^2 vector in adjoint of $so(12)$ \otimes \left( \psi^{1-2}_{-1/2},\overline{\psi}^{1-2}_{-1/2} \right)$ $\left( \lambda^7_{-1/2} \lambda^8_{-1/2}, vectors in adjoint of $U(1)^2$ \overline{\lambda}^7_{-1/2} \overline{\lambda}^8_{-1/2} \right) \otimes \left( \psi^{1-2}_{-1/2},\overline{\psi}^{1-2}_{-1/2} \right)$ $\left( \lambda^7_{-1/2} \overline{\lambda}^7_{-1/2}, vectors in adjoint of $U(1)^2$ \lambda^8_{-1/2} \overline{\lambda}^8_{-1/2} \right) \otimes \left( \psi^{1-2}_{-1/2},\overline{\psi}^{1-2}_{-1/2} \right)$ $\lambda^8_{-1/2}\left( \lambda^{1-6}_{-1/2}, \overline{\lambda}^{1-6}_{-1/2} half-hypermultiplet in ${\bf 12}$ of $so(12)$ \right) \otimes \left( \psi^3_{-1/2}, \overline{\psi}^4_{-1/2} \right)$ $\overline{\lambda}^8_{-1/2} \left( \lambda^{1-6}_{-1/2}, half-hypermultiplet in ${\bf 12}$ of $so(12)$ \overline{\lambda}^{1-6}_{-1/2}\right) \otimes \left( \overline{\psi}^3_{-1/2}, \psi^4_{-1/2} \right)$ $\lambda^7_{-1/2} \left( \lambda^{1-6}_{-1/2}, \overline{\lambda}^{1-6}_{-1/2} half-hypermultiplet in ${\bf 12}$ of $so(12)$ \right) \otimes \left( \overline{\psi}^3_{-1/2}, \psi^4_{-1/2} \right)$ $\overline{\lambda}^7_{-1/2}\left( half-hypermultiplet in ${\bf 12}$ of $so(12)$ \lambda^{1-6}_{-1/2}, \overline{\lambda}^{1-6}_{-1/2} \right) \otimes \left( \psi^3_{-1/2}, \overline{\psi}^4_{-1/2} \right)$ $\left( \overline{\partial} X^3, \overline{\partial} \overline{X}^4 \right) 1 singlet hypermultiplet \otimes \left( \overline{\psi}^3_{-1/2}, \psi^4_{-1/2} \right)$ $\left( \overline{\partial} \overline{X}^3, \overline{\partial} X^4 \right) 1 singlet hypermultiplet \otimes \left( \psi^3_{-1/2}, \overline{\psi}^4_{-1/2} \right)$ $\lambda^7_{-1/2} \overline{\lambda}^8_{-1/2} \otimes \left( 1/2 singlet hypermultiplet \psi^3_{-1/2}, \overline{\psi}^4_{-1/2} \right)$ $\overline{\lambda}^7_{-1/2} \lambda^8_{-1/2} \otimes \left( 1/2 singlet hypermultiplet \overline{\psi}^3_{-1/2}, \psi^4_{-1/2} \right)$ -------------------------------------------------------------------------------------------------------------------------------- There are no massless states in the untwisted (R,NS) sector, and no massless states in $k=1$, $k=2$ sectors. The $k=3$, $4$, $5$ sectors are copies of the $k=0$, $1$, $2$ sectors, respectively. Thus, altogether, the spectrum is two copies of the states above. Note that since there are no (R,NS) states, the nonabelian gauge symmetry is only $so(12)$; it is not enhanced to $e_7$. Also, since the spectrum is two copies of the states above, the spectrum contains two gravitons, and hence would be a likely candidate for decomposition. Unfortunately, the spectrum is also anomalous. The gauge symmetry is $so(12) \times e_8 \times u(1)^4$ (including the second $E_8$, which until now has been suppressed), so the total number of vector multiplets is 318, and the number of hypermultiplets is 27. Clearly $n_H - n_V \neq 244$, so this model is anomalous in six dimensions. Potential refinements of anomaly cancellation {#sect:possible-anomcanc} --------------------------------------------- So far we have described some consistent (0,2) SCFT’s of the class III form, and also illustrated in detail how theories of this form cannot be consistently used in supersymmetric heterotic string compactifications. This begs the question of whether there exists a criterion, perhaps a generalization of anomaly cancellation, that can be used to distinguish theories of this form. In this section, we will examine one such possibility. In appendix \[app:chern-reps\] we describe a modified notion of Chern classes and characters, labelled $c^{\rm rep}$ and ${\rm ch}^{\rm rep}$, that contain extra information in twisted sectors. It is tempting to speculate that one might be able to use these to obtain additional finite-group anomaly constraints on theories by demanding matching ${\rm ch}_2^{\rm rep}$’s. Let us check this by studying GLSM’s, for which anomaly cancellation conditions are more or less well understood. We will argue that although ${\rm ch}^{\rm rep}$’s play a vital role in index theory, confusingly they do not seem to define any new anomaly-cancellation conditions. Consider a (0,2) theory over the hypersurface $\mathfrak{X} = {\mathbb P}^n_{[k,k,\cdots,k]}[d]$, with gauge bundle ${\cal E}$: $$0 \: \longrightarrow \: {\cal E} \: \longrightarrow \: \oplus_a {\cal O}(n_a) \: \longrightarrow \: {\cal O}(m) \: \longrightarrow \: 0.$$ It is straightforward to compute that $$c_1^{\rm rep}(T\mathfrak{X})|_{\alpha} \: = \: (n+1) \frac{k}{k} J \: - \: \frac{d}{k} \alpha^{-d} J,$$ $$\begin{aligned} {\rm ch}_2^{\rm rep}(T\mathfrak{X})|_{\alpha} & = & {\rm ch}_2^{\rm rep}( \oplus_{n+1} {\cal O}(k) )|_{\alpha} \: - \: {\rm ch}_2^{\rm rep}( {\cal O}(d) )|_{\alpha}, \\ & = & \frac{1}{2} (n+1) \left( \frac{k}{k} J \right)^2 \: - \: \frac{1}{2} \left( \frac{d}{k} J \right)^2 \alpha^{-d},\end{aligned}$$ and for the bundle ${\cal E}$, $$c_1^{\rm rep}({\cal E})|_{\alpha} \: = \: \sum_a \frac{n_a}{k} J \alpha^{-n_a} \: - \: \frac{m}{k} J \alpha^{-m},$$ $$\begin{aligned} {\rm ch}_2^{\rm rep}({\cal E})|_{\alpha} & = & {\rm ch}_2^{\rm rep}( \oplus_a {\cal O}(n_a) )|_{\alpha} \: - \: {\rm ch}_2^{\rm rep}( {\cal O}(m) )|_{\alpha}, \\ & = & \frac{1}{2} \sum_a \left( \frac{n_a}{k} J \right)^2 \alpha^{-n_a} \: - \: \frac{1}{2} \left( \frac{m}{k} J \right)^2 \alpha^{-m}.\end{aligned}$$ By contrast, anomaly cancellation in the GLSM is merely the statement that $$\sum_a n_a^2 \: - \: m^2 \: = \: (n+1) k^2 \: - \: d^2,$$ a much weaker statement than demanding ${\rm ch}_2^{\rm rep}({\cal E}) = {\rm ch}_2^{\rm rep}(T \mathfrak{X})$ in each sector $\alpha$. Anomaly cancellation in the GLSM is well-understood – in the present case, this is just the gauge anomaly in a $U(1)$ gauge theory, which is under extremely good control. Demanding matching ${\rm ch}^{\rm rep}$’s gives a stronger condition – some theories that would satisfy GLSM anomaly cancellation, would not satisfy the constraint of matching ${\rm ch}^{\rm rep}$’s. For this reason, we do not believe that one should demand matching ${\rm ch}_2^{\rm rep}$’s. This is a somewhat puzzling conclusion, as these are not only the most natural notion of Chern classes on stacks, but they are also vital in index theory, which ordinarily would be a route to [*deriving*]{} their utility. (On the other hand, we briefly remark on a possible application of $c_1^{\rm rep}$ in appendix \[app:spectra:fockconstraints\].) Combinations {#sect:combos} ============ So far we have discussed three fundamental classes of examples of heterotic string compactifications on gerbes. Those three classes do not exhaust all possibilities; rather, one should think of them as ‘building blocks’ that can be used to assemble more complicated possibilities. For one example, consider a string on a ${\mathbb Z}_4$ gerbe, of which a ${\mathbb Z}_2$ subgroup acts on a rank 8 bundle, but the ${\mathbb Z}_2$ coset leaves the bundle invariant. A version of the decomposition conjecture should apply here, relating this (0,2) SCFT to a disjoint union of two (0,2) SCFT’s, each of which would involve a heterotic string on a ${\mathbb Z}_2$ gerbe with a nontrivial action on the gauge bundle. Those individual SCFT’s might be dual to a different string compactification (class II), or might not define a consistent heterotic string compactification (class III). It is straightforward to assemble more complicated possibilities, following similar patterns. Conclusions =========== In this paper we have examined general aspects of heterotic string compactifications on generalized spaces known as stacks, focusing on the particularly interesting special case of stacks that are gerbes. Briefly, we have described how heterotic string compactifications on gerbes are built from three basic classes: - In the special case that the gauge bundle on the gerbe is a pullback from an underlying space, the heterotic theory on the gerbe is equivalent to a heterotic theory on a disjoint union of spaces, the same sort of decomposition as type II strings on gerbes [@summ]. - In the special case that the gauge bundle on the gerbe is twisted in such a way as to locally duplicate a different ten-dimensional GSO projection, the gerbe compactification seems to be dual to a compactification of the corresponding different heterotic string. - In other cases in which the gauge bundle is different from a pullback from the base, although at least sometimes one can define consistent (0,2) SCFT’s, there do not seem to be any viable perturbative heterotic string compactifications. There are several open questions that would be interesting to pursue. For example, - We have not identified a complete set of sufficient conditions for a stack $\mathfrak{X}$ with bundle ${\cal E} \rightarrow \mathfrak{X}$ to define a consistent heterotic string compactification. We have identified a number of necessary conditions, such as anomaly cancellation on the cover and level-matching in orbifolds, we have derived additional necessary conditions from well-definedness of Fock vacua, but we have also observed that these conditions do not suffice in general. We have speculated on some enhancements of anomaly cancellation (involving the ${\rm ch}^{\rm rep}$’s that can be defined on stacks), but do not at this time have any definitive statements to make. - We have discussed a heterotic analogue of the decomposition conjecture for banded gerbes, with bundle a pullback from the base. We do not at this time have an analogue for nonbanded gerbes. These questions are left for future work. Acknowledgements ================ Various parts of this work have been in progress for approximately five years, and so we have a number of people to thank, including M. Ando (for discussions of elliptic genera with twisted bundles), P. Clarke (for assistance constructing Distler-Kachru examples), K. Dienes (for discussions of free fermion models and sufficiency of level matching), J. Distler, D. Freed (for discussions of uses and analogues of ${\rm ch}^{\rm rep}$’s), J. Gray, S. Hellerman, I. Melnikov, E. Scheidegger, K. Wendland (for discussions of the nonsupersymmetric $SO(8) \times SO(24)$ string), and especially T. Pantev for very many useful discussions and collaborations revolving around stacks. L. Anderson was supported by the Fundamental Laws Initiative of the Center for the Fundamental Laws of Nature, Harvard University. B. Ovrut was supported in part by the DOE under contract DE-AC02-76-ER-03071 and the NSF under grant 1001296. Over the course of this work, E. Sharpe was partially supported by NSF grants DMS-0705381, PHY-0755614, and PHY-1068725. Massless spectra of heterotic strings on stacks {#app:spectra} =============================================== Basic definitions ----------------- In this section, we will describe the computation of the massless spectrum of a perturbative heterotic $E_8 \times E_8$ string compactified on a smooth Deligne-Mumford stack $\mathfrak{X}$ with suitable gauge bundle ${\cal E} \rightarrow \mathfrak{X}$, following (in spirit when not detail) [@dist-greene] and [@manion-toappear]. Not only will this be useful for computations, but the existence of such a computational method is a good consistency check for the existence of heterotic string compactifications on stacks. Let $\mathfrak{X}$ be a smooth Deligne-Mumford stack of complex dimension[^18] $n \leq 4$, and ${\cal E}$ a holomorphic vector bundle over $\mathfrak{X}$ of rank $r$, satisfying suitable anomaly-cancellation conditions. We will embed the bundle in one of the $E_8$’s of the ten-dimensional heterotic string, so we will assume that $r < 8$. As in [@dist-greene], all our computations will be in a right-moving R sector (hence, spacetime fermions), but spacetime supersymmetry can be used to derive the NS sector (spacetime bosons) in principle. Let $I_{\mathfrak{X}}$ denote the inertia stack associated to $\mathfrak{X}$. Roughly speaking, the inertia stack is a geometric mechanism for encoding twisted sectors; it has multiple components, each of which corresponds to a twisted sector in a standard global orbifold. For example, if $\mathfrak{X} = [{\mathbb C}^2/{\mathbb Z}_2]$, where the ${\mathbb Z}_2$ acts by sign flips, then $$I_{\mathfrak{X}} \: = \: [{\mathbb C}^2/{\mathbb Z}_2] \amalg [{\rm point}/{\mathbb Z}_2].$$ For another example, if $\mathfrak{X} \: = \: [ {\mathbb C}/{\mathbb Z}_3]$, where the ${\mathbb Z}_3$ acts by multiplying by phases, then $$I_{\mathfrak{X}} \: = \: [ {\mathbb C}/{\mathbb Z}_3] \amalg [{\rm point}/{\mathbb Z}_3] \amalg [{\rm point}/{\mathbb Z}_3].$$ For yet another example, if $\mathfrak{X} \: = \: [{\mathbb C}/{\mathbb Z}_2]$, where the ${\mathbb Z}_2$ acts trivially (so that all of ${\mathbb C}$ is fixed), then $$I_{\mathfrak{X}} \: = \: [{\mathbb C}/{\mathbb Z}_2] \amalg [{\mathbb C}/{\mathbb Z}_2].$$ (See [*e.g.*]{} [@vistoli; @gomez; @lmb; @bx; @metzler1; @noohi1; @noohi2; @noohi3; @heinloth2; @bss1] for more information on the inertia stack.) In general, points in the inertia stack are pairs $(x,\alpha)$, where $x$ is a point of $\mathfrak{X}$, and $\alpha$ is an automorphism of $x$, which for an orbifold $[Y/G]$ by $G$ a finite group, would define the twisted sectors. In the $[{\mathbb C}^3/{\mathbb Z}_3]$ example, if $g$ generates ${\mathbb Z}_3$, then the two copies of $[{\rm point}/{\mathbb Z}_3]$ correspond to $\alpha = g, g^2$. The inertia stack $I_{\mathfrak{X}}$ always contains a copy of $\mathfrak{X}$ as one component, corresponding to $\alpha = {\rm Id}$. Let us describe how to compute the spectrum on each component $\alpha$ of $I_{\mathfrak{X}}$. (We will use $\alpha$ to denote both a component of $I_{\mathfrak{X}}$ and the automorphism defining that component.) First, let $q: I_{\mathfrak{X}} \rightarrow \mathfrak{X}$ denote the natural projection onto a single component, and for $\alpha \neq {\rm Id}$, decompose the pullback bundles into eigenbundles[^19] of $\langle \alpha \rangle$: $$\begin{aligned} q^* T\mathfrak{X}|_{\alpha} & = & \oplus_n T_n^{\alpha}, \\ q^* {\cal E} |_{\alpha} & = & \oplus_n {\cal E}_n^{\alpha}.\end{aligned}$$ Define $t_{\alpha}$ to be the order of the corresponding automorphism, and take $T_n^{\alpha}$ and ${\cal E}_n^{\alpha}$ to be associated with character $$\exp(2 \pi i n / t_{\alpha}).$$ By this we mean that the (R-sector) worldsheet fermions corresponding to $T_n^{\alpha}$ and ${\cal E}_n^{\alpha}$ have boundary conditions of the form $$\psi(\sigma + 2 \pi) \: = \: \exp(2 \pi i n / t_{\alpha}) \psi(\sigma).$$ We will denote fermions couplings to $T_n^{\alpha}$ (respectively, ${\cal E}_n^{\alpha}$) by $\psi_{+,n}$ (respectively, $\lambda_{-,n}$). Let us pause to briefly discuss some concrete examples, to illuminate these abstract definitions. For global orbifolds by finite groups, it should hopefully be clear that the description above is an abstraction of the standard prescription for distinguishing various worldsheet fermions with different boundary conditions. Let us turn to an example which does not have such a realization, but which is relevant to (0,2) GLSMs. Take $\mathfrak{X} = {\mathbb P}^4_{[1,1,1,2,2]}$, with bundle $$0 \: \longrightarrow \: {\cal E} \: \longrightarrow \: \oplus_a {\cal O}(n_a) \: \stackrel{F_a}{\longrightarrow} \: {\cal O}(m) \: \longrightarrow \: 0$$ where $\det {\cal E}^* \cong K_{\mathfrak{X}}$: $$\sum n_a \: - \: m \: = \: 7,$$ and second Chern classes match: $$\sum n_a^2 \: - \: m^2 \: = \: 11.$$ This is not Calabi-Yau, so it would not be directly useful for a string compactification, but can help illuminate some general aspects. This stack has a ${\mathbb P}^1$ of ${\mathbb Z}_2$ orbifolds, so the inertia stack has the form $$I_{\mathfrak{X}} \: = \: \mathfrak{X} \amalg {\mathbb P}^1_{[2,2]}.$$ On the nontrivial component ${\mathbb P}^1_{[2,2]}$, call it $\alpha$, we can work out the decomposition of the gauge bundle. Suppose, for example, that $m$ is odd. For any given $a$, if $n_a$ is even, then $F_a$ is odd, so $F_a = 0$; if $n_a$ is even on the other hand, there is no constraint on $F_a$. In this case, we can decompose $$q^* {\cal E} |_{\alpha} \: = \: {\cal E}_+ \oplus {\cal E}_-,$$ where ${\cal E}_+$ is invariant, ${\cal E}_-$ anti-invariant under ${\mathbb Z}_2$, and specifically $$\begin{aligned} {\cal E}_+ & = & \oplus {\cal O}(n_a \, {\rm even}), \\ {\cal E}_- & = & {\rm ker}\left( \oplus {\cal O}(n_a \, {\rm odd}) \: \longrightarrow \: {\cal O}(m) \right).\end{aligned}$$ A closely related decomposition exists for $m$ even. Now that we have illuminated the definitions, let us return to our description of the general procedure for spectrum computation. At this point, the computation of spectra becomes more or less identical to that in an ordinary global orbifold by a finite group, if we think of $\alpha$ as denoting a twisted sector. We will walk through the details, as there are a few important subtleties for general cases not usually discussed in the literature, especially regarding Fock vacua, but the rest of the computation is nearly standard, once one masters the description. Vacuum energies --------------- We need to compute left- and right-moving zero point energies in each twisted sector. Recall that a complex worldsheet fermion $\psi$ with boundary conditions $$\psi(\sigma + 2 \pi) \: = \: \exp(i(\pi + \theta)) \psi(\sigma), \: \: \: - \pi \leq \theta \leq \pi$$ contributes $$- \frac{1}{24} \: + \: \frac{1}{8}\left( \frac{\theta}{\pi} \right)^2$$ to the vacuum energy, and a complex boson with the same boundary conditions contributes with the opposite sign. Let $\theta^{T,\alpha}_n$ denote the $\theta$ corresponding to worldsheet fermions associated with $T_n^{\alpha}$, and $\theta^{{\cal E},\alpha}_n$ denote the $\theta$ corresponding to worldsheet fermions associated with ${\cal E}_n^{\alpha}$. For the moment, we will assume that we are in an (R,R) sector (meaning, left-moving fermions in the first $E_8$ and right-moving fermions in an R sector, second $E_8$ will be held fixed in an NS sector). In an (NS,R) sector (left-moving fermions in the first $E_8$ in an NS sector instead), we would modify the $\theta$’s for left-moving worldsheet fermions to take into account an extra sign in boundary conditions. Then, in an (R,R) sector, the left-moving vacuum energy is $$\begin{aligned} E_{{\rm (R,R)}, {\rm Id}} & = & 8\left(- \frac{1}{24} \right) \: + \: 8\left(+\frac{1}{12}\right) \: + \: 4\left( - \frac{1}{12} \right), \\ & = & 0,\end{aligned}$$ in the untwisted sector ($\alpha = {\rm Id}$) and in twisted sectors, $$\begin{aligned} E_{{\rm (R,R)},\alpha} & = & 8\left(- \frac{1}{24} \right) \: + \: \sum_n ({\rm rk}\, {\cal E}^{\alpha}_n)\left( - \frac{1}{24} \: + \: \frac{1}{8}\left( \frac{ \theta^{{\cal E},\alpha}_n }{\pi} \right)^2 \right) \: + \: (8-r)\left( + \frac{1}{12} \right) \\ & & \: + \: \sum_n ( {\rm rk}\, T^{\alpha}_n )\left( + \frac{1}{24} \: - \: \frac{1}{8} \left( \frac{ \theta^{T,\alpha}_n }{\pi} \right)^2 \right) \: + \: (4-n) \left( - \frac{1}{12} \right), \\ & = & \frac{n-r}{8} \: + \: \frac{1}{8} \sum_n ({\rm rk}\, {\cal E}^{\alpha}_n) \left( \frac{ \theta^{{\cal E},\alpha}_n }{\pi} \right)^2 \: - \: \frac{1}{8} \sum_n ( {\rm rk}\, T^{\alpha}_n )\left( \frac{ \theta^{T,\alpha}_n }{\pi} \right)^2.\end{aligned}$$ In all cases the right-moving vacuum energy vanishes, since the right-moving bosons and fermions make equal and opposite contributions. Vacuum energies in (NS,R) sectors (meaning, left-moving fermions of the first $E_8$ in an NS sector) can be computed similarly. For completeness, we list them below: in an untwisted sector, $$\begin{aligned} E_{{\rm (NS,R)}, {\rm Id}} & = & 8\left(- \frac{1}{24} \right) \: + \: 8\left( - \frac{1}{24} \right) \: + \: 4\left( - \frac{1}{12} \right), \\ & = & -1,\end{aligned}$$ and in a twisted sector, $$\begin{aligned} E_{{\rm (NS,R)}, \alpha} & = & 8\left(- \frac{1}{24} \right) \: + \: \sum_n ({\rm rk}\, {\cal E}^{\alpha}_n)\left( - \frac{1}{24} \: + \: \frac{1}{8}\left( \frac{ \tilde{\theta}^{{\cal E},\alpha}_n }{\pi} \right)^2 \right) \: + \: (8-r)\left( - \frac{1}{24} \right) \\ & & \: + \: \sum_n ( {\rm rk}\, T^{\alpha}_n )\left( + \frac{1}{24} \: - \: \frac{1}{8} \left( \frac{ \theta^{T,\alpha}_n }{\pi} \right)^2 \right) \: + \: (4-n) \left( - \frac{1}{12} \right), \\ & = & -1 \: + \: \frac{n}{8} \: + \: \frac{1}{8} \sum_n ({\rm rk}\, {\cal E}^{\alpha}_n)\left( \frac{ \tilde{\theta}^{{\cal E},\alpha}_n }{\pi} \right)^2 \: - \: \frac{1}{8} \sum_n ( {\rm rk}\, T^{\alpha}_n )\left( \frac{ \theta^{T,\alpha}_n }{\pi} \right)^2,\end{aligned}$$ where $\tilde{\theta}$ denotes $\theta$’s as modified to include a sign in the boundary conditions. Vacuum energies in (NS,R) sectors (meaning, left-moving fermions of the first $E_8$ in an NS sector) can be computed similarly. Fock vacua ---------- The fractional charges of the Fock vacua can and should be understood in terms of coupling to nontrivial bundles. Recall (see [*e.g.*]{} [@kw]) that a complex left-moving fermion $\lambda$ with boundary conditions $$\lambda(\sigma + 2 \pi) \: = \: e^{-i \theta} \lambda(\sigma)$$ contributes fractional fermion number $$\frac{\theta}{2\pi} \: - \: \left[ \frac{\theta}{2 \pi} \right] \: - \: \frac{1}{2}$$ and a complex right-moving fermion $\psi$ with the same boundary conditions contributes fractional fermion number $$- \left( \frac{\theta}{2\pi} \: - \: \left[ \frac{\theta}{2 \pi} \right] \: - \: \frac{1}{2} \right)$$ In the present case, in the sector defined by automorphism $\alpha$, we have complex left-moving fermions $\lambda_{-,n}$ coupling to bundle ${\cal E}^{\alpha}_n$, with boundary conditions $$\lambda_{-,n}(\sigma + 2 \pi) \: = \: \exp\left( 2 \pi i n / t_{\alpha} \right) \lambda_{-,n}(\sigma)$$ and complex right-moving fermions $\psi_{+,n}$ coupling to bundle $T^{\alpha}_n$, with boundary conditions $$\psi_{+,n}(\sigma + 2 \pi) \: = \: \exp\left( 2 \pi i n/t_{\alpha}\right) \psi_{+,n}(\sigma)$$ Putting this together, we see that from each set of $\lambda_{-,n}$, the Fock vacuum couples to $$\label{eq:left-Fock-nonzero} \left( \det {\cal E}^{\alpha}_n \right)^{- \frac{n}{t_{\alpha}} \: - \: \left[ - \frac{n}{t_{\alpha}} \right] \: - \: \frac{1}{2} }$$ and from each set of $\psi_{+,n}$, the Fock vacuum couples to $$\label{eq:right-Fock-nonzero} \left( \det T^{\alpha}_n \right)^{\frac{n}{t_{\alpha}} \: + \: \left[ - \frac{n}{t_{\alpha}} \right] \: + \: \frac{1}{2} }$$ Since the $\alpha$-sector has components which are $t_{\alpha}$ gerbes, $t_{\alpha}$-th roots of bundles might exist, though not necessarily. (See appendix \[app:canonical-roots\] for examples of bundles on ${\mathbb Z}_n$-gerbes which do and do not admit $n$th roots.) Existence of these roots is a necessary condition for the existence of the physical theories. When multiple roots exist, as will happen if the components are not simply-connected, the roots must be specified as part of the data defining the sigma model. When there are periodic fermions, there are multiple Fock vacua, each with different (fractional) charges. The different Fock vacua are defined by which subset of the fermi zero modes annihilate. In our case, we will work in conventions in which our Fock vacuum $| 0 \rangle$ has the properties $$\lambda_{-,0}^{a} | 0 \rangle \: = \: 0 \: = \: \psi_{+,0}^{\overline{\imath}} | 0 \rangle .$$ As before, reflecting the fact that the $\lambda$’s and $\psi$’s couple to nontrivial bundles, this Fock vacuum is itself a section of a line bundle. From those periodic fermions, the Fock vacuum behaves as a section of a square root of the determinant of the periodic modes, specifically, $$\label{eq:squareroot} \sqrt{ K_{\alpha} \otimes \det {\cal E}^{\alpha}_0 },$$ (square root chosen with periodic boundary conditions), where $$K_{\alpha} \: = \: \det (T^{\alpha}_0)^*$$ [*i.e.*]{} the canonical bundle of the $\alpha$ component of $I_{\mathfrak{X}}$. (Note that in an (NS,R) sector, the ‘invariant’ subbundle ${\cal E}_0$ is defined to be invariant under the combination of spacetime group action and spin state boundary condition, and hence will be different from the ${\cal E}_0$ in an (R,R) sector.) If the square root above does not exist, then the orbifold is not well-defined, which we shall come back to after we derive the expression above. We can derive the result above for periodic fermions as follows. Different choices of Fock vacua act as sections of different line bundles, related by fermions acting as raising and lowering operators. Just as in fractional charges, the square root and bundles above are constrained by the fact that the set of Fock vacua must be consistent with those raising and lowering operations. For example, the ‘opposite’ Fock vacuum $| 0 \rangle^{\rm op}$ is defined by applying raising operators maximally: $$| 0 \rangle^{\rm op} \: = \: \lambda_{-,0}^{\overline{a}_1} \cdots \lambda_{-,0}^{\overline{a}_r} \psi_{+,0}^{i_1} \cdots \psi_{+,0}^{i_d} | 0 \rangle ,$$ (where $r$ is the rank of ${\cal E}_0^{\alpha}$ and $d$ the rank of $T_0^{\alpha}$), so if our Fock vacuum $| 0 \rangle$ couples to a line bundle ${\cal L}$, then the opposite or dual Fock vacuum above must couple to $$\left( \det {\cal E}_0^{\alpha *} \right) \otimes \left( \det T_0^{\alpha } \right) \otimes {\cal L},$$ which, by symmetry, should also be the same as ${\cal L}^*$. In other words, $$\left( \det {\cal E}_0^{\alpha *} \right) \otimes \left( \det T_0^{\alpha } \right) \otimes {\cal L} \: \cong \: {\cal L}^*$$ or more simply $${\cal L}^2 \: \cong \: \left( \det {\cal E}_0^{\alpha} \right) \otimes \left( \det T_0^{\alpha *} \right) \: = \: K_{\alpha} \otimes \det {\cal E}_0^{\alpha},$$ from which our claim is derived. In particular, taking ${\cal L} = {\cal O}$ will not, in general, be consistent. In passing, note that the set of all Fock vacua in sector $\alpha$ form a vector bundle $$\left( \wedge^{\bullet} {\cal E}_0^{\alpha *} \right) \otimes \left( \wedge^{\bullet} T_0^{\alpha} \right) \otimes \sqrt{ K_{\alpha} \otimes \det {\cal E}^{\alpha}_0 } \otimes \otimes_{n > 0} \left( \left( \det {\cal E}^{\alpha}_n \right) \left( \det T^{\alpha}_n \right)^{-1}\right)^{- \frac{n}{t_{\alpha}} \: - \: \left[ - \frac{n}{t_{\alpha}} \right] \: - \: \frac{1}{2} }$$ over $I_{\mathfrak{X}}|_{\alpha}$, taking into account contributions from all boundary conditions. The phenomenon of Fock vacua coupling to nontrivial bundles has also been noted in this context in [@manion-toappear], [@ando-s]\[section 2.1\]. However, aside from those two sources, we are not aware of many discussions of Fock vacua as sections of line bundles over target spaces[^20] in the literature, so it is perhaps useful to elaborate on this point. As we shall see in the present case and also in [@manion-toappear], it plays a crucial role in closing the spectrum under Serre duality of the sheaf cohomology groups, a basic symmetry of the spectra discussed in [@dist-greene]. The same behavior also arises elsewhere. For example, in open string theories, the Fock vacuum also transforms as a section of a line bundle, a square root of the canonical bundle of the D-brane worldvolume $B$ (assumed Spin), if the D-brane worldvolume is not Calabi-Yau. This can be understood simply from the matter representations: a spinor in the worldvolume theory can be represented mathematically in the form [@lawson-m] $$\left( \wedge^{\bullet} TB \right) \otimes \sqrt{K_B}.$$ In terms of the worldsheet RNS formalism, perturbative modes realize the $TB$ factors, and the $\sqrt{K_B}$ is implemented by the Fock vacuum itself. This phenomenon is also reminiscent of factors arising from the Freed-Witten anomaly [@ks-ext; @s-branes], though we shall not pursue that direction here. Consistency conditions derived from existence of Fock vacua {#app:spectra:fockconstraints} ----------------------------------------------------------- In some cases, the $t_{\alpha}$th roots (\[eq:left-Fock-nonzero\]), (\[eq:right-Fock-nonzero\]) or the square root (\[eq:squareroot\]) might not[^21] exist as honest equivariant line bundles. In such a case, the heterotic string on the stack is not well-defined. In an ordinary orbifold, this is the case that the Fock vacua (and hence perturbative states built from them) form a merely projective representation of the orbifold group, instead of an honest representation, and the projection operator built implicitly in the structure of the string one-loop partition function no longer functions. This condition represents a new (to our knowledge) consistency condition, so let us take a few paragraphs to elaborate on this point. At least morally, this condition is a generalization to stacks of the old requirement that “$c_1 \equiv 0$ mod 2” for bundles embedded in $E_8$ in the standard fashion. That constraint could be understood in two ways: - In low-energy supergravity, this is ultimately the statement that the $U(n)$ bundle can be lifted to ${\rm Spin}(16)$, realized by the left GSO projection, whose embeddeding into $E_8$ then factors through Spin$(16)/{\mathbb Z}_2$, - On the worldsheet, this is the statement that the Fock vacua are well-defined in a left R sector. The Fock vacua couple to a square root of the gauge bundle; that square root will exist if and only if “$c_1 \equiv 0$ mod 2.” (For another recent discussion of constraints of this form, see for example [@enstx].) In toroidal orbifolds, this constraint is very mild, but illustrates an important point: not only the bundle must admit a square root, but also the equivariant structure. For a typical toroidal orbifold, the bundle factors are all trivial, only the equivariant structures are nontrivial. In typical such orbifolds, $K_{\alpha}$ is the trivial line bundle with trivial connection, but although $\det {\cal E}^{\alpha}_0$ is a trivial bundle, the equivariant structure may be nontrivial. In left R sector, ${\cal E}^{\alpha}_0$ describes couples to fermions that are both periodic and invariant under the orbifold group, so the equivariant structure is trivial. In a left NS sector, on the other hand, ${\cal E}^{\alpha}_0$ describes periodic fermions, which are anti-invariant under the orbifold group. In a left NS sector, if the rank of ${\cal E}^{\alpha}_0$ is even, the induced equivariant structure on $\det {\cal E}^{\alpha}_0$ is trivial; if the rank of ${\cal E}^{\alpha}_0$ is odd, then the induced equivariant structure is nontrivial, and does not admit a square root, hence there is an obstruction to the existence of the orbifold in this case. We can build an example of a toroidal orbifold in which this condition appears nontrivially as follows. Consider an $E_8 \times E_8$ string on a $[T^4/{\mathbb Z}_6]$ orbifold, in which the generator $g$ of ${\mathbb Z}_6$ act on $T^4$ by multiplication by $-1$. Define a rank 4 bundle over this stack by taking ${\cal O}^{\oplus 4}$ over $T^4$, and let $g$ act with eigenvalues $$\left(\exp(6 \pi i/6) = -1, \exp(4 \pi i/6) = \exp(2 \pi i/3), \exp(2 \pi i/3), \exp(- 2 \pi i/6) \right).$$ It is straightforward to check that this satisfies level-matching, in the sense of [@freedvafa]. In the $g$-twisted left NS sector, there will be one periodic fermion, which is problematic as above. It is tempting to speculate that a necessary condition for the existence of the square root (\[eq:squareroot\]) can be written in the form $$c_1^{\rm rep}({\cal E}) \: \equiv \: c_1^{\rm rep}(T \mathfrak{X}) \mbox{ mod } 2$$ applying the Chern-rep’s discussed in sections \[sect:possible-anomcanc\] and appendix \[app:chern-reps\]. We will leave such an interpretation to future work. Spectrum result and Serre duality --------------------------------- Finally, we are ready to associate sheaf cohomology groups to elements of the spectrum. A general element of the spectrum will have the form $$\lambda_-^{a_1} \cdots \lambda_-^{a_m} \psi_+^{\overline{\imath}_1} \cdots \psi_+^{\overline{\imath}_k} | 0 \rangle ,$$ where each $\lambda$ and $\psi$ has some unspecified moding, such that the sum of the modings equals the vacuum energy computed earlier. Canonical commutation relations descend to statements of the form $$\{ \lambda_{p}^a, \lambda_{-p}^{\overline{b}} \} \: \propto \: h^{a \overline{b}}, \: \: \: \{ \psi_p^i, \psi^{\overline{\jmath}}_{-p} \} \: \propto \: g^{i \overline{\jmath}} ,$$ where $p$ is a moding. So long as the modings are all negative, both holomorphic and antiholomorphic-indexed fermions can appear in states. For zero modes, our Fock vacuum conventions are such that only $\lambda_{-,0}^{\overline{a}}$ and $\psi_{+,0}^i$ contribute. In any event, it should now be clear, following [@dist-greene], that on component $\alpha$, states of the form[^22] $$\prod_n \left( \lambda_{-,n}^{a_1} \cdots \lambda_{-,n}^{a_{m_n}} \lambda_{-,n}^{\overline{b}_1} \cdots \lambda_{-,n}^{\overline{b}_{p_n}} \psi_{+,n}^{j_1} \cdots \psi_{+,n}^{j_{\ell_n}} \psi_{+,n}^{\overline{\imath}_1} \cdots \psi_{+,n}^{\overline{\imath}_{k_n}} \right) | 0 \rangle ,$$ (where the fermion modings add up to the vacuum energy in the $\alpha$ sector) are counted by the sheaf cohomology group $$\label{eq:countstates1} H^{k_0}\left( I_{\mathfrak{X}}|_{\alpha}, \left( \wedge^{m_0} {\cal E}_0^{\alpha *} \right) \otimes_{n>0}\left( \wedge^{m_n} {\cal E}_n^{\alpha } \otimes \wedge^{p_n} {\cal E}_n^{\alpha *} \otimes \wedge^{\ell_n} T_n^{\alpha} \otimes \wedge^{k_n} T_n^{\alpha *} \right) \otimes {\cal F} \right) ,$$ where $${\cal F}^{\alpha} \: = \: \sqrt{ K_{\alpha} \otimes \det {\cal E}^{\alpha}_0 } \otimes_{n>0} \left( \left( \det {\cal E}^{\alpha}_n \right) \left( \det T^{\alpha}_n \right)^{-1}\right)^{- \frac{n}{t_{\alpha}} \: - \: \left[ - \frac{n}{t_{\alpha}} \right] \: - \: \frac{1}{2} }$$ (reflecting the Fock vacuum). Strictly speaking, not all states need be of the form above – for example, one might also be able to multiply in bosonic $\partial \phi$ modes. As their inclusion is standard and their treatment should now be clear, for reasons of brevity we shall move on. For example, if $I_{\mathfrak{X}}|_{\alpha} = [ {\rm point}/{\mathbb Z}_2 ]$, then this becomes $$H^{k_0}\left( {\rm point}, \left( \wedge^{m_0} {\cal E}_0^{\alpha *} \right) \otimes_{n>0}\left( \wedge^{m_n} {\cal E}_n^{\alpha *} \otimes \wedge^{p_n} {\cal E}_n^{\alpha *} \otimes \wedge^{\ell_n} T_n^{\alpha} \otimes \wedge^{k_n} T_n^{\alpha *} \right) \otimes {\cal F}^{\alpha} \right)^{\mathbb{Z}_2}.$$ (Taking group invariants is encoded implicitly in taking sheaf cohomology on the quotient stack.) This group vanishes if $k_0 \neq 0$, and when $k_0 = 0$, is the dimension of the ${\mathbb Z}_2$-invariant part of the vector space fibers. Finally, in a physical computation, one must impose the left- and right- GSO projections. For states of the form above, this will amount to a chirality constraint on $k_0$ and $m_0$. As the procedure is standard, we will say no more. One of the central observations of the heterotic spectrum computation on smooth manifolds in [@dist-greene] is that it is closed under Serre duality. The same is true here. First, for any component of the inertia stack indexed by an automorphism $\alpha$, there is another (not necessarily distinct) component indexed by $\alpha^{-1}$, which is isomorphic: $$I_{\mathfrak{X}}|_{\alpha} \: \cong \: I_{\mathfrak{X}} |_{\alpha^{-1}}.$$ Eigenbundle decompositions are closely related: $$\begin{aligned} T_n^{\alpha^{-1}} & \cong & T_{-n}^{\alpha}, \: \: \: T_0^{\alpha^{-1}} \: \cong \: T_0^{\alpha}, \\ {\cal E}_n^{\alpha^{-1}} & \cong & {\cal E}_{-n}^{\alpha}, \: \: \: {\cal E}^{\alpha^{-1}}_0 \: \cong \: {\cal E}^{\alpha}_0,\end{aligned}$$ (in conventions where $-n$ denotes the component associated to the character of the inverse). Let us now consider the following factor in the Fock vacuum bundle, $${\cal F}^{\alpha}_+ \: = \: \otimes_{n>0} \left( \left( \det {\cal E}^{\alpha}_n \right) \left( \det T^{\alpha}_n \right)^{-1}\right)^{- \frac{n}{t_{\alpha}} \: - \: \left[ - \frac{n}{t_{\alpha}} \right] \: - \: \frac{1}{2} }$$ (where the tensor product runs over all nontrivial representations of ${\mathbb Z}_{t^{\alpha}}$). Using relations such as ${\cal E}^{\alpha^{-1}}_n \cong {\cal E}^{\alpha}_{-n}$, we see that each factor in ${\cal F}_+^{\alpha^{-1}}$ is equivalent to a factor in ${\cal F}_+^{\alpha}$, but with an exponent of the opposite sign, hence $$\label{eq:pos-Fock-duality} {\cal F}_+^{\alpha^{-1}} \: \cong \: \left( {\cal F}_+^{\alpha} \right)^* .$$ As the combinatorics in these exponents is slightly complicated, let us consider some special cases to explicitly confirm this prediction. When $t_{\alpha} = 2$, $$\begin{aligned} {\cal F}_+^{\alpha} & = & \left( \left( \det {\cal E}_1^{\alpha} \right) \left( \det T_1^{\alpha} \right)^{-1} \right)^{- \frac{1}{2} - \left[ - \frac{1}{2}\right] - \frac{1}{2} }, \\ & = & \left( \left( \det {\cal E}_1^{\alpha} \right) \left( \det T_1^{\alpha} \right)^{-1} \right)^{0} \: \cong \: {\cal O} \: \cong \: \left( {\cal F}_+^{\alpha^{-1}} \right)^* .\end{aligned}$$ When $t_{\alpha}=3$, $$\begin{aligned} {\cal F}_+^{\alpha} & = & \left( \left( \det {\cal E}_1^{\alpha} \right) \left( \det T_1^{\alpha} \right)^{-1} \right)^{-\frac{1}{3} - \left[ - \frac{1}{3} \right] - \frac{1}{2}} \otimes \left( \left( \det {\cal E}_2^{\alpha} \right) \left( \det T_2^{\alpha} \right)^{-1} \right)^{- \frac{2}{3} - \left[ - \frac{2}{3} \right] - \frac{1}{2} }, \\ & = & \left( \left( \det {\cal E}_1^{\alpha} \right) \left( \det T_1^{\alpha} \right)^{-1} \right)^{+1/6} \otimes \left( \left( \det {\cal E}_2^{\alpha} \right) \left( \det T_2^{\alpha} \right)^{-1} \right)^{-1/6} ,\end{aligned}$$ and $$\begin{aligned} {\cal F}_+^{\alpha^{-1}} & = & \left( \left( \det {\cal E}_1^{\alpha^{-1}} \right) \left( \det T_1^{\alpha^{-1}} \right)^{-1} \right)^{+1/6} \otimes \left( \left( \det {\cal E}_2^{\alpha^{-1}} \right) \left( \det T_2^{\alpha^{-1}} \right)^{-1} \right)^{-1/6}, \\ & = & \left( \left( \det {\cal E}_2^{\alpha} \right) \left( \det T_2^{\alpha} \right)^{-1} \right)^{+1/6} \otimes \left( \left( \det {\cal E}_1^{\alpha} \right) \left( \det T_1^{\alpha} \right)^{-1} \right)^{-1/6}, \\ & = & \left( {\cal F}_+^{\alpha} \right)^* .\end{aligned}$$ In this fashion we confirm equation (\[eq:pos-Fock-duality\]) explicitly. Vacuum energies are invariant: if a fermion boundary condition in sector $\alpha$ is determined by $\theta$, then in $\alpha^{-1}$ it is determined by $- \theta$, but vacuum energies only depend upon $(\theta)^2$, and so are invariant. Contributions to the spectrum from sector $\alpha$ are matched by Serre duals in sector $\alpha^{-1}$. In terms of global quotients by finite groups, this means the untwisted sector closes into itself under Serre duality, but twisted sectors are exchanged. For example, the Serre duals to (\[eq:countstates1\]) are given by $$\begin{aligned} \lefteqn{ H^{{\rm dim}-k_0} \Bigl( I_{\mathfrak{X}}|_{\alpha}, \left( \wedge^{m_0} {\cal E}_0^{\alpha } \right) \otimes_{n>0}\left( \wedge^{m_n} {\cal E}_n^{\alpha *} \otimes \wedge^{p_n} {\cal E}_n^{\alpha } \otimes \wedge^{\ell_n} T_n^{\alpha *} \otimes \wedge^{k_n} T_n^{\alpha } \right) } \\ & & \hspace*{3.25in} \left. \otimes ({\cal F}^{\alpha}_+)^* \otimes \sqrt{ K_{\alpha}^* \otimes \det {\cal E}^{\alpha *}_0 } \otimes K_{\alpha} \right)^* \\ & = & H^{{\rm dim}-k_0}\Bigl( I_{\mathfrak{X}}|_{\alpha^{-1}}, \left( \wedge^{{\rm rk} - m_0} {\cal E}_0^{\alpha^{-1} } \right) \otimes_{n>0}\left( \wedge^{m_n} {\cal E}_n^{\alpha^{-1} *} \otimes \wedge^{p_n} {\cal E}_n^{\alpha^{-1} } \otimes \wedge^{\ell_n} T_n^{\alpha^{-1} *} \otimes \wedge^{k_n} T_n^{\alpha^{-1} } \right) \\ & & \hspace*{3.5in} \left. \otimes {\cal F}^{\alpha^{-1}}_+ \otimes \sqrt{ K_{\alpha^{-1}} \otimes \det {\cal E}^{\alpha^{-1} }_0 } \right)^*,\end{aligned}$$ which is of the same form as equation (\[eq:countstates1\]), as desired. Note that the Fock vacuum contribution is essential for the spectrum to close under Serre duality in this fashion: otherwise, Serre duality would generate a factor of $K_{\alpha}$ in the coefficients, unmatched by anything else, and which is nontrivial if the $\alpha$ component is not Calabi-Yau[^23]. Our computations so far have focused on the (R,R) sector, but one should note that identical considerations hold in the (NS,R) sector as well. In the special case that the stack $\mathfrak{X}$ is a smooth Calabi-Yau manifold $X$, these computational methods reduce to those of [@dist-greene]. In this case, the inertia stack $I_{\mathfrak{X}}$ has no nontrivial components: $I_{\mathfrak{X}} = X$. Furthermore, we typically take $\det {\cal E}$ to be trivial, so the Fock vacuum is a section of a trivial line bundle. In the special case that the stack $\mathfrak{X}$ is a toroidal orbifold, again these methods reduce to known results. In this case, all of the bundles involved are trivial, so sheaf cohomology is nontrivial only in degree zero, and sheaf cohomology on a stack just takes group invariants of the coefficients. A less trivial example is discussed in section \[sect:class3-caution1\]. Further examples and computational techniques will appear in [@manion-toappear]. Just as in [@dist-greene], in principle the number of generations can be computed as an index based on the spectrum. We shall not work through details here, but appendix \[app:chern-reps\] contains general results on index theory computations on stacks. A/2 model spectra ----------------- In this appendix we have focused on physical heterotic string spectra. It is possible to apply the same methods to the A/2 model to formulate a mathematical theory of sheaf cohomology of orbifolds, and this has been done in [@manion-toappear]. Briefly, the A/2 model is a heterotic analogue of the A model topological field theory. If $X$ is a smooth space and ${\cal E} \rightarrow X$ a holomorphic vector bundle, then the A/2 model is well-defined if both[^24] $${\rm ch}_2({\cal E}) \: = \: {\rm ch}_2(TX) \mbox{ and } \det {\cal E}^* \: \cong \: K_X.$$ See [*e.g.*]{} [@katz-s; @ade; @s-b; @jock-ilarion; @dgks1; @dgks2; @mss; @tan1; @tan2] for more information on the A/2 and B/2 models. As this is no longer a physical theory, constraints on the dimension of $X$ and rank of ${\cal E}$ are dropped. When $X$ is smooth, the massless spectrum consists of sheaf cohomology groups of the form $$H^{\bullet} ( X, \wedge^{\bullet} {\cal E}^* )$$ When $X$ is a stack $\mathfrak{X}$, reference [@manion-toappear] applies methods similar to those in this appendix (modulo restricting to (R,R) sector states and omitting the GSO projections) to define a generalization, which broadly speaking adds in various sheaf cohomology groups associated to twisted sectors (nontrivial components of the inertia stack). Line bundles on gerbes over projective spaces {#app:linebundles} ============================================= For any stack $\mathfrak{X}$ presented as $\mathfrak{X} = [X/G]$ for some space $X$ and group $G$, a vector bundle (sheaf) on $\mathfrak{X}$ is the same as a $G$-equivariant vector bundle (sheaf) on $X$. Suppose that $G$ is an extension $$1 \: \longrightarrow \: K \: \longrightarrow \: G \: \longrightarrow \: H \: \longrightarrow \: 1,$$ where $K$ acts trivially on $X$, and $G/K \cong H$ acts effectively. In this case, $\mathfrak{X} = [X/G]$ is a $K$-gerbe. A vector bundle on $\mathfrak{X}$ is a $G$-equivariant vector bundle on $X$, and as such, the $K$ action is defined by a representation of $K$ on the fibers of that vector bundle. In this section, we will discuss in greater detail the special case of line bundles on gerbes over projective spaces. Generalities {#generalities} ------------ Let us first review some basic properties of line bundles on gerbes over projective spaces, and then we will outline their sheaf cohomology. First, let us consider some simple explicit examples. The total space of the line bundle ${\cal O}(-m)$ over the projective space ${\mathbb P}^n$ can be described[^25] by a gauged linear sigma model with fields of $U(1)$ charges $x_1$ $\cdots$ $x_{n+1}$ $p$ ------- ---------- ----------- ------ $1$ $\cdots$ $1$ $-m$ Now, a ${\mathbb Z}_k$ gerbe over ${\mathbb P}^n$ can be described by a gauged linear sigma model in which the $n+1$ fields/homogeneous coordinates have weight $k$ instead of weight $1$, as discussed in [*e.g.*]{} [@glsm]. Then, for example, the GLSM with fields and $U(1)$ charges $x_1$ $\cdots$ $x_{n+1}$ $p$ ------- ---------- ----------- ------ $k$ $\cdots$ $k$ $-k$ is surely going to be the pullback of ${\cal O}(-1) \rightarrow {\mathbb P}^n$ to the gerbe. However, how does one interpret GLSM’s defined by, for example: $x_1$ $\cdots$ $x_{n+1}$ $p$ ------- ---------- ----------- ------ $k$ $\cdots$ $k$ $-1$ This is the total space of what is sometimes referred to as the “${\cal O}(1/k)$” line bundle over the ${\mathbb Z}_k$ gerbe ${\mathbb P}^n_{[k,\cdots,k]}$. It is an example of a line bundle on the gerbe that is not a pullback of a line bundle on the base space – the gerbe has more bundles than the base space. More to the point, it can only be understood as the total space of a line bundle on a gerbe – so a physicist who was very careful in a study of GLSM’s would eventually be forced to discover gerbes in order to make sense of this example. In addition to being a line bundle over the stack, the total space of the ${\cal O}(1/k)$ line bundle is also a fibered orbifold over the projective space ${\mathbb P}^n$ – it is a type of fiber bundle over ${\mathbb P}^n$, in which the fibers are the orbifolds $[ {\mathbb C}/{\mathbb Z}_k ]$. For this reason, these are sometimes known as ‘orbibundles;’ see [*e.g.*]{} [@qureshi-szendroi] for references to the literature under this name. (This same structure has also been discussed in connection with interpreting hybrid Landau-Ginzburg models, see [*e.g.*]{} [@alg22].) Not all ${\mathbb Z}_k$ gerbes on projective spaces are of the form of weighted projective stacks. A more general class was discussed in [*e.g.*]{} [@glsm]\[section 3.3\], and, roughly, are given by ${\mathbb C}^{\times}$ quotients of principal ${\mathbb C}^{\times}$ bundles over ${\mathbb P}^n$. Specifically, consider a GLSM with fields $x_i$, $z$, and two ${\mathbb C}^{\times}$ actions, as follows: $x_i$ $z$ ----------- ------- ------ $\lambda$ $1$ $-n$ $\mu$ $0$ $k$ The first ${\mathbb C}^{\times}$, $\lambda$, defines the total space of a line bundle on ${\mathbb P}^n$ of degree $-n$. The second ${\mathbb C}^{\times}$, $\mu$, quotients out the fibers, leaving a ${\mathbb Z}_k$ kernel. The result is a ${\mathbb Z}_k$ gerbe over ${\mathbb P}^n$, of characteristic class $-n$ mod $k$. The weighted projective stacks we have been discussing correspond to an alternative presentation in the special case that $n=1$. One can define line bundles over these gerbes in the obvious fashion. The notation ${\cal O}(1/k)$, while initially catchy, is unfortunately ambiguous – for example, it does not distinguish a twisted bundle of $c_1=k$ over the gerbe from the pullback from ${\mathbb P}^n$ of an ordinary line bundle of $c_1=1$. Let us introduce a more precise notation. We will use “${\cal O}_{\Lambda}(m)$” to denote a line bundle defined by a superfield of charge $m$. For bundles on, say, ordinary projective spaces, the $k=1$ case, a superfield of charge $m$ couples to the line bundle ${\cal O}(m)$. To understand the meaning of this notation, let us first consider a ${\mathbb Z}_2$ gerbe over ${\mathbb P}^n$ defined by the weighted projective stack ${\mathbb P}^n_{[2,2,\cdots,2]}$. Let $G{\mathbb P}^n$ denote the gerbe, and $\pi: G{\mathbb P}^n \rightarrow {\mathbb P}^n$ the natural projection from the gerbe onto the underlying projective space. Now, coherent sheaves on the gerbe decompose into twisted sheaves on the underlying space (see section \[sect:twisting\] or [@summ]). Formally, if $\alpha \in H^2({\mathbb P}^n,{\mathbb Z}_2)$ is the characteristic class of the gerbe, then $$\mbox{Coh}(G {\mathbb P}^n) \: = \: \mbox{Coh}({\mathbb P}^n, 1(\alpha)) \cup \mbox{Coh}({\mathbb P}^n, \chi(\alpha)),$$ where $\mbox{Coh}(X, \lambda)$ denotes coherent sheaves on $X$ twisted by a 2-cocycle $\lambda$. In the notation above, $1$ and $\chi$ are the two irreducible representations of ${\mathbb Z}_2$, so $1(\alpha)$ is the vanishing 2-cocycle and $\chi(\alpha)$ is a cocycle that does not vanish identically. Note that [*both*]{} cocycles are cohomologous to the identity – both components of $\mbox{Coh}(G{\mathbb P}^n)$ are isomorphic to ordinary coherent sheaves $\mbox{Coh}({\mathbb P}^n)$. (This resolves a potential contradiction, in that the rank of a bundle twisted by a cohomologically nontrivial cocycle, must be divisible by the order of the cocycle, and so here would need to be divisible by $k$ – truly twisted line bundles do not exist.) In this language, we can immediately read off that $${\cal O}_{\Lambda}(k) \: = \: \left\{ \begin{array}{cl} \mbox{Coh}({\mathbb P}^n, 1(\alpha)) & k \mbox{ even}, \\ \mbox{Coh}({\mathbb P}^n, \chi(\alpha)) & k \mbox{ odd}. \end{array} \right.$$ In other words, if $k$ is even, then ${\cal O}_{\Lambda}(k)$ is a pullback to the gerbe from a line bundle on the base. For other values of $k$, the bundle is twisted by an action of the ${\mathbb Z}_2$. Now, the projection map $\pi: G {\mathbb P}^n \rightarrow {\mathbb P}^n$ defines a functor $$\pi^*: \: \mbox{Coh}( {\mathbb P}^n ) \:\stackrel{\sim}{\longrightarrow} \: \mbox{Coh}({\mathbb P}^n, 1(\alpha) ).$$ In addition, there is another functor $$\pi_1^* \: \equiv \: \pi^* \otimes {\cal O}_{\Lambda}(1): \: \mbox{Coh}({\mathbb P}^n) \: \stackrel{\sim}{\longrightarrow} \: \mbox{Coh}({\mathbb P}^n, \chi(\alpha) ).$$ (In fact, there is an analogue of $\pi_1^*$ for every ${\cal O}_{\Lambda}( \mbox{odd})$.) To determine $\pi^* {\cal O}(m)$ in terms of ${\cal O}_{\Lambda}$’s, consider the commutative diagram $$\xymatrix{ \frac{ {\mathbb C}^{n+1} - 0 }{ {\mathbb C}^{\times} } \ar[r] \ar[d] & \frac{ {\mathbb C}^{n+1} - 0 }{ {\mathbb C}^{\times} } \ar[d] \\ G {\mathbb P}^n \ar[r] & {\mathbb P}^n }$$ The line bundle ${\cal O}(k)$, defined by weights $1, \cdots, 1, k$, pulls back to weights $2, \cdots, 2, 2k$, from which we deduce that $$\pi^* {\cal O}(k) \: = \: {\cal O}_{\Lambda}(2k),$$ which implies $$\pi_1^* {\cal O}(k) \: = \: {\cal O}_{\Lambda}(2k+1).$$ Note that although $\pi^*$ preserves tensor products, $\pi_1^*$ does [*not*]{} preserve tensor products: $$\begin{aligned} \pi_1^* \left( {\cal O}(k) \otimes {\cal O}(m) \right) & \cong & \pi_1^* {\cal O}(k+m), \\ & \cong & {\cal O}_{\Lambda}(2k+2m+1), \\ & \not\cong & {\cal O}_{\Lambda}(2k+2m+2) \: \cong \: \left( \pi_1^* {\cal O}(k) \right) \otimes \left( \pi_1^* {\cal O}(m) \right).\end{aligned}$$ Indeed, this is an immediate consequence of the definition of $\pi_1^*$. In addition, for the same reason, $\pi_1^*$ does not commute with duality of bundles $$\pi_1^* \left( {\cal L}^{\vee} \right) \: \not\cong \: \left( \pi_1^* {\cal L} \right)^{\vee}.$$ Now, for any finite gerbe over any space, the tangent bundle of the gerbe is just the pullback (under $\pi$) of the tangent bundle to the space. One way to see this is to work locally on the atlas, which is just a finite cover, and so the tangent bundle should be the same. We can see this explicitly in the present case as follows. For the ${\mathbb Z}_2$ gerbe $G {\mathbb P}^n = {\mathbb P}^n_{[2,\cdots,2]}$, the tangent bundle seen by the gauged linear sigma model is $$0 \: \longrightarrow \: {\cal O}_{\Lambda} \: \longrightarrow \: {\cal O}_{\Lambda}(2)^{n+1} \: \longrightarrow \: T G {\mathbb P}^n \: \longrightarrow \: 0.$$ Using the isomorphisms above, we see this short exact sequence is the same as $$0 \: \longrightarrow \: \pi^* {\cal O} \: \longrightarrow \: \pi^* {\cal O}(1)^{n+1} \: \longrightarrow \: T G {\mathbb P}^n \: \longrightarrow \: 0,$$ which is just $\pi^*$ of the Euler sequence for the tangent bundle $$0 \: \longrightarrow \: {\cal O} \: \longrightarrow \: {\cal O}(1)^{n+1} \: \longrightarrow \: T {\mathbb P}^n \: \longrightarrow \: 0.$$ For ${\mathbb Z}_k$ gerbes over ${\mathbb P}^n$ built as the weighted projective stack ${\mathbb P}^n_{[k,\cdots,k]}$, there is a closely analogous story. Here, coherent sheaves on $G {\mathbb P}^n$ decompose as $$\mbox{Coh}(G{\mathbb P}^n) \: = \: \cup_{\chi} \mbox{Coh}({\mathbb P}^n, \chi(\alpha)),$$ where the union is over irreducible representations of ${\mathbb Z}_k$, and there are $k$ different pullbacks, first the canonical $$\pi^*: \: \mbox{Coh}({\mathbb P}^n) \: \stackrel{\sim}{\longrightarrow} \: \mbox{Coh}({\mathbb P}^n, 1(\alpha)),$$ followed by $\pi_i^*(-) \equiv \pi^*(-) \otimes {\cal O}_{\Lambda}(i)$. Identifying $\pi_0^*$ with $\pi^*$, we have the general relation $$\pi_i^* {\cal O}(m) \: = \: {\cal O}_{\Lambda}(km+i).$$ An argument nearly identical to the one above shows that the tangent bundle $T G {\mathbb P}^n$ seen by a gauged linear sigma model is given by $\pi^* T {\mathbb P}^n$, exactly as must be true on general grounds. Sheaf cohomology ---------------- On a global quotient stack $\mathfrak{X} = [X/G]$, for $G$ finite, given a vector bundle ${\cal E} \rightarrow \mathfrak{X}$, (equivalently, a $G$-equivariant bundle on $X$,) $$H^{\bullet}(\mathfrak{X}, {\cal E}) \: = \: H^{\bullet}(X, {\cal E})^G.$$ In our discussion of massless spectra of heterotic strings on stacks, this is ultimately the reason why in orbifolds one gets $G$-invariants. Now, nontrivial gerbes over projective spaces have a global quotient description as some $[X/G]$ for $G$ nonfinite, and the simple description of sheaf cohomology above in terms of $G$-invariants is only valid for $G$ finite, so for general cases a different approach is required. For example, let $\mathfrak{X} = {\mathbb P}^n_{[k,\cdots,k]}$, and ${\cal O}_{\mathfrak{X}}(m)$ as above, then $$H^i(\mathfrak{X}, {\cal O}_{\mathfrak{X}}(m)) \: = \: \left\{ \begin{array}{cl} 0 & k \nmid m, \\ H^i({\mathbb P}^n, {\cal O}_{ {\mathbb P}^n }(m/k)) & k \mid m. \end{array} \right.$$ For $m \geq 0$, we can check this as follows. First, $$H^i\left(\mathfrak{X}, {\cal O}_{\mathfrak{X}}(m) \right) \: = \: H^i_{ {\mathbb C}^{\times} }\left( {\mathbb C}^{n+1}-\{0\}, {\cal O} \right),$$ where the ${\cal O}$ coefficients have weight $m$ under the ${\mathbb C}^{\times}$. In principle, there is a spectral sequence converging to the right-hand side, with level-two terms $$H^p\left( {\mathbb C}^{\times}, H^q\left( {\mathbb C}^{n+1} - \{0\}, {\cal O}\right) \right),$$ but $H^q({\mathbb C}^{n+1}-\{0\}, {\cal O}) = 0$ for $q \neq 0, n$, and $$H^0\left( {\mathbb C}^{n+1}-\{0\}, {\cal O} \right) \: = \: {\mathbb C}[x_0, \cdots, x_n].$$ (The degree $n$ cohomology is also nonzero and infinite-dimensional, but it will not contribute any invariants for $m \geq 0$, only for $m < 0$, so we omit it from this discussion.) For $\lambda \in {\mathbb C}^{\times}$, the representation $$\rho: \: {\mathbb C}^{\times} \: \longrightarrow \: {\rm GL}( {\mathbb C}[x_0,\cdots, x_n])$$ is defined by $$\rho_{\lambda}(f(x)) \: = \: \lambda^{-m}f(\lambda^k x_0, \cdots, \lambda^k x_n).$$ The group $H^p({\mathbb C}^{\times}, ( {\mathbb C}[x_0,\cdots,x_n],\rho))$ is zero unless $p=0$, since it is a reductive group, and for $p=0$ is given by the invariants. Next, let us compute the invariants. Decompose $$f \: = \: f_0 \: + \: \cdots \: + \: f_N,$$ where $f_d$ denotes a homogeneous polynomial of degree $d$. Under the ${\mathbb C}^{\times}$ action, $$\rho_{\lambda}(f) \: = \: \lambda^{-m} f_0 \: + \: \lambda^{-m + k}f_1 \: + \: \cdots \: + \: \lambda^{-m + kN} f_N.$$ Thus, ${\mathbb C}^{\times}$ invariants only exist in the case that $k$ divides $m$, and in that case, are counted by degree $m/k$ polynomials in $n+1$ variables. Now, let us compare to the original claim. It is a standard result that for $\ell > 0$, $$H^i({\mathbb P}^n, {\cal O}_{{\mathbb P}^n}(\ell)) \: = \: \left\{ \begin{array}{cl} 0 & i \neq 0, \\ {\rm Sym}^{\ell} \mathbb{C}^{n+1} & i=0. \end{array} \right.$$ In other words, the only nonzero cohomology is in degree zero, and in that degree, it is counted by homogeneous polynomials of degree $\ell$ in $n+1$ variables. The desired result follows. Chern classes on the inertia stack {#app:chern-reps} ================================== As we are manipulating bundles on stacks, it is worth spending a little time reviewing corresponding Chern classes. It is possible to define Chern classes on a stack itself; for example, Chern classes of a vector bundle ${\cal E}$ on a quotient stack $[X/G]$ are simply $G$-equivariant Chern classes of ${\cal E}$ on $X$. However, these Chern classes do not always behave well under mathematical manipulations, and in any event a different notion of Chern classes and Chern characters, denoted $c^{\rm rep}$ and ${\rm ch}^{\rm rep}$, exists and is relevant for index theory. These alternative notions of Chern classes do not live in the cohomology of the original stack, but rather of the inertia stack, which encodes twisted sectors of string orbifolds. (See appendix \[app:spectra\] for more information on the inertia stack.) In this section, we will illustrate how to compute such Chern classes and characters (denoted $c^{\rm rep}$ and ${\rm ch}^{\rm rep}$) and describe their appearance in index theory in some examples. It is tempting to wonder whether one could derive extra anomaly constraints on orbifolds from these stack Chern classes over nontrivial components of the inertia stack, but we argue that does not seem to happen in heterotic compactifications in section \[sect:possible-anomcanc\] (though see section \[app:spectra:fockconstraints\] for a possible application of $c_1^{\rm rep}$). For any stack $\mathfrak{X}$, let $V$ be a vector bundle over $\mathfrak{X}$, and $I_{\mathfrak{X}}$ the inertia stack of $\mathfrak{X}$. Let $q: I_{\mathfrak{X}} \rightarrow \mathfrak{X}$ denote the natural projection operator onto one component. We define Chern classes of $V$ as follows. First, pullback $V$ to $I_{\mathfrak{X}}$ along $q$. Then, on each component $\alpha$ of $I_{\mathfrak{X}}$, $q^* V$ will decompose into eigenbundles of the action of the stabilizer for that component: $$q^* V|_{\alpha} \: = \: \oplus_{\chi} V_{\alpha, \chi}.$$ (When $\alpha$ is the identity, our conventions are that there is only one component, associated to the trivial character.) Define ${\rm ch}^{\rm rep}(V)$ over a component $\alpha$ to be $${\rm ch}^{\rm rep}(V)|_{\alpha} \: \equiv \: \bigoplus_{\chi} {\rm ch}(V_{\alpha, \chi}) \otimes \chi,$$ where $\chi$ is the eigenvalue of that component of $q^* V$ under the stabilizer, and ${\rm ch}$ denotes the naive notion of Chern classes, living in equivariant cohomology pertinent to the stack itself. (These seem to be the same as the Chern classes in “delocalized cohomology” described in [*e.g.*]{} [@at-seg; @bbmp; @baum-fete], though our starting point is different.) Intuitively, the idea is that on any component of the inertia stack determined by some generic automorphism, the bundle should decompose into eigenbundles, and $\chi$ is the eigenvalue associated with the action of that automorphism on the bundle. Slightly more generally, one can define a “diagonalization map” $$d: \: K^0(I_{\mathfrak{X}}) \otimes {\mathbb C} \: \longrightarrow \: K^0(I_{\mathfrak{X}}) \otimes {\mathbb C},$$ which on a component $\alpha$ maps a sheaf ${\cal F}$ to its isotypic decomposition, weighted by characters: $$d( [{\cal F}] ) |_{\alpha} \: = \: \sum_{\chi} {\cal F}_{\alpha, \chi} \otimes \chi.$$ In this language, $${\rm ch}^{\rm rep}(V) \: = \: {\rm ch}( d( q^* V) ).$$ To clarify these ideas, let us work through some examples. First, we shall consider a vector bundle on a trivial gerbe. Consider a vector bundle $V \rightarrow \mathfrak{X} \equiv X \times B {\mathbb Z}_k$, so $V = p_1^* E \otimes p_2^* \zeta$ for some bundle $E \rightarrow X$ and representation $\zeta \in {\mathbb Z}_k^{\vee}$. The inertia stack $I_{\mathfrak{X}}$ is given by $$I_{\mathfrak{X}} \: = \: \coprod_{g \in {\mathbb Z}_k} X \times B{\mathbb Z}_k \times \{ g \}.$$ There is a forgetful map $q: I_{\mathfrak{X}} \rightarrow X \times B {\mathbb Z}_k$. Consider $$q^* V \: = \: \oplus_{\chi \in {\mathbb Z}_k^{\vee} } V_{\chi},$$ where $V_{\chi}$ is the $\chi$ eigenspace for the $g$ action on $q^* V$: $$q^* V|_{X \times B {\mathbb Z}_k \times \{ g \} } \: = \: V,$$ $$V_{\chi}|_{X \times B{\mathbb Z}_k \times \{ g \} } \: = \: \left\{ \begin{array}{cl} V & \mbox{if } \chi(g) = \zeta(g), \\ 0 & \rm{else}. \end{array} \right.$$ Now, we want to compute ${\rm ch}^{\rm rep}(V) \in H^{\bullet}(I_{\mathfrak{X}}, {\mathbb C})$. $$V \: \mapsto \: q^* V \: = \: \oplus_{\chi} V_{\chi} \: \mapsto \: \oplus_{\chi} V_{\chi} \otimes \chi,$$ where $V_{\chi} \otimes \chi \in K^0(I_{\mathfrak{X}}) \otimes {\mathbb C}$. (We think of $V_{\chi} \in K^0(I_{\mathfrak{X}})$, and $\chi \in {\mathbb C}$.) Then, $${\rm ch}^{\rm rep}(V) \: = \: {\rm ch}\left( \oplus_{\chi} V_{\chi} \otimes \chi \right) \in H^{\bullet}(I_{\mathfrak{X}},{\mathbb C}) \: = \: \oplus_g H^{\bullet}(X),$$ $$V_{\chi} \otimes \chi |_{X \times B {\mathbb Z}_k \times \{ g \} } \: = \: \left\{ \begin{array}{cl} V \otimes \chi & \mbox{if } \chi(g) = \zeta(g), \\ 0 & {\rm else}. \end{array} \right.$$ Putting this together, we find $${\rm ch}^{\rm rep}(V) \: = \: \left( {\rm ch}^{\rm rep}(V)|_{(g)} \right)_{ g \in {\mathbb Z}_k},$$ where $${\rm ch}^{\rm rep}(V)|_{(g)} \: = \: \oplus_{\chi \: {\rm s.t.} \: \chi(g) = \zeta(g) } {\rm ch}(V) \otimes \chi.$$ Similarly, $${\rm ch}^{\rm rep}(T\mathfrak{X})|_{(g)} \: = \: \oplus_{\chi \: {\rm s.t.} \: \chi(g)=1 } {\rm ch}(T\mathfrak{X}) \otimes \chi.$$ For $g=1$, $${\rm ch}^{\rm rep}(V)|_{(1)} \: = \: \oplus_{\chi} {\rm ch}(V) \otimes \chi,$$ and similarly for ${\rm ch}^{\rm rep}(TX)^{(1)}$. Now, suppose $k$ is prime. Then $\chi(g)=1$ implies $\chi = 1$. Thus, $${\rm ch}^{\rm rep}(V)|_{(g)} \: = \: {\rm ch}(V) \otimes \zeta(g),$$ $${\rm ch}^{\rm rep}(T\mathfrak{X})|_{(g)} \: = \: {\rm ch}(T\mathfrak{X}) \otimes 1,$$ for all $g$. Next, let us consider a line bundle on a nontrivial gerbe. Consider the prototypical example of a ${\mathbb Z}_k$ gerbe on ${\mathbb P}^n$: $\mathfrak{X} = {\mathbb P}^n_{[k,k,\cdots,k]}$. Let ${\cal O}_{\mathfrak{X}}(m)$ denote the holomorphic line bundle defined by ${\mathbb C}^{\times}$ weight $-m$. In other words, if $m$ is divisible by $k$, then ${\cal O}_{\mathfrak{X}}(m)$ is the pullback of ${\cal O}_{ {\mathbb P}^n}(m/k)$ under the projection map from the gerbe $\mathfrak{X}$ to the underlying space ${\mathbb P}^n$. The components of the inertia stack are labelled by $k$th roots of unity (not characters, but group elements). The Chern classes ch$^{\rm rep}$ have $k$ components, each component in a cohomology class (with complex coefficients) on the stack. If we let $\alpha$ denote a $k$th root of unity, then on that component of the inertia stack, $$c_1^{\rm rep}({\cal O}_{\mathfrak{X}}(m))|_{\alpha} \: = \: \frac{m}{k} \alpha^{-m} J,$$ where $J$ is the pullback to the gerbe of the hyperplane class, and the total Chern character is $${\rm ch}^{\rm rep}( {\cal O}_{\mathfrak{X}}(m))|_{\alpha} \: = \: \alpha^{-m} \exp\left( \frac{m}{k} J \right).$$ To derive this, remember that for a line bundle $L$ over the stack $\mathfrak{X}$, if $\pi: I_{\mathfrak{X}} \rightarrow \mathfrak{X}$ denotes the projection from the inertia stack to $\mathfrak{X}$, then the Chern characters are $${\rm ch}^{\rm rep}(L)|_{\mathfrak{X} \times \{\alpha\} } \: = \: \pi^* \left. {\rm ch}\left(L\right)\right|_{\mathfrak{X} \times \{\alpha\} } \otimes \chi,$$ where $\chi$ is the eigenvalue of the stabilizer $\alpha$ on $\pi^* L|_{\mathfrak{X} \times \{\alpha\} }$. Here, $\chi = \alpha^{-m}$. More generally, over all components, we write $$c_1^{\rm rep}({\cal O}_{\mathfrak{X}}(m)) \: = \: \left( \frac{m}{k} J, \cdots, \frac{m}{k} \alpha^{-m} J, \cdots \right).$$ Multiplication of components of ch$^{\rm rep}$ multiplies not only the cohomology classes, but also the coefficients. For example, $$\left( c_1^{\rm rep}({\cal O}(m))|_{\mathfrak{X} \times \{\alpha\} } \right)^2 \: = \: \left( \frac{m}{k} J \right)^2 \alpha^{-2m}.$$ Now, for a line bundle $L$ on an ordinary space, $${\rm ch}_2(L) \: = \: (1/2) c_1^2(L),$$ but here, by contrast, $$\begin{aligned} {\rm ch}_2^{\rm rep}({\cal O}(m))|_{\mathfrak{X} \times\{\alpha\}} & = & \frac{1}{2} \left( \frac{m}{k} J \right)^2 \alpha^{-m}, \\ & = & \alpha^{+m} \frac{1}{2} \left( c_1^{\rm rep}({\cal O}(m))|_{\mathfrak{X} \times \{\alpha\} } \right)^2,\end{aligned}$$ so that the usual relation between Chern classes and Chern characters is modified on a stack. (In fact, if we were computing Chern classes of a bundle that split as several different eigenbundles, the relation would be much more complicated than just an additional complex phase.) As a consistency check, let us compute the index of this line bundle, using Hirzebruch-Riemann-Roch. For any bundle ${\cal E} \rightarrow \mathfrak{X}$, the Hirzebruch-Riemann-Roch index theorem says $$\chi({\cal E}) \: = \: \int_{I_{\mathfrak{X}}} {\rm ch}^{\rm rep}({\cal E}) {\rm Td}(\mathfrak{X})$$ where $$\chi({\cal E}) \: = \: \sum_i (-)^i h^i(\mathfrak{X}, {\cal E}),$$ and $${\rm Td}(\mathfrak{X}) \: = \: \alpha_{\mathfrak{X}}^{-1} {\rm Td}( T I_{\mathfrak{X}} ),$$ where $$\alpha_{\mathfrak{X}} \: = \: {\rm ch}( d( \lambda_q) ), \: \: \: \lambda_q \: = \: \sum_k (-)^k \wedge^k N_q^*,$$ for $N_q$ the normal bundle. (As $\lambda_q$ is not a pullback from $\mathfrak{X}$, but rather is defined intrinsically on $I_{\mathfrak{X}}$, ${\rm ch}^{\rm rep}(\lambda_q)$ is not well-defined, so instead the pertinent Chern character is defined via the diagonalization map $d$.) In the present case, since each component of the inertia stack $I_{\mathfrak{X}}$ is isomorphic to the original stack $\mathfrak{X}$, the normal bundle $N_q$ vanishes, and each component of ${\rm ch}(d(\lambda_q))$ is $1$. Furthermore, as $\mathfrak{X}$ is essentially a $k$-fold quotient of ${\mathbb P}^n$, $$\int_{\mathfrak{X}} \: = \: \frac{1}{k} \int_{{\mathbb P}^n}.$$ Plugging into the index formula, $$\begin{aligned} \int_{I_{\mathfrak{X}}} {\rm ch}^{\rm rep}({\cal O}_X(m)) {\rm Td}(TI_{\mathfrak{X}}) & = & \sum_{\alpha} \int_{\mathfrak{X}} \alpha^{-m} {\rm ch}({\cal O}_{\mathfrak{X}}(m)) {\rm Td}(T\mathfrak{X}), \\ & = & \sum_{\alpha} \alpha^{-m} \int_{\mathfrak{X}} \sum_i {\rm ch}_i({\cal O}_{\mathfrak{X}}(m)) {\rm Td}_{n-i}(T\mathfrak{X}).\end{aligned}$$ Now, since $\alpha$ is a $k$th root of unity, the sum $$\sum_{\alpha} \alpha^{-m}$$ will vanish unless $m$ is divisible by $k$. Thus, if $m$ is not divisible by $k$, we find that $\chi({\cal O}_{\mathfrak{X}}(m))$ vanishes. Next, suppose that $m=n k$ for some integer $n$. Then, $$\begin{aligned} \int_{I_{\mathfrak{X}}} {\rm ch}^{\rm rep}({\cal O}_{\mathfrak{X}}(m)) {\rm Td}(I_{\mathfrak{X}}) & = & \sum_{\alpha} \int_{\mathfrak{X}} \alpha^{-m} {\rm ch}({\cal O}_{\mathfrak{X}}(m)) {\rm Td}(T\mathfrak{X}), \\ & = & \sum_{\alpha} \int_{\mathfrak{X}} \pi^* {\rm ch}({\cal O}_{{\mathbb P}^n}(n)) {\rm Td}(T {\mathbb P}^n), \\ & = & \sum_{\alpha} \frac{1}{k} \int_{ {\mathbb P}^n } {\rm ch}({\cal O}_{{\mathbb P}^n}(n)) {\rm Td}(T {\mathbb P}^n), \\ & = & \int_{ {\mathbb P}^n } {\rm ch}({\cal O}_{{\mathbb P}^n}(n)) {\rm Td}(T {\mathbb P}^n), \\ & = & \chi\left({\mathbb P}^n, {\cal O}_{ {\mathbb P}^n }(n) \right).\end{aligned}$$ Now, let us compare to expectations. In the present case, if $m$ is not divisible by $k$, then all the sheaf cohomology groups of ${\cal O}_{\mathfrak{X}}(m)$ should vanish, so the Euler class $\chi({\cal O}_{\mathfrak{X}}(m))$ should vanish, exactly as we have computed. If $m$ is divisible by $k$, then $\chi({\cal O}_{\mathfrak{X}}(m)) = \chi({\cal O}_{ {\mathbb P}^n }(m/k))$, again matching the result of the computation. Another example[^26] will be handy to understand. Take $\mathfrak{X} = [T^4/{\mathbb Z}_2]$, where the ${\mathbb Z}_2$ acts by sign flips (and so has 16 fixed points). Let us compute $$\chi\left( {\cal O}_{\mathfrak{X}}[0] \right), \: \: \: \chi\left( {\cal O}_{\mathfrak{X}}[1/2] \right),$$ where ${\cal O}_{\mathfrak{X}}[0]$ denotes the structure sheaf with trivial ${\mathbb Z}_2$-equivariant structure, and ${\cal O}_{\mathfrak{X}}[1/2]$ denotes the structure sheaf with nontrivial equivariant structure. For this $\mathfrak{X}$, $I_{\mathfrak{X}}$ has 17 components: one copy of $\mathfrak{X}$, and 16 copies of $[{\rm pt}/{\mathbb Z}_2]$. From the definition $${\rm ch}^{\rm rep}(L)|_{\alpha} \: = \: \pi^* {\rm ch}(L) |_{\alpha} \otimes \chi,$$ where $\alpha$ is a component of $I_{\mathfrak{X}}$ and $\chi$ the eigenvalue of $\alpha$’s stabilizer on $\pi^* L$, it is straightforward to compute that $$\begin{aligned} {\rm ch}^{\rm rep}( {\cal O}[0] ) & = & (1, \vec{0}, 0; 1, \cdots 1), \\ {\rm ch}^{\rm rep}( {\cal O}[1/2] ) & = & (1, \vec{0}, 0; -1, \cdots -1),\end{aligned}$$ where the leading three entries are for the $\mathfrak{X}$ component, corresponding to elements of $H^0(\mathfrak{X}) = {\mathbb C}$, $H^2(\mathfrak{X}) = {\mathbb C}^6$, $H^4(\mathfrak{X}) = {\mathbb C}$, respectively, and the remaining sixteen entries are each for a copy of $[{\rm pt}/{\mathbb Z}_2]$. The normal bundle $N$ is $0$ for the trivial component $[T^4/{\mathbb Z}_2]$ of $I_{\mathfrak{X}}$, and is ${\mathbb C}^2$ with ${\mathbb Z}_2$ acting by sign flips for the other components of $I_{\mathfrak{X}}$. From that, we read off that $$\begin{aligned} {\rm ch}(d(\wedge^0 N)) & = & (1, \vec{0}, 0; 1, \cdots, 1), \\ {\rm ch}(d(N)) & = & (0, \vec{0}, 0; -2, \cdots, -2), \\ {\rm ch}(d(\wedge^2 N)) & = & (0, \vec{0}, 0; 1, \cdots, 1), \\ {\rm ch}(d(\wedge^k N)) & = & 0 \: \mbox{ for } k > 2.\end{aligned}$$ From this we find $$\alpha_{\mathfrak{X}} \: = \: {\rm ch}(d(\lambda_q)) \: = \: {\rm ch}(d( \sum_i (-)^i \wedge^i N^*) \: = \: (1, \vec{0}, 0; 4, \cdots, 4).$$ In addition, $${\rm ch}^{\rm rep}({\rm Td}(TI_{\mathfrak{X}})) \: = \: (1, \vec{0}, 0; 1, \cdots, 1),$$ hence $${\rm Td}(\mathfrak{X}) \: = \: \alpha_{\mathfrak{X}}^{-1} {\rm Td}(T I_{\mathfrak{X}}) \: = \: (1, \vec{0}, 0; 1/4, \cdots, 1/4).$$ Putting this together, we find $$\begin{aligned} \chi\left( {\cal O}_{\mathfrak{X}}[0] \right) & = & \int_{I_{\mathfrak{X}}} {\rm ch}^{\rm rep}( {\cal O}_{\mathfrak{X}}[0]) {\rm Td}(\mathfrak{X}) \\ & = & \int_{ [T^4/{\mathbb Z}_2] } (1) (1) \: + \: 16 \int_{ [{\rm pt}/{\mathbb Z}_2] } (1) (1/4),\\ & = & 0 \: + \: 4 \int_{ [ {\rm pt}/{\mathbb Z}_2] } 1, \\ & = & 4 \left(\frac{1}{2}\right) \: = \: 2, \\ \chi\left( {\cal O}_{\mathfrak{X}}[1/2] \right) & = & \int_{I_{\mathfrak{X}}} {\rm ch}^{\rm rep}( {\cal O}_{\mathfrak{X}}[1/2]) {\rm Td}(\mathfrak{X}) \\ & = & \int_{ [T^4/{\mathbb Z}_2] } (1) (1) \: + \: 16 \int_{ [{\rm pt}/{\mathbb Z}_2] } (-1) (1/4), \\ & = & 0 \: - \: 4 \int_{ [{\rm pt}/{\mathbb Z}_2] } 1, \\ & = & -4 \left( \frac{1}{2}\right) \: = \: -2.\end{aligned}$$ Let $Y$ denote a minimal resolution of $T^4/{\mathbb Z}_2$. Applying the McKay correspondence [@bkr], it can be shown [@tonypriv] that the bundle ${\cal O}_{\mathfrak{X}}[0]$ maps to ${\cal O}_Y$, and ${\cal O}_{\mathfrak{X}}[1/2]$ maps to ${\cal O}_Y( - (1/2) \sum E_a)$ where the $E_a$ are the exceptional divisors. Furthermore, it can be shown that on $Y$, $\chi( {\cal O}_Y) = +2$ and $\chi( {\cal O}_Y( - (1/2) \sum E_a )) = -2$, matching the Euler characteristics above. So far we have discussed the index of the operator $\overline{\partial}$. We are not aware of rigorous results concerning the Dirac index, which would be of direct relevance for physics. That said, it is very natural to conjecture that, by analogy with smooth manifolds, the Dirac index is computed by a closely analogous expression, except that ${\rm Td}(TI_{\mathfrak{X}})$ is replaced by $${\rm Td}(TI_{\mathfrak{X}}) \exp\left( - \frac{1}{2} c_1^{\rm rep}(TI_{\mathfrak{X}}) \right),$$ following the usual pattern that $$\hat{A}(M) = {\rm Td}(M) \exp( - (1/2) c_1(M) )$$ for a smooth manifold $M$. See also [*e.g.*]{} [@edidin1; @reid-icecream; @tt1] and references therein for more information on index theorems on stacks. Roots of canonical bundles {#app:canonical-roots} ========================== On a ${\mathbb Z}_k$ gerbe, sometimes there exist $k$th roots of the canonical bundle, and sometimes not, depending upon the gerbe. Let us work through some examples. First, consider a nontrivial ${\mathbb Z}_k$ gerbe over ${\mathbb P}^1$. In particular, let us consider the gerbe defined by the quotient $$\frac{ {\mathbb C}^2 - 0 }{ {\mathbb C}^{\times} },$$ where the ${\mathbb C}^{\times}$ acts with weight $k$. We will show that the pullback of any line bundle on ${\mathbb P}^1$ to this gerbe does admit a $k$th root. A line bundle over this gerbe will have a total space of the form $$\frac{ ( {\mathbb C}^2 - 0 ) \times {\mathbb C} }{ {\mathbb C}^{\times} },$$ where ${\mathbb C}^{\times}$ acts on $([x,y],z)$ as $$([x,y],z) \: \mapsto \: ( [ \lambda^k x, \lambda^k y], \lambda^n z),$$ and $n$ classifies the line bundle. The pullback of ${\cal O}(m)$ on ${\mathbb P}^1$ to the gerbe has $n = km$, so a line bundle on the gerbe with $n=m$ has the property that its $k$th tensor power with itself is the pullback of ${\cal O}(m)$. Thus, on this ${\mathbb Z}_k$ gerbe, $k$th roots of pullbacks of any line bundle on the base space do exist. Next, let us consider the trivial ${\mathbb Z}_k$ gerbe over ${\mathbb P}^1$. 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[^1]: Mathematically, the second description, as a gauge theory in which a finite subgroup acts trivially, implies the first, together with a small amount of additional information, a certain trivialization, which we have suppressed from the description of the first, so we should be slightly careful in claiming that they are precisely the same. [^2]: Meaning, only instantons with degrees satisfying certain divisibility properties are included. [^3]: The papers [@nr; @msx; @glsm] contain consistency checks of this computation. Ultimately, demanding modular invariance forces the spectrum to contain multiple dimension zero operators. [^4]: Another thrust of the same papers is a modern discussion of Fayet-Iliopoulos parameters in supergravity – it is argued that they can exist and are quantized. See [*e.g.*]{} [@dienes-thomas] for an excellent discussion of old lore on the subject, which is circumvented in the works above. [^5]: We would like to thank J. Gray for pointing this out to us. [^6]: In addition to the references above on the physics of nonlinear sigma models, there is also an extensive discussion of Gromov-Witten invariants of stacks in the math literature, see for example [@cr; @agv; @cclt; @mann] for a few representative examples. [^7]: With minor caveats, as discussed in [*e.g.*]{} [@msx]. [^8]: In addition, stacks can have presentations of other forms. However, realizing other types of presentations in physics would require a significant generalization of Faddeev-Popov and Batalin-Vilkovisky gauge-fixing procedures, which we do not claim to understand, so we do not claim that physics can be associated to all presentations. [^9]: Historically, this was one of several confusing points in understanding whether strings could be consistently defined on stacks. [^10]: Restricting the allowed instanton sectors ordinarily breaks cluster decomposition, and understanding how this can be consistent was, historically, another confusing issue that had to be straightened out to make sense of strings on stacks. Briefly, the answer is that the theory decomposes into a union of theories on ordinary spaces, see [*e.g.*]{} [@summ; @cdhps; @sugrav-g] for discussions in two and four-dimensional theories. We will return to this in section \[sect:het-gsomods\]. [^11]: Only if $K$ lies in the center of $G$ would the tangent bundle have a trivial extension over $\hat{K}$. [^12]: We have not been able to locate this particular duality in the literature, but would not be surprised if it has been discussed somewhere previously, presumably in a different context. The closest of which we are aware is old work on T-duality in toroidally compactified heterotic strings, relating Spin$(32)/{\mathbb Z}_2$ strings and $E_8 \times E_8$ strings after the gauge group has been Higgsed to a common subgroup, see for example [@ginsparghet]. [^13]: We would link to thank J. Distler for suggesting this construction to us. [^14]: This gerbe is the obstruction to lifting the principal ${\mathbb Z}_2$ bundle $T^4 \rightarrow [T^4/{\mathbb Z}_2]$ to a principal ${\mathbb Z}_4$ bundle on $[T^4/{\mathbb Z}_2]$. But a principal ${\mathbb Z}_k$ bundle on any space $X$ is the same thing as a homomorphism $\pi_1(X) \rightarrow {\mathbb Z}_k$. Therefore, we can study nontriviality of the gerbe as a question about lifts of group homomorphisms. The bundle $T^4 \rightarrow [T^4/{\mathbb Z}_2]$ corresponds to a homomorphism $$\phi: \: \pi_1\left( [T^4/{\mathbb Z}_2] \right) \: \longrightarrow \: {\mathbb Z}_2.$$ (In particular, since $T^4 \rightarrow [T^4/{\mathbb Z}_2]$ is a principal ${\mathbb Z}_2$ bundle, we have a long exact sequence with relevant part $$\pi_1(T^4) \: \longrightarrow \: \pi_1([T^4/{\mathbb Z}_2) \: \stackrel{\phi}{\longrightarrow} \: \pi_0\left( {\mathbb Z}_2 \right) \: \left( \cong \: {\mathbb Z}_2 \right) \: \longrightarrow \: \pi_0(T^4),$$ and as $T^4$ is connected, we see that $\phi$ is surjective.) We want to understand whether $\phi$ lifts to a homomorphism $$\psi: \: \pi_1\left( [T^4/{\mathbb Z}_2] \right) \: \longrightarrow \: {\mathbb Z}_4.$$ First note $$\pi_1\left( [T^4/{\mathbb Z}_2] \right) \: = \: {\mathbb Z}_2 \rtimes {\mathbb Z}^4,$$ where the nontrivial element in ${\mathbb Z}_2$ acts as multiplication by $-1$ on ${\mathbb Z}^4$. The homomorphism $\phi$ is the projection to ${\mathbb Z}_2$. The maximal 2-group quotient of ${\mathbb Z}_2 \rtimes {\mathbb Z}^4$ is ${\mathbb Z}_2 \times ({\mathbb Z}_2)^4$, so any homomorphism ${\mathbb Z}_2 \rtimes {\mathbb Z}^8 \rightarrow {\mathbb Z}_4$ will factor through ${\mathbb Z}_2 \times ({\mathbb Z}_2)^4$. But in the map ${\mathbb Z}_4 \rightarrow {\mathbb Z}_2$, the generator of ${\mathbb Z}_4$ maps onto the generator of ${\mathbb Z}_2$. Since ${\mathbb Z}_2 \times ({\mathbb Z}_2)^4$ does not contain any element of order 4, there is no map ${\mathbb Z}_2 \times ({\mathbb Z}_2)^4 \rightarrow {\mathbb Z}_4$ that lifts the projection onto the first factor. Therefore, the ${\mathbb Z}_2$ gerbe is nontrivial. More generally, if $[T^4/{\mathbb Z}_{2k}]$ is a ${\mathbb Z}_k$ gerbe over ${\mathbb Z}_2$, where the $Z_{2k}$ acts by first projecting to ${\mathbb Z}_2$, then it is nontrivial. [^15]: For further examples of Calabi-Yau threefolds with this property, see [*e.g.*]{} [@cd]. Examples include ${\mathbb P}^7[2,2,2,2]$ and $({\mathbb P}^1)^4$ with a degree (2,2,2,2) hypersurface. For both, the restriction of an ambient hyperplane class to the Calabi-Yau defines a line bundle which is invariant but not equivariant. [^16]: The dimension of this sheaf cohomology group can be determined from index theory, and applies to any stable irreducible rank 4 bundle ${\cal E}$ on a K3 surface. [^17]: It is a standard result that the moduli in an irreducible rank $r$ vector bundle ${\cal E}$ on K3 with $c_1({\cal E}) = 0$, $c_2({\cal E}) = c_2(T K3)$ is encoded in $24r + 1 - r^2$ hypermultiplets, or $2(24r+1-r^2$ half-hypermultiplets. Here, $r=4$. [^18]: For simplicity, as we wish to work in light-cone gauge, we will assume that the complex dimension is bounded by 4. [^19]: Since $\alpha$ leaves the points invariant, this component of the inertia stack must have a $\langle \alpha \rangle$ gerbe structure, and bundles on such gerbes have an eigenbundle decomposition as given here. [^20]: Fock vacua have been much more commonly described in terms of sections of bundles over CFT moduli spaces, see [*e.g.*]{} [@distler-trieste; @bcov], but descriptions as sections of bundles over target spaces are much more rare. [^21]: Since the $\alpha$-sector has components which are $t_{\alpha}$ gerbes, $t_{\alpha}$-th roots of bundles might exist, though not necessarily. (See appendix \[app:canonical-roots\] for examples of bundles on ${\mathbb Z}_n$-gerbes which do and do not admit $n$th roots.) [^22]: We have omitted modings for reasons of notational sanity. [^23]: To make it clear that this condition is nontrivial, here is an example of a global orbifold in which a twisted sector has support on a non-Calabi-Yau subvariety. Let $X$ be a branched double cover of ${\mathbb P}^n$, branched over a degree $2n+2$ locus. Now, orbifold by the globally-acting ${\mathbb Z}_2$ that exchanges the sheets of the cover. This leaves invariant the degree $2n+2$ branch locus, which is not Calabi-Yau. [^24]: The second condition arises from the need to make the path integral measure a scalar, ultimately. On stacks, one might wonder whether one should impose an analogous condition in each individual twisted sector, something of the form $$\det {\cal E}_0^{\alpha *} \: \cong \: K_{\alpha}.$$ Reference [@manion-toappear] does not impose a stronger condition of this sort. One reason is that there is no analogue of such a condition in GLSM’s (whereas the original condition $\det {\cal E}^* \cong K_{\mathfrak{X}}$ on the entire stack does manifest in GLSM’s). In terms of making sense of path integral measures, in twisted sectors one must insert twist fields to get nonzero results, and which would modify any such constraint one wished to impose on individual twisted sectors. [^25]: For $m>0$. The total spaces of line bundles of positive degree over projective spaces do not seem to admit a GLSM description, even though they are toric varieties – they can be described as GIT quotients of open subsets of ${\mathbb C}^{n+2}$ by ${\mathbb C}^{\times}$, but not as a GIT quotient of the full complex vector space, and they naturally compactify to ${\mathbb P}^{n+1}_{[1,\cdots,1,m]}$. We would like to thank D. Skinner for asking a question that made this manifest. [^26]: We would like to thank T. Pantev for explaining this example to us.

--- abstract: 'The top quark pair production and decay are considered in the framework of the smeared-mass unstable particles model. The results for total and differential cross sections in vicinity of $t\bar{t}$ threshold are in good agreement with the previous ones in the literature. The strategy of calculations of the higher order corrections in the framework of the model is discussed. Suggested approach significantly simplifies calculations compared to the standard perturbative one and can serve as a convenient tool for fast and precise preliminary analysis of processes involving intermediate time-like top quark exchanges in the near-threshold region.' address: - 'Institute of Physics, Southern Federal University, Rostov-on-Don 344090, Russia' - | Theoretical High Energy Physics, Department of Astronomy and Theoretical Physics,\ Lund University, SE 223-62 Lund, Sweden - 'Institute of Physics, Southern Federal University, Rostov-on-Don 344090, Russia' author: - 'V. I. KUKSA[^1]' - 'R. S. PASECHNIK[^2]' - 'D. E. VLASENKO[^3]' title: MASS SHELL SMEARING EFFECTS IN TOP PAIR PRODUCTION --- Introduction ============ The top pair production and decay are the key processes for precision tests of the Standard Model (SM) (see e.g. Ref. \[\] and references therein). They were intensively studied in the framework of the Quantum Chromodynamics (QCD) and Electro-Weak (EW) perturbation theory during last two decades, and various methods and schemes were proposed. The major goal of these investigations is to define the basic physical parameters of the top quark, such as its mass, width and couplings with other SM particles. In the past, the top quark physics was one of the primary research objectives at Tevatron. Nowadays, the biggest attention is paid to the process of the top quark production at the LHC (see e.g. Refs. \[\]). However, the highest precision measurements of the top quark properties can best be reached at the future Linear Collider (LC) which supposedly operates in a clean experimental environment. The top quark physics is one of the most interesting and challenging targets for future $e^+e^-$ or $\mu^+\mu^-$ LC experiments \[\]. The top pair production is followed by a decay chain with intermediate gauge boson states, i.e. the full process under consideration is $e^+e^-\to t^*\bar{t^*}\to b\bar{b}W^+W^-\to b\bar{b}4f$. The widths of both the top quark and the $W$-boson are large, and one necessarily needs to take into account corresponding Finite-Width Effects (FWE). In the framework of the standard perturbative approach, these effects are typically described by means of dressed propagators which are regularized by the total decay width. In order to analyze the full process of the top pair production relevant for phenomenological studies, we also have to take into account the background contribution coming from many other topologically different diagrams leading to the same six-fermion final states, which is a rather non-trivial task. The Born-level cross-sections of the processes $e^+e^-\to b\bar{b}u\bar{d}\mu^-\bar{\nu}_{\mu}$ and $e^+e^-\to b\bar{b}4q$ were calculated in Refs. \[\] and \[\], respectively. Other exclusive reactions with $b\bar{b}d\bar{u}\mu^+\nu_{\mu},\,b\bar{b}c\bar{s}d\bar{u}$ and $b\bar{b}\mu^+\nu_{\mu}\tau^-\bar{\nu}_{\tau}$ final states were considered in Ref. \[\]. In particular, it was shown that the contribution of the top-pair signal $e^+e^-\to t^*\bar{t}^*\to b\bar{b}4f$ is dominant, but the background (caused by one-resonant or non-resonant diagrams) can be quite significant too. However, it can be drastically decreased by applying certain kinematical cuts on the appropriate invariant masses. The QCD corrections for the reaction $e^+e^-\to t\bar{t}$ in the continuum above the threshold were previously obtained in Refs. \[\]. As well as the one-loop EW corrections were calculated in many papers (for corresponding references, see e.g. Introduction in Ref. \[\]). Concerning radiative corrections (RC) to reaction $e^+e^-\to b\bar{b}4f$ with six-fermion final states, the situation is more complicated and less clear \[\]. At the tree level, any of the reactions receives contributions from several hundreds of diagrams. The calculations of the full $O(\alpha)$ radiative corrections are very complicated, and different approximation schemes are typically applied. The most detailed analysis of the exclusive reactions $e^+e^-\to b\bar{b}\mu^+\nu_{\mu}\mu^-\bar{\nu}_{\mu}$ and $e^+e^-\to b\bar{b}d\bar{u}\mu^-\bar{\nu}_{\mu}$ was performed in Ref. \[\]. In this paper, the cross-sections were calculated taking into account the leading radiative corrections, such as the initial state radiation (ISR) and factorizable EW corrections to the on-shell top-pair production, to the decay of the top quark into $bW$ and to the subsequent decays of the $W$-bosons. Usually, such calculations are carried out automatically by Monte Carlo techniques (see Ref. \[\] and references therein). In this work, we consider reactions like $e^+e^-\to t^*\bar{t}^*\to b\bar{b}4f$ with any four-fermion final states $4f$. The analysis is performed in the framework of the smeared mass unstable particles model (below, SMUP model) \[\]. Due to exact factorization at intermediate $t,\bar{t}$ and $W^+,W^-$ states, the cross-section can be represented in a simple analytical form which is convenient for analytical and numerical analysis. So far, we have applied the SMUP approach only for unstable gauge boson production and decay (see e.g. Refs. \[\]). As a continuation of our earlier studies, in this work we test the SMUP approach for the case of unstable fermions, i.e., specifically, top quarks. In our calculations, we take into account NLO radiative EW and QCD factorizable corrections which dominate close to $t\bar{t}$ threshold. Also, we illustrate the influence of the mass smearing effects and various radiative corrections (RC’s) on the differential cross-sections. The results are compared with ones calculated by using the standard perturbative methods \[\], where cross-sections were represented for case of full $2\to 6$ process and, separately, for the top signal contribution alone. It was shown that in the Born approximation the results coincide with a rather high precision, and deviations of the higher-order corrected results from the standard ones are at the percentage level. So, the suggested approach can be applied in a fast preliminary analysis of various complicated processes involving intermediate top quark exchanges in the Standard Model and beyond. Note, here we do not consider the near-threshold effects caused by the generation of the coupled $t\bar{t}$ state, which were considered in detail in many previous studies (see, for instance, Ref. \[\] and references therein). We postpone this issue for a forthcoming study. The model cross-section of the top-pair production and decay at the tree level ============================================================================== The process of top-pair production with subsequent decay $e^+e^-\to t^*\bar{t^*}\to b\bar{b}W^+W^-\to b\bar{b}4f$ is schematically represented in Fig. \[fig1\]. The full process contains two steps with unstable intermediate time-like states, namely, $t,\bar{t}$ and $W^+,W^-$ states. In this case, as was shown in Ref. \[\], the double factorization takes place and can be described in the framework of the SMUP model \[\]. Due to this factorisation, the full process can be divided into three stages: $e^+e^-\to t^*\bar{t}^*$, $t^*\bar{t}^*\to b\bar{b}W^+W^-$ and $W^+W^-\to 4f$. Here, the top-quarks and $W$-bosons are treated as unstable particles, and finite-width effects should be taken into account. ![Feynman diagram of the top quark signal process $e^+e^-\to t^*\bar{t^*}\to b\bar{b}W^+W^-\to b\bar{b}4f$.[]{data-label="fig1"}](tbart1.eps){width=".6\textwidth"} The SMUP model cross-section of the first reaction $e^+e^-\to t^*\bar{t}^*$ can be written as \[\] $$\label{2.1} \sigma(e^+e^-\to t^*\bar{t}^*)=\int_{m^2_0}^s\int_{m^2_0}^{(\sqrt{s}-m_1)^2} \sigma(e^+e^-\to t(m_1)\bar{t}(m_2))\rho_t(m_1)\rho_t(m_2)dm^2_1 dm^2_2,$$ where $m_0\approx 2M_b$ ($M_b$ is the bottom quark mass) is the threshold value of the top mass variable, $\sigma(e^+e^-\to t(m_1)\bar{t}(m_2))$ is the cross-section of top pair production with random masses $m_1$ and $m_2$ and $\rho_t(m)$ is the probability density which describes the mass smearing of top quarks. In our calculations we take it in the Lorentzian form as \[\] $$\label{2.2} \rho_t(m)=\frac{1}{\pi}\frac{m\Gamma_t(m)}{(m^2-M^2_t)^2+m^2\Gamma^2_t(m)},$$ where $\Gamma_t(m)$ is the total decay width of the top quark with mass $m$. The decay mode $t\to bW$ has a very large branching ratio $\mathrm{Br}(t\to bW) \approx 0.999$, so formula (\[2.1\]) almost exactly describes the cascade process $e^+e^-\to t^*\bar{t}^*\to b\bar{b}W^+W^-$ in the stable $W$-boson approximation. In order to take into account the instability of $W$-bosons we have to express the top quark width $\Gamma_t(m)\approx \Gamma(t\to bW)$ in Eq. (\[2.2\]) as a function of smeared $W$-boson mass $\Gamma(t\to bW(M_W))$ with averaging over $M_W$. Thus, the model cross-section of the full inclusive process $e^+e^-\to t^*\bar{t}^*\to b\bar{b}W^+W^- \to b\bar{b}\sum_f 4f$ depicted in Fig. \[fig1\] has the following convolution form: $$\begin{aligned} \label{2.3} \sigma(e^+e^-\to b\bar{b}\sum_f 4f)=&\int_{m^2_0}^s\int_{m^2_0}^{(\sqrt{s}-m_1)^2} \sigma(e^+e^-\to t(m_1)\bar{t}(m_2))\times \notag\\ &\int_{(m_0-M_b)^2}^{(m_1-M_b)^2}\rho_t(m_1,m_{W^+})\rho_W(m_{W^+})dm^2_{W^+}\times\\ &\int_{(m_0-M_b)^2}^{(m_2-M_b)^2}\rho_t(m_2,m_{W^-})\rho_W(m_{W^-})dm^2_{W^-}dm^2_1 dm^2_2,\notag\end{aligned}$$ where $\rho_W(m)$ is defined by Eq. (\[2.2\]). In order to describe an exclusive reaction $e^+e^-\to t^*\bar{t}^*\to b\bar{b}f_1f_2f_3f_4$ we have to replace the total decay widths of $W$-bosons, which enter the numerator in Eq. (\[2.2\]), by corresponding exclusive ones (see Section 4). The same result can be obtained exactly if one calculates the cross-section of this process explicitly in the framework of the SMUP model by using dressed propagators of unstable particles (UP’s). In Ref. \[\] it was shown that exact factorization of a decay chain process with UP’s in an intermediate state takes place when we exploit the model effective propagators for fermion and vector UP’s in the following form $$\label{2.4} \hat{D}(q)=i\frac{\hat{q}+q}{P_F(q)},\qquad D_{\mu\nu}(q)=-i\frac{g_{\mu\nu}-q_{\mu}q_{\nu}/q^2}{P_V(q)},$$ where $P_F(q)$ and $P_V(q)$ are the denominators of the fermion and vector boson dressed propagators, which contain corresponding total decay widths. The structure of numerators in Eq. (\[2.4\]) provides exact factorization and leads to a convolution-like expression (\[2.3\]) for the cross-section. So, there is a self-consistency between the model and UP effective theory description of the processes with UP in intermediate states. Thus, the process with a six-particles final state shown in Fig. \[fig1\] is described by a simple analytical expression (\[2.3\]) with four integrations over smeared unstable top and $W$-boson masses. Note, that the standard perturbative treatment of the six-particle final states in general case leads to $N=3\cdot 6-4=14$ independent parameters, from which 13 parameters have to be integrated over \[\]. Such a complicated problem can be solved by using involved Monte Carlo numerical simulations only. The results of the SMUP model calculations are presented in Fig. \[fig2\]. Here, the dotted line represents the cross-section of the top-pair production in the stable particle approximation (SPA), i.e. without smearing of the top mass. The dashed line is the cross-section incorporating the top mass smearing or top finite-width effects (FWE) only, and the solid line gives the full mass smearing result, both top-quarks and $W$-bosons. Note, that the second case corresponds to the standard treatment of the process $e^+e^-\to t^*\bar{t}^*\to b\bar{b}W^+W^-$ in the stable $W$-boson approximation, and the third case – to the full process shown in Fig. \[fig1\]. ![The cross-sections of the processes $e^+e^-\to t\bar{t}$, $e^+e^-\to b\bar{b}W^+W^-$ and $e^+e^-\to b\bar{b}\sum_f4f$.[]{data-label="fig2"}](SmearMass.eps){width=".6\textwidth"} From Fig. \[fig2\], one can see that the contribution of the top quarks’ FWE’s is significant (up to a few percents in the near-threshold region), while the contribution of $W$-bosons’ FWEs is small. The comparison of our results with ones in the standard perturbative treatment shows that deviations are typically very small. For instance, it was obtained in Ref. \[\], that $\sigma(e^+e^-\to t^*\bar{t}^*\to b\bar{b}W^+W^-)$ for $\sqrt{s}=500\,\mbox{GeV}$ is equal to 629 fb for $M_t=150\,\mbox{GeV}$ and 553 fb for $M_t=180\,\mbox{GeV}$. For the same input data, we have obtained 630 fb and 554 fb, respectively, which are in a good agreement with the result mentioned above. This comparison proves the applicability of the SMUP model fermion propagator given by the first expression in Eq. (\[2.4\]). In Section 4, we make such a comparison for exclusive processes as well where both SMUP model fermion and boson propagators Eq. (\[2.4\]) are used. It should be noticed also that we consider the FWE’s, which are significant in the near-threshold region, but we do not include near-threshold effects caused by possible intermediate $t\bar{t}$ bound states. Since the top mean lifetime is considerably shorter than the hadronisation time, the bound state effect has no sharp resonant nature. However, it can be comparable with FWE’s or mass-smearing effects under consideration, and this problem will be considered in more detail elsewhere. Factorizable corrections to the cross-section ============================================= As it was shown in previous papers \[\], the EW and QCD corrections give large contributions to the cross-section of the top-pair production at energy scales close to its threshold. In this Section, we describe the strategy of our model calculations and give the total cross-section including the principal part of NLO EW and QCD corrections. Note, that the strategy of calculations and the choice of input parameters are mainly caused and defined by the effective character of the model treatment. In the framework of the SMUP model, the instability (or finite width) of unstable particles is accounted for by the smearing of their masses, i.e. by the probability density function $\rho(m)$. In turn, this function contains momentum dependent parameters $M(q)$ and $\Gamma(q)$ in analogy with the standard perturbative treatment which uses dressed propagators. So, in that sense the corrections of self-energy type are already included at the “effective” tree level, and it is reasonable to use an effective couplings, such as running coupling, absorbing the major part of vertex-type corrections. In our calculations we have used the following input data \[\]: $$\begin{aligned} \label{3.1} &\alpha(M_Z)=0.00781763,\,\,\,\alpha_s(M_Z)=0.118,\,\,\,\sin^2\theta_W(M_Z)=\hat{s}^2_Z=0.2313,\notag\\ &M_Z=91.1876\,\mbox{GeV},\,\,\,M_W=80.399\,\mbox{GeV},\,\,\,M_t=172.9\,\mbox{GeV}.\end{aligned}$$ The running coupling constants $\alpha_k(Q^2),\,k=1,2,3$ were used in the one-loop approximation: $$\label{3.2} \alpha_k(Q^2)=\frac{\alpha_k(M_Z)}{1-(\beta_k/2\pi) \ln(Q^2/M^2_Z)},\,\,\,\beta_k=(4.1,\,-19/6,\,-7).$$ The cross-sections are calculated including the following corrections: - Vertex and self-energy type corrections for stable particles are mainly included into running couplings (\[3.2\]).\ - Self-energy corrections for unstable particles are included into the probability density function $\rho(m)$, which describes the smearing of UP’s masses.\ - Initial state radiation (ISR) is described by the photon radiation spectrum \[\], and the bremsstrahlung from the final $t$-quark states – by vertex $Q$-dependent factor \[\].\ - QCD corrections to the top production and decay are described by the vertex multiplicative factor \[\].\ - Contribution of the box diagrams to the total cross section was evaluated at energy scales close to the threshold by using numerical FormCalc v7.3 \[\] routines. The higher order corrected cross-sections of the inclusive process $e^+e^-\to t^*\bar{t}^*\to b\bar{b}\sum_f4f$ are shown in Fig. \[fig3\]. There, the dotted line represents the Born model cross-section, the dashed line – the cross-section with ISR and the solid line – the cross-section with total factorisable corrections (without box diagrams contribution). From the figure, one can see that the main contribution is given by ISR correction, which significantly reduces the cross-section in the near-threshold energy range and increases it at energy scales above $\sim$0.6 TeV. At large energies ($\sqrt{s}>0.5\,\mbox{TeV}$) the contribution of EW and QCD corrections becomes significant and has to be properly taken into account. ![The higher order corrected cross-sections of the process $e^+e^-\to b\bar{b}\sum_f4f$.[]{data-label="fig3"}](Corrections.eps){width=".6\textwidth"} In Fig. \[fig4\] we present the invariant mass distribution and illustrate the influence of various corrections on it. One can see that the corrections which we have taken into account according to the procedure above give noticeable contribution into this distribution in the peak area. We, also, illustrate such influence on the angular differential cross-section presented in Fig. \[fig5\]. Again, we notice that this influence is quite significant and should be taken into account. The cross-sections of exclusive processes ========================================= So far, we have considered the cross-section of inclusive process $e^+e^-\to t^*\bar{t}^*\to b\bar{b}\sum_f4f$ where the final state is summed up over all possible fermion flavors. As was noticed in the second Section, in order to get the cross-section of exclusive process $e^+e^-\to t^*\bar{t}^*\to b\bar{b}f_1f_2f_3f_4$ we can include the corresponding branching ratios $\mathrm{Br}(W\to f_1f_2)$ and $\mathrm{Br}(W\to f_3f_4)$. Acting this way we obtain $$\label{4.1} \sigma(e^+e^-\to b\bar{b}f_1f_2f_3f_4)=\sigma(e^+e^-\to b\bar{b}\sum_f4f)\mathrm{Br}(W\to f_1f_2)\mathrm{Br}(W\to f_3f_4)\,.$$ where we omit intermediate virtual $t^*\bar{t}^*$ state for simplicity. This relation directly follows from the Eq. (\[2.3\]) when one substitutes a partial decay width of the $W$-boson into numerator of the probability distribution function $\rho_W(m)$ instead of the total width. It can be also derived by straightforward calculation of the $\sigma(e^+e^-\to b\bar{b}f_1f_2f_3f_4)$ in the framework of the effective theory (see Ref. \[\]). The expressions for the branchings ratios $\mathrm{Br}(W\to f_1f_2)$ were considered in detail in Ref. \[\]. Here, we use very simple but sufficiently precise formulae which incorporate QCD corrections: $$\label{4.2} \mathrm{Br}(W\to l\bar{\nu}_l)=\frac{1}{9(1+2\alpha_s(M_Z)/3\pi)},\,\,\,\mathrm{Br}(W\to u_i\bar{d}_k)=\frac{|V_{ik}|^2(1+\alpha_s(M_Z)/\pi)}{3(1+2\alpha_s(M_Z)/3\pi)},$$ where $V_{ik}$ are elements of the Cabibbo-Kobayashi-Maskawa mixing matrix. We, also, employ the QCD corrected expression for the top quark width \[\]: $$\label{4.3} \Gamma(t\to bW)=\frac{1}{16}\alpha_2(M_t)|V_{tb}|^2\,\eta_{QCD}\,M_t\, f(M_t,M_W,M_b),$$ where $$\begin{aligned} \label{4.4} &&f(M_t,M_W,M_b)=\lambda(M^2_b,M^2_W;M^2_t)\left(\frac{(M^2_t-M^2_b)^2}{M^2_t M^2_W} +\frac{M^2_t+M^2_b-2M^2_W}{M^2_t}\right);\\ &&\lambda(M^2_b,M^2_W;M^2_t)=\left(1-2\frac{M^2_b+M^2_W}{M^2_t} +\frac{(M^2_W-M^2_b)^2}{M^4_t}\right)^{1/2};\nonumber \\ &&\eta_{QCD}=1-\frac{2\alpha_s(M_t)}{3\pi}\left(\frac{2\pi^2}{3}-\frac{5}{2}\right). \nonumber\end{aligned}$$ Using Eqs. (\[4.1\])–(\[4.3\]) we can calculate the exclusive cross-section for an arbitrary six-fermion final state $(b\bar{b}f_1f_2f_3f_4)$. Such calculations taking into account the factorizable EW corrections were performed within the standard perturbative approach for the case of $(b\bar{b}\mu^+\nu_{\mu}\mu^-\bar{\nu}_{\mu})$ and $(b\bar{b}\mu^+\nu_{\mu}d\bar{u})$ final states in Ref. \[\]. In this work, the full set of topologically different Born diagrams leading to the same six-fermion final state was considered. It was shown, that certain cuts on invariant masses of the $Wb$ and $f_if_k$ pairs, which correspond to intermediate $t,\bar{t}$ and $W^+,W^-$ states for signal diagrams, significantly reduce the relative contribution of the background (see Table \[tab1\]). In Tables \[tab1\] and \[tab2\] the cross-sections are given for two distinct reactions $$\label{4.5} (1):\quad e^+e^-\to b\bar{b}\mu^+\nu_{\mu}\mu^-\bar{\nu}_{\mu},\qquad (2):\quad e^+e^-\to b\bar{b}\mu^+\nu_{\mu}d\bar{u}.$$ for the energies $\sqrt{s}=430,\,500,\,1000,\,\mbox{GeV}$. In Table \[tab1\] the cross-sections are presented in the Born approximation for total set of diagrams ($\sigma_{Born}^{(k)}$(total)) and for the signal diagrams ($\sigma_{Born}^{(k)}(t^*\bar{t}^*)$), where $k=1,2$ denotes the first and second reactions in Eq. (\[4.5\]), respectively. These values (in fb) are taken from Table 1 in Ref. \[\] and are calculated with the kinematical cuts $\delta_i<0.1$, where $\delta_i$ is the deviation of the ratio $m_i^{inv}/M_i$ from unity and index $i$ is related to different $t,\bar{t},W^+,W^-$ states (for more details, see Ref. \[\]). [@|c|c|c|c|c|@]{} $\sqrt{s}$, GeV &$\sigma_{Born}^{(1)}$(total)&$\sigma_{Born}^{(1)}(t^*\bar{t}^*)$& $\sigma_{Born}^{(2)}$(total)&$\sigma_{Born}^{(2)}(t^*\bar{t}^*)$\ 430&5.9117&5.8642&17.727&17.592\ 500&5.3094&5.2849&15.950&15.855\ 1000&1.6387&1.6369&4.9134&4.9106\ \[tab1\] In Table \[tab2\] the results for the total cross sections (in fb) of processes $(1)$ and $(2)$ from Eq. (\[4.5\]) in the Born approximation are shown in the second column. The cross-sections with separate ISR and factorizable EW (FEWC) corrections are presented in the third and forth columns, respectively, and the cross-section with both the FEWC and ISR corrections included – in the fifth column. All values are calculated with the kinematical cuts mentioned above. [@|c||c|c|c|c|@]{} $\quad\sqrt{s},\,\rm{GeV}\quad$& $\quad\sigma^{t^*\bar{t}^*}_{\rm Born}\quad$ & $\quad\sigma_{\rm Born+ISR}\quad$ & $\quad\sigma_{\rm Born+FEWC}\quad$ & $\quad\sigma_{\rm Born+ISR+FEWC}\quad$\ \ 430 & $5.8642(45)$ & $5.2919(91)$ & $5.6884(55)$ & $5.0978(53)$\ 500 & $5.2849(43)$ & $5.0997(51)$ & $4.9909(49)$ & $4.8085(48)$\ 1000 & $1.6369(15)$ & $1.8320(18)$ & $1.4243(14)$ & $1.6110(16)$\ \ 430 & $5.86476$ & $5.27613$ & $5.77727$ & $5.19941$\ 500 & $5.27352$ & $5.08651$ & $5.18407$ & $5.00291$\ 1000 & $1.63061$ & $1.83508$ & $1.58925$ & $1.79079$\ \ 430 & $17.592(13)$ & $15.857(20)$ & $17.052(16)$ & $15.283(16)$\ 500 & $15.855(13)$ & $15.311(15)$ & $14.977(16)$ & $14.438(14)$\ 1000 & $4.9106(46)$ & $5.4949(55)$ & $4.2697(40)$ & $4.8287(47)$\ \ 430 & $17.8163$ & $16.0351$ & $17.5540$ & $15.8019$\ 500 & $16.0203$ & $15.4517$ & $15.7516$ & $15.1979$\ 1000 & $4.95397$ & $5.57465$ & $4.82889$ & $5.44011$\ \[tab2\] From Table \[tab2\], it follows that the differences of the model and standard Born cross-sections are of an order of 0.1 percent and ISR correction increases it only slightly. In principle, these deviations can be further reduced. The situation becomes worse, when we take into account all major corrections. The deviations increase and become up to a few percents. This discrepancy is caused by the fact that in Ref. \[\] an additional contribution from the non-signal (background) diagrams was included while we consider the signal contribution only. Moreover, we do not include the contribution of the box diagrams which becomes very important at large energies far from the threshold. According to estimations in the framework of the standard perturbative treatment, the box diagrams contribution is of an order of a few percents in the near-threshold energy range. Rough estimations in the framework of the SMUP model give the box contribution equal to $1.5 - 2$ percents in the energy region under consideration, and these estimations decrease the deviations. However, in the framework of the SMUP model, as well as in the effective theory of UP, the higher order corrections have an effective character, and a consistent formulation of the perturbative treatment with this model is required. This problem leads to using the model propagators inside loop diagrams, but the validity of such a procedure has still to be justified theoretically. In particular, one should first analyze the asymptotic properties of the propagators. The analysis is not carried out yet, but it is in progress now. Note, that a good agreement of the SMUP model and standard Born-level results, illustrated in Table \[tab2\], provides a good basis for such an analysis. Conclusion ========== The production of the $t\bar{t}$ pair and its subsequent decay into six fermion final states in $e^+e^-$ annihilation has been previously analyzed within the standard perturbative treatment in a vast literature. In this work, we performed the corresponding analysis in the framework of SMUP model. So far, this approach was applied mainly to the gauge boson production, where the structure of the model boson propagators was successfully tested \[\]. In the present work, we have tested the structure of the model fermion propagator, and the top quark production mechanism has been chosen as an important example. It was shown that the results of Born-level calculations are in a good agreement with the standard perturbative ones, providing the applicability of the SMUP approach to the top-quark production and decay processes. The SMUP model provides simple analytical expressions for the total cross-sections of inclusive and exclusive processes with top quark pair production and its subsequent decay. It is a convenient and simple instrument for description of complicated multi-step processes with unstable particles participation. 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--- bibliography: - 'sample.bib' --- [**Detecting Memory and Structure in Human Navigation Patterns Using Markov Chain Models of Varying Order** ]{}\ Philipp Singer$^{1,\ast}$, Denis Helic$^{2}$, Behnam Taraghi$^{3}$, Markus Strohmaier$^{1,4}$\ **[1]{} GESIS - Leibniz Institute for the Social Sciences, Cologne, Germany\ **[2]{} Technical University of Graz, Knowledge Technologies Institute, Graz, Austria\ **[3]{} Technical University of Graz, Institute for Information Systems and Computer Media, Graz, Austria\ **[4]{} University Koblenz-Landau, Institute for Web Science and Technologies, Koblenz, Germany\ $\ast$ E-mail: philipp.singer@gesis.org******** Abstract {#abstract .unnumbered} ======== One of the most frequently used models for understanding human navigation on the Web is the Markov chain model, where Web pages are represented as states and hyperlinks as probabilities of navigating from one page to another. Predominantly, human navigation on the Web has been thought to satisfy the memoryless Markov property stating that the next page a user visits only depends on her current page and not on previously visited ones. This idea has found its way in numerous applications such as Google’s PageRank algorithm and others. Recently, new studies suggested that human navigation may better be modeled using higher order Markov chain models, i.e., the next page depends on a longer history of past clicks. Yet, this finding is preliminary and does not account for the higher complexity of higher order Markov chain models which is why the memoryless model is still widely used. In this work we thoroughly present a diverse array of advanced inference methods for determining the appropriate Markov chain order. We highlight strengths and weaknesses of each method and apply them for investigating memory and structure of human navigation on the Web. Our experiments reveal that the complexity of higher order models grows faster than their utility, and thus we confirm that the memoryless model represents a quite practical model for human navigation on a page level. However, when we expand our analysis to a topical level, where we abstract away from specific page transitions to transitions between topics, we find that the memoryless assumption is violated and specific regularities can be observed. We report results from experiments with two types of navigational datasets (goal-oriented vs. free form) and observe interesting structural differences that make a strong argument for more contextual studies of human navigation in future work. Introduction {#sec:intro .unnumbered} ============ Navigation represents a fundamental activity for users on the Web. Modeling this activity, i.e., understanding how predictable human navigation is and whether regularities can be detected has been of interest to researchers for nearly two decades – an example of early work would be work by Catledge and Pitkow [@catledge]. Another example would be [@xing], who focused on trying to understand preferred user navigation patterns in order to reveal users’ interests or preferences. Not only has our community been interested in gaining deeper insights into human behavior during navigation, but also in understanding how models of human navigation can improve user interfaces or information network structures [@borges1999]. Further work has focused on understanding whether models of human navigation can help to predict user clicks in order to prefetch Web sites (e.g., [@bestavros]) or enhance a site’s interface or structure (e.g., [@perkowitz]). More recently, such models have also been deployed in the field of recommender systems (e.g., [@rendle]). However, models of human navigation can only be useful to the extent human navigation itself exhibits regularities that can be exploited. An early study on user navigation in the Web by Huberman et al. [@huberman], for example, already identified interesting regularities in the distributions of user page visits on a Web site. More recently, Wang and Huberman [@wang2012] confirmed these observations and Song et al. [@song] argued that the regularities in human activities might be based on the inherent regularities of human behavior in general. The most prominent model for describing human navigation on the Web is the Markov chain model (e.g., [@pirolli]), where Web pages are represented as states and hyperlinks as probabilities of navigating from one page to another. Predominantly, the Markov chain model has been memoryless in a wide range of works (e.g., Google’s PageRank [@brin]) indicating that the next state only depends on the current state of a user’s Web trail. Recently, a study [@chierichetti] suggested that human navigation might be better modeled with memory – i.e., the next page depends on a longer history of past clicks. However, this finding is preliminary and does not account for the higher complexity of higher order Markov chain models which is why the memoryless model is still widely used. ![**Example of a navigation sequence in the WikiGame dataset.** Bottom row of nodes: A user navigates a series of Wikipedia articles, which can be represented as a sequence of Web pages. Top row of nodes: Each Wikipedia article can be mapped to a corresponding topic through Wikipedia’s system of categories. This results in a sequence of topics.[]{data-label="fig:pathexample"}](paths_categories_cropped){width="85.00000%"} [**Research questions.**]{} In this paper, we are interested in shedding a deeper light on regularities in human navigation on the World Wide Web by studying memory and structure in human navigation patterns. We start by investigating memory of human navigational paths over Web sites by determining the order of corresponding Markov chains. We are specifically interested in detecting if the benefit of a larger memory (or higher order Markov chain) can compensate for the higher complexity of the model. In order to understand whether and to what extent human navigation exhibits memory on a topical level, we abstract away from specific page transitions and study memory effects on a topical level by representing click streams as sequences of topics[^1] (cf. Figure \[fig:pathexample\]). This enables us to (i) move up from the page to topical level and (ii) significantly reduce the complexity of higher order models and therefore (iii) gain deeper insights into memory and structure of human navigational patterns. Finally, we discuss our findings and demonstrate interesting differences between human navigation in free browsing vs. more goal-oriented settings. [**Methods and Materials.**]{} We study memory and structure in human navigation patterns on three similarly structured datasets: WikiGame (a navigation dataset with known navigation goals), Wikispeedia (another goal-oriented navigation dataset) and MSNBC (a free navigation dataset). [[ For analyzing memory, we use Markov chains to model human behavior and analyze the appropriate Markov chain order – i.e., we investigate whether human navigation is memoryless or not. For model selection – i.e., the process of finding the most appropriate Markov chain order – we resort to a highly diverse array of methods stemming from distinct statistical schools: (i) likelihood [@stigler2002statistics; @tong1975], (ii) Bayesian [@Strelioff] and (iii) information-theoretic methods [@akaike; @katz; @murphy; @schwarz; @tong1975]. We supplement these with a (iv) cross validation approach for a prediction task [@murphy]. We thoroughly elaborate each method, put them into relation to each other and also highlight strengths and weaknesses of each. Such detailed derivation of model parameters and the model comparison is, for example, missing in previous work [@chierichetti], which prevents us from drawing definite conclusions. We apply these methods to our human navigational data in order to get an exhaustive picture about memory in human navigation. Finally, we identify structural aspects by analyzing transition matrices produced by our Markov chain analyses. ]{}]{} [**Contributions.**]{} The main contributions of this work are three-fold: - First, we deploy four different, yet complementary, approaches for order selection of Markov chain models (likelihood, Bayesian, information-theoretic and cross validation methods) and elaborate their strengths and weaknesses. [[Hence, our work extends existing studies that model human navigation on the Web using Markov chain models [@chierichetti].]{}]{} By applying these methods on navigational Web data, our work presents – to the best of our knowledge – the most comprehensive and systematic evaluation of Markov model orders for human navigational sequences on the Web to date. Furthermore, we make our methods in the form of an open source framework available online[^2] to aid future work [@github]. - Our empirical results confirm what we inferred from theory: It is difficult to make plausible statements about the appropriate Markov chain order having insufficient data but a vast amount of states, which is a common situation for Web page navigational paths. All evaluation approaches would favor a zero or first order because the number of parameters grows exponentially with the chain order and the available data is too sparse for proper parameter inferences. Thus, we show further evidence that the memoryless model seems to be a quite practical and legitimate model for human navigation on a page level. - By abstracting away from the page level to a topical level, the results are different. By representing all datasets as navigational sequences of topics that describe underlying Web pages (cf. Figure \[fig:pathexample\]), we find evidence that topical navigation of humans is not memoryless at all. On three rather different datasets of navigation – free navigation (MSNBC) and goal-oriented navigation (WikiGame and Wikispeedia) – we find mostly consistent memory regularities on a topical level: In all cases, Markov chain models of order two (respectively three) best explain the observed navigational sequences. We analyze the structure of such navigation, identify strategies and the most salient common sequences of human navigational patterns and provide visual depictions. Amongst other structural differences between goal-oriented and free form navigational patterns, users seem to stay in the same topic more frequently for our free form navigational dataset (MSNBC) compared to both of the goal oriented datasets (Wikigame and Wikispeedia). Our analysis thereby provides new insights into the memory and structure that users employ when navigating the Web that can e.g., be useful to improve recommendation algorithms, web site design or faceted browsing. The paper is structured as follows: In the section entitled “” we review the state-of-the-art in this domain. Next, we present our methodology and experimental setup in the sections called “” and “”. We present and discuss our results in the section named “”. In the section called “ we provide a final discussion and the section called ”" concludes our paper. Related Work {#sec:related .unnumbered} ============ In the late 1990s, the analysis of user navigational behavior on the Web became an important and wide-spread research topic. Prominent examples are models by Huberman and Adamic [@huberman2] that determine how users choose new sites while navigating, or the work by Huberman et al. [@huberman] who have shown that strong regularities in human navigation behavior exist and that, for example, the length of navigational paths on the Web is distributed as an inverse Gaussian distribution. These first models of human navigation on the Web set a standard modeling framework for future research - the majority of navigation models have been stochastic henceforth. Common stochastic models of human navigation are Markov chains. For example, the Random Surfer model in Google’s PageRank algorithm can be seen as a special case of a Markov chain [@brin]. Some further examples of the application of Markov chains as models of Web navigation can be found in [@borges; @deshpande; @lempel; @pirolli; @sen; @anderson; @cadez; @zukerman; @pitkow]. In a Markov chain, Web pages are represented as states and links between the pages are modeled as probabilistic transitions between the states. The dynamics of a user’s navigation session, in which she visits a number of pages by following the links between them, can thus be represented as a sequence of states. Specific configurations of model parameters – such as transition probabilities or model orders – have been used to reflect different assumptions about navigation behavior. One of the most influential assumptions in this field to date is the so-called Markovian property, which postulates that the next page that a user visits depends only on her current page, and not on any other page leading to the current one. This assumption is adopted in a number of prevalent models of human navigation in information networks, for example also in the Random Surfer model [@brin]. However, this property is neglecting the observations stated above that human navigation exhibits strong regularities which hints towards longer memory patterns in human navigation. We argue, that the more consistency human navigation in information networks displays the higher the appropriate Markov chain order should be. *The Markovian assumption might be wrong:* The principle that human navigation might exhibit longer memory patterns than the first order Markov chain captures has been investigated in the past (see e.g., [@borges1999; @pirolli] or [@rosvall2013networks] for a more general approach of looking at memory in network flows). However, higher order Markov chains have been often disputed for modeling human navigation because the gain of a higher order model did not compensate for the additional complexity introduced by the model [@pirolli]. Therefore, it was a common practice to focus on a first order model since it was a reasonable but extremely simple approximation of user navigation behavior (e.g., [@cadez; @sarukkai; @sen; @zukerman]). The discussion about the appropriate Markov chain order was just recently picked up again by Chierichetti et al. [@chierichetti]. While the authors’ results again show indicators that users on the World Wide Web are not Markovian, the study does not account for the higher complexity of such models and the possible lack of statistically significant gains of these models. Technically, the authors analyzed Markov chain models of different orders by measuring the likelihood of real navigational sequences given a particular model. In the next step, the authors compared the models by their likelihoods and found that the Markovian assumption does not hold for their given data and, thus, higher order Markov chain models seem to be more appropriate. As a result, the authors argue that users on the World Wide Web are not Markovian. However, their results come with certain limitations, such as the fact that choosing the model with the highest likelihood is biased towards models with more parameters. [[Because lower order models are always nested within higher order models and as higher order Markov chains have exponentially more parameters than lower order models (potential overfitting), they are always a better fit for the data [@murphy].]{}]{} Thus, higher order models are naturally favored by their improvements in likelihoods. A more comprehensive view on this issue shows that there exists a broad range of established model comparison techniques that also take into the account the complexity of a model in question [@akaike; @bartlett; @gates1976; @katz; @schwarz; @Strelioff; @tong1975]. Moreover, the principle objects of interest in the majority of the past studies are transitions between Web pages. Only a few studies [@cadez; @kumar2010; @west] investigate navigation as transitions between Web page features, such as the content or context of those Web pages. Methods {#sec:methodology .unnumbered} ======= [[ In the following, we briefly introduce Markov chains before discussing an expanded set of methods for order selection, including *likelihood*, *Bayesian*, *information-theoretic* and *cross validation* model selection techniques. ]{}]{} Markov Chains {#subsec:markovchains .unnumbered} ------------- Formally, a discrete (time and space) finite Markov chain is a stochastic process which amounts to a sequence of random variables $X_1, X_2, ..., X_n$. For a Markov chain of the first order, i.e., for a chain that satisfies the memoryless Markov property the following holds: $$\begin{aligned} \nonumber P(X_{n+1} = x_{n+1} | X_1 = x_1, X_2 = x_2, ..., X_n = x_n) & = \\ P(X_{n+1} = x_{n+1} | X_n = x_n)\end{aligned}$$ [[ This classic first order Markov chain model is usually also called a *memoryless model* as we only use the current information for deriving the future and do not look into the past.]{}]{} For all our models we assume *time-homogeneity* – the probabilities do not change as a function of time. To simplify the notation we denote data as a sequence $D=(x_1, x_2, ..., x_n)$ with states from a finite set $S$. With this simplified notation we write the Markov property as: $$p(x_{n+1}|x_1, x_2, ..., x_n)=p(x_{n+1}|x_n)$$ [[ As we are also interested in higher order Markov chain models in this article – i.e., memory models – we now also define a Markov chain for an arbitrary order $k$ with $k \in \mathbb{N}$ – or a chain with memory $k$. ]{}]{} In a Markov chain of $k$-th order the probability of the next state depends on $k$ previous states. Formally, we write: $$p(x_{n+1}|x_1, x_2, ..., x_n)=p(x_{n+1}|x_n, x_{n-1}, ..., x_{n-k+1})$$ Markov chains of a higher order can be converted into Markov chains of order one in a straightforward manner – the set of states for a higher order Markov chain includes all sequences of length $k$ (resulting in a state set of size $|S|^k|S|$). The transition probabilities are adjusted accordingly. A Markov model is typically represented by a transition (stochastic) matrix $P$ with elements $p_{ij}=p(x_j|x_i)$. Since $P$ is a stochastic matrix it holds that for all $i$: $$\sum_j p_{ij} = 1$$ Please note, that for a Markov chain of order $k$ the current state $x_i$ can be a compound state of length $k$ – it is a sequence of past $k$ states. [[Throughout this paper we use this simpler notation, but one should keep in mind that $x_i$ differs for distinct orders $k$.]{}]{} [[ For the sake of completeness, we also allow $k$ to be zero. In such a *zero order* Markov chain model the next state does not depend on any current or previous events, but simply can be seen as a *weighted random selection* – i.e., the probability of choosing a state is defined by how frequently it occurs in the navigational paths. This should serve as a baseline for our evaluations.]{}]{} Next, we want to estimate the vector $\theta$ of parameters of a particular Markov chain that generated observed data $D$ as well as determine the appropriate Markov chain order. For a Markov chain the model parameters are the elements $p_{ij}$ of the transition matrix $P$, i.e., $\theta = P$. Model Selection {#subsec:modelselection .unnumbered} --------------- [[ In this article our main goal is to determine the appropriate order of a Markov chain – i.e., the appropriate length of the memory. For doing so, we resort to well established statistical methods. As we want to provide a preferably complete array of methods for doing so, we present and apply methods from distinct statistical schools: (i) likelihood, (ii) Bayesian and (iii) information-theoretic methods[^3]. We also supplement the methods coming from these three schools by providing a model selection technique usually known from machine learning: (iv) cross validation. We provide an overall ample view of methods and discuss advantages and limitations of each in the following sections. ]{}]{} ### Likelihood Method {#subsubsec:mle .unnumbered} [[ The term *likelihood* was coined and popularized by R. A. Fisher in the 1920’s (see e.g, [@stigler2002statistics] for a historic recap of the developments). Likelihood can be seen as a central element of statistics and we will also see in the following sections that other methods also resort to the concept. ]{}]{} The likelihood is a function of the parameters $\theta$ and it equals to the probability of observing the data given specific parameter values: $$\begin{aligned} \nonumber P(D|\theta) &=& p(x_n|x_{n-1})p(x_{n-1}|x_{n-2})...p(x_2|x_1)p(x_1)\\ &=& p(x_1) \prod_i \prod_j p_{ij}^{n_{ij}},\end{aligned}$$ where $n_{ij}$ is the number of transition from state $x_i$ to state $x_j$ in $D$. Fisher also popularized the so-called *maximum likelihood estimate (MLE)* which has a very intuitive interpretation. This is the estimation of the parameters $\theta$ – i.e., transition probabilities – that most likely generated data $D$. Concretely, the maximum likelihood estimate $\hat{\theta}_{MLE}$ are the values of the parameters $\theta$ that maximize the likelihood function, i.e., $\hat{\theta}_{MLE}=\arg\max_\theta P(D|\theta)$ (a thorough introduction to MLE can be found in [@royall1997statistical]). The maximum likelihood estimation for Markov chains is an example of an optimization problem under constraints. Such optimization problems are typically solved by applying Lagrange multipliers. To simplify the calculus we will work with the log-likelihood function $\mathcal{L(P(D|\theta))}=log P(D|\theta)$. Because the $log$ function is a monotonic function that preserves order, maximizing the log-likelihood is equivalent to maximizing the likelihood function. Thus, we have: $$\begin{aligned} \nonumber \mathcal{L(P(D|\theta))} &=& log \left(p(x_1) \prod_i \prod_j p_{ij}^{n_{ij}}\right) \\ &=& log p(x_1) + \sum_i \sum_j n_{ij} log p_{ij} \end{aligned}$$ Our constraints capture the fact that each transition matrix row sums to $1$: $$\sum_j p_{ij} = 1$$ We have $n$ rows and therefore we need $n$ Lagrange multipliers $\lambda_1, \lambda_2, ..., \lambda_n$. We can rewrite the constraints using Lagrange multipliers as: $$\lambda_i\left(\sum_j p_{ij} - 1\right) = 0$$ Now, the new objective function is: $$f(\mathbf{\lambda, \theta}) = \mathcal{L}(P(D|\theta)) - \sum_i\lambda_i\left(\sum_j p_{ij} - 1\right)$$ To maximize the objective function we set partial derivatives with respect to $\lambda_i$ to $0$, which gives back the original constraints. Further, we set partial derivatives with respect to $p_{ij}$ to $0$ and solve the equation system for $p_{ij}$. This gives: $$p_{ij} = \frac{n_{ij}}{\sum_j n_{ij}}$$ Thus, the maximum likelihood estimate for a specific $p_{ij}$ is the number of transitions from state $x_i$ to state $x_j$ divided by the total number of transitions from state $x_i$ to any other state. For example, in a navigation scenario the maximum likelihood estimate for a transition from page $A$ to page $B$ is the number of clicks on a link leading to page $B$ from page $A$ divided by the total number of clicks on page $A$. Our concrete goal is to determine the appropriate order of a Markov chain. [[Using the log-likelihoods of the specific order models is not enough, as we will always get a better fit to our training data using higher order Markov chains. The reason for this is that lower order models are nested within higher order models. Also, the number of parameters increases exponentially with $k$ which may result in overfitting [@murphy] since we can always produce better fits to the data with more model parameters.]{}]{} To demonstrate this behavior, we produced a random navigational dataset by randomly (uniformly) picking a next click state out of a list of arbitrary states. One of these states determines that a path is finished and a new one begins. With this process we could generate a random path corpus that is close to one main dataset of this work (Wikigame topic dataset explained in the section called “”). Concretely, we as well chose 26 states and the same number of total clicks. Purely from our intuition, such a process should produce navigational patterns with an appropriate Markov chain order of zero or at maximum one. However, if we look at the log-likelihoods depicted in Figure \[fig:randomloglikelihood\] we can observe that the higher the order the higher the corresponding log likelihoods are. This strongly suggests that – as previously explained – looking at the log-likelihoods is not enough for finding the appropriate Markov chain order. Hence, we first resort to a well-known statistical likelihood tool for comparing two models – the so-called *likelihood ratio test*. This test is suited for comparing the fit of two composite hypothesis where one model – the so-called *null model* $k$ – is a special case of the *alternative model* $m$. The test is based on the log likelihood ratio, which expresses how much more likely the data is with the alternative model than with the null model. We follow the notation provided by Tong [@tong1975] and denote the ratio as ${_k}\eta{_m}$: $${_k}\eta{_m} = -2(\mathcal{L(P(D|}{\theta}_k)) - \mathcal{L(P(D|\theta}_m))) \label{eq:lr}$$ To address the overfitting problem we perform a significance test on this ratio. The significance test recognizes whether a better fit to data comes only from the increased number of parameters. The test calculates the p-value of the likelihood ratio distribution. Whenever the null model is nested within the alternative model the likelihood ratio approximately follows a $\chi^2$ distribution with degrees of freedom specified by $(|S|^m-|S|^k)(|S|-1)$. If the p-value is below a specific significance level we can reject the null hypothesis and prefer the alternative model [@bartlett][^4]. [[ Likelihood ratios and corresponding tests have been shown to be a very understandable approach of specifying evidence [@perneger]. They also have the advantage of specifying a clear value (i.e., the likelihood ratio) with can give us intuitive meaning about the advantage of one model over the other. However, the likelihood-ratio test also has limitations like that it only works for nested models, which is fine for our approach but may be problematic for other use cases. It also requires us to use elements from frequentist approaches (i.e., the p-value) for deciding between two models which have been criticized in the past (e.g., [@morrison2006significance]). Furthermore, we are only able to compare two models with each other at a time. This makes it difficult to choose one single model as the most likely one as we may end up with several statistical significant improvements. Also, as we increase the number of hypothesis in our test, we as well increase the probability that we find at least one significant result (Type 1 error)[^5]. ]{}]{} ![**Log-likelihoods for random path dataset.** Simple log-likelihoods of varying Markov chain orders would suggest higher orders as the higher the order the higher the corresponding log-likelihoods are. This suggests that looking at these log-likelihoods is not enough for finding the appropriate Markov chain order as methods are necessary that balance the goodness-of-fit against the number of model parameters.[]{data-label="fig:randomloglikelihood"}](paths_wikigame_random_loglikelihoods){width="0.7\columnwidth"} ### Bayesian Method {#subsubsec:bayesianinference .unnumbered} [[ Bayesian inference is a statistical method utilizing the Bayes’ rule – Rev. Thomas Bayes started to talk about the Bayes theorem in 1764 – for updating prior believes with additional evidence derived from data. A general introduction to Bayesian inference can e.g., be found in [@box2011bayesian]; in this article we focus on explaining the application for deriving the appropriate Markov chain order (see [@Strelioff] for further details). ]{}]{} In Bayesian inference data and the model parameters are treated as random variables [[(cf. MLE where parameters are unknown constants)]{}]{}. We start with a joint probability distribution of data $D$ and parameters $\theta_k$ given a model $M$; that is given a Markov chain of a specified order $k$. Thus, we are interested in $P(D, \theta_k | M_k)$. The joint distribution $P(D, \theta_k | M_k)$ can be written as the product of the conditional probability of data $D$ given the parameters $\theta_k$ and the marginal distribution of the parameters, or we can write this joint distribution as the product of the conditional probability of the parameters given the data and the marginal distribution of the data. Solving then for the posterior distribution of parameters given data and a model we obtain the famous Bayes rule: $$P(\theta_k | D, M_k) = \frac{P(D | \theta_k, M_k)P(\theta_k | M_k)}{P(D | M_k)},$$ where $P(\theta_k | M_k)$ is the prior probability of model parameters, $P(D | \theta_k, M_k)$ is the likelihood function; that is the probability of observing the data given the parameters, and $P(D | M_k)$ is the evidence (marginal likelihood). $P(\theta_k | D, M_k)$ is the posterior probability of the parameters, which we obtain after we update the prior with the data. For a more detailed and an in-depth technical analysis of Bayesian inference of Markov chains we point to an excellent discussion of the topic in [@Strelioff]. *Likelihood.* As previously, we have: $$P(D | \theta_k, M_k) = p(x_1) \prod_i \prod_j p_{ij}^{n_{ij}}$$ *Prior.* The prior reflects our (subjective or objective) belief about the parameters before we see the data. In Bayesian inference, conjugate priors are of special interest. Conjugate priors result in posterior distributions from the same distribution family. In our case, each row of the transition matrix follows a categorical distribution. The conjugate prior for categorical distribution is the Dirichlet distribution. Further information on applying Dirichlet conjugate prior and dealing with Dirichlet process can be found in [@Huelsenbeck]. The Dirichlet distribution is defined as $Dir(\alpha)$: $$Dir(\alpha) = \frac{\Gamma(\sum_j \alpha_{j})}{\prod_j \Gamma(\alpha_{j})}\prod_j x_{j}^{\alpha_{j} - 1},$$ where $\Gamma$ is the gamma function, $\alpha_j > 0$ for each $j$ and $\sum_j x_j = 1$ is a probability simplex. The probability outside of the simplex is $0$. The *hyperparameters* $\alpha$ reflect our assumptions about the parameters $\theta$ before we have observed the data. We can think about the hyperparameters as fake counts in the transition matrix of a Markov chain. A standard uninformative selection for hyperparameters is a uniform prior – for example, we set $\alpha_j = 1$ for each $j$. Thus, for row $i$ of the transition matrix we have the following prior: $$Dir(\alpha_{i}) = \frac{\Gamma(\sum_j \alpha_{ij})}{\prod_j \Gamma(\alpha_{ij})}\prod_j p_{ij}^{\alpha_{ij} - 1}$$ As before, it holds that: $$\sum_j p_{ij}=1$$ The prior for the complete transition matrix is the product of the Dirichlet distributions for each row: $$P(\theta_k | M_k) = \prod_i\frac{\Gamma(\sum_j \alpha_{ij})}{\prod_j \Gamma(\alpha_{ij})}\prod_j p_{ij}^{\alpha_{ij} - 1}$$ *Evidence.* To calculate the evidence we take a weighted average over all possible values of the parameters $\theta_k$. Thus, we need to integrate out the parameters $\theta_k$. $$P(D | M_k) = \int P(D | \theta_k, M_k)P(\theta_k | M_k)d\theta_k$$ $$\begin{aligned} \nonumber P(D | M_k) &=& \int P(D | \theta_k, M_k)P(\theta_k | M_k)d\theta_k\\ \nonumber &=& \int p(x_1) \prod_i \prod_j p_{ij}^{n_{ij}} \prod_i\frac{\Gamma(\sum_j \alpha_{ij})}{\prod_j \Gamma(\alpha_{ij})}\prod_j p_{ij}^{\alpha_{ij} - 1} d\theta_k\\ \nonumber &=& p(x_1)\prod_i\frac{\Gamma(\sum_j \alpha_{ij})}{\prod_j \Gamma(\alpha_{ij})} \int \prod_j p_{ij}^{n_{ij}} \prod_j p_{ij}^{\alpha_{ij} - 1} d\theta_k\\ \nonumber &=& p(x_1)\prod_i\frac{\Gamma(\sum_j \alpha_{ij})}{\prod_j \Gamma(\alpha_{ij})} \int \prod_j p_{ij}^{n_{ij}+\alpha_{ij} - 1} d\theta_k\end{aligned}$$ Please note, that: $$\begin{aligned} \nonumber \int \frac{\Gamma(\sum_j \alpha_{j})}{\prod_j \Gamma(\alpha_{j})}\prod_j x_{j}^{\alpha_{j} - 1} dx &=& 1\\ \nonumber \frac{\Gamma(\sum_j \alpha_{j})}{\prod_j \Gamma(\alpha_{j})} \int \prod_j x_{j}^{\alpha_{j} - 1} dx &=& 1\\ \nonumber \int \prod_j x_{j}^{\alpha_{j} - 1} dx &=& \frac{\prod_j \Gamma(\alpha_{j})}{\Gamma(\sum_j \alpha_{j})}\end{aligned}$$ Thus, we have $$\int \prod_j p_{ij}^{n_{ij}+\alpha_{ij} - 1} d\theta_k = \frac{\prod_j \Gamma(n_{ij}+\alpha_{ij})}{\Gamma(\sum_j (n_{ij}+\alpha_{ij}))}$$ And thus, $$P(D | M_k) = p(x_1)\prod_i\frac{\Gamma(\sum_j \alpha_{ij})}{\prod_j \Gamma(\alpha_{ij})} \frac{\prod_j \Gamma(n_{ij}+\alpha_{ij})}{\Gamma(\sum_j (n_{ij}+\alpha_{ij}))} \label{eq:evidence}$$ *Posterior.* For the posterior distribution over the parameters $\theta_k$ we obtain: $$\begin{aligned} \nonumber P(\theta_k | D, M_k) &=& \prod_i \prod_j p_{ij}^{n_{ij}} \prod_i \prod_j p_{ij}^{\alpha_{ij} - 1} \frac{\Gamma(\sum_j (n_{ij}+\alpha_{ij}))}{\prod_j \Gamma(n_{ij}+\alpha_{ij})}\\ \nonumber &=& \prod_i \prod_j p_{ij}^{n_{ij} + \alpha_{ij} - 1} \frac{\Gamma(\sum_j (n_{ij}+\alpha_{ij}))}{\prod_j \Gamma(n_{ij}+\alpha_{ij})}\end{aligned}$$ This equation is the product of the Dirichlet distributions for each row with parameters $n_{j} + \alpha_{j}$: $$P(\theta_k | D, M_k) = \prod_i Dir(n_i + \alpha_{i})$$ The posterior distribution is a combination of our prior belief and the data that we have observed. In fact, the expectation and the variance of the posterior distribution are: $$E[p_{ij}] = \frac{n_{ij} + \alpha_{ij}}{\sum_j (n_{ij} + \alpha_{ij})}$$ $$Var[(p_{ij}] = \frac{(n_{ij} + \alpha_{ij})(\sum_j (n_{ij} + \alpha_{ij}) - (n_{ij} + \alpha_{ij}))}{(\sum_j (n_{ij} + \alpha_{ij}))^2(\sum_j (n_{ij} + \alpha_{ij}) + 1)}$$ We can rewrite the expectation as: $$E[p_{ij}] = \frac{1}{\sum_j (n_{ij} + \alpha_{ij})}\left(\sum_j n_{ij}\frac{n_{ij}}{\sum_j n_{ij}} + \sum_j \alpha_{ij}\frac{\alpha_{ij}}{\sum_j \alpha_{ij}}\right)$$ Setting $c=\frac{\sum_j n_{ij}}{\sum_j (n_{ij} + \alpha_{ij})}$, we can rewrite the expectation of the posterior distribution as: $$E[p_{ij}] = c\frac{n_{ij}}{\sum_j n_{ij}} + (1-c)\frac{\alpha_{ij}}{\sum_j \alpha_{ij}}$$ Thus, the posterior expectation is a *convex combination* of the MLE and the prior. When the number of the observation becomes large ($n_{ij} \gg \alpha_{ij}$) then $c$ tends to $1$, and the posterior expectation tends to the MLE. By setting $\alpha_{ij} = 1$ for each $i$ and $j$ we effectively obtain Laplace’s prior; that is we apply Laplace smoothing [@murphy]. For model selection we adopt once more the Bayesian inference (again see [@Strelioff] for a thorough discussion). We have a set $M$ of models $M_k$ with varying order $k$ and are interested in deciding between several models (c.f. [@kass1995bayes]). We are interested in the joint probability distribution $P(D, M_k)$ of data $D$ and a model $M_k$. We can write the joint distribution as a product of a conditional probability (of data given a model, or of a model given the data) and a prior marginal distribution (of data or a model) and by solving for the posterior distribution of a model given the data we again obtain the Bayes rule: $$P(M_k | D) = \frac{P(D | M_k) P(M_k)}{P(D)},$$ where $P(D)$ is the weighted average over all models $M_k$: $$P(D) = \sum_k P(D | M_k) P(M_k).$$ The likelihood of data $D$ given a model $M_k$ is the evidence $P(D | M_k)$ given by Equation \[eq:evidence\], which is the weighted average over all possible model parameters $\theta_k$ given the model $M_k$. Following Strelioff et al. [@Strelioff], we select two priors over the model set $M$ – a uniform prior and a prior with an exponential penalty for the higher order models [@Strelioff]. The uniform prior assigns the identical probability for each model: $$P(M_k) = \frac{1}{|M|}.$$ With the uniform prior we obtain the following expression for the posterior probability of a model $M_k$ given the data: $$P(M_k | D) = \frac{P(D | M_k)}{\sum_k P(D | M_k)}.$$ The prior with the exponential penalty can be defined as: $$P(M_k) = \frac{e^{-|S_k|}}{\sum_k e^{-|S_k|}},$$ where $|S_k|$ is the number of states of the model $M_k$ and can be calculated as: $$|S_k| = |S|^k(|S| - 1),$$ with $|S|$ being the number of states of the model of order $1$. After solving for the posterior distribution for the prior with the exponential penalty we obtain: $$P(M_k | D) = \frac{P(D | M_k) e^{-|S_k|}}{\sum_k P(D | M_k) e^{-|S_k|}}.$$ The calculations are best implemented with log-evidence and logarithms of the gamma function to avoid underflow since the numbers are extremely small. To implement the sum for the normalizing constant in the denominator we apply the so-called *log-sum-exp trick* [@Durbin]. First, we calculate the log-evidence: $log P(D | M_k)$ and then calculate the logarithm of the normalizing constant $log(C)$: $$log(C) = log (\sum_k e^{log(P(D | M_k))}).$$ A direct calculation of $e^{log(P(D | M_k))}$ results in an underflow, and thus we pull the largest log-evidence $E_{max} = max(log(P(D | M_k))$ out of the sum: $$log(C) = E_{max} + log (\sum_k e^{log(P(D | M_k)) - E_{max}}).$$ One downside of using Bayesian model selection is that it is frequently difficult to calculate Bayes factors. Concretely, it is often complicated to calculate the necessary integral analytically and one needs to resort to various alternatives in order to avoid this problem. Nowadays, several such methods exist: e.g., asymptotic approximation or sampling from the posterior (MCMC, Gibbs) [@kass1995bayes]. Also, we need to specify prior distributions for the parameters of each model. As elaborated by Kass and Raftery [@kass1995bayes], one approach is to use the BIC (see the next section entitled “”) which gives an appropriate approximation given one specific prior. Compared to the likelihood ratio test (see section entitled ), the Bayesian model selection technique does not require the models to be nested. The main benefit of Bayesian model selection is that it includes a natural *Occam’s razor* – i.e., a penalty for too much complexity – which helps us to avoid overfitting [@kass1995bayes; @mackay1992bayesian; @murray2005note; @mackay2003information]. The Occam’s razor is a principle that advises to prefer simpler theories over more complex ones. Based on this definition there is no need to include extra complexity control as we e.g., additionally did for our exponential penalty. We see this though as a nice further control mechanism for cautiously penalizing model complexity and for validating the natural Occam’s razor. ### Information-theoretic Methods {#subsubsec:informationtheoretic .unnumbered} [[ Information-theoretic methods are based on concepts and ideas derived from information theory with a specific focus on *entropy*. In the following we will provide a description of the two probably most well-known methods; i.e., AIC and BIC. A thorough overview of information-theoretic methods can e.g., be found in various work by K. P. Burnham [@burnham2002model; @burnham2004multimodel]. ]{}]{} *Akaike information criterion (AIC).* Akaike [@akaike] introduced in 1973 a one dimensional statistic for determining the optimal model from a class of competing models. The criterion is based on Kullback-Leibler divergence [@KLD] and the asymptotic properties of the likelihood ratio statistics described in the section entitled “”. The approach is based on minimization of AIC (minimum AIC estimate – MAICE) amongst several competing models [@gates1976] and has been first used for Markov chains by Tong [@tong1975]. Hence, we define the AIC based on the choice of a loss function proposed by Tong [@tong1975]: $$AIC(k) = {_k}\eta{_m} - 2(|S|^m-|S|^k)(|S|-1) \label{eq:aic}$$ The test represents an asymptotic version of the likelihood ratio test defined in Equation \[eq:lr\] for composite hypothesis. The idea is to choose $m$ reasonably high and test lower order models until an optimal order is found. MAICE chooses the order $k$ which exhibits the minimum AIC score and tries to balance between overfitting and underfitting [@gates1976]. *Bayesian Information Criterion (BIC).* In 1978 Schwarz [@schwarz] introduced this criterion which can be seen as an approximation of the Bayes factor for Bayesian model selection (see the previous section entitled “”). It is similar to the AIC introduced above with the difference that it penalizes higher order models even more by adding an additional penalization for the number of observations [@katz]: $$BIC(k) = {_k}\eta{_m} - (|S|^m-|S|^k)(|S|-1)ln(n) \label{eq:bic}$$ Again we choose $m$ reasonably high and test lower order models against it. The penalty function is the degree of freedom multiplied with the natural logarithm of the number of observations $n$. This function converges to infinity at a still slow enough rate and hence, grants a consistent estimator of the Markov chain order [@katz]. Frequently, both AIC and BIC suggest the same model. However, there are certain cases, where they might slightly disagree. In model selection literature there is a still ongoing debate of whether one should prefer AIC or BIC over each other – e.g., see [@weakliem1999critique] for a critique of the BIC for model selection. However, as pointed out by Burnham and Anderson [@burnham2004multimodel], each has its strength and weaknesses in distinct domains. The authors emphasize that both can be seen as either frequentist or Bayesian procedures. In case of inequality, Katz [@katz] suggests to investigate the patterns further by simulating observations and investigate distinct sample sizes. In this paper we instead apply additional model comparison techniques to further analyze the data. The performance of AIC and BIC has also been investigated in the terms of determining the appropriate Markov chain order which is the main goal of this article. R. W. Katz [@katz] pointed out that by using AIC there is the possibility of overestimating the true order independent of how large the data is. Hence, he points out that AIC is an inconsistent method. Contrary, he emphasizes that BIC is a consistent estimator – i.e., if there is a true underlying model BIC will select it with enough data. Alas, it does not perform well for small sample sizes (see also [@csiszar2000consistency]). Nonetheless, AIC is the most used estimator for determining the appropriate order, maybe due to higher efficiency for smaller data samples, as elaborated by Baigorri et al. [@baigorri2009markov]. While both AIC and BIC seem at first to be very similar to the likelihood ratio test (see section entitled “) there are some elementary differences. First and foremost, they can also be applied for non-nested models [@burnham2002model]. Moreover, they do not need to resort to hypothesis testing. BIC is also closely related to Bayesian model selection techniques; specifically to the Bayes factor (see section called ”"). Kass and Raftery [@kass1995bayes] emphasize the advantages of BIC over the Bayes factor by pointing out that it can be applied even when the priors are hard to set. Also, it can be a rough approximation to the logarithm of the Bayes factor if the number of observations is large. BIC is also declared as being well suited for scientific reporting. Finally, we want to point out that one could also see AIC as being best for prediction, while BIC might be better for explanation. Also, as pointed out by M. Stone [@stone], AIC is asymptotically equivalent to cross validation (see the section entitled “”) if both use maximum likelihood estimation. ### Cross Validation Method {#subsubsec:prediction .unnumbered} Another – quite natural – way of determining the appropriate order of a Markov chain is cross-validation [@chierichetti; @murphy]. The basic idea is to estimate the parameters on a training set and validate the results on an independent testing set. In order to reduce variance we perform a stratified 10-fold cross-validation. In difference to a classic machine learning scenario, we refer to stratified as a way of keeping approximately the equal amount of observations in each fold. Thus, we keep approximately 10% of all clicks in a single fold. With this method we focus on prediction of the next user click. Markov chains have been already used to prefetch the next page that the user most probably will visit on the next click. In the simplest scenario, this prefetched page is the page with the highest transition probability from the current page. To measure the prediction accuracy we measure the average rank of the actual page in sorted probabilities from the transition matrix. Thus, we determine the rank of the next page $x_{n+1}$ in the sorted list of transition probabilities (expectations of the Bayesian posterior) of the current page $x_{n}$ (see the section named “”). We then average the rank over all observations in the testing set. Hence, we can formally define the average rank $\overline{r(D_f)}$ of a fold $D_f$ for some arbitrary model $M_k$ the following way: $$\overline{r(D_f)} = \frac{\sum_i \sum_j n_{ij} r_{ij}}{\sum_i \sum_j n_{ij}},$$ where $n_{ij}$ is the number of transition from state $x_i$ to state $x_j$ in $D_f$ and $r_{ij}$ denotes the rank of $x_j$ in the $i$-th row of the transition matrix. For ranking the states in a row of the matrix, we resort to *modified competition ranking*. This means that if there is a tie between two or more values, we assign the maximum rank of all ties to each corresponding one; i.e., we leave the gaps before a set of ties (e.g., “14445” ranking). By doing so, we assign the worst possible ranks to ties. One important implication of this methodology is that we include a natural penalty (a natural Occam’s razor) for higher order Markov chains. The reason for this is that the transition matrices generally become sparser the higher the order. Hence, we come up with many more ties and the chance is higher that we assign higher ranks for observed transitions in the testing data. The most extreme case happens when we do not have any information available for observations in the testing set (which frequently happens for higher orders); then we assign the maximum rank (i.e., the number of states) to all states. We finally average the ranks over all folds for a given order and suggest the model with the lowest average rank.[^6] This method requires priors (i.e., fake counts; see the section named “”) – otherwise prediction of unseen states is not possible. It also resorts to the maximum likelihood estimate for calculating the parameters of the models as described in the section entitled “”. Also, as shown in the previous section called “” cross validation has asymptotic equivalence to AIC. One disadvantage of cross validation methods usually is that the results are dependent on how one splits the data. However, by using our stratified k-fold cross validation approach, we counteract this problem as it matters less of how the data is divided. Yet, by doing so we need to rerun the complete evaluation k times, which leads to high computational expenses compared to the other model selection techniques described earlier and we have to manually decide of which k to use. One main advantage of this method is that eventually each observation is used for both training and testing. -- ----------------------------------------------------- -- -- **[Wikigame]{} & **[Wikispeedia]{} & **[MSNBC]{}\ \#Page Ids & 360,417 & n/a & n/a\ \#Topics & 25 & 15 & 17\ \#Paths & 1,799,015 & 43,772 & 624,383\ \#Visited nodes & 10,758,242 & 259,019 & 4,333,359\ ****** -- ----------------------------------------------------- -- -- : **[Dataset statistics]{}** \[tab:datasetfacts\] Materials {#sec:experimental .unnumbered} ========= In this paper, we perform experiments on three datasets. While the first two datasets (WikiGame and Wikispeedia) are representatives of goal-oriented navigation scenarios (where the target node for each navigation sequence is known beforehand), the third dataset (MSNBC) is representative of free navigation on the Web (where we have no knowledge about the targets of navigation). #### Wikigame dataset This dataset is based on the online game *TheWikiGame*[^7]. The game platform offers a multiplayer game, where users navigate from a randomly selected Wikipedia page (the start page) to another randomly selected Wikipedia page (the target page). All pairs of start and target pages are connected through Wikipedia’s underlying network. The users are only allowed to click on Wikipedia links or on the browser back button to reach the target page, but they are not allowed to use search functionality. In this study, we only considered click paths of length two or more going through the main article namespace in Wikipedia. Table \[tab:datasetfacts\] shows some main characteristics of our Wikigame dataset. As motivated in Section “”, we will represent the navigational paths through Wikipedia twofold: (a) each node in a path is represented by the corresponding Wikipedia page ID – we refer to this as the *Wikigame page* dataset – and (b) each node in a path is represented by a corresponding Wikipedia category (representing a specific topic) – we call this the *Wikigame topic* dataset. For the latter dataset we determine a corresponding top level Wikipedia category[^8] in the following way. The majority of Wikipedia pages belongs to one or more Wikipedia categories. For each of these categories we find a shortest path to the top level categories and select a top level category with the shortest distance. In the case of a tie we pick a top level category uniformly at random. Finally, we replace all appearances of that page with the chosen top level category. Thus, in this new dataset we replaced each navigational step over a page with an appropriate Wikipedia category (topic) and the dataset contains paths of topics which users visited during navigation (see Figure \[fig:pathexample\]). Figure \[fig:histograms\] illustrates the distinct topics and their corresponding occurrence frequency (A). #### Wikispeedia dataset This dataset is based on a similar online game as the Wikigame dataset called *Wikispeedia*[^9]. Again, the players are presented with two randomly chosen Wikipedia pages and they are as well connected via the underlying link structure of Wikipedia. Furthermore, users can also select their own start and target page instead of getting randomly chosen ones. Contrary to the Wikigame, this game is no multiplayer game and you do not have a time limit. Again, we only look at navigational paths with at least two nodes in the path. The main difference to the Wikigame dataset is that Wikispeedia is played on a limited version of Wikipedia (Wikipedia for schools[^10]) with around 4,600 articles. Some main characteristics are presented in Table \[tab:datasetfacts\]. Conducted research and further explanations of the dataset can be found in [@west; @west2; @west3; @scaria2014last]. As we want to look at transitions between topics we determine a corresponding top level category (topic) for each page in the dataset. We do this in similar fashion as for our Wikigame dataset, but the Wikipedia version used for Wikispeedia has distinct top level categories compared to the full Wikipedia. Figure \[fig:histograms\] illustrates the distinct categories and their corresponding occurrence frequency (B). ![**Topic frequencies.** Frequency of categories (in percent) of all paths in (A) the Wikigame topic dataset (B) the Wikispeedia dataset and (C) the MSNBC dataset. The colors indicate the categories we will investigate in detail later and are representative for a single dataset – this means that the same color in the datasets does not represent the same topic. The Wikigame topic dataset consists of more distinct categories than the Wikispeedia and MSNBC dataset. Furthermore, the most frequently occuring topic in the Wikigame topic dataset is Culture with around 13%. The Wikispeedia dataset is dominated by the two categories the most Science and Geography each making up for almost 25% of all clicks. Finally, the most frequent topic in the MSNBC dataset is the frontpage with a frequency of around 22%.[]{data-label="fig:histograms"}](histograms){width="\textwidth"} #### MSNBC dataset This dataset[^11] consists of Web navigational paths from MSNBC[^12] for a complete day. Each single path is a sequence of page categories visited by a user within a time frame of 24 hours. The categories are available through the structure of the site and include categories such as *news*, *tech*, *weather*, *health*, *sports*, etc. In this dataset we also eliminate all paths with just a single click. Table \[tab:datasetfacts\] shows the basic statistics for this dataset and in Figure \[fig:histograms\] the frequency of all categories of this dataset are depicted (C). #### Data preparation Each dataset $D$ consists of a set of paths $\mathbb{P}$. A single path contains a single game in the Wikigame and Wikispeedia dataset or a single navigation session in the MSNBC dataset. A path $p$ is defined as a $n$-tuple $(v_1,\ldots,v_n)$ with $v_i \in V, 1\leq i\leq n$ and $(v_i, v_{i+1}) \in E, 1 \leq i \leq n-1$ where $V$ is the set of all nodes in $\mathbb{P}$ and $E$ is the set of all observed transitions in $\mathbb{P}$. We also define the length of a path $len(p)$ as the length of the corresponding tuple $(v_1,\ldots,v_n)$. Additionally, we want to define ${\bf p} = \left\{ v_k | k =1 \ldots n \right\}$ as the set of nodes in a path $p$. Note that $|{\bf p}| \leq n$. The finite state set $S$ needed for Markov chain modeling is originally the set of vertices $V$ in a set of paths $\mathbb{P}$ given a specific dataset $D$. To prepare the paths for estimation of parameters of a Markov chain of order $k$, we separate single paths by prepending a sequence of $k$ generic *RESET* states to each path, and also by appending one *RESET* state at the end of each path. This enables us to connect independent paths and – through the addition of the *RESET* state – to forget the history between different paths. Hence, we end up with an ergodic Markov chain (see [@chierichetti]). With this artificial *RESET* state, the final number of states is $|S|+1$. Results {#sec:results .unnumbered} ======= In this section we present the results obtained from analyzing human navigation patterns based on our datasets at hand introduced in Section “”. We begin by presenting the results of our investigations of memory – i.e., appropriate Markov chain order using the Markov chain methods thoroughly explained in the section called “” – of user navigation patterns in the section entitled “”. Based on these calculations and observations we dig deeper into the structure of human navigation and try to find consistent patterns – i.e., specific sequences of navigated states – in the section named “”. Memory {#subsec:memory .unnumbered} ------ We start by analyzing human navigation over Wikipedia pages on the Wikigame page dataset. Afterwards, we will focus on our topic datasets for getting insights on a topical level. ### Page navigation {#subsubsec:pagenavi .unnumbered} #### Wikigame page dataset [[ The initial Markov chain model selection results (see Figure \[fig:paths\_all\]) obtained from experiments on the Wikigame page dataset confirm our theoretical considerations. We observe that the likelihoods are rising with higher Markov chain orders (confirming what [@chierichetti] found) which intuitively would indicate a better fit to the data using higher order models. However, the likelihood grows per definition with increasing order and number of model parameters and therefore, the likelihood based methods for model selection fail to penalize the increasing model complexity (c.f. Section “”). All other applied methods take the model complexity into account. ]{}]{} First, we can imply already from the likelihood statistics (B) that there might be no improvement over the most basic zero order Markov chain model as we can not find any statistically significant improvements of higher orders. Both AIC (C) and BIC (D) results confirm these observations and also agree with each other. Even though we can see equally low values for a zero, first and second order Markov chain, we would most likely prefer the most simple model in such a case – further following the ideas of the Occam’s razor. In order to extend these primary observations we used a uniform Laplace prior and Bayesian inference and henceforth, we obtain the results illustrated in the first two figures of the bottom row in Figure \[fig:paths\_all\]. The Bayesian inference results again suggest a zero order Markov chain model as the most appropriate as indicated by the highest evidence (E) and the highest probability obtained using Bayesian model selection with and without a further exponential penalty for the number of parameters (F). The observations and preference of using a zero order model are finally confirmed by the results obtained from using 10-fold cross-validation and a prediction task (G). We can see that the average position is the lowest for a zero order model approving our observations made above. ![ **Model selection results for the Wikigame page dataset.** The top row shows results obtained using likelihood and information theoretic results: (A) likelihoods, (B) likelihood ratio statistics (\* statistically significant at the 1% level; \*\* statistically significant at the 0.1% level) as well as AIC (C) and BIC (D) statistics. The bottom row illustrates results obtained from Bayesian Inference: (E) evidence and (F) Bayesian model selection. Finally, the figure presents the results from (G) cross validation. The overall results suggest a zero order Markov chain model.[]{data-label="fig:paths_all"}](paths_wikigame_all){width="\textwidth"} **Summary:** [[ Our analysis of the Wikigame page dataset thereby reveals a clear trend towards a zero order Markov chain model. This is imminent when looking at all distinct model selection techniques introduced and applied in this article, as they all agree on the choice of weighted random selection as the statistically significant most approvable model. This is a strong approval of our initial hypothesis stating it is highly difficult to make plausible statements about the appropriate Markov chain order having insufficient data but a vast amount of states. The higher performance of higher order chains can not compensate the necessary additional complexity in terms of statistically significant improvements. However, this may be purely an effect of the data sparsity in our investigation (i.e., the limited number of observations compared to the huge amount of distinct states). One can argue that real human navigation always can be better modeled by at least an order of one, because – as soon as we have enough data – links play a vital role in human navigation as humans by definition follow links when they navigate[^13]. Consequently, we believe that the memoryless Markov chain model is a plausible model for human navigation on a page level. Yet, further detailed studies are necessary to confirm this. ]{}]{} At the same time, one could argue that memory is best studied on a topical level, where pages are represented by topics. Consequently, we focus on studying transitions between topics next, which yields a reduced state space that allows analysis of the memory and structure of human navigation patterns on a topical level. ![ **Model selection results for the Wikigame topic dataset.** The top row shows results obtained using likelihood and information theoretic results: (A) likelihoods, (B) likelihood ratio statistics (\* statistically significant at the 1% level; \*\* statistically significant at the 0.1% level) as well as AIC (C) and BIC (D) statistics. The bottom row illustrates results obtained from Bayesian Inference: (E) shows evidence and (F) Bayesian model selection. (G) presents the results from cross validation. The overall results suggest that higher order chains seem to be more appropriate for our navigation paths consisting of topics. In detail, we find that a second order Markov chain model for our Wikigame topic dataset best explains the data.[]{data-label="fig:paths_cat"}](paths_wikigame){width="\textwidth"} ![ **Model selection results for the Wikispeedia dataset.** The top row shows results obtained using likelihood and information theoretic results: (A) likelihoods, (B) likelihood ratio statistics (\* statistically significant at the 1% level; \*\* statistically significant at the 0.1% level) as well as AIC (C) and BIC (D) statistics. The bottom row illustrates results obtained from Bayesian Inference: (E) shows evidence and (F) Bayesian model selection. (G) presents the results from cross validation. The overall results suggest that higher order chains seem to be more appropriate for our navigation paths consisting of topics. Concretely, we find that a second order Markov chain model for our Wikispeedia topic dataset best explains the data.[]{data-label="fig:paths_wikispeedia"}](paths_wikispeedia){width="\textwidth"} ![ **Model selection results for the MSNBC dataset.** The top row shows results obtained using likelihood and information theoretic results: (A) likelihoods, (B) likelihood ratio statistics (\* statistically significant at the 1% level; \*\* statistically significant at the 0.1% level) as well as AIC (C) and BIC (D) statistics. The bottom row illustrates results obtained from Bayesian Inference: (E) shows evidence and (F) Bayesian model selection. (G) presents the results from cross validation. The overall results suggest that higher order chains seem to be more appropriate for our navigation paths consisting of topics. Specifically, the results suggest a third order Markov chain model.[]{data-label="fig:paths_msnbc"}](paths_msnbc){width="\textwidth"} ### Topics navigation {#subsubsec:topicalnavi .unnumbered} #### Wikigame topic dataset Performing our analyses by representing Wikipedia pages by their topical categories shows a much clearer and more interesting picture as one can see in Figure \[fig:paths\_cat\]. Similar to above we can see (A) that the log likelihoods are rising with higher orders. However, in contrast to the Wikigame page dataset, we can now see (B) that several higher order Markov chain models are significantly better than lower orders. In detail, we can see that the appropriate Markov chain order is at least of order one and we can also observe a trend towards an order of two or three. Nevertheless, as pointed out in the section entitled “”, it is hard to concretely suggest one specific Markov chain order from these pairwise comparisons which is why we resort to this extended repertoire of model selection techniques described next. The AIC (C) and BIC (D) statistics show further indicators – even though they are disagreeing – that the appropriate model is of higher order. Concretely, the suggest an order of three or two respectively by exhibiting the lowest values at these points. Not surprisingly, AIC suggests a higher order compared to BIC as the latter model selection method additionally penalized higher orders by the number of observations as stated in the section called “”. The Bayesian inference investigations (E, F) exhibit a clear trend towards a Markov chain of order two. The results in (F) nicely illustrate the inherent Occam’s razor of the Bayesian model selection method as both priors – (a) no penalty and (b) exponential penalty for higher orders – suggest the same order[^14]. Finally, the cross validation results (G) confirm that a second order Markov chain produces the best results, while a third order model is nearly as good. **Summary:** Overall, we can see that representing Wikigame paths as navigational sequences of corresponding topics leads to more interesting results: Higher order Markov chains exhibit statistically significant improvements, thereby suggesting that memory effects are at play. Overall, we can suggest that a second order Markov chain model seems to be the most appropriate for modeling the corresponding data as it gets suggested by all methods except for AIC which is known for slightly overestimating the order. This means, that humans remember their topical browsing patterns – in other words, the next click in navigational trails is dependent on the previous two clicks on a topical level. #### Wikispeedia dataset This section presents the results obtained from the Wikispeedia dataset introduced in the section entitled “”. Similar to the Wikigame topic dataset we look at navigational paths over topical categories in Wikipedia and present the results in Figure \[fig:paths\_wikispeedia\]. Again we can observe that the likelihood statistics suggest higher order Markov chains to be appropriate (B). Yet, further analyses are necessary for a clear choice of the appropriate order. The AIC (C) and BIC (D) statistics agree to prefer a second order model; however, we need to note that all orders from zero to four have similarly low values. The Bayesian inference investigations (E, F) show a much clearer trend towards a second order model. [[ The prediction results (G) agree on these observations by also showing the best results for a second order model. This time we can also observe a clear consilience between the cross validation and AIC results which are – as described in the section called “” – asymptotically equivalent. ]{}]{} **Summary:** This dataset is similar to the Wikigame topic dataset and the results are comparable to the previous results on the first goal-oriented dataset (Wikigame topic). Hence, even though the game is played on a much smaller set of Wikipedia articles and also the dataset consists of distinct categories, we can see the exact same behavior which strongly indicates that human navigation is not memoryless on a topical level and can be best modeled by a second order Markov chain model. This strongly suggests that humans follow common topical strategies while navigating in a goal-oriented scenario. #### MSNBC dataset In this section we present the results obtained from the MSNBC dataset introduced in the section called “”. Again we look at navigational paths over topical categories and henceforth, we only look at categorical information of nodes and present the results in Figure \[fig:paths\_msnbc\]. Similar to the experiments conducted for the Wikigame and Wikispeedia topic datasets we can again see, based on the likelihood ratio statistics (B), that a higher order Markov chain seems to be appropriate. The AIC (C) and BIC (D) statistics suggest an order of three and two respectively. To further investigate the behavior we illustrate the Bayesian inference results (E, F) that clearly suggest a third order Markov chain model. Finally, this is also confirmed by the cross validation prediction results (G) which again is in accordance with the AIC. **Summary:** By and large, almost all methods for order selection suggest a Markov chain of order three for the topic sequence in the MSNBC dataset. Again, we can observe that the navigational patterns are not memoryless. Even though this dataset is not a goal-oriented navigation dataset, but is based on free navigation on MSNBC, we can identify similar memory effects as above. Structure {#subsec:structure .unnumbered} --------- In the previous section we observed memory patterns in human navigation over topics in information networks. We are now interested in digging deeper into the structure of human navigational patterns on a topical level. Concretely, we are interested in detecting common navigational sequences and in investigating structural differences between goal-oriented and free form navigation. First, we want to get a global picture of common transition patterns for each of the datasets. We start with the Markov chain transition matrices, but instead of normalizing the row vectors, we normalize each cell by the complete number of transitions in the dataset. We illustrate these matrices as heatmaps to get insights into the most common transitions in the complete datasets. Due to tractability, we focus on a first order analysis and will focus on higher order patterns later on. {width="\textwidth"} The heatmaps are illustrated in Figure \[fig:heatmaps\]. Predominantly, we can observe that self transitions seem to be very common as we can see from the high transition counts in the diagonals of the matrices. This means, that users regularly seem to stay in the same topic while they navigate the Web[^15]. For the Wikigame (A) we can observe that the categories *Culture* and *Politics* are the most visited topics throughout the navigational paths. Most of the time the navigational paths start with a page belonging to the *People* topic which is visible by the dark red cell from *RESET* to *People* (remember that the *RESET* state marks both the start and end of a path - see Section “”). However, as this is a game-based goal-oriented navigation scenario, the start node is always predefined. In our second goal-oriented navigation dataset (B) we can see that the paths are dominated by transitions from and to the categories *Science* and *Geography* and there are fewer transitions between other topics. In our MSNBC dataset (C) we can observe that most of the time users remain in the same topic while they navigate and globally no topic changes are dominant. This may be an artifact of the free navigation users practice on MSNBC. Perhaps unsurprisingly, users start with the frontpage most of the time while navigating but do not necessarily come back to it in the end. ![**Local structure of navigation for the Wikigame topic dataset.** The graphs above illustrate selected state transitions from the Wikigame topic dataset for different $k$ values. The nodes represent categories and the links illustrate transitions between categories. The link weight corresponds to the transition probability from the source to the target node determined by MLE. The node size corresponds to the sum of the incoming transition probabilities from all other nodes to that source node. In the left figure the top four categories with the highest incoming transition probabilities are illustrated for an order of $k=1$. For those nodes we draw the four highest outgoing transition probabilities to other nodes. In the middle figure we visualize the Markov chain of order $k=2$ by setting the top topic (*Culture*) as the first click; this diagram shows transition probabilities from top four categories given that users first visited the *Culture* topic. For example, the links from the red node (*Society*) in the bottom-right part of the diagram represent the transition probabilities from the sequence (*Culture*, *Society*). Similarly, we visualize order $k=3$ in the right figure by selecting a node with the highest incoming probability (*Culture*, *Culture*) of order $k=2$. We then show transition probabilities from other nodes given that users already visited (*Culture*, *Culture*). For example, the links from the brown node (*Politics*) at the top represent the transition probabilities from the sequence (*Culture*, *Culture*, *Politics*).[]{data-label="fig:local_WIKI_all"}](wiki_local_3){width="\columnwidth"} ![**Self transition structure of navigation for the Wikigame topic dataset.** The number of times users stay within the same topic vs. the number of times they change the topic during navigation for different orders $k$ for our Wikigame dataset. Only the top three categories with the highest transition probabilities are shown. With high consistency, the transition probabilities to the same topic increase while those to other categories decrease with ascending order $k$.[]{data-label="fig:sameornot_WIKI"}](sameornot_WIKI){width="\columnwidth"} As we have now identified global navigational patterns on the first order transition matrices we turn our attention to models of higher order. Furthermore, we are now interested in investigating local transition probabilities – e.g., being at topic *Science*, what are the transition probabilities to other states. The transition weights directly correspond to the transition probabilities from the source to the target state determined by the MLE (see the section called “”). We illustrate these local transitional patterns for our Wikigame dataset in Figure \[fig:local\_WIKI\_all\] (the investigations on the other goal-oriented Wikispeedia dataset exhibit similar patterns, but are omitted due to space limitations). Similar to the observations in Figure \[fig:heatmaps\] we can observe that *Culture* is the most visited topic in our Wikigame dataset. We can now also identify specific prominent topical transition trails. For example, users seem to navigate between *Culture* and *Politics* quite frequently and also vice versa. Contrary, there seem to be specific unidirectional patterns too, e.g., users frequently navigate from *People* to *Politics* but not vice versa. Higher order chains also show similar structure, but on a more detailed level. As previously, the figure also depicts that the vast amount of transitions is between same categories. However, we can now observe that this is also the case for higher order Markov chains – this suggests, that the probability that users stay in the same topic increases with each new click on that topic. To further look into this structural pattern, we illustrate the number of times users stay within the same topic vs. the number of times they change the topic during navigation in Figure \[fig:sameornot\_WIKI\]. We can see that the longer the history – i.e., the higher the order of the Markov chain – the more likely people tend to stay in the same topic instead of switching to another topic. We can also see differences regarding this behavior between distinct categories; e.g., users are more likely to stay in the topic *Chronology* than in the topic *Politics* the higher the order is. For our Wikispeedia dataset we can observe similar patterns – i.e., the higher the order the higher the chance to stay in the same topic. In order to contrast goal-oriented and free-form navigation, we also depict state transitions in similar fashion derived from the MSNBC dataset in Figure \[fig:local\_msnbc\_all\]. In this figure we can see that the topic *business* is the most used. To give a navigational example: users frequently navigate from *business* to *news* and vice versa. However, there are also navigational patterns just going one direction. For example, users seem to frequently navigate from *business* to *sports* but not in the opposite direction. Again, higher order chains show similar patterns. Like in the Wikigame topic dataset we can as well observe that most of the transitions seem to be between similar categories. In Figure  \[fig:sameornot\_msnbc\] we depict the number of times a user stays in the same topic vs. the number of times she switches the topic for the categories with the highest transition probabilities. We can again observe that the higher the Markov chain the more likely people tend to stay in the same topic while navigating. Nevertheless, an interesting difference to the Wikigame topic dataset can be observed. Concretely, we can see that the probability of staying in the same topic is much higher for the MSNBC dataset. Especially, the topic *weather* exhibits a very high probability of staying in the same topic ($0.9$ for $k=1$). A possible explanation is that users navigate on a semantically more narrow path on MSNBC. If you are interested about the weather you just check the specific pages on MSNBC while on Wikipedia you might get distracted by different categories at a higher probability. So these concrete observations seem to be very specific for the Web site and domains of the site users navigate on while the general patterns seem to be applicable for both of our datasets at hand. ![**Local structure of navigation for the MSNBC dataset.** The graphs above illustrate selected state transitions from the MSNBC dataset for different $k$ values. The nodes represent categories and the links illustrate transitions between categories. The link weight corresponds to the transition probability from the source to the target node determined by MLE. The node size represents the global importance of a node in the whole dataset and corresponds to the sum of the outgoing transition probabilities from that node to all other nodes. For visualization reasons we primarily focus on the top four categories with the highest sum of outgoing transition probabilities – i.e., those with the largest node sizes – for an order of $k=1$. For those nodes we draw the four highest outgoing transition probabilities to other nodes. In the middle figure we visualize the Markov chain of order $k=2$ by setting the top topic (frontpage) from order $k=1$ as the first click; this diagram shows transition probabilities from top four categories given that users first visited the frontpage topic (represented by the dashed transitions in the left figure representing $k=1$). For example, the links from the blue node (news) in the top-left corner of the diagram represent the transition probabilities from the sequence (frontpage, news) to other nodes. Similarly, we visualize order $k=3$ in the right figure by selecting a node with the highest sum of outgoing transition probabilities (frontpage, frontpage) and its four highest outgoing transition probabilities from order $k=2$ (represented by the dashed transitions in the middle figure representing $k=2$). We then show transition probabilities from other nodes given that users already visited (frontpage, frontpage). For example, the links from the red node (sports) at the top represent the transition probabilities from the sequence (frontpage, frontpage, sports) to other nodes.[]{data-label="fig:local_msnbc_all"}](MSNBC_local_3){width="\columnwidth"} ![**Self transition structure of navigation for the MSNBC dataset.** The number of times users stay within the same topic vs. the number of times they change the topic during navigation for different values of $k$. Only the top three categories with the highest transition probabilities are shown. With high consistency, the transition probabilities to the same topic increase while those to other categories decrease with ascending order $k$.[]{data-label="fig:sameornot_msnbc"}](sameornot_MSNBC){width="\columnwidth"} Discussion {#subsec:discussion .unnumbered} ---------- Our findings and observations in this article show that simple likelihood investigations (see e.g., [@chierichetti]) may not be sufficient to select the appropriate order of Markov chains and to prove or falsify whether human navigation is memoryless or not. To ultimately answer this, we think it is inevitable to look deeper into the results obtained and to investigate them with a broader spectrum of model selection methods starting with the ones presented in this work. By applying these methods to human navigational data, the results suggest that on the Wikigame page dataset a zero order model should be preferred. This is due to the rising complexity of higher order models and indicates that it is difficult to derive the appropriate order for finite datasets with a huge amount of distinct pages having only limited observations of human navigational behavior. In this article we presented and applied a variety of distinct model selection that all include (necessary) ways of penalizing the large number of parameters needed for higher order models. [[Yet, we do not necessarily know what would happen if we would apply the models to a much larger number of navigational paths over pages. Perhaps higher order models would then outperform lower ones. As it is unlikely to get hands on such an amount of data for large websites, a starting point to further test this could be to analyze a sub-domain with rich data; i.e., a large number of observations over just a very limited number of distinct pages. However, due to no current access to such data, we leave this open for future work.]{}]{} On the other hand, the results on a topical level are intriguing and show a much clearer picture: They suggest that the navigational patterns are not memoryless. Higher order Markov chains – i.e., second or third order – seem to be the most appropriate. Henceforth, the navigation history of users seem to span at least two or three states on a topical level. This gives high indications that common strategies (at least on a topical level) exist among users navigating information networks on the Web. It is certainly intriguing to see similar memory patterns in both goal-oriented navigation (Wikigame and Wikispeedia) and free form navigation (MSNBC), and different kinds of systems (encylopedia vs. news portal). In order to confirm that these observed memory effects are based on the actual human navigation patterns we again look at our random path dataset introduced in the section entitled “” with the log-likelihoods visualized in Figure \[fig:randomloglikelihood\]. We can recapitalize, that these simple log-likelihoods would suggest a higher order model for the randomly produced navigational patterns. However, if we apply our various model selection techniques the results suggest a zero or at maximum a first order Markov chain model which is the logic conclusion for this random process. Hence, this confirms that our observations on the real nature navigational data are based on human navigational memory patterns and would not be present in a random process. Finally, we showed in the section called “” that common structure in the navigational trails exist among many users – i.e., common sequences of navigational transitions. First of all, we could observe that transitions between the same topic are common among all three datasets. However, they occur more frequently in our free form navigational data (MSNBC) than in the goal-oriented navigation datasets (Wikigame and Wikispeedia). Furthermore, users also seem to be more likely to stay longer in the same topic while navigating MSNBC while they seem to switch categories more frequently in both the Wikigame and Wikispeedia datasets. A possible explanation for this user behavior might be that users on MSNBC are more driven by specific information needs regarding one topic. For example, a user might visit the website to get information about the weather only. Contrary, exact information goals on Wikipedia might not always be in the same topic. Suppose, you are located on *Seoul* which belongs to the *Geography* topic and you want to know more about important inventions made in *Seoul*. A possible path then could be that you navigate over a *People* topic page and finally reach a *Science* topic page. However, we need to keep in mind that our goal-oriented datasets are based on game data with predefined start and target nodes. This means, that if the target nodes regularly lie in distinct categories, the user might be forced to switch categories more frequently. To rule this out, we illustrate the heatmap of our Wikigame dataset (cf. Figure \[fig:heatmaps\]) again by splitting the path corpus into two parts (see Figure \[fig:heatmaps\_same\_diff\]): (A) only considering paths where the start and target node lie in the same topic and (B) only taking paths with distinct start and target categories. If the bias of given start and target nodes would influence our observations for specific structural properties of goal-oriented navigational patterns, Figure \[fig:heatmaps\_same\_diff\] would show strong dissimilarities between both illustrations which is not the case. Hence, we can state with strong confidence that the differences between goal-oriented and free form navigation stated in this section are truly based on the distinct strategies and navigational scenarios. Nevertheless, we also need to keep in mind that the website design and inherent link structure (Wikipedia vs. MSNBC) might also influence this behavior. For example, a reason could be that Wikipedia has more direct links between distinct categories in comparison to MSNBC or that Wikipedia’s historical coverage steers user behavior to specific kinds of navigational patterns. To explicitly rule this possibility out, we would need to investigate the underlying link networks in greater detail, which we leave open for future work. We also plan on looking at data capturing navigational paths over distinct platforms of the Web (e.g., from toolbar data) which may allow us to make even more generic statements about human navigation on the Web. ![**Common global transition patterns of navigational behavior on the Wikigame topic dataset.** The results should be compare with Figure \[fig:heatmaps\]. The results are split by only looking at a corpus of paths where each path starts with the same topic as it ends (A) and by looking at a corpus with distinct start and target categories (B). []{data-label="fig:heatmaps_same_diff"}](wikigame_same_diff){width="\textwidth"} Conclusions {#sec:conclusions .unnumbered} =========== This work presented an extensive view on detecting memory and structure in human navigational patterns. We leveraged Markov chain models of varying order for detecting memory of human navigation and took a thorough look at structural properties of human navigation by investigating Markov chain transition matrices. We developed an open source framework[^16] [@github] for detecting memory of human navigational patterns by calculating the appropriate Markov chain order using four different, yet complementary, approaches (likelihood, Bayesian, information-theoretic and cross validation methods). In this article we thoroughly present each method and emphasize strengths, weaknesses and relations between them. By applying this framework to actual human navigational data we find that it is indeed difficult to make plausible statements about the appropriate order of a Markov chain having insufficient data but a vast amount of states which results in too complex models. However, by representing pages by their corresponding topic we could identify that navigation on a topical level is not memoryless – an order of two and respectively three best explain the observed data, independent whether the navigation is goal-oriented or free-form. Finally, our structural investigations illustrated that users tend to stay in the same topic while navigating. However, this is much more frequent for our free form navigational dataset (MSNBC) as compared to both of the goal-oriented datasets (Wikigame and Wikispeedia). Future attempts of modeling human behavior in the Web can benefit from the methodological framework presented in this work to thoroughly investigate such behavior. If one wants to resort to a single model selection technique, we would recommend to use the Bayesian approach if computationally feasible. Our work strongly indicates memory effects of human navigational patterns on a topical level. Such observations as well as detailed insights into structural regularities in human navigation patterns can e.g., be useful for improving recommendation systems, web site design as well as faceted browsing. In future work, we want to extend our ideas of representing Web pages with categories by looking at further features for representation. We also plan on tapping into the usefulness of further Markov models like the hidden Markov model, varying order Markov model or semi Markov model. Also, we want to improve recommendation algorithms by the insights generated in this work and explore the implications higher order Markov chain models may have on ranking algorithms like PageRank. Acknowledgments {#acknowledgments .unnumbered} =============== We want to thank Alex Clemesha (Wikigame) and Robert West (Wikispeedia) for granting us access to their navigational datasets as well as the anonymous reviewers for the highly valuable inputs. [^1]: Note that the terms “topic” and “category” should be seen as synonyms throughout this work."’ [^2]: <https://github.com/psinger/PathTools> [^3]: [[ Note that no official classification of statistical schools is available; some may also argue that there are only the two competing schools of frequentists (which we do not explicitly discuss in this article) and Bayesians. The categorization used here is motivated by a short blog post (see <http://labstats.net/articles/overview.html>). ]{}]{} [^4]: [[Note that this method also utilizes mechanisms usually known from the frequentist school; i.e., hypothesis testing.]{}]{} [^5]: [[We could tackle this problem by e.g., applying the *Bonferroni correction* which we leave open for future work]{}]{} [^6]: [[In order to confirm our findings we also applied an additional way of determining the accuracy which is motivated by a typical evaluation technique known from link predictors [@liben]. Concretely, it counts how frequently the true next click is present in the TopK (k=5) states determined by the probabilities of the transition matrix. In case of ties in the TopK elements we randomly draw from the ties. By applying this method to our data we can mirror the evaluation results obtained by using the described and used ranking technique. Note that we do not explicitly report the additional results of this evaluation method throughout the paper.]{}]{} [^7]: <http://thewikigame.com/> [^8]: <http://en.wikipedia.org/wiki/Category:Main_topic_classifications> [^9]: <http://www.cs.mcgill.ca/~rwest/wikispeedia/> [^10]: <http://schools-wikipedia.org/> [^11]: <http://kdd.ics.uci.edu/databases/msnbc/msnbc.html> [^12]: <http://msnbc.com> [^13]: Except for teleportation which we do not model in this work. [^14]: Both priors agree throughout all our investigations in this article. [^15]: [[Consequently, we might get better representations of the data by using Markov chain models that, instead modeling state transitions in equal time steps, additionally stochastically model the duration times in states (e.g., semi Markov or Markov renewal models). However, we leave these investigations open for future work.]{}]{} [^16]: <https://github.com/psinger/PathTools>

--- author: - | Sho Tanaka[^1]\ Kurodani 33-4, Sakyo-ku, Kyoto 606-8331, Japan title: 'Holographic Relation in Yang’s Quantized Space-Time Algebra and Area-Entropy Relation in $D_0$ Brane Gas System' --- addtoreset[equation]{}[section]{} In the preceding paper, we derived a kind of kinematical holographic relation (KHR) in the Lorentz- covariant Yang’s quantized space-time algebra (YSTA). It essentially reflects the fundamental nature of the noncommutative geometry of YSTA and its representation, that is, a definite kinematical reduction of spatial degrees of freedom in comparison with the ordinary lattice space. On the basis of the relation and its extension to various spatial dimensions, we derive a new area-entropy relation in a simple $D_0$ brane gas system subject to YSTA, following the idea of M-theory. Furthermore, we make clear its inner relation with the Bekenstein-Hawking area-entropy relation in connection with Schwarzschild black hole. Key words: Yang’s quantized space-time algebra(YSTA); kinematical reduction of spatial degrees of freedom; holographic relation in YSTA; area-entropy relation; Schwarzschild black hole; $D_0$ brane gas model. Introduction ============ In the preceding paper, $^{[1]}$ referred hereafter as I, we derived a kind of holographic relation in the Lorentz-covariant Yang’s quantized space-time algebra(YSTA),$^{[1],[2],[3]}$ which we called the kinematical holographic relation (KHR). As was emphasized in I, the relation essentially reflects the fundamental nature of the noncommutative geometry of YSTA, that is, a definite kinematical reduction of spatial degrees of freedom in comparison with the ordinary lattice space. As will be shown in the present paper, this relation seems also to give an important clue to resolve the long-pending problem encountered in the Bekenstein-Hawking area-entropy relation$^{[4]}$ or the holographic principle,$^{[5]}$ that is, the apparent gap between the degrees of freedom of any bounded spatial region associated with entropy and of its boundary area. In addition to the last problem, in the arguments of the holographic principle, the limit of the present local field theory has been discussed, as seen, for instance, in the unified regularization or cutoff of UV/IR divergences. With respect to this problem, as was emphasized in refs. \[1\], \[2\], YSTA which is intrinsically equipped with short- and long-scale parameters, $\lambda$ and $R$, gives a finite number of spatial degrees of freedom for any finite spatial region and provides a basis for the field theory free from ultraviolet- and infrared-divergences. In fact, we found in I, the following form of kinematical holographic relation (KHR) in YSTA, $$\begin{aligned} \hspace{-3cm} [KHR] \hspace{2cm} n^L_{\rm dof}= {\cal A} / G, \nonumber\end{aligned}$$ that is, the proportional relation between $n^L_{\rm dof}$ and ${\cal A}$ with proportional constant $G$, where $n^L_{\rm dof}$ and ${\cal A}$, respectively, denote the number of degrees of freedom of any spherical bounded spatial region with radius $L$ in Yang’s quantized space-time and the boundary area in unit of $\lambda.$ In this paper, we derive a new area-entropy relation \[AER\] on the basis of the above \[KHR\] and make clear its inner relation with the ordinary Bekenstein-Hawking area-entropy relation in connection with Schwarzschild black hole. It will be made through a simple $D_0$ brane gas model$^{[6]}$ on Yang’s quantized space-time according to the idea of M-theory,$^{[7]}$ with the aid of a kind of Gedanken-experiment on the present static toy model. The present paper is organized as follows. In Sec. 2, we briefly recapitulate Yang’s quantized space-time algebra (YSTA) and its representations. Sec. 3 is devoted to the recapitulation of the kinematical holographic relation (KHR) and to its extension to the lower-dimensional bounded regions, $V_d^L$. In section 4, we introduce a simple $D_0$ brane (D-particle) gas model on $V_d^L$ and find a new area-entropy relation in the system in connection with Schwarzschild black hole. In the final section, we discuss the inner relation between our area-entropy relation based on KHR in YSTA and the ordinary Bekenstein-Hawking area-entropy relation and point out our future task beyond the present simple $D_0$ brane gas model. Yang’s Quantized Space-Time Algebra (YSTA) and Its Representations ================================================================== Yang’s Quantized Space-Time Algebra (YSTA) ------------------------------------------- Let us first recapitulate briefly the Lorentz-covariant Yang’s quantized space-time algebra (YSTA). $D$-dimensional Yang’s quantized space-time algebra is introduced$^{[1],[2]}$ as the result of the so-called Inonu-Wigner’s contraction procedure with two contraction parameters, $R$ and $\lambda$, from $SO(D+1,1)$ algebra with generators $\hat{\Sigma}_{MN}$; $$\begin{aligned} \hat{\Sigma}_{MN} \equiv i (q_M \partial /{\partial{q_N}}-q_N\partial/{\partial{q_M}}),\end{aligned}$$ which work on $(D+2)$-dimensional parameter space $q_M$ ($M= \mu,a,b)$ satisfying $$\begin{aligned} - q_0^2 + q_1^2 + \cdots + q_{D-1}^2 + q_a^2 + q_b^2 = R^2.\end{aligned}$$ Here, $q_0 =-i q_D$ and $M = a, b$ denote two extra dimensions with space-like metric signature. $D$-dimensional space-time and momentum operators, $\hat{X}_\mu$ and $\hat{P}_\mu$, with $\mu =1,2,\cdots,D,$ are defined in parallel by $$\begin{aligned} &&\hat{X}_\mu \equiv \lambda\ \hat{\Sigma}_{\mu a} \\ &&\hat{P}_\mu \equiv \hbar /R \ \hat{\Sigma}_{\mu b}, \end{aligned}$$ together with $D$-dimensional angular momentum operator $\hat{M}_{\mu \nu}$ $$\begin{aligned} \hat{M}_{\mu \nu} \equiv \hbar \hat{\Sigma}_{\mu \nu}\end{aligned}$$ and the so-called reciprocity operator $$\begin{aligned} \hat{N}\equiv \lambda /R\ \hat{\Sigma}_{ab}.\end{aligned}$$ Operators $( \hat{X}_\mu, \hat{P}_\mu, \hat{M}_{\mu \nu}, \hat{N} )$ defined above satisfy the so-called contracted algebra of the original $SO(D+1,1)$, or Yang’s space-time algebra (YSTA): $$\begin{aligned} &&[ \hat{X}_\mu, \hat{X}_\nu ] = - i \lambda^2/\hbar \hat{M}_{\mu \nu} \\ &&[\hat{P}_\mu,\hat{P}_\nu ] = - i\hbar / R^2\ \hat{M}_{\mu \nu} \\ &&[\hat{X}_\mu, \hat{P}_\nu ] = - i \hbar \hat{N} \delta_{\mu \nu} \\ &&[ \hat{N}, \hat{X}_\mu ] = - i \lambda^2 /\hbar \hat{P}_\mu \\ &&[ \hat{N}, \hat{P}_\mu ] = i \hbar/ R^2\ \hat{X}_\mu,\end{aligned}$$ with familiar relations among ${\hat M}_{\mu \nu}$’s omitted. Quasi-Regular Representation of YSTA ------------------------------------ Let us further recapitulate briefly the representation$^{[1],[2]}$ of YSTA for the subsequent consideration in section 4. First, it is important to notice the following elementary fact that ${\hat\Sigma}_{MN}$ defined in Eq.(2.1) with $M, N$ being the same metric signature have discrete eigenvalues, i.e., $0,\pm 1 , \pm 2,\cdots$, and those with $M, N$ being opposite metric signature have continuous eigenvalues, $\footnote{The corresponding eigenfunctions are explicitly given in ref. [9].}$ consistently with covariant commutation relations of YSTA. This fact was first emphasized by Yang$^{[3]}$ in connection with the preceding Snyder’s quantized space-time.$^{[8]}$ This conspicuous aspect is well understood by means of the familiar example of the three-dimensional angular momentum in quantum mechanics, where individual components, which are noncommutative among themselves, are able to have discrete eigenvalues, consistently with the three-dimensional rotation-invariance. This fact implies that Yang’s space-time algebra (YSTA) presupposes for its representation space to take representation bases like $$\begin{aligned} | t/\lambda,n_{12}, \cdots> \equiv |{\hat{\Sigma}}_{0a} =t/\lambda> |{\hat{\Sigma}}_{12}=n_{12}> \cdots|{\hat{\Sigma}}_{910}=n_{910}>,\end{aligned}$$ where $t$ denotes [*time*]{}, the continuous eigenvalue of $\hat{X}_0 \equiv \lambda\ \hat{\Sigma}_{0 a}$ and $n_{12}, \cdots$ discrete eigenvalues of maximal commuting set of subalgebra of $SO(D+1,1)$ which are commutative with ${\hat{\Sigma}}_{0a}$, for instance, ${\hat{\Sigma}}_{12}$, ${\hat{\Sigma}}_{34},\cdots , {\hat{\Sigma}}_{910}$, when $D=11$.$^{[9],[1],[2]}$ Indeed, an infinite dimensional linear space expanded by $|\ t/\lambda, n_{12},\cdots>$ mentioned above provides a representation space of unitary infinite dimensional representation of YSTA. It is the so-called “quasi-regular representation”$^{[10]}$ of SO(D+1,1),[^2] and is decomposed into the infinite series of the ordinary unitary irreducible representations of $SO(D+1,1)$ constructed on its maximal compact subalgebra, $SO(D+1)$. It means that there holds the following form of decomposition theorem, $$\begin{aligned} | t/\lambda, n_{12},\cdots>= \sum_{\sigma 's}\ \sum_{l,m}\ C^{\sigma's, n_{12}, \cdots }_{l,m}(t/\lambda)\ | \sigma 's ; l,m>,\end{aligned}$$ with expansion coefficients $C^{\sigma's, n_{12}, \cdots}_{l,m}(t/\lambda).^{[9],[1]}$ In Eq.(2.13), $|\sigma 's ; l, m>'s$ on the right hand side describe the familiar unitary irreducible representation bases of $SO(D+1,1)$, which are designated by $\sigma 's$ and $(l,m),$ [^3] denoting, respectively, the irreducible unitary representations of $SO(D+1,1)$ and the associated irreducible representation bases of $SO(D+1)$, the maximal compact subalgebra of $SO(D+1,1)$, mentioned above. It should be noted here that, as remarked in I, $l$’s are limited to be integer, excluding the possibility of half-integer, because of the fact that generators of $SO(D+1)$ in YSTA are defined as differential operators on $S^D$, i.e., ${q_1}^2 + {q_2}^2 + \cdots + {q_{D-1}}^2 + {q_a}^2 + {q_b}^2 = 1.$ In what follows, let us call the infinite dimensional representation space introduced above for the representation of YSTA, Hilbert space I, in distinction to Hilbert space II which is Fock-space constructed dynamically by creation-annihilaltion operators of second-quantized fields on YSTA, such as $D_0$ brane field,$^{[9]}$ discussed in section 4. Kinematical Holographic Relation \[KHR\] in YSTA ================================================ Recapitulation of Kinematical Holographic Relation \[KHR\] ---------------------------------------------------------- First, let us remember that the following kinematical holographic relation[^4] $$\begin{aligned} \hspace{-3cm} [KHR] \hspace{2cm} n^L_{\rm dof}= {\cal A} / G,\end{aligned}$$ with the proportional constant $G$ $$\begin{aligned} G\ \sim {(2 \pi)^{D/2} \over 2}\ (D-1)!! &&for\ D\ even \\ \sim (2 \pi)^{(D-1)/2}(D-1)!! &&for\ D\ odd,\end{aligned}$$ was derived in I for the $D$-dimensional space-like region with finite radius $L$ in D-dimensional Yang’s quantized space-time in the unit of $\lambda$. Let us denote the region hereafter as $V_D^L$, which was defined by $$\begin{aligned} \sum_{K \neq 0}{\Sigma_{aK}}^2 = \sum_{\mu \neq 0}{\Sigma_{a \mu}}^2 + {\Sigma_{ab}}^2 = (L/\lambda)^2,\end{aligned}$$ or $$\begin{aligned} {X_1}^2 + {X_2}^2 + \cdots + {X_{D-1}}^2 + R^2\ N^2 = L^2.\end{aligned}$$ Here, $\Sigma_{MN}$’s are presumed to be given in terms of Moyal star product formalism applied to the expression, $\Sigma_{MN}= ( -q_M p_n + q_N p_M)$, as was treated in detail in I. ${\cal A}$ in \[KHR\] (3.1) simply denotes the boundary surface area of $V_D^L$, that is, $$\begin{aligned} {\cal A} = ({\rm area\ of}\ S^{D-1}) ={(2 \pi)^{D/2} \over {(D-2)!!}} (L/\lambda)^{D-1} &&for\ D\ even, \nonumber\\ =2 {(2\pi)^{(D-1)/2} \over {(D-2)!!}} (L/\lambda)^{D-1} &&for\ D\ odd. \end{aligned}$$ On the other hand, $n^L_{\rm dof}$ in \[KHR\] (3.1), which denotes, by definition, the number of spatial degrees of freedom of YSTA inside $V_D^L$, was given in I as follows, $$\begin{aligned} &&n^L_{\rm dof} = dim\ ( \rho_{[L/\lambda]}) = {2 \over (D-1)!}{([L/\lambda]+ D-2)! \over ([L/\lambda]-1)!} \nonumber\\ &&\hskip3.5cm \sim {2 \over (D-1)!} [L/\lambda]^{D-1}.\end{aligned}$$ Indeed, the derivation of the above equation (3.7) was the central task in I. In fact, we emphasized that the number of degrees of freedom $n^L_{\rm dof}$ inside $V_D^L$, which is subject to noncommutative algebra, YSTA, should be, logically and also practically, found in the structure of representation space of YSTA, that is, Hilbert space I defined in section 2. Let us here recapitulate in detail the essence of the derivation in order to make the present paper as self-contained as possible. In fact, one finds that the representation space needed to calculate $n^L_{\rm dof}$ is prepared in Eq.(2.13), where any “quasi-regular” representation basis,\ $ | t/\lambda, n_{12}, \cdots>$, is decomposed into the infinite series of the ordinary unitary representation bases of $SO(D+1,1)$, $| \sigma 's ; l,m>.$ As was stated in subsection 2.2, the latter representation bases, $| \sigma 's ; l,m>'s$ are constructed on the familiar finite dimensional representations of maximal compact subalgebra of YSTA, $SO(D+1)$, whose representation bases are labeled by $(l,m)$ and provide the representation bases for spatial quantities under consideration, because $SO(D+1)$ just involves those spatial operators $( \hat{X}_u, R \hat{N})$. In order to arrive at the final goal of counting $n^L_{\rm dof}$, therefore, one has only to find mathematically a certain irreducible representation of $SO(D+1)$, which [*properly*]{} describes (as seen in what follows) the spatial quantities $( \hat{X}_u, R \hat{N})$ inside the bounded region with radius $L$, then one finds $n^L_{\rm dof}$ through counting the dimension of the representation. At this point, it is important to note that, as was remarked in advance in subsection 2.2, any generators of $SO(D+1)$ in YSTA are defined by the differential operators on the $D-$dimensional unit sphere, $S^D$, i.e., ${q_1}^2 + {q_2}^2 + \cdots + {q_{D-1}}^2 + {q_a}^2 + {q_b}^2 = 1,$ limiting its representations with $l$ to be integer. On the other hand, it is well known that the irreducible representation of arbitrary high-dimensional $SO(D+1)$ on $S^D = SO(D+1)/SO(D)$ is derived in the algebraic way, $^{[11]}$ irrelevantly to any detailed knowledge of the decomposition equation (2.13), but solely in accord with the fact that $SO(D+1)$ in YSTA is defined originally on $S^D$, as mentioned above. One can choose, for instance, $SO(D)$ with generators $\hat{\Sigma}_{MN} (M,N=b, u)$, while $SO(D+1)$ with generators $\hat{\Sigma}_{MN}(M,N=a,b,u)$. Then, it turns out that any irreducible representation of $SO(D+1)$, denoted by $\rho_l$, is uniquely designated by the maximal integer $l$ of eigenvalues of ${\hat \Sigma}_{ab}$ in the representation, where ${\hat \Sigma}_{ab}$ is known to be a possible Cartan subalgebra of the so-called compact symmetric pair $(SO(D+1),SO(D))$ of rank $1$.$^{[11]}$ According to the so-called Weyl’s dimension formula, the dimension of $\rho_l$ is given by$^{[11],[1],[2]}$ $$\begin{aligned} dim\ (\rho_l)= {(l+\nu) \over \nu} {(l+2\nu-1)! \over {l!(2\nu -1)!}},\end{aligned}$$ where $ \nu \equiv (D-1)/2$ and $D \geq 2$.[^5] Finally, we can find a certain irreducible representation of $SO(D+1)$ among those $\rho_l 's $ given above, which [*properly*]{} describes (or realizes) the spatial quantities inside the bounded region $V_D^L$. Now, let us choose tentatively $l = [L/\lambda]$ with $[L/\lambda]$ being the integer part of $L/\lambda$. In this case, one finds out that the representation $\rho_{[L/\lambda]}$ just [*properly*]{} describes all of generators of $SO(D+1)$ inside the above bounded spatial region $V_D^L$, because $[L/\lambda ]$ indicates also the largest eigenvalue of any generators of $SO(D+1)$ in the representation $\rho_{[L/\lambda]}$ on account of its $SO(D+1)-$invariance and hence eigenvalues of spatial quantities $( \hat{X}_u, R \hat{N})$ are well confined inside the bounded region with radius [L]{}. As the result, one finds that the dimension of $\rho_{[L/\lambda]}$ just gives the number of spatial degrees of freedom inside $V_D^L$, $n^L_{\rm dof}$, as shown in (3.7). KHR in the lower-dimensional spatial region $V_d^L$ ---------------------------------------------------- According to the argument given for $V_D^L$ in the preceding subsection, let us study the kinematical holographic relation in the lower-dimensional bounded spatial region $V_d^L$ for the subsequent argument of the area-entropy relation in section 4. In fact, it will be given through a simple $D_0$ brane gas system formed inside $d\ (\leq{D-1})$-dimensional bounded spatial region, $V_d^L$, which is defined by $$\begin{aligned} {X_1}^2 + {X_2}^2 + \cdots + {X_d}^2 = L^2,\end{aligned}$$ instead of (3.5). In this case, the boundary area of $V_d^L$, that is, ${\cal A}\ (V_d^L)$ is given by $$\begin{aligned} {\cal A}\ (V_d^L) = ({\rm area\ of}\ S^{d-1}) ={(2 \pi)^{d/2} \over {(d-2)!!}} (L/\lambda)^{d-1} &&for\ d\ even \nonumber\\ =2 {(2\pi)^{(d-1)/2} \over {(d-2)!!}} (L/\lambda)^{d-1} &&for\ d\ odd, \end{aligned}$$ corresponding to Eq. (3.6). On the other hand, the number of degrees of freedom of $V_d^L$, let us denote it $n_{dof} (V_d^L)$, is calculated by applying the arguments given for derivation of $n_{dof}^L$ in (3.7). In fact, it is found in a certain irreducible representation of $SO(d+1)$, a minimum subalgebra of YSTA, which includes the $d$ spatial quantities, $\hat{X}_1, \hat{X}_2, \cdots, \hat{X}_d$ needed to properly describe $V_d^L$, and is really constructed by the generators $\hat{\Sigma}_{MN}$ with $M,N$ ranging over $a,1,2, \cdots,d$. The representation of $SO(d+1)$, let us denote it $\rho_l\ (V_d^L)$ with suitable integer $l = [L/\lambda]$, is given on the representation space $S^d =SO(d+1)/SO(d)$, taking the subalgebra $SO(d)$, for instance, $\hat{\Sigma}_{MN}$ with $M,N$ ranging over $1,2,\cdots,d$, entirely in accord with the argument on the irreducible representation of $SO(D+1)$ given in the preceding subsection 3.1. One immediately finds that $$\begin{aligned} &&n_{\rm dof}\ (V_d^L) = dim\ ( \rho_{[L/\lambda]}\ (V_d^L)) = {2 \over (d-1)!}{([L/\lambda]+ d-2)! \over ([L/\lambda]-1)!} \nonumber\\ &&\hskip3.5cm \sim {2 \over (d-1)!} [L/\lambda]^{d-1}.\end{aligned}$$ corresponding to (3.7), and there holds, from (3.10) and (3.11), the following kinematical holographic relation for $V_d^L$ in general $$\begin{aligned} \hspace{-3cm} [KHR] \hspace{2cm} n^L_{\rm dof}\ (V_d^L)= {\cal A}\ (V_d^L) / G_d,\end{aligned}$$ with the proportional constant $G_d$ $$\begin{aligned} G_d \sim {(2 \pi)^{d/2} \over 2}\ (d-1)!! &&for\ d\ even \\ \sim (2 \pi)^{(d-1)/2}(d-1)!! &&for\ d\ odd,\end{aligned}$$ corresponding to Eqs. (3.1)- (3.3) for $V_D^L$. Area-Entropy Relation in $D_0$ Brane Gas subject to YSTA ======================================================== $D_0$ Brane Gas Model in $V_d^L$ and Its Mass and Entropy --------------------------------------------------------- Now, let us consider the central problem of the present paper, that is, the derivation of a possible area-entropy relation through a simple $D_0$ brane gas$^{[6]}$ model formed inside $V_d^L$ according to the idea of M-theory. This implies that one has to deal with the dynamical system of the second-quantized $D_0$ brane field ${\hat D}_0$ inside $V_d^L$. In the present toy model of the $D_0$ brane gas, however, we avoid to enter into detail of the dynamics of $D_0$ brane system, but treat it as an ideal gas, only taking into consideration that the system is developed on $V_d^L$ subject to YSTA and its representation discussed above, but neglecting interactions of $D_0$ branes, for instance, with strings, as well as possible self-interactions among themselves. First of all, according to the argument given in the preceding subsection 3.2, the spatial structure of $V_d^L$ is described through the specific representation $ \rho_{[L/\lambda]}\ (V_d^L)$. Let us denote its orthogonal basis-vector system in Hilbert space I, as follows $$\begin{aligned} \rho_{[L/\lambda]}\ (V_d^L): \quad |\ m >, \qquad m= 1,2,\cdots, n_{\rm dof}(V_d^L). \end{aligned}$$ In the above expression, $n_{\rm dof}\ (V_d^L)$ denotes the dimension of the representation $\rho_{[L/\lambda]}\ (V_d^L)$, as defined in (3.11). At this point, one should notice that the [*second quantized*]{} ${\hat D}_0$-brane field$^{[12]}$ on $V_d^L$ must be the linear operators operating on Hilbert space I, and described by $n_{\rm dof}(V_d^L) \times n_{\rm dof}(V_d^L)$ matrix under the representation $\rho_{[L/\lambda]}\ (V_d^L)$ like $< m\ |{\hat D}|\ n >$ on the one hand, and on the other hand each matrix element must be operators operating on Hilbert space II, playing the role of creation-annihilation of $D_0$ branes. On the analogy of the ordinary quantized local field, let us define those creation-annihilation operators through the diagonal parts in the following way:[^6] $$\begin{aligned} < m\ | {\hat D}|\ m >\ \sim\ {\bf a}_m\ {\rm or}\ {\bf a}_m^\dagger.\end{aligned}$$ In the above expression, ${\bf a}_m$ and ${\bf a}_m^\dagger$, respectively, denote annihilation and creation operators of $D_0$ brane, satisfying the familiar commutation relations, $$\begin{aligned} &&[{\bf a}_m, {\bf a}_n^\dagger]= \delta_{mn}, \\ &&[{\bf a}_m, {\bf a}_n]= 0 .\end{aligned}$$ One notices that the labeling number $m$ of basis vectors, which ranges from $1$ to $n_{\rm dof}(V_d^L)$ plays the role of [*spatial coordinates*]{} of $V_d^L$ in the present noncommutative YSTA, corresponding to the so-called lattice point in the lattice theory. Let us denote the [*point*]{} hereafter $[site]$ or $[site\ m]$ of $V_d^L$. Now, let us focus our attention on quantum states constructed dynamically in Hilbert space II by the creation-annihilation operators ${\bf a}_m$ and ${\bf a}_m^\dagger$ of $D_0$ branes introduced above at each \[site\] inside $V_d^L$. One should notice here the important fact that in the present simple $D_0$ brane gas model neglecting all interactions of $D_0$ branes, each $[site]$ can be regarded as independent quantum system and described in general by own statistical operator, while the total system of gas is described by their direct product. In fact, the statistical operator at each $[site\ m]$ denoted by ${\hat W}[m]$, is given in the following form, $$\begin{aligned} {\hat W}[m] = \sum_ k w_k\ |\ [m]: k >\ < k :[m]\ |,\end{aligned}$$ with $$\begin{aligned} |\ [m]: k > \equiv {1 \over \sqrt{k!}}({\bf a}_m^\dagger)^{k}|\ [m]:0 >.\end{aligned}$$ That is, $|\ [m]: k > ( k=0, 1, \cdots)$ describes the normalized quantum-mechanical state in Hilbert space II with $k$ $D_0$ branes constructed by ${\bf a}_m^\dagger$ on $|\ [m]:0 >,$ i.e. the vacuum state of $[site\ m]$. [^7] And $w_k$’s denote the realization probability of state with occupation number $k$, satisfying $\sum_k w_k = 1.$ We assume here that the statistical operator at each $[site\ m]$ is common to every \[site\] in the present $D_0$ brane gas under equilibrium state, with the common values of $w_k$’s and the statistical operator of total system on $V_d^L$ , ${\hat W}(V_d^L)$, is given by $$\begin{aligned} {\hat W}(V_d^L) = {\hat W}[1] \otimes {\hat W}[2] \cdots \otimes {\hat W}[m] \cdots \otimes {\hat W}[n_{dof}].\end{aligned}$$ Consequently, one finds that the entropy of the total system, $S(V_d^L)$, is given by $$\begin{aligned} S(V_d^L) = - {\rm Tr}\ [{\hat W}(V_d^L)\ {\rm ln} {\hat W}(V_d^L)] = n_{dof}(V_d^L)\times S[site], \end{aligned}$$ where $S[site]$ denotes the entropy of each \[site\] assumed here to be common to every \[site\] and given by $$\begin{aligned} S[site] = - {\rm Tr}\ [ {\hat W}[site]\ {\rm ln}{\hat W}[site]] = - \sum_k w_k\ {\rm ln} w_k.\end{aligned}$$ Comparing this result (4.8) with \[KHR\] (3.12) derived in the preceding section, we find an important fact that the entropy $S(V_d^L)$ is proportional to the surface area ${\cal A}\ (V_d^L)$, that is, a kind of area-entropy relation (\[AER\]) of the present system: $$\begin{aligned} \hspace{-3cm} [AER] \hspace{2cm} S(V_d^L) = {\cal A}\ (V_d^L)\ {S[site] \over G_d},\end{aligned}$$ where $G_d$ is given by (3.13)-(3.14). Next, let us introduce the total energy or mass of the system, $M(V_d^L)$. If one denotes the average energy or mass of the individual $D_0$ brane inside $V_d^L$ by $\mu$, it may be given by $$\begin{aligned} M(V_d^L) = \mu {\bar N}[site]\ n_{\rm dof}(V_d^L) \sim \mu {\bar N}[site] {2 \over (d-1)!} [L/\lambda]^{d-1},\end{aligned}$$ where ${\bar N}[site]$ denotes the average occupation number of $D_0$ brane at each $[site]$ given by $$\begin{aligned} {\bar N}[site] \equiv \sum_k k w_k.\end{aligned}$$ Comparing this expression (4.11) with (4.8) and (3.12), respectively, we obtain a kind of mass-entropy relation (\[MER\]) $$\begin{aligned} \hspace{-2cm} [MER] \hspace{2cm} M(V_d^L) / S(V_d^L) = \mu {\bar N}[site] / S[site],\end{aligned}$$ and a kind of area-mass relation (\[AMR\]) $$\begin{aligned} \hspace{-3cm} [AMR] \hspace{2cm} M(V_d^L) = {\cal A}(V_d^L)\ {\mu {\bar N}[site] \over G_d}.\end{aligned}$$ Schwarzschild Black Hole and Area-Entropy Relation In $D_0$ brane Gas System ---------------------------------------------------------------------------- In the preceding subsection 4.1, we have studied $D_0$ brane gas system and derived area-entropy relation $[AER]$ (4.10), mass-entropy and area-mass relations, $[MER]$ (4.13) and $[AMR]$ (4.14), which are essentially based on the kinematical holographic relation in YSTA studied in section 3. At this point, it is quite important to notice that these three relations explicitly depend on the following ${\it static}$ factors of the gas system, $\mu$, ${\bar N}[site]$ and $S[site]$, that is, the average energy of individual $D_0$ brane, the average occupation number of $D_0$ branes and the entropy at each \[site\], which are assumed to be common to every \[site\], while these factors turn out to play an important role in arriving finally at the area-entropy relation in connection with black holes, as will be seen below. Now, let us investigate how the present gas system tends to a black hole. We assume for simplicity that the system is under $d=3$, and becomes a Schwarzschild black hole, in which the above factors acquire certain limiting values, $ \mu_S$, ${\bar N}_S[site]$ and $S_S [site]$, while the size of the system, $L$, becomes $R_S$, that is, the so-called Schwarzschild radius given by $$\begin{aligned} R_S \equiv 2 G M(V_3^{R_S})/c^2,\end{aligned}$$ where $G$ and $c$ denote Newton ’s constant and the light velocity, respectively, and $M(V_3^{R_S})$ is given by Eq.(4.11) with $L= R_S$, $\mu = \mu_S$ and ${\bar N}[site] = {\bar N}_S[site]$. Indeed, inserting the above values into Eq.(4.11), we arrive at the important relation, called hereafter the black hole condition \[BHC\], $$\begin{aligned} \hspace{-1cm} [BHC] \hspace{2cm} M(V_3^{R_S}) = {\lambda^2 \over 4\mu_S {\bar N}_S [site]}{c^4 \over G^2} = {M_P^2 \over 4 \mu_S {\bar N}_S[site]}.\end{aligned}$$ In the last expression, we assumed that $\lambda$, i.e., the small scale parameter in YSTA is equal to Planck length $l_P = [G \hbar / c^3]^{1/2} = \hbar /( c M_P )$, where $M_P$ denotes Planck mass. On the other hand, we simply obtain the area-entropy relation \[AER\] under the Schwarzschild black hole by inserting the above limiting values into \[AER\] (4.10) $$\begin{aligned} S(V_3^{R_S}) = {\cal A}\ (V_3^{R_S})\ {S_S[site] \over 4 \pi},\end{aligned}$$ noting that $G_3 =4 \pi.$ At this point, one finds that it is a very important problem how to relate the above area-entropy relation under a Schwarzschild black hole with \[AER\] (4.10) of $D_0$ brane gas system in general, which is derived irrelevantly of the detail whether the system is a black hole or not. As was mentioned in the beginning of this subsection, however, the problem seems to exceed the applicability limit of the present toy model of $D_0$ brane gas, where the system is treated solely as a [*static*]{} state under [*given*]{} values of parameters, $\mu$, ${\bar N}[site]$ and $S[site]$, while the critical behavior around the formation of Schwarzschild black hole must be hidden in a possible ${\it dynamical}$ change of their values. In order to supplement such a defect of the present static toy model, let us try here a Gedanken-experiment, in which one increases the entropy of the gas system $S(V_3^L)$, keeping its size $L$ at the initial value $L_0$, until the system tends to a Schwarzschild black hole, where Eqs.(4.16) and (4.17) with $R_S = L_0$ hold. Then, one finds that according to \[AER\] (4.10), the entropy of $[site]$, $S[site]$ increases proportionally to $S(V_3^L)$ and reaches the limiting value $S_S [site]$, starting from any initial value $S_0[site]$ prior to formation of the black hole, because ${\cal A}(V_d^L)$ in Eq.(4.10) is invariant during the process. Namely, one finds a very simple fact that $S_0[site] \leq S_S [site].$ However, this simple fact combined with \[AER\] (4.10) leads us to the following form of a new area-entropy relation which holds throughout for the $D_0$ brane gas system up to the formation of Schwarzschild black hole,[^8] $$\begin{aligned} \hspace{-3cm} [AER] \hspace{2cm} S(V_3^L) \leq {{\cal A} (V_3^L) S_S[site] \over 4 \pi},\end{aligned}$$ where the equality holds for Schwarzschild black hole, as seen in Eq. (4.17). Concluding Remarks ================== We have derived the area-entropy relation \[AER\] (4.18) together with (4.10) in our toy model of $D_0$ brane gas subject to Yang’s quantized space-time algebra, YSTA. Indeed, it is essentially based on the fundamental nature of the noncommutative geometry of YSTA, that is, the kinematical reduction of spatial degrees of freedom and holographic relation in YSTA, which was pointed out in I and now extended to the lower dimensional region, $V_d^L$, as shown by \[KHR\] (3.12) in section 3. In addition, it should be noted that \[AER\] (4.18) has been derived with aid of a crude Gedanken-experiment on $D_0$ brane gas system. Before entering into discussions on the implication of the relation, let us consider \[AER\] (4.18) in comparison with the Bekenstein-Hawking area-entropy relation or holographic principle, which has been discussed in various ways during the past decades and typically shown in the simple form as $S \leq {\cal A} /4$ in accord with the expression in \[AER\] (4.18). First of all, one finds out that the latter relation provides an important knowledge on the present \[AER\] (4.18), that is, $$\begin{aligned} S_S[site] /4 \pi = 1/4\end{aligned}$$ or $$\begin{aligned} S_S[site] = \pi .\end{aligned}$$ If one remembers that $S[site]$ can be expressed in general by $$\begin{aligned} S[site] = {\rm ln}\ {\cal N}[site]\end{aligned}$$ with ${\cal N}[site]$ the number of independent quantum states in $[site]$ or, roughly speaking, the maximal limit of occupation number $k$ in Eqs. (4.5)-(4.6), one finds out that Eq. (5.2) implies $$\begin{aligned} {\cal N}_S [site] = e^\pi.\end{aligned}$$ At this point, there arises a fundamental question why our present approach is unable to derive such a condition (5.2) or (5.4). It is certainly clear, because of the applicability of the present toy model of $D_0$ brane gas where the system is treated as a kind of ideal gas of $D_0$ branes neglecting all of their interactions, as was stated in the beginning of section 4. It is well-known that the factor [1/4]{} on the right hand side of (5.1) comes from the consideration of Hawking radiation of black holes. Furthermore, the present approach does not answer the question what happens when $S[site]$ exceeds $S_S [site]$. It is surely concerned with the problem of gravitational collapse or evaporation of black hole. In order to answer these questions, therefore, it is clear that one has to take into consideration at least the interactions of $D_0$ branes with gravitation- or radiation- fields, which are expected to come from interactions of $D_0$ branes with open and closed strings. All of them, however, exceed the scope of the present paper. We have tried here solely to give an outline to arrive at area-entropy relation in our present scheme, leaving the satisfactory treatment of the quantum field theory of $D_0$ brane$^{[12]}$ subject to YSTA to the forthcoming paper, in which we expect further that the theory is ultimately free from UV- and IR-divergences.$^{[1],[2],[13]}$ [99]{} S. Tanaka, ”Kinematical reduction of spatial degrees of freedom and holographic relation in Yang’s quantized space-time algebra," [*Found. Phys.*]{} [**39**]{} 510 (2009), hep-th/0710.5438. 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S. Tanaka, “Space-time quantization and matrix model," [*Nuovo Cim. **114 B***]{} 49 (1999), hep-th/9808064;\ S. Tanaka, “Space-time quantization and nonlocal field theory - Relativistic second quantization of matrix model," hep-th/0002001.\ S. Tanaka, “From Yukawa to M-theory," [Proc. of Int. Symp. on Hadron Spectroscopy, Chiral Symmetry and Relativistic Description of Bound Systems,]{} 3 (2003), hep-th/0306047. [^1]: Em. Professor of Kyoto University, E-mail: st-desc@kyoto.zaq.ne.jp [^2]: It corresponds, in the case of unitary representation of Lorentz group $SO(3,1)$, to taking $K_3\ (\sim \Sigma_{03})$ and $J_3\ (\sim \Sigma_{12})$ to be diagonal, which have continuous and discrete eigenvalues, respectively, instead of ${\bf J}^2$ and $J_3$ in the familiar representation. [^3]: In the familiar unitary irreducible representation of $SO(3,1)$, it is well known that $\sigma$’s are represented by two parameters, $(j_0, \kappa)$, with $j_0$ being $1,2, \cdots \infty$ and $\kappa$ being purely imaginary number, for the so-called principal series of representation. With respect to the associated representation of $SO(3)$, when it is realized on $S^2$, as in the present case, $l$’s denote positive integers, $l= j_0, j_0+1, j_0+2,\cdots,\infty$, and $m$ ranges over $\pm l, \pm(l-1) , \cdots,\pm1, 0.$ [^4]: The argument in this subsection was given in I, ref. \[1\], on the basis of the following form of $D_0$ brane field equation: $[({X_\sigma}^2 +R^2 N^2 )( (\partial/\partial {X_\mu})^2 + R^{-2} (\partial/\partial {N})^2)) - ( X_\mu \partial/\partial {X_\mu} + N \partial/\partial{N})^2 - (D-1)( X_\mu \partial/\partial {X_\mu}+N \partial/\partial{N})\ ]\ D ( X_\nu, N) = 0,$ which was derived in ref. \[9\] from the following $D_0$ brane field action after M-theory,$^{[7]}$ $ \bar{\hat L} = A\ {\rm tr}\ \{ [\hat {\Sigma}_{KL}, \hat {D}^\dagger]\ [\hat {\Sigma}_{KL}, \hat {D}]\}= A'\ {\rm tr}\ \{ 2\ (R^2 /\hbar^2)\ [{\hat P}_\mu, \hat {D}^\dagger ]\ [ \hat {P}_\mu, \hat {D}] - {\lambda}^{-4 }\ [\ [\hat {X}_\mu, \hat {X}_\nu], \hat {D}^\dagger] [ [\hat {X}_\mu,\hat {X}_\nu], \hat {D}]\},$ with $K, L = (\mu, b),$ by means of the Moyal star product method. [^5]: This equation just gives the familiar result $dim\ (\rho_l)= 2l +1,$ in the case $SO(3)$ taking $D=2.$ [^6]: On the other hand, the non-diagonal parts, $< m\ |{\hat D}|\ n >,$ are to be described in terms like ${\bf a}_m {\bf a}_n^\dagger$ or ${\bf a}_m^\dagger {\bf a}_n$ in accord with the idea of M-theory where they are conjectured to be concerned with the interactions between $[site\ m]$ and $[site\ n].$ The details must be left to the rigorous study of the second quantization of $D_0$-brane field.$^{[12]}$ [^7]: The proper vacuum state in Hilbert space II is to be expressed by their direct product. [^8]: Similarly, by the second Gedanken-experiment, in which one increases the total mass of gas system $M(V_3^L)$ with the fixed size $L_0$ in connection with \[AMR\] (4.14), in place of the increase of the entropy of gas system $S(V_3^L)$ in the first Gedanken-experiment, one obtains a new area-mass relation \[AMR\], $M(V_3^L) \leq {\cal A}(V_3^L) \mu_S {\bar N}_S[site] / 4 \pi$.

--- abstract: 'The PVLAS collaboration has recently reported the observation of a rotation of the polarization plane of light propagating through a transverse static magnetic field. Such an effect can arise from the production of a light, $m_A\sim$ meV, pseudoscalar coupled to two photons with coupling strength $g_{A\gamma}\sim 5\times 10^{-6}$ GeV$^{-1}$. Here, we review these experimental findings, discuss how astrophysical and helioscope bounds on this coupling can be evaded, and emphasize some experimental proposals to test the scenario.' address: 'Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D–22607 Hamburg, Germany' author: - Andreas Ringwald title: '[-1cmDESY 05-229]{} Axion interpretation of the PVLAS data?[^1]' --- There are various proposals in the literature in favour of the existence of light pseudoscalar particles beyond the Standard Model which have, so far, remained undetected, due to their weak coupling to ordinary matter. Such light particles would arise if there was a global continuous symmetry in the theory that is spontaneously broken in the vacuum. A well known example is the axion [@Weinberg:1978ma], which arises from a natural solution to the strong $CP$ problem. It appears as a pseudo Nambu-Goldstone boson of a spontaneously broken Peccei-Quinn symmetry [@Peccei:1977hh], whose scale $f_A$ determines its mass, ${m_A} = [z^{1/2}/(1+z)]\, m_\pi f_\pi/ f_A= { 0.6\, {\rm meV}} \times ( 10^{10}\, {\rm GeV}/{ f_A} ) $ in terms of the mass $m_\pi$ and decay constant $f_\pi$ of the pion and the current quark mass ratio $z=m_u/m_d$. Only invisible axion models [@Kim:1979if; @Zhitnitsky:1980tq], where $f_A\gg 247$ GeV, are viable experimentally [@Eidelman:2004wy]. Clearly, it is of great interest to set stringent constraints on the properties of such light pseudoscalars. The interactions of axions and similar light pseudoscalars with Standard Model particles are model dependent, i.e. not a function of $1/f_A$ only. The most stringent constraints to date come from their coupling to photons, $g_{A\gamma}$, which arises via the axial anomaly [@Bardeen:1977bd], $$\label{eq:ax_ph} {\mathcal L}_{\rm int} = -\frac{1}{4}\,{ g_{A\gamma}}\,A\ F_{\mu\nu} \tilde{F}^{\mu\nu} = { g_{A\gamma}}\,A\ {\mathbf E}\cdot {\mathbf B}\, ; \hspace{5ex} { g_{A\gamma}} = -\frac{\alpha}{2\pi { f_A}} \left( { \frac{E}{N}} - \frac{2}{3}\,\frac{4+z}{1+z}\right) \,,$$ where $A$ is the pseudoscalar field, $F_{\mu\nu}$ ($\tilde{F}^{\mu\nu}$) the (dual) electromagnetic field strength tensor, $\alpha$ the fine-structure constant, and $E/N$ the ratio of electromagnetic over color anomalies. As illustrated in Fig. \[fig:ax\_ph\], two quite distinct invisible axion models, namely the KSVZ [@Kim:1979if] (or hadronic) and the DFSZ [@Zhitnitsky:1980tq] (or grand unified) one, lead to quite similar $g_{A\gamma}$. The strongest constraints currently involve cosmological and astrophysical considerations. Only the laser experiments in Fig. \[fig:ax\_ph\] aim also at the production of axions in the laboratory. ![Exclusion region in mass $m_A$ vs. axion-photon coupling $g_{A\gamma}$ for various current and future experiments. The laser experiments [@Cameron:mr; @Zavattini:2005tm; @Ringwald:2001cp; @Ringwald:2003ns] aim at axion production and detection in the laboratory. The galactic dark matter experiments [@Eidelman:2004wy] exploit microwave cavities to detect axions under the assumption that axions are the dominant constituents of our galactic halo, and the solar experiments search for axions from the sun [@Andriamonje:2004hi]. The constraint from horizontal branch (HB) stars [@Eidelman:2004wy; @Raffelt:1999tx] arises from a consideration of stellar energy losses through axion production. \[fig:ax\_ph\]](ax_ph_lim_taup05.eps){width="15.5cm"} Let us discuss such laser experiments in some detail. The most straightforward ones exploit photon regeneration. They are based on the idea [@Sikivie:ip] to send a polarized laser beam, with average power $\langle P\rangle$ and frequency $\omega$, along a superconducting dipole magnet of length $\ell$, such that the laser polarization is parallel to the magnetic field. In the latter, the photons may convert into axions via a Primakoff process. If another identical dipole magnet is set up in line with the first magnet, with a sufficiently thick wall between them to absorb the incident laser photons, then photons may be regenerated from the pure axion beam in the second magnet and detected with an efficiency $\epsilon$. The expected counting rate of such an experiment is given by $$\label{eq:ax_counting_rate} \frac{{\rm d}N_\gamma}{{\rm d}t} = {\frac{\langle P\rangle }{\omega}}\ \frac{N_r+2}{2}\, \frac{1}{16} \left( g_{A\gamma}\,{B}\,\ell\right)^4 \sin^2 \left( \frac{m_A^2\,\ell }{4\,\omega } \right) \left( \frac{m_A^2\,\ell }{4\,\omega } \right)^2 \approx { \frac{\langle P\rangle }{\omega}}\ \frac{N_r+2}{2}\, \frac{1}{16} \left( g_{A\gamma}\,{B}\,\ell\right)^4 \eta \,,$$ if one makes use of the possibility of putting the first magnet into an optical cavity with a total number $N_r$ of reflections. For $m_A \ll \sqrt{2\,\pi\,\omega/\ell} = 4\times 10^{-4}\, {\rm eV} \sqrt{({ \omega}/1\, {\rm eV}) (10\, {\rm m}/\ell ) }$, the approximate sign in (\[eq:ax\_counting\_rate\]) applies and the expected counting rate for a photon regeneration experiment is independent of the axion mass. A pilot photon regeneration experiment was performed by the Brookhaven-Fermilab-Rutherford-Trieste (BFRT) collaboration [@Cameron:mr]. It employed an optical laser of wavelength $\lambda =2\pi/\omega = 514$ nm and power $\langle P\rangle = 3$ W for $t=220$ minutes in an optical cavity with $N_r=200$, and used two superconducting dipole magnets with $B = 3.7$ T and $\ell = 4.4$ m. No signal of photon regeneration was found, which leads, taking into account a detection efficiency of $\eta =0.055$, to a $2\,\sigma$ upper limit of $g_{A\gamma}<6.7\times 10^{-7}$ GeV$^{-1}$ for axion-like pseudoscalars with mass $m_A< 10^{-3}$ eV. Another possibility to probe $g_{a\gamma}$ is to measure changes in the polarization state when photons have traversed a transverse magnetic field [@Maiani:1986md]. In particular, the real production of axions leads to a rotation of the polarization plane of an initially linearly polarized laser beam by an angle $$\begin{aligned} \label{ax_rot} \epsilon &=& N_r\,\frac{g_{A\gamma}^2\,B^2\,\omega^2}{m_A^4}\, \sin^2\left( \frac{m_A^2\,\ell}{4\,\omega}\right)\, \sin 2\,\theta \approx \frac{N_r}{16} \left( g_{A\gamma}\,B\,\ell \right)^2\,\sin 2\,\theta \,,\end{aligned}$$ where $\theta$ is the angle between the light polarization direction and the magnetic field component normal to the light propagation vector. The BFRT collaboration has also performed a pilot polarization experiment along these lines, with the same laser and magnets described before. For $\ell = 8.8$ m, $B=2$ T, and $N_r=254$, an upper limit on the rotation angle $\epsilon< 3.5\times 10^{-10}$ rad was set, leading to a limit $g_{A\gamma}< 3.6\times 10^{-7}$ GeV$^{-1}$ at the 95% confidence level, provided $m_A<1$ meV [@Cameron:mr]. Similar limits have been set from the absence of ellipticity in the transmitted beam. The overall envelope of the constraints from the BFRT collaboration [@Cameron:mr] is shown in Fig. \[fig:ax\_ph\] and labelled by “Laser (BFRT)” (cf. Ref. [@Eidelman:2004wy]). Recently, the PVLAS experiment [@Zavattini:2005tm], consisting of a Fabry-Pérot cavity of very high finesse ($N_r\approx 44\, 000$), immersed in a magnetic dipole with $\ell =1$ m and $B=5$ T, reported the observation of a rotation of the polarization plane of light propagating through a transverse static magnetic field [@Zavattini:2005tm]. If interpreted in terms of the production of a light neutral pseudoscalar, the PVLAS collaboration finds a region $1.7\times 10^{-6}\, {\rm GeV}^{-1}\,{\mbox{\,\raisebox{.3ex} {$<$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}\,}}\,g_{A\gamma}\,{\mbox{\,\raisebox{.3ex} {$<$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}\,}}\,1.0\times 10^{-5}\, {\rm GeV}^{-1}$ for $0.7\, \textrm{meV}\,{\mbox{\,\raisebox{.3ex} {$<$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}\,}}\, m_A\,{\mbox{\,\raisebox{.3ex} {$<$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}\,}}\, 2.0\, \textrm{meV}$, from a combination of the $g_{A\gamma}$ vs. $m_A$ curve corresponding to the PVLAS rotation signal (cf. Eq. (\[ax\_rot\])) with the BFRT limits on the same quantities. Clearly, a pseudoscalar with these properties is hardly compatible with a genuine QCD axion. For the latter, a mass $m_A\sim 1$meV implies a symmetry breaking scale $f_A\sim 6\times 10^{9}$GeV. According to (\[eq:ax\_ph\]), one needs then an extremely large ratio $|E/N|\sim 3\times 10^7$ of electromagnetic and color anomalies in order to arrive at an axion-photon coupling in the range suggested by PVLAS. This is far away from the predictions of any model conceived so far [@Cheng:1995fd]. Moreover, such a pseudoscalar must have very peculiar properties in order to evade the strong constraints on $g_{A\gamma}$ from stellar energy loss considerations (“HB stars" in Fig. \[fig:ax\_ph\]) and from its non-observation in helioscopes such as the CERN Axion Solar Telescope (“Solar (CAST)" in Fig. \[fig:ax\_ph\]) [@Raffelt:2005mt]. Pseudoscalar production in stars may be hindered, for example, if the $A\gamma\gamma$ vertex is suppressed at keV energies due to low scale compositeness of $A$ [@Masso:2005ym] or if, in stellar interiors, $A$ acquires an effective mass larger than the typical photon energy, $\sim$ keV [@jaeckel:tbp]. In any case, an independent and decisive experimental test of the finding of PVLAS is urgently needed. One opportunity is offered by high luminosity $e^+e^-$ colliders, e.g. a possible super-$B$ factory at KEK, where one may search for events with a single photon plus missing transverse energy in the final state [@Masso:1997ru]. The best and most timely possibilities, however, are offered by dedicated photon regeneration experiments, either based on ordinary optical lasers (e.g. [@Duvillaret:2005sv]) or on (soft) X-rays from free-electron lasers (FEL) at DESY and SLAC [@Ringwald:2001cp]. In fact, as can be seen in Fig. \[fig:ax\_ph\], the region of parameter space implied by PVLAS could be probed in a matter of minutes if one sets up a photon regeneration experiment exploiting the already operating FEL at DESY’s TESLA Test Facility, which provides tunable radiation from the vacuum-ultraviolet (VUV) to soft X-rays, $\omega=10$–$200$ eV, with an average power $\langle P\rangle = 20$–$40$ W, together with two superconducting dipole magnets of the type used in DESY’s electron proton collider HERA ($B=5$ T, $\ell =10$ m) [@Ringwald:2001cp]. The tuning of the FEL for fixed photon flux would allow a precision determination of $m_A$. Such an experiment could also serve as a test facility for an ambitious large scale photon regeneration experiment with sensitivity exceeding CAST [@Ringwald:2003ns], based on the recycling of all the 400 dipole magnets of HERA after its decommissioning in mid of 2007. 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--- abstract: | The energies and wave functions of stationary many-body states are analyzed to look for the signatures of quantum chaos. Shell model calculations with the Wildenthal interaction are performed in the $J-T$ scheme for 12 particles in the $sd$-shell. The local level statistics are in perfect agreement with the GOE predictions. The analysis of the amplitudes of the eigenvectors in the shell model basis with the aid of the informational entropy and moments of the distribution function shows evidence for local chaos with a localization length reaching 90% of the total dimension in the middle of the spectrum. The degree of chaoticity is sensitive to the the strength of the residual interaction as compared to the single particle energy spacing. [**[PACS numbers:]{}**]{} 24.60.-k, 24.60.Lz, 21.10.-k, 21.60.Cs --- **CHAOS AND ORDER IN THE SHELL MODEL EIGENVECTORS\ Vladimir Zelevinsky$^{1,2}$, Mihai Horoi$^{1,3}$ and B. Alex Brown$^{1}$\ ** [*$^{1}$National Superconducting Cyclotron Laboratory, East Lansing, MI 48824\ $^{2}$Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia\ $^{3}$Institute of Atomic Physics, Bucharest, Romania\ *]{} Quantum chaos in many-body systems was studied mostly from the viewpoint of level statistics which displays a clear relation to the notion of classical chaos [@Haake]. Presumably much more information could be obtained from an analysis of the wave functions and transition amplitudes. Here one expects to encounter the transition from the simple picture of almost independent elementary excitations to extremely mixed compound states which would display new specific features as, for example, so called dynamic enhancement of weak interactions [@SF]. To perform such an analysis and to check various hypotheses concerning complicated quantum dynamics, one needs a rich set of data which would allow one to make statistically reliable conclusions. Realistic nuclear shell model calculations are one of the most promising candidates for studying this largely unknown structure of quantum chaotic states. We studied the behavior of the basis-state amplitudes of the shell model eigenvectors produced in the $J-T$ scheme for 12 particles in the $sd$ shell. Our model hamiltonian describing a many-body system of valence particles within a major shell contains a one-body part, which is due to an existing core (e.g. $^{16}$O for the $sd$ shell) and a two-body antisymmetrized interaction of the valence particles $$H = \sum \epsilon_{\mu} a^{\dagger}_{\mu} a_{\mu} + \frac{1}{4} \sum V_{\mu \nu \lambda \rho} a^{\dagger}_{\mu} a^{\dagger}_{\nu} a_{\lambda} a_{\rho}\ . \label{eq:ham}$$ In our calculations the Wildenthal interaction along with the well known procedure to project out of the $m$-scheme the states with correct values of the total angular momentum $J$ and isospin $T$ were utilized [@Wild; @OXBA]. The $J-T$ projected states $\mid k\rangle$ are used to build the matrix of the many-body hamiltonian, $H_{k k'} = \langle J T;\ k \mid H \mid J T;\ k' \rangle$, which is eventually diagonalized producing the eigenvalues $E_{\alpha}$ and the eigenvectors $$\mid J T ;\ \alpha \rangle = \sum_{k} C_{k}^{\alpha} \mid J T ;\ k\rangle . \label{eq:eigv}$$ They represent the object of our investigation. The matrix dimension for the $J^{\pi}T = 2^{+}0$ states is 3273. The density of states steeply increases along with excitation energy, reaches its maximum and then decreases again for the highest energy. This high-energy behavior, as well as the approximate symmetry with respect to the middle of the spectrum, are artificial features of models with finite Hilbert space in contrast to actual many-body systems. For the analysis of the level statistics we used levels 200 - 3000. Fig. 1 shows the standard quantities which define the chaoticity of a quantum system [@Haake; @Brody], the unfolded distribution of the nearest neighbor spacings $P(s)$ and the spectral rigidity $\Delta_{3}$, for this class of states. The solid lines in both parts of the figure describe the random matrix results for the Gaussian Orthogonal Ensemble (GOE). The dashed line on the right corresponds to the Poisson level distribution which is characteristic of an ordered system. The closeness of $\Delta_{3}$ to the random matrix results even for very large values of $L$ is remarkable. Previous to this study the largest value of $L$ considered was 80 [@Arve]. Thus, the level statistics manifest generic chaotic behavior. We next look to the structure of the wave functions which could reveal in more detail how close to chaoticity we are. The appropriate quantities to measure the degree of complexity of a given eigenstate $|\alpha\rangle$, eq.(2), with respect to the original shell model basis are, for instance, the informational entropy [@Izr; @Reichl], $$S^{\alpha} = - \sum_{k}\mid C^{\alpha}_{k} \mid^{2} \ln \mid C^{\alpha}_{k} \mid^{2}, \label{eq:end}$$ or the moments of the distribution of amplitudes $|C^{\alpha}_{k}|^{2}$. The second moment determines the number of principal components ($NPC$) of an eigenvector $|\alpha \rangle$, $$(NPC)^{\alpha} = \left(\sum_{k}\mid C^{\alpha}_{k} \mid^{4} \right)^{-1}. \label{eq:npc}$$ In the GOE all basis states are completely mixed so that the resulting eigenvectors are totally delocalized and cover uniformly the $N$-dimensional sphere of radius 1 [@Brody; @Perc; @Berry]. Gaussian fluctuations with zero mean, $\overline{C^{\alpha}_{k}}=0$, and width $\overline{|C^{\alpha}_{k}|^{2}} = 1/N$ lead to the values $\ln (0.48N)$ and $N/3$ for the quantities (3) and (4) respectively. Here $N$ is the total dimension of the model space. In reality, the incomplete mixing of basis states determined by specific properties of the hamiltonian can coexist with the GOE-type level correlations. The left upper part of Fig. 2 presents the $\exp (S^{\alpha})$ quantity for the $2^{+}0$ states. On the $x$-axis are the eigenstates numbered in order of their energies. This simple “numbered” scale is equivalent to the “unfolding” procedure described for example by Brody [*et al*]{} [@Brody]. “Unfolding” is introduced to separate local correlations and fluctuations from the global spectral properties. The solid line represents the GOE result ($0.48 N$). One observes a semicircle-type behavior and a 12% deviation from the GOE even for the maximum entropy in the middle of the spectrum. It is interesting to study the role of single-particle energies (see eq. (1)) for the chaotic behavior of the amplitudes. The upper right part of Fig. 2 shows the $\exp(S^{\alpha})$ quantity for the $2^{+}0$ states for the same hamiltonian but with all single particle energies, $\epsilon_{\mu}$ in eq. (1), set to zero. In this degenerate case the GOE limit (solid line) is attained and the chaotic regime extends over a larger part of the spectrum. To further quantify these effects, we look to the distribution $P(l_{S})$ of $l_{S}^{\alpha} = \exp (S^{\alpha}) / 0.48 N$. One can interprete $l_{S}^{\alpha}$ as a delocalization length $N^{\alpha}/N$. It is expected to be shaped around $l_S = 1$ in the chaotic limit. The lower left panel of Fig. 3 presents the results for the values of $l_{S}$ calculated for the normal hamiltonian and shown on the upper left panel. Here the limit of $l_S = 1$ is not reached. On the other hand, for degenerate single-particle orbitals (upper and lower panels on the right side), the distribution of localization lengths is more narrow and the full chaotic limit is reached. This is related to the fact that the mean field in general tends to smooth out the chaotic aspects of many-body dynamics [@Zel93]. The number of principal components (4) behaves in a very similar way gradually increasing from the edges of the spectrum to the middle, Fig.3 (left). Even the most complicated states are shifted down from the GOE limit of complete mixing. However, for the ratio $\exp S^{\alpha}/(NPC)^{\alpha}$ one obtains the results in the right part of Fig. 3. For a Gaussian distribution of amplitudes $C^{\alpha}_{k}$ of a given eigenvector $|\alpha\rangle$, this ratio would be given by the universal ($N$-independent) random matrix result equal to 1.44 (solid line). The flattened region indicates that the chaotic dynamics, even if not complete, extends far beyond the region nearby the maximum of the informational entropy. Again we use the “unfolded” numbered scale rather than the energy scale. The “unfolding” reveals the presence of “local” chaos: in a given small energy range, the eigenstates are characterized by a typical delocalization length $N^{\alpha} < N$ and by a Gaussian distribution of the amplitudes $C^{\alpha}_{k}$ with zero mean value and variance $(N^{\alpha})^{-1}$. This length cancels in the ratio $\exp (S^{\alpha}) /(NPC)^{\alpha}$ for the majority of states in the middle of the spectrum. The flatness of this ratio, as compared to strong $\alpha$-dependence of $\exp (S^{\alpha})$ and $(NPC)^{\alpha}$ separately, indicates the existence of the local chaotic properties scaling with $N^{\alpha}$. The edge regions with this ratio larger than 1 clearly correspond to relatively weakly mixed states with a reduced $NPC$. We note the very narrow dispersion of points in Figs. 2 and 3 for our measures of complexity. Using the same tools we also analyzed the $J^{\pi}T = 0^{+}0$ states of the model (the dimension of this subspace is 839) and obtained very similar results. In conclusion, we have studied the chaotic properties of a many-body quantum system which consists of 12 valence particles interacting in the $sd$-shell. We have shown that standard signatures of chaos like nearest neighbor spacing distribution or spectral rigidity are not sensitive enough to show deviations from the random matrix results. The informational entropy or moments of the $\mid C^{\alpha}_{k} \mid^{2}$ distribution like $(NPC)$ are much more suited to reveal these details. Arguments are given in favor of local chaos characterized by a Gaussian distribution of the components of the wave functions with the variance related to the localization length. This length grows as the level density increases but does not reach the GOE limit. Finally, we have shown that the effect of the core (given by the single-particle energies) diminishes the maximal degree of chaoticity which can be obtained when the system consists of interacting valence particles only. We point out that our signatures of complexity, $l_{S}$ and ($NPC$), are basis-dependent, they reflect mutual properties of the eigenbasis of the problem and the original “simple” basis $|k\rangle$. In the basis of eigenvectors of a random matrix, unrelated to the mean field of the problem, the results have been checked to coincide with those for the GOE as it should be due to the orthogonal invariance. The basis dependence can give additional physical information and it should be studied separately. The mean field basis is in some sense exceptional since it can be shown that the mean field itself is generated by averaging out the most chaotic components of many-body dynamics [@Zel93]. Therefore it can be considered as a preferential representation for our purpose. It is remarkable that the “natural” choice of the shell model basis sheds a detailed light on the global and local chaotic properties of the wave functions in the many-body system with strong interaction. The authors would like to acknowledge support from the NSF grant 94-03666. [99]{} F. Haake, Quantum signatures of chaos (Berlin, Heidelberg, New York: Springer, 1991). O.P. Sushkov and V.V. Flambaum, Sov. J. Nucl. Phys. [**33**]{}, 31 (1981); Sov. Phys. Usp. [**25**]{}, 1 (1982). B.A. Brown and B.H. Wildenthal, Ann. Rev. Nucl. Part. Sci. [**38**]{}, 29 (1988). B.A. Brown [*et al*]{}., OXBASH-code, MSUNSCL Report [**524**]{}, 1988. T.A. Brody [*et al*]{}., Rev. Mod. Phys. [**53**]{}, 385 (1981). H.-D. Gräf [*et al*]{}., Phys. Rev. Lett. [**69**]{}, 1296 (1992). F.M. Izrailev, Phys. Rep. [**196**]{}, 299 (1990). J. Reichl, Europhys. Lett. [**6**]{}, 669 (1988). J. Percival, J. Phys. [**B6**]{}, L229 (1973). M.V. Berry, in [*Les Houches LII, Chaos and Quantum Physics*]{}, M.-J. Giannoni, A. Voros and J. Zinn-Justin, eds. (North Holland, Amsterdam,1991). V.G. Zelevinsky, Nucl. Phys. [**A555**]{}, 109 (1993). [**Figure captions**]{} [**Figure 1**]{} Unfolded distribution of the nearest neighbor spacings, $P(s)$, left, and the rigidity of the spectrum, $\Delta_{3}$, right, for $2^{+}0$ states.\ [**Figure 2**]{} Left panel: exponential of entropy (upper part), and the distribution of $l_{S} = \exp S/0.48 N$ for $2^{+}0$ states calculated with the full hamiltonian of the model (lower part); right panel: the same quantities for the degenerate model with $\epsilon_{\mu} = 0$.\ [**Figure 3**]{} The Number of Principal Components, eq.(4), of $2^{+}0$ states, left, and the ratio $\exp S/(NPC)$ for the same states, right.

--- abstract: 'While Jeffreys priors usually are well-defined for the parameters of mixtures of distributions, they are not available in closed form. Furthermore, they often are improper priors. Hence, they have never been used to draw inference on the mixture parameters. We study in this paper the implementation and the properties of Jeffreys priors in several mixture settings, show that the associated posterior distributions most often are improper, and then propose a noninformative alternative for the analysis of mixtures.' author: - 'Clara Grazian[^1]' - 'Christian P. Robert[^2]' title: 'Jeffreys priors for mixture estimation: properties and alternatives' --- Introduction {#intro} ============ Bayesian inference in mixtures of distributions has been studied quite extensively in the literature. See, e.g., [@maclachlan:peel:2000] and [@fruhwirth:2006] for book-long references and [@lee:marin:mengersen:robert:2008] for one among many surveys. From a Bayesian perspective, one of the several difficulties with this type of distribution, $$\label{eq:theMix} \sum_{i=1}^k p_i\,f(x|\theta_i)\,,\quad \sum_{i=1}^k p_i=1\,,$$ is that its ill-defined nature (non-identifiability, multimodality, unbounded likelihood, etc.) leads to restrictive prior modelling since most improper priors are not acceptable. This is due in particular to the feature that a sample from may contain no subset from one of the $k$ components $f(\cdot|\theta_i)$ (see. e.g., [@titterington:smith:makov:1985]). Albeit the probability of such an event is decreasing quickly to zero as the sample size grows, it nonetheless prevents the use of independent improper priors, unless such events are prohibited [@diebolt:robert:1994]. Similarly, the exchangeable nature of the components often induces both multimodality in the posterior distribution and convergence difficulties as exemplified by the [*label switching*]{} phenomenon that is now quite well-documented [@celeux:hurn:robert:2000; @stephens:2000b; @jasra:holmes:stephens:2005; @fruhwirth:2006; @geweke:2007; @puolamaki:kaski:2009]. This feature is characterized by a lack of symmetry in the outcome of a Monte Carlo Markov chain (MCMC) algorithm, in that the posterior density is exchangeable in the components of the mixture but the MCMC sample does not exhibit this symmetry. In addition, most MCMC samplers do not concentrate around a single mode of the posterior density, partly exploring several modes, which makes the construction of Bayes estimators of the components much harder. When specifying a prior over the parameters of , it is therefore quite delicate to produce a manageable and sensible non-informative version and some have argued against using non-informative priors in this setting (for example, [@maclachlan:peel:2000] argue that it is impossible to obtain proper posterior distribution from fully noninformative priors), on the basis that mixture models were ill-defined objects that required informative priors to give a meaning to the notion of a component of . For instance, the distance between two components needs to be bounded from below to avoid repeating the same component over and over again. Alternatively, the components all need to be informed by the data, as exemplified in [@diebolt:robert:1994] who imposed a completion scheme (i.e., a joint model on both parameters and latent variables) such that all components were allocated at least two observations, thereby ensuring that the (truncated) posterior was well-defined. [@wasserman:2000] proved ten years later that this truncation led to consistent estimators and moreover that only this type of priors could produce consistency. While the constraint on the allocations is not fully compatible with the i.i.d. representation of a mixture model, it naturally expresses a modelling requirement that all components have a meaning in terms of the data, namely that all components genuinely contributed to generating a part of the data. This translates as a form of weak prior information on how much one trusts the model and how meaningful each component is on its own (by opposition with the possibility of adding meaningless artificial extra-components with almost zero weights or almost identical parameters). While we do not seek Jeffreys priors as the ultimate prior modelling for non-informative settings, being altogether convinced of the lack of unique reference priors [@robert:2001; @robert:chopin:rousseau:2009], we think it is nonetheless worthwile to study the performances of those priors in the setting of mixtures in order to determine if indeed they can provide a form of reference priors and if they are at least well-defined in such settings. We will show that only in very specific situations the Jeffreys prior provides reasonable inference. In Section \[sec:jeffreys\] we provide a formal characterisation of properness of the posterior distribution for the parameters of a mixture model, in particular with Gaussian components, when a Jeffreys prior is used for them. In Section \[sec:prosper\] we will analyze the properness of the Jeffreys prior and of the related posterior distribution: only when the weights of the components (which are defined in a compact space) are the only unknown parameters it turns out that the Jeffreys prior (and so the relative posterior) is proper; on the other hand, when the other parameters are unknown, the Jeffreys prior will be proved to be improper and in only one situation it provides a proper posterior distribution. In Section \[sec:alternative\] we propose a way to realize a noninformative analysis of mixture models and introduce improper priors for at least some parameters. Section \[sec:concl\] concludes the paper. Jeffreys priors for mixture models {#sec:jeffreys} ================================== We recall that the Jeffreys prior was introduced by [@jeffreys:1939] as a default prior based on the Fisher information matrix $$\pi^\text{J}(\theta) \propto |I(\theta)|^{{\nicefrac{1}{2}}}\,,$$ whenever the later is well-defined; $I(\cdot)$ stand for the expected Fisher information matrix and the symbol $|\cdot|$ denotes the determinant. Although the prior is endowed with some frequentist properties like matching and asymptotic minimal information [@robert:2001 Chapter 3], it does not constitute the ultimate answer to the selection of prior distributions in non-informative settings and there exist many alternative such as reference priors [@berger:bernardo:sun:2009], maximum entropy priors [@rissanen:2012], matching priors [@ghosh:carlin:srivastava:1995], and other proposals [@kass:wasserman:1996]. In most settings Jeffreys priors are improper, which may explain for their conspicuous absence in the domain of mixture estimation, since the latter prohibits the use of most improper priors by allowing any subset of components to go “empty" with positive probability. That is, the likelihood of a mixture model can always be decomposed as a sum over all possible partitions of the data into $k$ groups at most, where $k$ is the number of components of the mixture. This means that there are terms in this sum where no observation from the sample brings any amount of information about the parameters of a specific component. Approximations of the Jeffreys prior in the setting of mixtures can be found, e.g., in [@figueiredo:jain:2002], where the Authors revert to independent Jeffreys priors on the components of the mixture. This induces the same negative side-effect as with other independent priors, namely an impossibility to handle improper priors. [@rubio:steel:2014] provide a closed-form expression for the Jeffreys prior for a location-scale mixture with two components. The family of distributions they consider is $$\dfrac{2\epsilon}{\sigma_1}f\left(\frac{x-\mu}{\sigma_1}\right)\mathbb{I}_{x<\mu}+ \dfrac{2(1-\epsilon)}{\sigma_2}f\left(\frac{x-\mu}{\sigma_2}\right) \mathbb{I}_{x>\mu}$$ (which thus hardly qualifies as a mixture, due to the orthogonality in the supports of both components that allows to identify which component each observation is issued from). The factor $2$ in the fraction is due to the assumption of symmetry around zero for the density $f$. For this specific model, if we impose that the weight $\epsilon$ is a function of the variance parameters, $ \epsilon=\nicefrac{\sigma_1}{\sigma_1+\sigma_2}, $ the Jeffreys prior is given by $ \pi(\mu,\sigma_1,\sigma_2) \propto \nicefrac{1}{\sigma_1\sigma_2\{\sigma_1+\sigma_2\}}. $ However, in this setting, [@rubio:steel:2014] demonstrate that the posterior associated with the (regular) Jeffreys prior is improper, hence not relevant for conducting inference. (One may wonder at the pertinence of a Fisher information in this model, given that the likelihood is not differentiable in $\mu$.) [@rubio:steel:2014] also consider alternatives to the genuine Jeffreys prior, either by reducing the range or even the number of parameters, or by building a product of conditional priors. They further consider so-called non-objective priors that are only relevant to the specific case of the above mixture. Another obvious explanation for the absence of Jeffreys priors is computational, namely the closed-form derivation of the Fisher information matrix is almost inevitably impossible. The reason is that integrals of the form $$-\int_{\mathcal{X}} \frac{\partial^2 \log \left[\sum_{h=1}^k p_h\,f(x|\theta_h)\right]}{\partial \theta_i \partial \theta_j}\left[\sum_{h=1}^k p_h\,f(x|\theta_h)\right]^{-1} d x$$ (in the special case of component densities with a single parameter) cannot be computed analytically. We derive an approximation of the elements of the Fisher information matrix based on Riemann sums. The resulting computational expense is of order $\mathrm{O}(d^2)$ if $d$ is the total number of (independent) parameters. Since the elements of the information matrix usually are ratios between the component densities and the mixture density, there may be difficulties with non-probabilistic methods of integration. Here, we use Riemann sums (with $550$ points) when the component standard deviations are sufficiently large, as they produce stable results, and Monte Carlo integration (with sample sizes of $1500$) when they are small. In the latter case, the variability of MCMC results seems to decrease as $\sigma_i$ approaches $0$. Properness for prior and posterior distributions {#sec:prosper} ================================================ Unsurprisingly, most Jeffreys priors associated with mixture models are improper, the exception being when only the weights of the mixture are unknown, as already demonstrated in [@bernardo:giron:1988]. We will characterize properness and improperness of Jeffreys priors and derived posteriors, when some or all of the parameters of distributions from location-scale families are unknown. These results are established both analytically and via simulations, with sufficiently large Monte Carlo experiments checking the behavior of the approximated posterior distribution. Characterization of Jeffreys priors {#subsec:priors} ----------------------------------- ### Weights of mixture unknown A representation of the Jeffreys prior and the derived posterior distribution for the weights of a 3-component mixture model is given in Figure \[weights-priorpost\]: the prior distribution is much more concentrated around extreme values in the support, i.e., it is a prior distribution conservative in the number of important components. ![Approximations (on a grid of values) of the Jeffreys prior (on the log-scale) when only the weights of a Gaussian mixture model with 3-components are unknown (on the top) and of the derived posterior distribution (with known means equal to -1, 0 and 2 respectively and known standard devitations equal to 1, 5 and 0.5 respectively). The red cross represents the true values.[]{data-label="weights-priorpost"}](weights-priorpost){width="6.5cm" height="7.5cm"} \[lem:weights\] When the weights $p_i$ are the only unknown parameters in , the corresponding Jeffreys prior is proper. Figure \[weights-boxplots\] shows the boxplots for the means of the approximated posterior distribution for the weights of a three-component Gaussian mixture model. ![Boxplots of the estimated means of the three-component mixture model $0.25\mathcal{N}(-10,1)+0.65\mathcal{N}(0,5) +0.10\mathcal{N}(15,0.5)$ for 50 simulated samples of size $100$, obtained via MCMC with $10^5$ simulations. The red crosses represent the true values of the weights.[]{data-label="weights-boxplots"}](weights-boxplot.pdf){width="6.5cm" height="7.5cm"} The generic element of the Fisher information matrix is (for $i,j=\{1,\ldots,k-1\}$) $$\int_\mathcal{X} \frac{(f_i(x)-f_k(x))(f_j(x)-f_k(x))}{\sum_{l=1}^k p_l f_l(x)} d x \label{eq:ww-prior}$$ when we consider the parametrization in $(p_1,\ldots,p_{k-1})$, with $$p_k=1-p_1-\cdots-p_{k-1}\,.$$ We remind that, since the Fisher information matrix is a positive semi-definite, its determinant is bounded by the product of the terms in the diagonal, thanks to the Hadamard’s inequality. Therefore, we may consider the diagonal term, $$\begin{aligned} \int_\mathcal{X} \frac{(f_i(x)-f_k(x))^2}{\sum\limits_{l=1}^k p_l f_l(x)} d x &= \int_{f_i(x)\ge f_k(x)} \frac{(f_i(x)-f_k(x))^2}{\sum\limits_{l=1}^k p_l f_l(x)} d x\\ &\quad + \int_{f_i(x)\le f_k(x)} \frac{|(f_i(x)-f_k(x))^2|}{\sum\limits_{l=1}^k p_l f_l(x)} d x\\ &= \int_{f_i(x)\ge f_k(x)} \frac{f_i(x)-f_k(x)}{\sum\limits_{l=1}^k p_l f_l(x)} \{f_i(x)-f_k(x)\}d x\\ &\quad + \int_{f_i(x)\le f_k(x)} \Big| \frac{f_i(x)-f_k(x)}{\sum\limits_{l=1}^k p_l f_l(x)} \Big| |f_i(x)-f_k(x)| d x\\ &= \frac{1}{p_i}\,\int_{f_i\ge f_k} \frac{p_i\{f_i(x)-f_k(x)\}}{p_i\{f_i(x)-f_k(x)\}+\sum\limits_{l\ne i,k} p_l \{f_l(x)-f_k(x)\}+f_k(x)}\\ &\qquad\qquad \{f_i(x)-f_k(x)\}d x\\ &\quad + \frac{1}{p_i}\,\int_{f_i\le f_k} \Big|\frac{p_i\{f_i(x)-f_k(x)\}}{p_i\{f_i(x)-f_k(x)\}+\sum\limits_{l\ne i,k} p_l \{f_l(x)-f_k(x)\}+f_k(x)} \Big| \\ &\qquad\qquad |f_i(x)-f_k(x)| d x\\ &\le \frac{1}{p_i}\int_{f_i(x)\ge f_k(x)} \{f_i(x)-f_k(x)\}d x + \frac{1}{p_i}\int_{f_i(x)\le f_k(x)} | f_i(x)-f_k(x) |d x\\ &= \frac{2}{p_i}\int_{f_i(x)\ge f_k(x)} \{f_i(x)-f_k(x)\}d x\end{aligned}$$ since both integrals are equal. Therefore, the Jeffreys prior will be bounded by the square root of the product of the terms in the diagonal of the Fisher information matrix $$\pi^J(\mathbf{p}) \propto \prod_{i=1}^k p_i^{-\frac{1}{2}}$$ which is a generalization to $k$ components of the prior provided in [@bernardo:giron:1988] for $k=2$ (however, [@bernardo:giron:1988] find the reference prior for the limiting case when all the components have pairwise disjoint supports, while for the opposite limiting case where all the components converge to the same distribution, the Jeffrey’s prior is the uniform distribution on the $k$-dimensional simplex). This reasoning leads [@bernardo:giron:1988] to conclude that the usual $\mathcal{D}(\lambda_1,\ldots,\lambda_k)$ Dirichlet prior with $\lambda_i \in [\nicefrac{1}{2},1]$ for $\forall i=1,\cdots,k$ seems to be a reasonable approximation. They also prove that the Jeffreys prior for the weights $p_i$ is convex, with a argument based on the sign of the second derivative. As a remark, the configuration shown in proof of Lemma \[lem:weights\] is compatible with the Dirichlet configuration of the prior proposed by [@rousseau:mengersen:2011]. The shape of the Jeffreys prior for the weights of a mixture model depends on the type of the components. Figure \[weights-GMM\], \[weights-GtMM\] and \[weights-GtMM-df\] show the form of the Jeffreys prior for a 2-component mixture model for different choices of components. It is always concentrated around the extreme values of the support, however the amount of concentration around $0$ or $1$ depends on the information brought by each component. In particular, Figure \[weights-GMM\] shows that the prior is much more symmetric as there is symmetry between the variances of the distribution components, while Figure \[weights-GtMM\] shows that the prior is much more concentrated around 1 for the weight relative to the normal component if the second component is a Student t distribution. Finally Figure \[weights-GtMM-df\] shows the behavior of the Jeffreys prior when the first component is Gaussian and the second is a Student t and the number of degrees of freedom is increasing. As expected, as the Student t is approaching a normal distribution, the Jeffreys prior becomes more and more symmetric. ![Approximations of the marginal prior distributions for the first weight of a 2-component Gaussian mixture model, $p\,\mathcal{N}(-10,1)+(1-p)\,\mathcal{N}(10,1)$ (black), $p\,\mathcal{N}(-1,1)+(1-p)\,\mathcal{N}(1,1)$ (red) and $p\,\mathcal{N}(-10,1)+(1-p)\,\mathcal{N}(10,10)$ (blue).[]{data-label="weights-GMM"}](weights-prior-comparison-GMM){width="6.5cm" height="7.5cm"} ![Approximations of the marginal prior distributions for the first weight of a 2-component mixture model where the first component is Gaussian and the second is Student t, $p\,\mathcal{N}(-10,1)+(1-p)\,\mathrm{t}(df=1,10,1)$ (black), $p\,\mathcal{N}(-1,1)+(1-p)\,\mathrm{t}(df=1,1,1)$ (red) and $p\,\mathcal{N}(-10,1)+(1-p)\,\mathrm{t}(df=1,10,10)$ (blue).[]{data-label="weights-GtMM"}](weights-prior-comparison-GtMM){width="6.5cm" height="7.5cm"} ![Approximations of the marginal prior distributions for the first weight of a 2-component mixture model where the first component is Gaussian and the second is Student t with an increasing number of degrees of freedom.[]{data-label="weights-GtMM-df"}](weights-prior-comparison-GtMM-df){width="6.5cm" height="7.5cm"} ### Location and scale parameters of a mixture model unknown If the components of the mixture model are distributions from a location-scale family and the location or scale parameters of the mixture components are unknown, this turns the mixture itself into a location-scale model. As a result, model may be reparametrized by following [@mengersen:robert:1996], in the case of Gaussian components $$\label{reparMix} p\mathcal{N}(\mu,\tau^2)+(1-p)\mathcal{N}(\mu+\tau\delta,\tau^2\sigma^2)$$ namely using a reference location $\mu$ and a reference scale $\tau$ (which may be, for instance, the location and scale of a specific component). Equation may be generalized to the case of $k$ components as $$\begin{aligned} \label{eq:k_reparMix} p\mathcal{N}(\mu,\tau^2)&+\sum_{i=1}^{k-2} (1-p) (1-q_1) \cdots (1-q_{i-1})q_i \mathcal{N}(\mu+\tau\theta_1+\cdots+\tau\cdots\sigma_{i-1}\theta_i,\tau^2\sigma_1^2\cdots\sigma_i^2) \nonumber \\ &\qquad {} + (1-p)(1-q_1)\cdots (1-q_{k-2})\mathcal{N}(\mu+\tau\theta_1+\cdots+\tau\cdots\sigma_{k-2}\theta_{k-1},\tau^2\sigma_1^2\cdots\sigma_{k-1}^2)\end{aligned}$$ In this way, the mixture model is more cleary a location-scale model, which implies that the Jeffreys prior is flat in the location and powered as $\tau^{-d/2}$ if $d$ is the total number of parameters of the components, respectively [@robert:2001 Chapter 3], as we will see in the following. \[lem:meansd-prior\] If the parameters of the components of a mixture model are either location or scale parameters, the corresponding Jeffreys prior is improper. In the proof of Lemma \[lem:meansd-prior\], we will consider a Gaussian mixture model and then extend the results to the general situation of components from a location-scale family. ### Unknown location parameters {#unknown-location-parameters .unnumbered} We first consider the case where the means are the only unknown parameters of a Gaussian mixture model $$g_X(x)=\sum_{l=1}^k p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)$$ The generic elements of the expected Fisher information matrix are, in the case of diagonal and off-diagonal terms respectively: $$\mathbb{E}\left[- \frac{\partial^2 \log g_X(X)}{\partial \mu_i^2}\right]=\frac{p_i^2}{\sigma_i^4} \bigintsss_{-\infty}^\infty \frac{\left[ (x-\mu_i) \mathfrak{n}(x|\mu_i,\sigma_i^2)\right]^2}{\sum_{l=1}^k p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x$$ $$\mathbb{E}\left[- \frac{\partial^2 \log g_X(X)}{\partial \mu_i \partial \mu_j}\right]=\frac{p_i p_j}{\sigma_i^2 \sigma_j^2} \bigintsss_{-\infty}^\infty \frac{(x-\mu_i) \mathfrak{n}(x|\mu_i,\sigma_i^2)(x-\mu_j) \mathfrak{n}(x|\mu_j,\sigma_j^2) }{\sum_{l=1}^k p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x$$ Now, consider the change of variable $t=x-\mu_i$ in the above integrals, where $\mu_i$ is thus the mean of the $i$-th Gaussian component ($i\in\{1,\cdots,k\}$). The above integrals are then equal to $$\begin{aligned} \mathbb{E}\left[- \frac{\partial^2 \log g_X(X)}{\partial \mu_j^2}\right] &= \frac{p_j^2}{\sigma_j^4} \bigintsss_{-\infty}^\infty \frac{\left[ (t-\mu_j+\mu_i) \mathfrak{n}(t|\mu_j-\mu_i,\sigma_i^2)\right]^2}{\sum_{l=1}^k p_l \mathfrak{n}(t|\mu_l-\mu_i,\sigma_l^2)} d x\\ \mathbb{E}\left[- \frac{\partial^2 \log g_X(X)}{\partial \mu_j \partial \mu_m}\right] &= \frac{p_j p_m}{\sigma_j^2 \sigma_m^2} \bigintsss_{-\infty}^\infty \frac{(t-\mu_j+\mu_i) \mathfrak{n}(x|\mu_j,\sigma_j^2)(t-\mu_m+\mu_i) \mathfrak{n}(t|\mu_m-\mu_i,\sigma_m^2) }{\sum_{l=1}^k p_l \mathfrak{n}(t|\mu_l-\mu_i,\sigma_l^2)} d x\\ \label{eq:means-prior}\end{aligned}$$ Therefore, the terms in the Fisher information only depend on the differences $\delta_j=\mu_i-\mu_j$ for $j \in \{1,\cdots,k \}$. This implies that the Jeffreys prior is improper since a reparametrization in ($\mu_i,\mathbf{\delta}$) shows the prior does not depend on $\mu_i$. This feature will reappear whenever the location parameters are unknown. When considering the general case of components from a location-scale family, this feature of improperness of the Jeffreys prior distribution is still valid, because, once reference location-scale parameters are chosen, the mixture model may be rewritten as $$\label{eq:mix-locscale} p_1 f_1(x|\mu,\tau)+\sum_{i=2}^k p_i f_i(\frac{a_i+ x}{b_i} |\mu,\tau,a_i,b_i).$$ Then the second derivatives of the logarithm of model behave as the ones we have derived for the Gaussian case, i.e. they will depend on the differences between each location parameter and the reference one, but not on the reference location itself. Then the Jeffreys prior will be constant with respect to the global location parameter. When considering the reparametrization , the Jeffreys prior for $\delta$ for a fix $\mu$ has the form: $$\pi^J(\delta|\mu)\propto \left[ \int_\mathfrak{X}\frac{\left[{(1-p)x\exp\{-\frac{x^2}{2}\}}\right]^2}{{p\sigma\exp\{-\frac{\sigma^2(x+\frac{\delta}{\sigma\tau})^2}{2}\}}+{(1-p)\exp\{-\frac{x^2}{2}\}}} d x \right]^{\frac{1}{2}}$$ and the following result may be demonstrated. The Jeffreys prior of $\delta$ conditional on $\mu$ when only the location parameters are unknown is improper. The improperness of the conditional Jeffreys prior on $\delta$ depends (up to a constant) on the double integral $$\begin{aligned} \int_\Delta \int_\mathfrak{X} c \frac{\left[(1-p)x\exp\{-\frac{x^2}{2}\}\right]^2}{p\sigma\exp\{-\frac{\sigma^2(x+\frac{\delta}{\sigma\tau})^2}{2}\}+(1-p)\exp\{-\frac{x^2}{2}\}} d x d\delta.\end{aligned}$$ The order of the integrals is allowed to be changed, then $$\begin{aligned} \int_\mathfrak{X} x^2 \int_\Delta \frac{\left[(1-p)\exp\{-\frac{x^2}{2}\}\right]^2}{p\sigma\exp\{-\frac{\sigma^2(x+\frac{\delta}{\sigma\tau})^2}{2}\}+(1-p)\exp\{-\frac{x^2}{2}\}} d\delta d x \end{aligned}$$ Define $f(x)=(1-p)e^{-\frac{x^2}{2}}=\frac{1}{d}$. Then $$\begin{aligned} \int_\mathcal{X} x^2 \int_\Delta \frac{1}{d^2 p\sigma\exp\{-\frac{\sigma^2(x+\frac{\delta}{\sigma\tau})^2}{2}\}+d} d\delta d x \end{aligned}$$ Since the behavior of $\left[d^2 p\sigma\exp\{-\frac{\sigma^2(x+\frac{\delta}{\sigma\tau})^2}{2}\}+d\right]$ depends on $\exp\{-\delta^2\}$ as $\delta$ goes to $\infty$, we have that $$\int_{-\infty} ^{+\infty} \frac{1}{\exp\{-\delta^2\}+d} d\delta > \int_{A} ^{+\infty} \frac{1}{\exp\{-\delta^2\}+d} d\delta$$ because the integrand function is positive. Then $$\int_{A} ^{+\infty} \frac{1}{\exp\{-\delta^2\}+d} d\delta > \int_{A} ^{+\infty} \frac{1}{\varepsilon+d} d\delta = +\infty$$ Therefore the conditional Jeffreys prior on $\delta$ is improper. Figure \[fig:priorpost-diff\] compares the behavior of the prior and the resulting posterior distribution for the difference between the means of a two-component Gaussian mixture model: the prior distribution is symmetric and it has different behaviors depending on the value of the other parameters, but it always stabilizes for large enough values; the posterior distribution appears to always concentrate around the true value. ![Approximations (on a grid of values) of the Jeffreys prior (on the natural scale) of the difference between the means of a Gaussian mixture model with only the means unknown (left) and of the derived posterior distribution (on the right, the red line represents the true value), with known weights equal to $(0.5,0.5)$ (black lines), $(0.25,0.75)$ (green and blue lines) and known standard deviations equal to $(5,5)$ (black lines), $(1,1)$ (green lines) and $(7,1)$ (blue lines).[]{data-label="fig:priorpost-diff"}](priorpost-diff.pdf){width="6.5cm" height="7.5cm"} ### Unknown scale parameters {#unknown-scale-parameters .unnumbered} Consider now the second case of the scale parameters being the only unknown parameters. First, consider a Gaussian mixture model and suppose the mixture model is composed by only two components; the Jeffreys prior for the scale parameters is defined as $$\begin{aligned} \pi^J(\sigma_1,\sigma_2)&\propto \left\{ \frac{p_1^2}{\sigma_1^2} \bigintsss_{-\infty}^\infty \frac{\left[ \left(\frac{(x-\mu_1)^2}{\sigma_1^2}-1\right) \mathfrak{n}(x|\mu_1,\sigma_1^2)\right]^2}{\sum_{l=1}^2 p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x \right. \nonumber \\ &\cdot \left.{} \frac{p_2^2}{\sigma_2^2} \bigintsss_{-\infty}^\infty \frac{\left[ \left(\frac{(x-\mu_2)^2}{\sigma_2^2}-1\right) \mathfrak{n}(x|\mu_2,\sigma_2^2)\right]^2}{\sum_{l=1}^2 p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x \right. \nonumber \\ &- \left.{} \left[\frac{p_1 p_2}{\sigma_1 \sigma_2} \bigintsss_{-\infty}^\infty \frac{ \left(\frac{(x-\mu_1)^2}{\sigma_1^2}-1\right) \left(\frac{(x-\mu_2)^2}{\sigma_2^2}-1\right) \mathfrak{n}(x|\mu_1,\sigma_1^2)\mathfrak{n}(x|\mu_2,\sigma_2^2)}{\sum_{l=1}^2 p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x \right]^2\right\}^\frac{1}{2} \end{aligned}$$ Since the Fisher information matrix is positive definite, it is bounded by the product on the diagonal, then we can write: $$\begin{aligned} \pi^J(\sigma_1,\sigma_2)&\leq c \frac{p_1 p_2}{\sigma_1\sigma_2}\left\{ \bigintsss_{-\infty}^\infty \frac{\left(\frac{(x-\mu_1)^2}{\sigma_1^2}-1\right)^2 \frac{1}{\sigma_1^2} \exp\left\{ -\frac{(x-\mu_1)^2}{\sigma_1^2}\right\}}{\frac{p_1}{\sigma_1}\exp\left\{-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right\}+\frac{p_2}{\sigma_2}\exp\left\{-\frac{(x-\mu_2)^2}{2\sigma_2^2}\right\}} d x \right. \nonumber \\ &\cdot \left.{} \bigintsss_{-\infty}^\infty \frac{\left(\frac{(x-\mu_2)^2}{\sigma_2^2}-1\right)^2 \frac{1}{\sigma_2^2} \exp\left\{ -\frac{(x-\mu_2)^2}{\sigma_2^2}\right\}}{\frac{p_1}{\sigma_1}\exp\left\{-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right\}+\frac{p_2}{\sigma_2}\exp\left\{-\frac{(x-\mu_2)^2}{2\sigma_2^2}\right\}} d x \right\}^\frac{1}{2} \end{aligned}$$ In particular, if we reparametrize the model by introducing $\sigma_1=\tau$ and $\sigma_2=\tau \sigma$ and study the behavior of the following integral $$\begin{aligned} \label{eq:scaleprior} \bigintsss_0^{\infty} \bigintsss_0^{\infty} & c \frac{p_1 p_2}{\tau\sigma}\left\{ \bigintsss_{-\infty}^\infty \frac{\left(z^2-1\right)^2 \exp\left\{ -z^2\right\}}{p_1 \exp\left\{-\frac{z^2}{2}\right\}+\frac{p_2}{\sigma}\exp\left\{-\frac{(z\tau+\mu_1-\mu_2)^2}{2\tau^2\sigma^2}\right\}} d z \right. \nonumber \\ &\cdot \left.{} \left\{ \bigintsss_{-\infty}^\infty \frac{\left(u^2-1\right)^2 \exp\left\{-u^2\right\}}{p_1\sigma \exp\left\{-\frac{(u\tau\sigma+\mu_2-\mu_1)^2}{2\tau^2}\right\}+p_2\exp\left\{-\frac{u^2}{2}\right\}}\right\} d u \right\}^\frac{1}{2} d \tau d \sigma\end{aligned}$$ where the internal integrals with respect to $z$ and $u$ converge with respect to $\sigma$ and $\tau$, then the behavior of the external integrals only depends on $\frac{1}{\tau\sigma}$. Therefore they do not converge. This proof can be easily extended to the case of $k$ components: the behavior of the prior depends on the inverse of the product of the scale parameters, which implies that the prior is improper. Moreover this proof may be easily extended to the general case of mixtures of location-scale distributions , because the second derivatives of the logarithm of the model will depend on factors $b_i^{-2}$ for $i \in {1,\cdots,k}$. When the square root is considered, it is evident that the integral will not converge. Figures \[fig:sd-priorpost-clm\] and \[fig:sd-priorpost-asym\] show the prior and the posterior distributions of the scale parameters of a two-component mixture model for some situations with different weights and different means. ![Same as Figure \[fig:sd-priorpost-farm\] but with known weights equal to $(0.25,0.75)$ and known means equal to $(-1,1)$.[]{data-label="fig:sd-priorpost-clm"}](lsd-priorpost-clm2){width="6.5cm" height="7.5cm"} ![Same as Figure \[fig:sd-priorpost-farm\] but with known weights equal to $(0.25,0.75)$ and known means equal to $(-2,7)$.[]{data-label="fig:sd-priorpost-asym"}](lsd-priorpost-farm2){width="6.5cm" height="7.5cm"} Summarized results of the posterior approximation obtained via a random-walk Metropolis-Hastings algorithm by exploring the posterior distribution associated with the Jeffreys prior on the standard deviations are shown in Figures \[fig:sd2-bxp\] and \[fig:sd3-bxp\], which display boxplots of the posterior means: provided a sufficiently high sample size, simulations exhibit a convergent behavior. ![Boxplots of posterior means of the standard deviations of the two-component mixture model $0.50\mathcal{N}(-1,1) + 0.50\mathcal{N}(2,0.5)$ for 50 replications of the experiment and a sample size equal to $10$, obtained via MCMC with $10^5$ simulations. The red cross represents the true values.[]{data-label="fig:sd2-bxp"}](sd2-boxplot){width="6.5cm" height="7.5cm"} ![Boxplots of posterior means of the standard deviations of the three-component mixture model $0.25\mathcal{N}(-1,1) + 0.65\mathcal{N}(0,0.5) + 0.10\mathcal{N}(2,5)$ for 50 replications of the experiment and a sample size equal to $50$, obtained via MCMC with $10^5$ simulations. The red cross represents the true values.[]{data-label="fig:sd3-bxp"}](sd3-boxplot){width="6.5cm" height="7.5cm"} ### Location and scale parameters unknown. Consider now the case where both location and scale parameters are unknown. Once again, each element of the Fisher information matrix is an integral in which a change of variable $x-\mu_i$ can be used, for some choice of $\mu_i,\ ,i=1,\cdots,k$ so that each term only depends on the difference $\delta_j=\mu_i-\mu_j$; the elements are $$\mathbb{E}\left[- \frac{\partial^2 log f_X(X)}{\partial \sigma_i^2}\right]=\frac{p_i^2}{\sigma_i^2} \int_{-\infty}^\infty \frac{\left[ \left(\frac{(x-\mu_i)^2}{\sigma_i^2}-1\right) \mathfrak{n}(x|\mu_i,\sigma_i^2)\right]^2}{\sum_{l=1}^k p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x$$ $$\mathbb{E}\left[- \frac{\partial^2 log f_X(X)}{\partial \sigma_i \partial \sigma_j}\right]=\frac{p_i p_j}{\sigma_i \sigma_j} \int_{-\infty}^\infty \frac{ \left(\frac{(x-\mu_i)^2}{\sigma_i^2}-1\right) \left(\frac{(x-\mu_j)^2}{\sigma_j^2}-1\right) \mathfrak{n}(x|\mu_i,\sigma_i^2)\mathfrak{n}(x|\mu_j,\sigma_j^2)}{\sum_{l=1}^k p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x$$ $$\mathbb{E}\left[- \frac{\partial^2 log f_X(X)}{\partial \mu_i \partial \sigma_i}\right]=\frac{p_i^2}{\sigma_i^3} \int_{-\infty}^\infty \frac{ \left(x-\mu_i\right)\left(\frac{(x-\mu_i)^2}{\sigma_i^2}-1\right) \left[\mathfrak{n}(x|\mu_i,\sigma_i^2)\right]^2}{\sum_{l=1}^k p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x$$ $$\mathbb{E}\left[- \frac{\partial^2 log f_X(X)}{\partial \mu_i \partial \sigma_j}\right]=\frac{p_i p_j}{\sigma_i \sigma_j} \int_{-\infty}^\infty \frac{ \frac{(x-\mu_i)}{\sigma_i^2 \sigma_j} \left(\frac{(x-\mu_j)^2}{\sigma_j^2}-1\right) \mathfrak{n}(x|\mu_i,\sigma_i^2)\mathfrak{n}(x|\mu_j,\sigma_j^2)}{\sum_{l=1}^k p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x$$ ### Location and scale parameters unknown {#location-and-scale-parameters-unknown .unnumbered} When considering all the parameters unknown, the form of the Jeffreys prior may be partly defined by considering the mixture model as a location-scale model, for which a general solution exists; see [@robert:2001]. When all the parameters of a Gaussian mixture model are unknown, the Jeffreys prior is constant in $\mu$ and powered as $\tau^{-d/2}$, where $d$ is the total number of components parameters. We have already proved the Jeffreys prior is constant on the global mean (first proof of Lemma \[lem:meansd-prior\]). Consider a two-component mixture model and the reparametrization . With some computations, it is straightforward to derive the Fisher information matrix for this model, partly shown in Table \[tab:FishInfo\_repar\], where each term is multiplied for a term which does not depend on $\tau$. . \[tab:FishInfo\_repar\] **$\sigma$** **$\delta$** **p** **$\mu$** **$\tau$** -------------- -------------- -------------- ------- ------------- ------------- **$\sigma$** 1 1 $\tau^{-1}$ $\tau^{-1}$ **$\delta$** 1 1 $\tau^{-1}$ $\tau^{-1}$ **p** 1 1 $\tau^{-1}$ $\tau^{-1}$ **$\mu$** $\tau^{-1}$ $\tau^{-1}$ $\tau^{-2}$ $\tau^{-2}$ **$\tau$** $\tau^{-1}$ $\tau^{-1}$ $\tau^{-2}$ $\tau^{-2}$ : Factors depending on $\tau$ of the Fisher information matrix for the reparametrized model Therefore, the Fisher information matrix considered as a function of $\tau$ is a block matrix. From well-known results in linear algebra, if we consider a block matrix $$M= \begin{bmatrix} A & B \\ C & D \end{bmatrix}$$ then its determinant is given by $\det(M)=\det(A-BD^{-1}C)\det(D)$. In the case of a two-component mixture model, $\det(D)\propto\tau^{-4}$, while $\det(A-BD^{-1}C)\propto 1$ (always seen as functions of $\tau$ only). Then the Jeffreys prior for a two-component location-scale mixture model is proportional to $\tau^{-2}$. This result may be easily generalized to the case of $k$ components. Posterior distributions of Jeffreys priors ------------------------------------------ We now derive analytical and computational characterizations of the posterior distributions associated with Jeffreys priors for mixture models. Simulated examples are used to support the analytical results. For this purpose, we have repeated simulations from the models $$0.50\mathcal{N}(\mu_1,1) + 0.50\mathcal{N}(\mu_2,0.5) \label{eq:2mix}$$ and $$0.25\mathcal{N}(\mu_1,1) + 0.65\mathcal{N}(\mu_0,0.5) + 0.10\mathcal{N}(\mu_2,5) \label{eq:3mix}$$ where $\mu_1$ and $\mu_2$ are chosen to be either close ($\mu_1=-1$, $\mu_2=2$) or well separated ($\mu_1=-10$, $\mu_2=15$) and $\mu_0=0$. The Tables shown in the following will analyze the behavior of simulated Markov chains with the goal to approximate the posterior distribution. Even if the output of a MCMC method is not conclusive to assess the properness of the target distribution, it may give a hint on improperness: if the target is improper, an MCMC chain cannot be positive recurrent but instead either null-recurrent or transient [@robert:casella:2004], then it should show convergence problems, as trends or difficulties to move from a particular region. Therefore, simulation studies will be used to support analytical results on properness or improperness of the posterior distribution. In the following, we will say that the results are stable if they show a convergent behavior, i.e. they move around the true values which have generated the data. In particular, an approximation is stable if the proportion of experiments for which the chains show no trend and acceptance rates around the expected values (20%-40%, which means that there are not regions where the chain have difficulties to move from) is 0. The following results are based on Gaussian mixture models, anyway, the Jeffreys prior has a behavior common to all the location-scale families, as shown in Section \[subsec:priors\], as well as the likelihood function; therefore the results may be generalized to any location-scale family. ### Location parameters unknown A first numerical study where the Jeffreys prior and its posterior are computed on a grid of parameter values confirms that, provided the means only are unknown, the prior is constant on the difference between the means and takes higher and higher values as the difference between them increases. However, the posterior distribution is correctly concentrated around the true values for a sufficiently high sample size and it exhibits the classical bimodal nature of such posteriors [@celeux:hurn:robert:2000]. In Figure \[fig:mean-priorpost\], the posterior distribution appears to be perfectly symmetric because the other parameters (weights and standard deviations) have been fixed as identical. ![Approximations (on a grid of values) of the Jeffreys prior (on the log-scale) when only the means of a Gaussian mixture model with two components are unknown (on the top) and of the derived posterior distribution (with known weights both equal to 0.5 and known standard deviations both equal to 5).[]{data-label="fig:mean-priorpost"}](means-prior){width="6.5cm" height="7.5cm"} Tables \[tab:post2means\] and \[tab:post3means\] show that, when considering a two-component Gaussian mixture model, the results are stabilizing for a sample size equal to $10$ if the components are close and they are always stable if the means are far enough; on the other hand, huge sample sizes (around $100$ observations) are needed to have always converging chains for a three-component mixture model (even if, when the components are well-separated a sample size equal to $10$ seems to be enough to have stable results). When $k=2$, the posterior distribution derived from the Jeffreys prior when only the means are unknown is proper. \[lem:mean-post\] The conditional Jeffreys prior for the means of a Gaussian mixture model is $$\begin{aligned} \pi^J(\mu|p,\sigma) &\propto \frac{p_1 p_2}{\sigma_1^2 \sigma_2^2}\left\{ \int_{-\infty}^{+\infty}\frac{\left[ t\mathfrak{n}(0,\sigma_1)\right]^2}{p_1\mathfrak{n}(0,\sigma_1)+p_2\mathfrak{n}(\delta,\sigma_2)} d t \right. \nonumber \\ &{} \left. \times \int_{-\infty}^{+\infty} \frac{\left[ u\mathfrak{n}(0,\sigma_2)\right]^2}{p_1\mathfrak{n}(-\delta,\sigma_1)+p_2\mathfrak{n}(0,\sigma_2)} d u \right. \nonumber \\ &{} \left. -\left(\int_{-\infty}^{+\infty} \frac{ t\mathfrak{n}(0,\sigma_1) (t-\delta)\mathfrak{n}(\delta,\sigma_2)}{p_1\mathfrak{n}(0,\sigma_1)+p_2\mathfrak{n}(\delta,\sigma_2)} d t\right)^2 \right\}^\frac{1}{2}\end{aligned}$$ where $\delta=\mu_2-\mu_1$. The posterior distribution is then defined as $$\prod_{j=1}^n \left[p_1\mathfrak{n}(\mu_1,\sigma_1)+p_2\mathfrak{n}(\mu_2,\sigma_2)\right]\pi^J(\mu_1,\mu_2|p,\sigma)$$ The likelihood may be rewritten (without loss of generality, by considering $\sigma_1=\sigma_2=1$, since they are known) as $$\begin{aligned} L(\theta)&=\prod_{j=1}^n \left[p_1\mathfrak{n}(\mu_1,1)+p_2\mathfrak{n}(\mu_2,1)\right] \nonumber \\ &= \frac{1}{(2\pi)^\frac{n}{2}}\left[p_1^n e^{-\frac{1}{2}\sum_{i=1}^n (x_i-\mu_1)^2}+\sum_{j=1}^n p_1^{n-1}p_2e^{-\frac{1}{2}\sum_{i\neq j} (x_i-\mu_1)^2-\frac{1}{2}(x_j-\mu_2)^2}\right. \nonumber \\ &{} \left. +\sum_{j=1}^n \sum_{k\neq j} p_1^{n-2}p_2^2 e^{-\frac{1}{2}\sum_{i\neq j,k} (x_i-\mu_1)^2-\frac{1}{2}\left[(x_j-\mu_2)^2+(x_k-\mu_2)^2 \right]} \right. \nonumber \\ &{} \left. +\cdots+p_2^n e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\mu_2)^2}\right] \label{eq:mixlik}\end{aligned}$$ Then, for $|\mu_1|\rightarrow\infty$, $L(\theta)$ tends to the term $p_2^n e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\mu_2)^2}$ that is constant for $\mu_1$. Therefore we can study the behavior of the posterior distribution for this part of the likelihood to assess its properness. This explains why we want the following integral to converge: $$\int_{\mathbb{R}\times\mathbb{R}} p_2^n e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\mu_2)^2} \pi^J(\mu_1,\mu_2) d\mu_1 d\mu_2$$ which is equal to (by the change of variable $\mu_2-\mu_1=\delta$) $$\int_{\mathbb{R}\times\mathbb{R}} p_2^n e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\mu_1-\delta)^2} \pi^J(\mu_1,\delta) d\mu_1 d\delta$$ We have seen that the prior distribution only depends on the difference between the means $\delta$: $$\begin{aligned} &\int_\mathbb{R} p_2^n \int_\mathbb{R} e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\mu_1-\delta)^2}d\mu_1 \pi^J(\delta)d\delta \nonumber \\ &\propto \int_\mathbb{R} \int_\mathbb{R} e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\delta)^2 +\mu_1\sum_{j=1}^n(x_j-\delta)-\frac{1}{2}n\mu_1^2} d\mu_1 \pi^J(\delta)d\delta \nonumber \\ &=\int_\mathbb{R} \left[\int_\mathbb{R} e^{\mu_1\sum_{j=1}^n(x_j-\delta)-\frac{1}{2}n\mu_1^2} d\mu_1\right] e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\delta)^2} \pi^J(\delta)d\delta \nonumber \\ &=\int_\mathbb{R} e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\delta)^2+\sum_{j=1}^n\frac{(x_j-\delta)}{2n}} \pi^J(\delta)d\delta \nonumber \\ &\approx \int_\mathbb{R} e^{-\frac{n-1}{2}\delta^2} \pi^J(\delta)d\delta \label{eq:postmean}\end{aligned}$$ The prior on $\delta$ depends on the determinant of the corresponding Fisher information matrix that is positive definite, then it is bounded by the product of the Fisher information matrix diagonal entries: $$\footnotesize{ \pi(\delta)\leq \frac{p_1 p_2}{\sigma_1 \sigma_2}\left\{ \bigintsss_{-\infty}^{+\infty} \frac{\left[t\mathfrak{n}(0,\sigma_1^2)\right]^2}{p_1 \mathfrak{n}(0,\sigma_1^2)+p_2\mathfrak{n}(\delta,\sigma_2^2)} d t \times \bigintsss_{-\infty}^{+\infty} \frac{\left[u\mathfrak{n}(0,\sigma_2^2)\right]^2}{p_1 \mathfrak{n}(-\delta,\sigma_1^2)+p_2\mathfrak{n}(0,\sigma_2^2)} d u \right\}^\frac{1}{2} } \label{eq:deltaprior}$$ where we have used the proof of lemma \[lem:meansd-prior\] and a change of variable $(t-\delta)=u$ in the second integral. As $\delta \rightarrow \pm \infty$ this quantity is constant with respect to $\delta$. Therefore the integral is convergent for $n \geq 2$. Unfortunately this result can not be extended to the general case of $k$ components. When $k>2$, the posterior distribution derived from the Jeffreys prior when only the means are unknown is improper. \[lem:meankcomp-post\] In the case of $k\neq 2$ components, the Jeffreys prior for the location parameters is still constant with respect to a reference mean (for example, $\mu_1$). Therefore it depends on the difference parameters $(\delta_2=\mu_2-\mu_1,\delta_3=\mu_3-\mu_1,\cdots,\delta_k=\mu_k-\mu_1)$. The Jeffreys prior will be bounded by the product on the diagonal, which is an extension of Equation : $$\begin{aligned} \pi^J(\delta_2,\cdots,\delta_k) &\leq c \left\{ \bigintsss_{-\infty}^\infty \frac{[t\mathfrak{n}(0,\sigma_1^2)]^2}{p_1\mathfrak{n}(0,\sigma_1^2)+\cdots+p_k \mathfrak{n}(\delta_k,\sigma_k^2)}d t \right. \nonumber \\ & \left. {} \cdots \bigintsss_{-\infty}^\infty \frac{[u \mathfrak{n}(0,\sigma_k^2)]^2}{p_1\mathfrak{n}(-\delta_k,\sigma_1^2)+\cdots+p_k \mathfrak{n}(0,\sigma_k^2)} d u \right\}^\frac{1}{2}.\end{aligned}$$ If we consider the case as in Lemma \[lem:mean-post\], where only the part of the likelihood depending on e.g. $\mu_2$ may be considered, the convergence of the following integral has to be studied: $$\int_\mathbb{R} \cdots \int_\mathbb{R} e^{-\frac{n-1}{2}\delta_2^2} \pi^J(\delta_2,\cdots,\delta_k) d \delta_2 \cdots d \delta_k$$ In this case, however, the integral with respect to $\delta_2$ may converge, nevertheless the integrals with respect to $\delta_j$ with $j\neq 2$ will diverge, since the prior tends to be constant for each $\delta_j$ as $|\delta_j| \rightarrow \infty$. This results confirms the idea that each part of the likelihood gives information about at most the difference between the location of the respective components and the reference locations, but not on the locations of the other components. [|cccc|ccc|]{} & & & & & &\ & ** --------- Ave. Accept. Rate --------- : $\mu$ unknown, k=2: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior (on the left) and a prior constant on the means (on the right).[]{data-label="tab:post2means"} & ** ---------------- Chains towards high values ---------------- : $\mu$ unknown, k=2: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior (on the left) and a prior constant on the means (on the right).[]{data-label="tab:post2means"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : $\mu$ unknown, k=2: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior (on the left) and a prior constant on the means (on the right).[]{data-label="tab:post2means"} & ** --------- Ave. Accept. Rate --------- : $\mu$ unknown, k=2: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior (on the left) and a prior constant on the means (on the right).[]{data-label="tab:post2means"} & ** ---------------- Chains towards high values ---------------- : $\mu$ unknown, k=2: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior (on the left) and a prior constant on the means (on the right).[]{data-label="tab:post2means"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : $\mu$ unknown, k=2: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior (on the left) and a prior constant on the means (on the right).[]{data-label="tab:post2means"} \ & 0.2505 & 0.88 & 1.8182 & 0.2709 & 0.72 & 1.9968\ & 0.2656 & 0.94 & 1.6804 & 0.2782 & 0.58 & 1.9613\ & 0.2986 & 0.56 & 1.3097 & 0.2812 & 0.18 & 1.9824\ & 0.2879 & 0.48 & 1.2918 & 0.2830 & 0.14 & 1.8358\ & 0.3066 & 0.16 & 1.1251 & 0.3090 & 0.00 & 1.9363\ & 0.3052 & 0.24 & 1.1205 & 0.3103 & 0.02 & 1.7994\ & 0.3181 & 0.02 & 1.0149 & 0.3521 & 0.00 & 1.3923\ & 0.3101 & 0.02 & 1.0244 & 0.3369 & 0.00 & 1.5219\ & 0.3460 & 0.00 & 0.9914 & 0.3627 & 0.00 & 1.2933\ & 0.3418 & 0.00 & 1.0097 & 0.3913 & 0.00 & 1.1970\ & 0.3881 & 0.00 & 0.9948 & 0.4097 & 0.00 & 1.1032\ & 0.4556 & 0.00 & 1.0005 & 0.4515 & 0.00 & 1.0303\ & 0.5090 & 0.00 & 1.0008 & 0.5090 & 0.00 & 1.0007\ & 0.5603 & 0.00 & 1.0006 & 0.5305 & 0.00 & 1.0002\ & 0.4915 & 0.00 & 1.0006 & 0.2327 & 0.00 & 1.0042\ & & & & & &\ & 0.2752 & 0.00 & 1.0838 & 0.2736 & 0.00 & 1.0474\ & 0.2692 & 0.00 & 1.0313 & 0.2546 & 0.00 & 1.0313\ & 0.2969 & 0.00 & 1.1385 & 0.3152 & 0.00 & 1.0167\ & 0.2938 & 0.00 & 1.0138 & 0.2920 & 0.00 & 0.9968\ & 0.3066 & 0.00 & 1.2207 & 0.3470 & 0.00 & 0.9975\ & 0.3350 & 0.00 & 1.1055 & 0.3473 & 0.00 & 0.9920\ & 0.3154 & 0.00 & 1.1374 & 0.3583 & 0.00 & 1.0092\ & 0.3309 & 0.00 & 1.1566 & 0.3512 & 0.00 & 0.9893\ & 0.3338 & 0.00 & 1.1820 & 0.3601 & 0.00 & 1.0112\ & 0.3579 & 0.00 & 1.1796 & 0.3840 & 0.00 & 1.0136\ & 0.3950 & 0.00 & 1.1615 & 0.4190 & 0.00 & 1.0096\ & 0.4879 & 0.00 & 1.1682 & 0.4659 & 0.00 & 1.0059\ & 0.5083 & 0.00 & 1.2123 & 0.4957 & 0.00 & 1.0017\ & 0.5570 & 0.00 & 1.1996 & 0.4777 & 0.00 & 0.9976\ & 0.3463 & 0.00 & 1.2161 & 0.1792 & 0.00 & 1.0010\ [|cccc|]{} & & &\ & ** --------- Ave. Accept. Rate --------- : $\mu$ unknown, k=3: as in Table \[tab:post2means\] for two three-component Gaussian mixture models, with close and far means, only for the Jeffreys prior.[]{data-label="tab:post3means"} & ** ---------------- Chains towards high values ---------------- : $\mu$ unknown, k=3: as in Table \[tab:post2means\] for two three-component Gaussian mixture models, with close and far means, only for the Jeffreys prior.[]{data-label="tab:post3means"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : $\mu$ unknown, k=3: as in Table \[tab:post2means\] for two three-component Gaussian mixture models, with close and far means, only for the Jeffreys prior.[]{data-label="tab:post3means"} \ & 0.2366 & 1.00 & 2.5175\ & 0.2608 & 1.00 & 2.8447\ & 0.2455 & 0.98 & 1.3749\ & 0.2446 & 1.00 & 1.3807\ & 0.2330 & 1.00 & 1.4062\ & 0.2480 & 0.98 & 1.2411\ & 0.2684 & 0.94 & 1.2535\ & 0.2784 & 0.98 & 1.2744\ & 0.2904 & 0.68 & 1.1168\ & 0.3214 & 0.74 & 1.1217\ & 0.3819 & 0.32 & 1.0616\ & 0.3774 & 0.10 & 1.0383\ & 0.4407 & 0.04 & 1.0108\ & 0.4935 & 0.00 & 1.0018\ & 0.5577 & 0.00 & 1.0068\ & 0.5511 & 0.00 & 1.0006\ & & &\ & 0.2641 & 1.00 & 2.1786\ & 0.2804 & 1.00 & 2.1039\ & 0.2813 & 0.82 & 1.1173\ & 0.2840 & 0.84 & 1.0412\ & 0.2887 & 0.84 & 1.1050\ & 0.2865 & 0.82 & 1.0840\ & 0.3248 & 0.66 & 1.0982\ & 0.3277 & 0.76 & 1.1177\ & 0.2998 & 0.00 & 1.2604\ & 0.3038 & 0.00 & 1.3149\ & 0.2869 & 0.00 & 1.3533\ & 0.3762 & 0.00 & 1.2479\ & 0.4283 & 0.00 & 1.3791\ & 0.5251 & 0.00 & 1.2585\ & 0.5762 & 0.00 & 1.4779\ & 0.4751 & 0.00 & 1.2161\ ### Scale parameters unknown The posterior distribution derived from the Jeffreys prior when only the standard deviations are unknown is improper. \[lem:sd-post\] Consider equation generalized to the case of $\sigma_1$ and $\sigma_2$ unknown: then when we integrate the posterior distribution with respect to $\sigma_1$ and $\sigma_2$, the complete integral may be split into several integrals then summed up. In particular, if we consider the first part of the likelihood (which only depends on the first component of the mixture) and use the change of variable used in , we have: $$\begin{aligned} \bigintsss_0^{\infty} \bigintsss_0^{\infty} & c \frac{p_1^n}{\tau^n}\frac{p_1 p_2}{\tau\sigma} \exp \left\{-\frac{1}{2\tau^2}\sum_{i=1}^n(x_i-\mu_1)^2\right\} \nonumber \\ & \times{} \left\{ \bigintsss_{-\infty}^\infty \frac{\left(z^2-1\right)^2 \exp\left\{ -z^2\right\}}{p_1 \exp\left\{-\frac{z^2}{2}\right\}+\frac{p_2}{\sigma}\exp\left\{-\frac{(z\tau+\mu_1-\mu_2)^2}{2\tau^2\sigma^2}\right\}} d z \right. \nonumber \\ &\times \left.{} \bigintsss_{-\infty}^\infty \frac{\left(u^2-1\right)^2 \exp\left\{-u^2\right\}}{p_1\sigma \exp\left\{-\frac{(u\tau\sigma+\mu_2-\mu_1)^2}{2\tau^2}\right\}+p_2\exp\left\{-\frac{u^2}{2}\right\}} d u \right\}^\frac{1}{2} d \tau d \sigma\end{aligned}$$ The integral with respect to $\tau$ in the previous equation converges, nevertheless the likelihood does not provide information for $\sigma$, then the integral with respect to $\sigma$ diverges and the posterior will be improper. This results may be easily extented to the case of $k$ components: there is a part of the likelihood which only depends on the global scale parameter and is not informative for the ay other components; the form of the integral will remain the same, with integrations with respect to $\sigma_1,\sigma_2,\cdots,\sigma_k$ which do not converge. When only the standard deviations are unknown, the Jeffreys prior is concentrated around $0$. Nevertheless, the posterior distribution shown in Figures \[fig:sd-priorpost-farm\] turns out to be concentrated around the true values of the parameters for a sufficient high sample size (in the figures, $n$ is always equal to $100$). ![Approximations (on a grid of values) of the Jeffreys prior (on the log-scale) when only the standard deviations of a Gaussian mixture model with 2 components are unknown (on the top) and of the derived posterior distribution (with known weights both equal to $0.5$ and known means equal to $(-5,5)$). The blue cross represents the maximum likelihood estimates.[]{data-label="fig:sd-priorpost-farm"}](lsd-priorpost2){width="6.5cm" height="7.5cm"} Figures \[fig:sd-priorpost-clm\] and \[fig:sd-priorpost-asym\] show the prior and the posterior distributions of the scale parameters of a two-component mixture model for some situations with different weights and different means. ![Same as Figure \[fig:sd-priorpost-farm\] but with known weights equal to $(0.25,0.75)$ and known means equal to $(-1,1)$.[]{data-label="fig:sd-priorpost-clm"}](lsd-priorpost-clm2){width="6.5cm" height="7.5cm"} ![Same as Figure \[fig:sd-priorpost-farm\] but with known weights equal to $(0.25,0.75)$ and known means equal to $(-2,7)$.[]{data-label="fig:sd-priorpost-asym"}](lsd-priorpost-farm2){width="6.5cm" height="7.5cm"} Summarized results of the posterior approximation obtained via a random-walk Metropolis-Hastings algorithm by exploring the posterior distribution associated with the Jeffreys prior on the standard deviations are shown in Figures \[fig:sd2-bxp\] and \[fig:sd3-bxp\], which display boxplots of the posterior means: provided a sufficiently high sample size, simulations exhibit a convergent behavior. ![Boxplots of posterior means of the standard deviations of the two-component mixture model $0.50\mathcal{N}(-1,1) + 0.50\mathcal{N}(2,0.5)$ for 50 replications of the experiment and a sample size equal to $10$, obtained via MCMC with $10^5$ simulations. The red cross represents the true values.[]{data-label="fig:sd2-bxp"}](sd2-boxplot){width="6.5cm" height="7.5cm"} ![Boxplots of posterior means of the standard deviations of the three-component mixture model $0.25\mathcal{N}(-1,1) + 0.65\mathcal{N}(0,0.5) + 0.10\mathcal{N}(2,5)$ for 50 replications of the experiment and a sample size equal to $50$, obtained via MCMC with $10^5$ simulations. The red cross represents the true values.[]{data-label="fig:sd3-bxp"}](sd3-boxplot){width="6.5cm" height="7.5cm"} Repeated simulations show that, for a Gaussian mixture model with two components, a sample size equal to $10$ is necessary to have convergent results, while for a three-component Gaussian mixture model with a sample size equal to $50$ is still possible to have chains stuck to values of standard deviations close to $0$. Table \[tab:post2sd\] and \[tab:post3sd\] show results for repeated simulations in the cases of two-component and three-component Gaussian mixture models with unknown standard deviations, respectively, where the means that generate the data may be close or far from one another. In Table \[tab:post2sd\] it seems that the chains tend to be convergent for sample sizes smaller than $10$, but in Table \[tab:post3sd\] one may see that even with a high sample size (equal to $50$) it may happens, for $k=3$, that the chains are stuck to very small values of standard deviations and this fact confirms what we have proved with Lemma \[lem:sd-post\]. [|cccc|]{} & & &\ & ** --------- Ave. Accept. Rate --------- : $\sigma$ unknown, $k=2$: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior.[]{data-label="tab:post2sd"} & ** ----------------- Chains stuck at small values of $\sigma$ ----------------- : $\sigma$ unknown, $k=2$: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior.[]{data-label="tab:post2sd"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : $\sigma$ unknown, $k=2$: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior.[]{data-label="tab:post2sd"} \ & 0.2414 & 0.02 & 1.2245\ & 0.1875 & 0.02 & 1.1976\ & 0.2403 & 0.00 & 1.0720\ & 0.2233 & 0.02 & 1.1269\ & 0.2475 & 0.00 & 1.0553\ & 0.2494 & 0.02 & 1.0324\ & 0.2465 & 0.00 & 1.0093\ & 0.2449 & 0.00 & 1.0026\ & 0.2476 & 0.00 & 0.9960\ & 0.2541 & 0.00 & 0.9959\ & 0.2480 & 0.00 & 0.9946\ & 0.2364 & 0.00 & 1.0052\ & 0.2510 & 0.00 & 0.9981\ & 0.3033 & 0.00 & 0.9994\ & 0.4314 & 0.00 & 0.9999\ & 0.4353 & 0.00 & 1.0001\ & & &\ & 0.2262 & 0.14 & 1.09202\ & 0.2384 & 0.10 & 1.0536\ & 0.2542 & 0.02 & 1.0281\ & 0.2502 & 0.04 & 0.9932\ & 0.2550 & 0.00 & 0.9981\ & 0.2554 & 0.00 & 0.9569\ & 0.2473 & 0.00 & 0.9929\ & 0.2481 & 0.00 & 0.9888\ & 0.2402 & 0.00 & 0.9969\ & 0.2431 & 0.00 & 0.9988\ & 0.2416 & 0.00 & 0.9998\ & 0.2453 & 0.04 & 1.0016\ & 0.2550 & 0.00 & 0.9992\ & 0.2359 & 0.00 & 0.9999\ & 0.3000 & 0.00 & 1.0001\ & 0.3345 & 0.00 & 1.0000\ [|cccc|]{} & & &\ & ** --------- Ave. Accept. Rate --------- : $\sigma$ unknown, $k=3$: as in table \[tab:post2sd\] for two three-components Gaussian mixture models, with close and far means.[]{data-label="tab:post3sd"} & ** ----------------- Chains stuck at small values of $\sigma$ ----------------- : $\sigma$ unknown, $k=3$: as in table \[tab:post2sd\] for two three-components Gaussian mixture models, with close and far means.[]{data-label="tab:post3sd"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : $\sigma$ unknown, $k=3$: as in table \[tab:post2sd\] for two three-components Gaussian mixture models, with close and far means.[]{data-label="tab:post3sd"} \ & 0.0441 & 0.88 & 0.1206\ & 0.0659 & 0.72 & 1.0638\ & 0.0621 & 0.70 & 1.1061\ & 0.1013 & 0.54 & 1.0655\ & 0.0781 & 0.52 & 1.0880\ & 0.0729 & 0.60 & 1.1003\ & 0.1506 & 0.26 & 1.0516\ & 0.1689 & 0.18 & 1.0493\ & 0.2322 & 0.10 & 1.0478\ & 0.2366 & 0.00 & 1.0125\ & 0.4407 & 0.02 & 1.0061\ & 0.2666 & 0.00 & 1.0021\ & 0.3871 & 0.00 & 1.0003\ & 0.4353 & 0.00 & 1.0001\ & & &\ & 0.0222 & 0.78 & 1.0045\ & 0.0610 & 0.44 & 1.0427\ & 0.0567 & 0.52 & 1.0317\ & 0.0779 & 0.46 & 1.0147\ & 0.0862 & 0.32 & 1.0244\ & 0.1312 & 0.26 & 1.0027\ & 0.1472 & 0.18 & 1.0350\ & 0.15884 & 0.14 & 1.0170\ & 0.2331 & 0.06 & 1.0092\ & 0.2464 & 0.04 & 1.0062\ & 0.2498 & 0.00 & 1.0017\ & 0.2567 & 0.00 & 1.0008\ & 0.2594 & 0.00 & 0.9999\ & 0.3073 & 0.00 & 1.2161\ ### Location and weight parameters unknown. Figure \[fig:MW-bxp\] shows the boxplots of repeated simulations when both the weights and the means are unknown. It is evident that the posterior chains are concentrated around the true values, neverthless some chains (the 14% of the replications) show a drift to very high values (in absolute value) and this behavior suggests improperness of the posterior distribution. ![Boxplots of posterior means of the weigths and the means of the three-component mixture model $0.25\mathcal{N}(-1,1) + 0.65\mathcal{N}(0,0.5) + 0.10\mathcal{N}(2,5)$ for 50 replications of the experiment, obtained via MCMC with $10^5$ simulations. The red cross represents the true value.[]{data-label="fig:MW-bxp"}](MW-boxplot){width="6.5cm" height="7.5cm"} ### All the parameters unknown {#sub:post} Improperness of the prior does not imply improperness of the posterior, obviously, but requires a careful checking of whether or not the posterior is proper, however the proof of Lemma \[lem:sd-post\] gives an hint about the actual properness of the posterior distribution when all the parameters are unknown. The posterior distribution derived from the Jeffreys prior when all the parameters are unknown is improper. \[lem:all-post\] Consider the elements on the diagonal of the Fisher information matrix; again, since the Fisher information matrix is positive definite, the determinant is bounded by the product of the terms in the diagonal. Consider a reparametrization into $\tau=\sigma_1$ and $\tau\sigma=\sigma_2$. Then it is straightforward to see that the integral of this part of the prior distribution will depend on a term $(\tau)^{-(d+1)}(\sigma)^{-d}$. Again, as in the proof of Lemma \[lem:sd-post\], when composing the prior with the part of the likelihood which only depends on the first component, this part does not provide information about the parameters $\sigma$ and the integral will diverge. In particular, the integral of the first part of the posterior distribution relative to the part of the likelihood dependent on the first component only and on the product of the diagonal terms of the Fisher information matrix for the prior when considering a two-component mixture model is $$\begin{aligned} \int_0^1 & \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \int_0^{\infty} \int_0^{\infty} c \frac{p_1^n}{\tau^n}\frac{p_1^2 p_2^2}{\tau^3\sigma^2} \exp \left\{-\frac{1}{2\tau^2}\sum_{i=1}^n(x_i-\mu_1)^2\right\} \nonumber \\ & \times{} \left\{ \int_{-\infty}^{\infty} \frac{\left[ \sigma\exp\left\{-\frac{(\tau\sigma y + \delta)^2}{2\tau^2} \right\} - \exp\left\{-\frac{y^2}{2} \right\}\right]^2}{p_1 \sigma \exp\left\{-\frac{(\tau \sigma y + \delta)^2}{2\tau^2}\right\}+p_2\exp\left\{-\frac{y^2}{2}\right\}} d y \right. \nonumber \\ & \times \left.{} \int_{-\infty}^{\infty} \frac{z^2 \exp(-z^2)}{p_1 \exp\left\{-\frac{z^2}{2}\right\}+\frac{p_2}{\sigma}\exp\left\{-\frac{(z\tau-\delta)^2}{2\tau^2\sigma^2}\right\}} d z \right. \nonumber \\ & \times \left.{} \int_{-\infty}^\infty \frac{w^2 \exp\left\{ -w^2\right\}}{p_1 \sigma \exp\left\{-\frac{(\tau \sigma w+\delta)^2}{2\tau^2\sigma^2}\right\}+p_2\exp\left\{-\frac{w^2}{2}\right\}} d w \right. \nonumber \\ & \times \left.{} \int_{-\infty}^\infty \frac{\left(z^2-1\right)^2 \exp\left\{ -z^2\right\}}{p_1 \exp\left\{-\frac{z^2}{2}\right\}+\frac{p_2}{\sigma}\exp\left\{-\frac{(z\tau+\mu_1-\mu_2)^2}{2\tau^2\sigma^2}\right\}} d z \right. \nonumber \\ &\times \left.{} \int_{-\infty}^\infty \frac{\left(u^2-1\right)^2 \exp\left\{-u^2\right\}}{p_1\sigma \exp\left\{-\frac{(u\tau\sigma+\mu_2-\mu_1)^2}{2\tau^2}\right\}+p_2\exp\left\{-\frac{u^2}{2}\right\}} d u \right\}^\frac{1}{2} d \tau d \sigma d \mu_1 d \mu_2 d p_1 \nonumber\end{aligned}$$ When considering the integrals relative to the Jeffreys prior, they do not represent an issue for convergence with respect to the scale parameters, because exponential terms going to $0$ as the scale parameters tend to $0$ are present. However, when considering the part out of the previous integrals, a factor $\sigma^-2$ whose behavior is not convergent is present. Then this particular part of the posterior distribution is not integrating. When considering the case of $k$ components, the integral will always inversily depends on $\sigma_1, \sigma_2,\cdots, \sigma_{k-1}$ and then the posterior will always be improper. As a note aside, it is worth noting that the usual separation between parameters proposed by Jeffreys himself in the multidimensional problems does not change the behavior of the posterior, because even if the Fisher information matrix is decomposed as $$I(\theta)=\left( \begin{matrix} I_1(\theta_1) & 0 \\ 0 & I_2(\theta_2) \end{matrix} \right)$$ for any possible combination of the parameters $\theta=(p,\mu_1,\mu_2,\sigma_1,\sigma_2)$ (note that $\theta_1$ and $\theta_2$ are vectors and $I(\theta_1)$ and $I(\theta_2)$ are diagonal or non-diagonal matrices), the product of the elements in the diagonal (considered in the proof) will be the same. A comparison with maximum likelihood estimation obtained via EM has shown that the Bayesian estimates obtained via MCMC and by using a Jeffreys prior seems to better identify the true values which have generated the data for a sufficient high sample size. Table \[tab:EMvsBayes\] shows the comparison between the ML and the Bayesian estimates (for repeated simulations, the initial values for the MCMC algorithm have been randomly chosen to have a sufficiently high likelihood level). The log-likelihood value of the ML estimates is always lower that the log-likelihood value of the Bayesian estimates. The better performance of the Bayesian algorithm is only shown for practical reasons, since we have already proved the posterior distribution is improper. Figure \[priorlik\] shows that the MCMC algorithm accepts moves with an increasing likelihood value, until this value stabilizes around $-210$. The same happens for the prior level. **Parameters** **ML** **Bayes** **True** ---------------- ------------------------ ------------------------ ------------------ -- -- $\mu$ (-7.245,13.308,14.999) (-10.003,0.307,14.955) (-10,0,15) $\sigma$ (0.547,5.028,0.154) (1.243,3.642,0.607) (1.0,5.0,0.5) w (0.350,0.016,0.634) (0.258,0.106,0.636) (0.25,0.10,0.65) : Comparison between ML estimates and Bayesian estimates obtained by using a Jeffreys prior for a 3-components Gaussian mixture model.[]{data-label="tab:EMvsBayes"} ![Values of (Jeffreys) prior (above) and likelihood function (below) for the accepted moves of the MCMC algorithm which estimate the posterior distribution of the parameters of a 3-components Gaussian mixture model and a sample size equal to 1000.[]{data-label="priorlik"}](wmeansd1-priorlik){width="6.5cm" height="7.5cm"} For small sample sizes, the chains tend to get stuck when very small values of standard deviations are accepted. Table \[tab:allunkn2\] and \[tab:allunkn3\] show the results for different sample sizes and different scenarios (in particular, the situations when the means are close or far from each other are considered) for a mixture model with two and three components respectively. The second and the third columns show the reason why the chain goes into trouble: sometimes the chains do not converge and tend towards very high values of means, sometimes the chains get stuck to very small values of standard deviations. [|ccccc|]{} & & & &\ & ** --------- Ave. Accept. Rate --------- : k=2, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior.[]{data-label="tab:allunkn2"} & ** ----------------- Chains stuck at small values of $\sigma$ ----------------- : k=2, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior.[]{data-label="tab:allunkn2"} & ** ---------------- Chains towards high values of $\mu$ ---------------- : k=2, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior.[]{data-label="tab:allunkn2"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : k=2, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior.[]{data-label="tab:allunkn2"} \ & 0.1119 & 0.54 & 0.74 & 3.5280\ & 0.1241 & 0.56 & 0.74 & 3.6402\ & 0.0927 & 0.56 & 0.70 & 3.2180\ & 0.0693 & 0.54 & 0.70 & 3.1380\ & 0.1236 & 0.42 & 0.72 & 3.3281\ & 0.1081 & 0.44 & 0.84 & 2.8173\ & 0.1172 & 0.40 & 0.78 & 2.1455\ & 0.1107 & 0.40 & 0.70 & 1.8998\ & 0.1273 & 0.44 & 0.74 & 1.8269\ & 0.1253 & 0.42 & 0.76 & 1.2876\ & 0.1218 & 0.36 & 0.82 & 1.2949\ & 0.1278 & 0.38 & 0.66 & 1.2587\ & & & &\ & 0.1650 & 0.18 & 0.30 & 3.7712\ & 0.2218 & 0.12 & 0.20 & 3.1400\ & 0.1836 & 0.12 & 0.36 & 3.1461\ & 0.2313 & 0.08 & 0.08 & 3.5102\ & 0.1942 & 0.14 & 0.12 & 3.5585\ & 0.2290 & 0.04 & 0.02 & 3.0718\ & 0.2320 & 0.04 & 0.02 & 2.9825\ & 0.2305 & 0.08 & 0.02 & 2.9122\ & 0.2264 & 0.06 & 0.00 & 2.9571\ & 0.2292 & 0.08 & 0.04 & 1.0612\ & 0.2005 & 0.12 & 0.04 & 1.0804\ & 0.2343 & 0.00 & 0.02 & 1.0146\ [|ccccc|]{} & & & &\ & ** --------- Ave. Accept. Rate --------- : k=3, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: as in table \[tab:allunkn2\] for two three-component Gaussian mixture models with close and far means.[]{data-label="tab:allunkn3"} & ** ----------------- Chains stuck at small values of $\sigma$ ----------------- : k=3, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: as in table \[tab:allunkn2\] for two three-component Gaussian mixture models with close and far means.[]{data-label="tab:allunkn3"} & ** ---------------- Chains towards high values of $\mu$ ---------------- : k=3, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: as in table \[tab:allunkn2\] for two three-component Gaussian mixture models with close and far means.[]{data-label="tab:allunkn3"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : k=3, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: as in table \[tab:allunkn2\] for two three-component Gaussian mixture models with close and far means.[]{data-label="tab:allunkn3"} \ & 0.0302 & 0.76 & 0.44 & 2.9095\ & 0.0368 & 0.76 & 0.48 & 3.2507\ & 0.0290 & 0.80 & 0.30 & 3.1318\ & 0.0578 & 0.62 & 0.54 & 3.0043\ & 0.0488 & 0.74 & 0.52 & 2.5798\ & 0.0426 & 0.70 & 0.44 & 2.3023\ & 0.0572 & 0.66 & 0.38 & 1.7497\ & 0.0464 & 0.66 & 0.48 & 1.4032\ & 0.0706 & 0.52 & 0.44 & 1.9303\ & 0.0556 & 0.66 & 0.36 & 1.3588\ & 0.0610 & 0.74 & 0.44 & 1.3588\ & 0.0654 & 0.48 & 0.46 & 1.2161\ & & & &\ & 0.0644 & 0.60 & 0.10 & 5.9707\ & 0.0631 & 0.64 & 0.18 & 2.0557\ & 0.0726 & 0.54 & 0.08 & 2.9351\ & 0.1745 & 0.22 & 0.12 & 2.9193\ & 0.1809 & 0.32 & 0.04 & 95.793\ & 0.1724 & 0.28 & 0.14 & 2.5938\ & 0.1948 & 0.24 & 0.14 & 3.1566\ & 0.1718 & 0.26 & 0.08 & 2.8595\ & 0.2110 & 0.16 & 0.06 & 1.8595\ & 0.1880 & 0.24 & 0.10 & 1.2165\ & 0.1895 & 0.20 & 0.12 & 1.2133\ & 0.2468 & 0.08 & 0.02 & 1.0146\ Since the improperness of the posterior distribution is due to the scale parameters, we may use a reparametrization of the problem as in Equation and use a proper prior on the parameter $\sigma$, for example, by following [@robert:mengersen:1999] $$p(\sigma)=\frac{1}{2}\mathcal{U}_{[0,1]}(\sigma)+\frac{1}{2}\frac{1}{\mathcal{U}_{[0,1]}(\sigma)}.$$ and the Jeffreys prior for all the other parameters $(\mathbf{p},\mu,\delta,\tau)$ conditionally on $\sigma$. Since the improperness of the posterior distribution is due to the scale parameters, we may use a reparametrization of the problem as in Equation and use a proper prior on the parameter $\sigma$, for example, by following [@robert:mengersen:1999] $$p(\sigma)=\frac{1}{2}\mathcal{U}_{[0,1]}(\sigma)+\frac{1}{2}\frac{1}{\mathcal{U}_{[0,1]}(\sigma)}.$$ and the Jeffreys prior for all the other parameters $(\mathbf{p},\mu,\delta,\tau)$ conditionally on $\sigma$. Actually, using a proper prior on $\sigma$ does not avoid convergence trouble, as demonstrated by Table \[tab:sigmaprop2\], which shows that, even if the chains with respect to the standard deviations are not stuck around $0$ when using a proper prior for $\sigma$ in the reparametrization proposed by [@robert:mengersen:1999], the chains with respect to the locations parameters demonstrate a divergent behavior. [|c|cccc|]{} & & & &\ & ** --------- Ave. Accept. Rate --------- : k=2, ($\mathbf{p}$, $\mu$, $\delta$, $\tau$,$\sigma$) unknown, proper prior on $\sigma$: results of 50 replications of the experiment by using a proper prior on $\sigma$ and the Jeffreys prior for the other parameters conditionally on it for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior. []{data-label="tab:sigmaprop2"} & ** ----------------- Chains stuck at small values of $\sigma$ ----------------- : k=2, ($\mathbf{p}$, $\mu$, $\delta$, $\tau$,$\sigma$) unknown, proper prior on $\sigma$: results of 50 replications of the experiment by using a proper prior on $\sigma$ and the Jeffreys prior for the other parameters conditionally on it for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior. []{data-label="tab:sigmaprop2"} & ** ---------------- Chains towards high values of $\mu$ ---------------- : k=2, ($\mathbf{p}$, $\mu$, $\delta$, $\tau$,$\sigma$) unknown, proper prior on $\sigma$: results of 50 replications of the experiment by using a proper prior on $\sigma$ and the Jeffreys prior for the other parameters conditionally on it for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior. []{data-label="tab:sigmaprop2"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : k=2, ($\mathbf{p}$, $\mu$, $\delta$, $\tau$,$\sigma$) unknown, proper prior on $\sigma$: results of 50 replications of the experiment by using a proper prior on $\sigma$ and the Jeffreys prior for the other parameters conditionally on it for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior. []{data-label="tab:sigmaprop2"} \ 5 & 0.2094 & 0.02 & 0.92 & 1.4440\ 6 & 0.2152 & 0.00 & 0.98 & 1.3486\ 7 & 0.2253 & 0.00 & 0.92 & 1.3290\ 8 & 0.2021 & 0.00 & 0.94 & 1.2258\ 9 & 0.1828 & 0.00 & 0.84 & 1.2666\ 10 & 0.2087 & 0.00 & 0.88 & 1.1770\ 11 & 0.1854 & 0.00 & 0.94 & 1.2088\ 12 & 0.1829 & 0.00 & 0.86 & 1.2153\ 13 & 0.1658 & 0.00 & 0.92 & 1.1682\ 14 & 0.2017 & 0.00 & 0.86 & 1.2043\ 15 & 0.1991 & 0.00 & 0.88 & 1.2002\ 20 & 0.1851 & 0.00 & 0.76 & 1.1688\ & & & &\ 5 & 0.2071 & 0.00 & 0.70 & 1.5741\ 6 & 0.2021 & 0.00 & 0.68 & 1.4384\ 7 & 0.1947 & 0.00 & 0.60 & 1.3597\ 8 & 0.2054 & 0.00 & 0.44 & 1.2869\ 9 & 0.2093 & 0.00 & 0.46 & 1.3064\ 10 & 0.2271 & 0.00 & 0.20 & 1.1618\ 11 & 0.2030 & 0.00 & 0.32 & 1.1996\ 12 & 0.2178 & 0.00 & 0.24 & 1.1494\ 13 & 0.2812 & 0.00 & 0.18 & 1.1215\ 14 & 0.1880 & 0.00 & 0.08 & 1.0717\ 15 & 0.2511 & 0.00 & 0.06 & 1.0594\ 20 & 0.2359 & 0.00 & 0.00 & 1.0166\ A noninformative alternative to Jeffreys prior {#sec:alternative} ============================================== The information brought by the Jeffreys prior does not seem to be enough to conduct inference in the case of mixture models. The computation of the determinant creates a dependence between the elements of the Fisher information matrix in the definition of the prior distribution which makes it difficult to find slight modifications of this prior that would lead to a proper posterior distribution. For example, using a proper prior for part of the scale parameters and the Jeffreys prior conditionally on them does not avoid impropriety, as we have demonstrated in Section \[sub:post\]. The literature covers attempts to define priors which add a small amount of information that is sufficient to conduct the statistical analysis without overwhelming the information contained in the data. Some of these are related to the computational issues in estimating the parameters of mixture models, as in the approach of [@casella:mengersen:robert:titterington:2002], who find a way to use perfect slice sampler by focusing on components in the exponential family and conjugate priors. A characteristic example is given by [@richardson:green:1997], who propose weakly informative priors, which are data-dependent (or empirical Bayes) and are represented by flat normal priors over an interval corresponding to the range of the data. Nevertheless, since mixture models belong to the class of ill-posed problems, the influence of a proper prior over the resulting inference is difficult to assess. Another solution found in [@mengersen:robert:1996] proceeds through the reparametrization and introduces a reference component that allows for improper priors. This approach then envisions the other parameters as departures from the reference and ties them together by considering each parameter $\theta_i$ as a perturbation of the parameter of the previous component $\theta_{i-1}$. This perspective is justified by the fact that the $(i-1)$-th component is not informative enough to absorb all the variability in the data. For instance, a three-component mixture model gets rewritten as $$\begin{aligned} p\mathcal{N}(\mu,\tau^2)&+(1-p)q\mathcal{N}(\mu+\tau\theta,\tau^2\sigma_1^2) \\ &\quad {} + (1-p)(1-q)\mathcal{N}(\mu+\tau\theta+\tau\sigma\epsilon,\tau^2\sigma_1^2\sigma_2^2)\end{aligned}$$ where one can impose the constraint $1 \geq \sigma_1 \geq \sigma_2$ for identifiability reasons. Under this representation, it is possible to use an improper prior on the global location-scale parameter $(\mu,\tau)$, while proper priors must be applied to the remaining parameters. This reparametrization has been used also for exponential components by [@gruet:philippe:robert:1999] and Poisson components by [@robert:titterington:1998]. Moreover, [@roeder:wasserman:1997] propose a Markov prior which follows the same resoning of dependence between the parameters for Gaussian components, where each parameter is again a perturbation of the parameter of the previous component $\theta_{i-1}$. This representation suggests to define a global location-scale parameter in a more implicit way, via a hierarchical model that considers more levels in the analysis and choose noninformative priors at the last level in the hierarchy. More precisely, consider the Gaussian mixture model $$\label{eq:hierarc1} g(x|\boldsymbol{\theta})=\sum_{i=1}^K p_i \mathfrak{n}(x|\mu_i,\sigma_i).$$ The parameters of each component may be considered as related in some way; for example, the observations have a reasonable range, which makes it highly improbable to face very different means in the above Gaussian mixture model. A similar argument may be used for the standard deviations. Therefore, at the second level of the hierarchical model, we may write $$\begin{aligned} \label{eq:hierarc2} \mu_i & \stackrel{iid}{\sim} \mathcal{N}(\mu_0, \zeta_0) \nonumber \\ \sigma_i & \stackrel{iid}{\sim} \frac{1}{2} \mathcal{U}(0,\zeta_0) + \frac{1}{2}\frac{1}{\mathcal{U}(0,\zeta_0)} \nonumber \\ \mathbf{p} & \sim Dir\left(\frac{1}{2},\cdots,\frac{1}{2}\right) \end{aligned}$$ which indicates that the location parameters vary between components, but are likely to be close, and that the scale parameters may be lower or bigger than $\zeta_0$, but not exactly equal to $\zeta_0$. The weights are given a Dirichlet prior (or in the case of just two components, a Beta prior) independently from the components’ parameters. At the third level of the hierarchical model, the prior may be noninformative: $$\begin{aligned} \label{eq:hierarc3} \pi(\mu_0,\zeta_0) \propto \frac{1}{\zeta_0}\end{aligned}$$ As in @mengersen:robert:1996 the parameters in the mixture model are considered tied together; on the other hand, this feature is not obtained via a representation of the mixture model itself, but via a hierarchy in the definition of the model and the parameters. The posterior distribution derived from the hierarchical representation of the Gaussian mixture model associated with , and is proper. Consider the composition of the three levels of the hierarchical model described in equations , and : $$\begin{aligned} \label{eq:hierarch_post} \pi(\boldsymbol{\mu},\boldsymbol{\sigma},\mu_0,\zeta_0;\mathbf{x}) & \propto L(\mu_1,\mu_2,\sigma_1,\sigma_2;\mathbf{x}) p^{-1/2} (1-p)^{-1/2} \nonumber \\ & {} \times \frac{1}{\zeta_0} \frac{1}{2\pi\zeta_0^2} \exp\left\{- \frac{(\mu_1-\mu_0)^2 (\mu_2-\mu_0)^2}{2\zeta_0^2}\right\} \nonumber \\ & {} \times \left[ \frac{1}{2}\frac{1}{\zeta_0} \mathbb{I}_{[\sigma_1\in(0,\zeta_0)]}(\sigma_1) + \frac{1}{2}\frac{\zeta_0}{\sigma_1^2} \mathbb{I}_{[\sigma_1\in(\zeta_0,+\infty)]}(\sigma_1) \right] \nonumber \\ & {} \times \left[ \frac{1}{2}\frac{1}{\zeta_0} \mathbb{I}_{[\sigma_2\in(0,\zeta_0)]}(\sigma_2) + \frac{1}{2}\frac{\zeta_0}{\sigma_2^2} \mathbb{I}_{[\sigma_2\in(\zeta_0,+\infty)]}(\sigma_2) \right]\end{aligned}$$ where $L(\cdot;\mathbf{x})$ is given by Equation . Once again, we can initialize the proof by considering only the first term in the sum composing the likelihood function for the mixture model. Then the product in may be split into four terms corresponding to the different terms in the scale parameters’ prior. For instance, the first term is $$\begin{aligned} \int_0^\infty & \int_{-\infty}^\infty \int_\mathbb{R}\int_\mathbb{R} \int_\mathbb{R^+} \int_\mathbb{R^+} \int_0^1 \frac{1}{\sigma_1^n} p_1^n \exp \left\{- \frac{\sum_{i=1}^n (x_i-\mu_1)^2}{2\sigma_1^2} \right\} \nonumber \\ & {} \times \frac{1}{\zeta_0^3} \exp\left\{-\frac{(\mu_1-\mu_0)^2 (\mu_2-\mu_0)^2}{2\zeta_0^2} \right\} \nonumber \\ & {} \times \frac{1}{4}\frac{1}{\zeta_0} \frac{1}{\zeta_0} \mathbb{I}_{[\sigma_1 \in (0,\zeta_0)]}(\sigma_1) \mathbb{I}_{[\sigma_2 \in (0,\zeta_0)]}(\sigma_2) d p d\sigma_1 d\sigma_2 d\mu_1 d\mu_2 d \mu_0 d\zeta_0 \end{aligned}$$ and the second one $$\begin{aligned} \int_0^\infty & \int_{-\infty}^\infty \int_\mathbb{R}\int_\mathbb{R} \int_\mathbb{R^+} \int_\mathbb{R^+} \int_0^1 \frac{1}{\sigma_1^n} p_1^n \exp \left\{- \frac{\sum_{i=1}^n (x_i-\mu_1)^2}{2\sigma_1^2} \right\} \nonumber \\ & {} \times \frac{1}{\zeta_0^3} \exp\left\{-\frac{(\mu_1-\mu_0)^2 (\mu_2-\mu_0)^2}{2\zeta_0^2} \right\} \nonumber \\ & {} \times \frac{1}{4}\frac{1}{\zeta_0} \frac{\zeta_0}{\sigma_2^2} \mathbb{I}_{[\sigma_1 \in (0,\zeta_0)]}(\sigma_1) \mathbb{I}_{[\sigma_2 \in (\zeta_0,\infty)]}(\sigma_2) d p d\sigma_1 d\sigma_2 d\mu_1 d\mu_2 d \mu_0 d\zeta_0 .\end{aligned}$$ The integrals with respect to $\mu_1$, $\mu_2$ and $\mu_0$ converge, since the data are carrying information about $\mu_0$ through $\mu_1$. The integral with respect to $\sigma_1$ converges as well, because, as $\sigma_1 \rightarrow 0$, the exponential function goes to $0$ faster than $\frac{1}{\sigma_1^n}$ goes to $\infty$ (integrals where $\sigma_1>\zeta_0$ are not considered here because this reasoning may easily extend to those cases). The integrals with respect to $\sigma_2$ converge, because they provide a factor proportional to $\zeta_0$ and $1/\zeta_0$ respectively which simplifies with the normalizing constant of the reference distribution (the uniform in the first case and the Pareto in second one). Finally, the term $1/\zeta_0^4$ resulting from the previous operations has its counterpart in the integrals relative to the location priors. Therefore, the integral with respect to $\zeta_0$ converges. The part of the posterior distribution relative to the weights is not an issue, since the weights belong to the corresponding simplex. Table \[tab:hierMM\] shows the results given by simulation from the posterior distribution of the hierarchical mixture model and confirms that the chains always converge. [|c|ccccccc|]{} & & & & & & &\ & ** --------- Ave. Accept. Rate --------- : Hierarchical Mixture model: results of 50 replications of the experiment for a two and a three-component Gaussian mixture model with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0, the mean and the median log-likelihood of the last accepted values and the mean and the median maximum log-likelihood of the accepted values.[]{data-label="tab:hierMM"} & ** ----------------- Chains stuck at small values of $\sigma$ ----------------- : Hierarchical Mixture model: results of 50 replications of the experiment for a two and a three-component Gaussian mixture model with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0, the mean and the median log-likelihood of the last accepted values and the mean and the median maximum log-likelihood of the accepted values.[]{data-label="tab:hierMM"} & ** ---------------- Chains towards high values of $\mu$ ---------------- : Hierarchical Mixture model: results of 50 replications of the experiment for a two and a three-component Gaussian mixture model with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0, the mean and the median log-likelihood of the last accepted values and the mean and the median maximum log-likelihood of the accepted values.[]{data-label="tab:hierMM"} & ** -------------------- Mean l($\theta^{fin}$)/ l($\theta^{true}$) -------------------- : Hierarchical Mixture model: results of 50 replications of the experiment for a two and a three-component Gaussian mixture model with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0, the mean and the median log-likelihood of the last accepted values and the mean and the median maximum log-likelihood of the accepted values.[]{data-label="tab:hierMM"} & ** -------------------- Median l($\theta^{fin}$)/ l($\theta^{true}$) -------------------- : Hierarchical Mixture model: results of 50 replications of the experiment for a two and a three-component Gaussian mixture model with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0, the mean and the median log-likelihood of the last accepted values and the mean and the median maximum log-likelihood of the accepted values.[]{data-label="tab:hierMM"} & ** -------------------- Mean max(l($\theta$))/ l($\theta^{true}$) -------------------- : Hierarchical Mixture model: results of 50 replications of the experiment for a two and a three-component Gaussian mixture model with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0, the mean and the median log-likelihood of the last accepted values and the mean and the median maximum log-likelihood of the accepted values.[]{data-label="tab:hierMM"} & ** -------------------- Median max(l($\theta$))/ l($\theta^{true}$) -------------------- : Hierarchical Mixture model: results of 50 replications of the experiment for a two and a three-component Gaussian mixture model with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0, the mean and the median log-likelihood of the last accepted values and the mean and the median maximum log-likelihood of the accepted values.[]{data-label="tab:hierMM"} \ 3 & 0.1947 & 0.00 & 0.00 & 1.1034 & 0.9825 & 0.0838 & 0.5778\ 4 & 0.2295 & 0.00 & 0.00 & 1.0318 & 1.0300 & 0.4678 & 0.5685\ 5 & 0.2230 & 0.00 & 0.00 & 0.9572 & 0.9924 & 0.8464 & 0.7456\ 6 & 0.2275 & 0.00 & 0.00 & 0.9870 & 0.9641 & 0.6614 & 0.6696\ 7 & 0.2112 & 0.00 & 0.00 & 1.0658 & 1.0043 & 0.8406 & 0.7848\ 8 & 0.2833 & 0.00 & 0.00 & 1.0077 & 1.0284 & 0.8268 & 0.8495\ 9 & 0.2696 & 0.00 & 0.00 & 1.0741 & 1.0179 & 0.8854 & 0.8613\ 10 & 0.2266 & 0.00 & 0.00 & 1.1446 & 0.9968 & 0.9589 & 0.8508\ 15 & 0.1982 & 0.00 & 0.00 & 1.0201 & 0.9959 & 0.9409 & 0.9280\ 20 & 0.2258 & 0.00 & 0.00 & 1.2023 & 1.0145 & 0.9172 & 0.9400\ 30 & 0.2073 & 0.00 & 0.00 & 0.9888 & 1.0022 & 1.0424 & 0.9656\ 50 & 0.2724 & 0.00 & 0.00 & 1.0493 & 1.0043 & 1.0281 & 0.9859\ 100 & 0.2739 & 0.00 & 0.00 & 1.0932 & 1.0025 & 1.0805 & 0.9932\ 200 & 0.3031 & 0.00 & 0.00 & 1.1610 & 1.0036 & 1.1519 & 0.9964\ 500 & 0.2753 & 0.00 & 0.00 & 1.1729 & 1.0023 & 1.1694 & 0.9989\ 1000 & 0.2317 & 0.00 & 0.00 & 1.1800 & 1.0021 & 1.1772 & 0.9994\ & & & & & & &\ 3 & 0.2840 & 0.00 & 0.00 & 1.1316 & 1.0503 & 0.3432 & 0.2950\ 4 & 0.2217 & 0.00 & 0.00 & 1.0326 & 0.9452 & 0.6699 & 0.6624\ 5 & 0.2144 & 0.00 & 0.00 & 1.0610 & 1.0421 & 0.6858 & 0.6838\ 6 & 0.2258 & 0.00 & 0.00 & 1.0908 & 0.9683 & 0.6472 & 0.6355\ 7 & 0.1843 & 0.00 & 0.00 & 1.0436 & 0.9915 & 0.7878 & 0.8008\ 8 & 0.2760 & 0.00 & 0.00 & 1.0276 & 1.0077 & 0.7996 & 0.7958\ 9 & 0.2028 & 0.00 & 0.00 & 1.0025 & 1.0145 & 0.7830 & 0.8016\ 10 & 0.2116 & 0.00 & 0.00 & 1.0426 & 1.0015 & 0.8752 & 0.8591\ 15 & 0.2023 & 0.00 & 0.00 & 1.0247 & 1.0063 & 0.8810 & 0.8871\ 20 & 0.2211 & 0.00 & 0.00 & 1.0281 & 1.0104 & 0.9290 & 0.9268\ 30 & 0.2242 & 0.00 & 0.00 & 1.1978 & 1.0123 & 1.0841 & 0.9508\ 50 & 0.2513 & 0.00 & 0.00 & 1.0543 & 1.0142 & 1.0148 & 0.9775\ 100 & 0.2768 & 0.00 & 0.00 & 1.0563 & 1.0206 & 1.0324 & 0.9955\ 200 & 0.2910 & 0.00 & 0.00 & 1.0325 & 1.0118 & 1.0200 & 0.9993\ 500 & 0.2329 & 0.00 & 0.00 & 1.0943 & 1.0079 & 1.0882 & 1.0002\ 1000 & 0.2189 & 0.00 & 0.00 & 1.1068 & 1.0105 & 1.1212 & 1.0110\ Figures \[fig:hierc\_densmean2\_3\_8\]–\[fig:hierc\_densmean3\_30\_1000\] show the results how a simulations study to approximate the posterior distribution of the means of a two or three-component mixture model, compared to the true values (red vertical lines) and for different sample sizes, from $n=3$ to $n=1000$. ![Distribution of the posterior means for the hierarchical mixture model with two components, global mean $\mu_0=0$ and global variance $\zeta_0=5$, based on $50$ replications of the experiment with different sample sizes, black and blue lines for the marginal posterior distribution of $\mu_1$ and $\mu_2$ respectively.[]{data-label="fig:hierc_densmean2_3_8"}](densmeans2_3_8) ![Same caption as in Figure \[fig:hierc\_densmean2\_3\_8\].[]{data-label="fig:hierc_densmean2_9_14"}](densmeans2_9_14) ![Same caption as in Figure \[fig:hierc\_densmean2\_3\_8\].[]{data-label="fig:hierc_densmean2_15_20"}](densmeans2_15_20) ![Same caption as in Figure \[fig:hierc\_densmean2\_3\_8\].[]{data-label="fig:hierc_densmean2_30_1000"}](densmeans2_30_1000) ![Distribution of the posterior means for the hierarchical mixture model with three components, global mean $\mu_0=0$ and global variance $\zeta_0=5$, based on $50$ replications of the experiment with different sample sizes (the red lines stands for the true values, black, green and blue lines for the marginal posterior distributions of $\mu_1$, $\mu_2$ and $\mu_3$ respectively).[]{data-label="fig:hierc_densmean3_3_8"}](densmeans3_3_8) ![Same caption as in Figure \[fig:hierc\_densmean3\_3\_8\].[]{data-label="fig:hierc_densmean3_9_14"}](densmeans3_9_14) ![Same caption as in Figure \[fig:hierc\_densmean3\_3\_8\].[]{data-label="fig:hierc_densmean3_15_20"}](densmeans3_15_20) ![Same caption as in Figure \[fig:hierc\_densmean3\_3\_8\].[]{data-label="fig:hierc_densmean3_30_1000"}](densmeans3_30_1000) Implementation features {#sec:implant} ======================= The computing expense due to derive the Jeffreys prior for a set of parameter values is in $\mathrm{O}(d^2)$ if $d$ is the total number of (independent) parameters. Each element of the Fisher information matrix is an integral of the form $$-\int_{\mathcal{X}} \frac{\partial^2 \log \left[\sum_{h=1}^k p_h\,f(x|\theta_h)\right]}{\partial \theta_i \partial \theta_j}\left[\sum_{h=1}^k p_h\,f(x|\theta_h)\right]^{-1} d x$$ which has to be approximated. We have applied both numerical integration and Monte Carlo integration and simulations show that, in general, numerical integration obtained via Gauss-Kronrod quadrature (see [@piessens:1983] for details), has more stable results. Neverthless, when one or more proposed values for the standard deviations or the weights is too small, the approximations tend to be very dependent on the bounds used for numerical integration (usually chosen to omit a negligible part of the density) or the numerical approximation may not be even applicable. In this case, Monte Carlo integration seems to have more stable, where the stability of the results depends on the Monte Carlo sample size. Figure \[fig:MCvsNUM\_incrN\] shows the value of the Jeffreys prior obtained via Monte Carlo integration of the elements of the Fisher information matrix for an increasing number of Monte Carlo simulations both in the case where the Jeffreys prior is concentrated (where the standard deviations are small) and where it assumes low values. The value obtained via Monte Carlo integration is then compared with the value obtained via numerical integration. The sample size relative to the point where the graph stabilizes may be chosen to perform the approximation. A similar analysis is shown in Figures \[fig:MCvsNUM\_bpl1\] and \[fig:MCvsNUM\_bpl2\] which provide the boxplots of $100$ replications of the Monte Carlo approximations for different numbers of simulations (on the *x*-axis); one can choose to use the number of simulations which lead to a reasonable or acceptable variability of the results. ![Jeffreys prior obtained via Monte Carlo integration (and numerical integration, in *red*) for the model $0.25\mathcal{N}(-10,1)+0.10\mathcal{N}(0,5)+0.65\mathcal{N}(15,7)$ (above) and for the model $\frac{1}{3}\mathcal{N}(-1,0.2)+\frac{1}{3}\mathcal{N}(0,0.2)+\frac{1}{3}\mathcal{N}(1,0.2)$ (below).[]{data-label="fig:MCvsNUM_incrN"}](MCvsNUM-increaN) ![Boxplots of 100 replications of the procedure which approximates the Fisher information matrix via Monte Carlo integration to obtain the Jeffreys prior for the model $0.25\mathcal{N}(-10,1)+0.10\mathcal{N}(0,5)+0.65\mathcal{N}(15,7)$ for sample sizes from $500$ to $3000$. The value obtained via numerical integration is represented by the red line.[]{data-label="fig:MCvsNUM_bpl1"}](MCvsNUM-boxpl1) ![Same caption as in Figure \[fig:MCvsNUM\_bpl1\] for the model $\frac{1}{3}\mathcal{N}(-1,0.2)+\frac{1}{3}\mathcal{N}(0,0.2)+\frac{1}{3}\mathcal{N}(1,0.2)$.[]{data-label="fig:MCvsNUM_bpl2"}](MCvsNUM-boxpl2) Since the approximation problem is one-dimensional, another numerical solution could be based on the sums of Riemann; Figure \[fig:MCvsNUMSRvsINTR\] shows the comparison between the results of the Gauss-Kronrod quadrature procedure and a procedure based on sums of Riemann for an increasing number of points considered in a region which contain the $99.999\%$ of the data density. Moreover, Figure \[fig:MCvsRIEMbxp\] shows the comparison between the approximation to the Jeffreys prior obtained via Monte Carlo integration and via the sums of Riemann: it is clear that the sums of Riemann lead to more stable results in comparison with Monte Carlo integration. On the other hand, they can be applied in more situations than the Gauss-Kromrod quadrature, in particular, in cases where the standard deviations are very small (of order $10^{-2}$). Nevertheless, when the standard deviations are smaller than this, one has to pay attention on the features of the function to integrate. In fact, the mixture density tends to concentrate around the modes, with regions of density close to 0 between them. The elements of the Fisher informtation matrix are, in general, ratios between the components’ densities and the mixture density, then in those regions an indeterminate form of type $\frac{0}{0}$ is obtained; Figure \[fig:FishInfoelem\] represents the behavior of one of these elements when $\sigma_i \rightarrow 0$ for $i=1,\cdots,k$. ![Comparison between the Jeffreys prior density obtained via integration in the Fisher information matrix via Gauss-Kronrod quadrature and sums of Riemann for the model $0.25\mathcal{N}(-10,1)+0.10\mathcal{N}(0,5)+0.65\mathcal{N}(15,7)$ (above) and $\frac{1}{3}\mathcal{N}(-1,0.2)+\frac{1}{3}\mathcal{N}(0,0.2)+\frac{1}{3}\mathcal{N}(1,0.2)$ (below).[]{data-label="fig:MCvsNUMSRvsINTR"}](SRvsINTR-increaN) ![Boxplots of 100 replications of the procedure based on Monte Carlo integration (above) and sums of Riemann (below) which approximates the Fisher information matrix of the model $0.25\mathcal{N}(-10,1)+0.10\mathcal{N}(0,5)+0.65\mathcal{N}(15,7)$ for sample sizes from $500$ to $1700$. The value obtained via numerical integration is represented by the red line (in the graph below, all the approximations obtained with more than $550$ knots give the same result, exactly equal to the one obtained via Gauss-Kronrod quadrature).[]{data-label="fig:MCvsRIEMbxp"}](MCvsRIEM-boxpl1) ![The first element on the diagonal of the Fisher information matrix relative to the first weight of the two-component Gaussian mixture model $0.5 \mathcal{N}(-1,0.01)+0.5 \mathcal{N}(2,0.01)$.[]{data-label="fig:FishInfoelem"}](elem) Thus, we have decided to use the sums of Riemann (with a number of points equal to $550$) to approximate the Jeffreys prior when the standard deviations are sufficiently big and Monte Carlo integration (with sample sizes of $1500$) when they are too small. In this case, the variability of the results seems to decrease as $\sigma_i$ approaches $0$, as shown in Figure \[fig:MCsmallsd\]. ![Approximation of the Jeffreys prior (in log-scale) for the two-component Gaussian mixture model $0.5 \mathcal{N}(-1,\sigma)+0.5\mathcal{N}(2,\sigma)$, where $\sigma$ is taken equal for both components and decreasing.[]{data-label="fig:MCsmallsd"}](MCsmallsigma) We have chosen to consider Monte Carlo samples of size equal to $1500$ because both the value of the approximation and its standard deviations are stabilizing. An adaptive MCMC algorithm has been used to define the variability of the kernel density functions used to propose the moves. During the burnin, the variability of the kernel distributions has been reduced or increased depending on the acceptance rate, in a way such that the acceptance rate stay between $20\%$ and $40\%$. The transitional kernel used have been truncated normals for the weights, normals for the means and log-normals for the standard deviations (all centered on the values accepted in the previous iteration). Conclusion {#sec:concl} ========== This thorough analysis of the Jeffreys priors in the setting of Gaussian mixtures shows that mixture distributions can also be considered as an ill-posed problem with regards to the production of non-informative priors. Indeed, we have shown that most configurations for Bayesian inference in this framework do not allow for the standard Jeffreys prior to be taken as a reference. While this is not the first occurrence where Jeffreys priors cannot be used as reference priors, the wide range of applications of mixture distributions weights upon this discovery and calls for a new paradigm in the construction of non-informative Bayesian procedures for mixture inference. 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[^1]: Corresponging Author: Memotef, Sapienza Università di Roma. Via del Castro Laurenziano, 9, 00161, Roma, Italy. CEREMADE Université Paris-Dauphine, Paris, France. e-mail: clara.grazian@uniroma1.it [^2]: CEREMADE Université Paris-Dauphine, University of Warwick and CREST, Paris. e-mail: xian@ceremade.dauphine.fr.

--- bibliography: - 'short.bib' title: Mathematical Formulae in Wikimedia Projects 2020 ---

[Jancar’s formal system for deciding bisimulation of first-order grammars\ and its non-soundness.]{}\ by Géraud Sénizergues\ LaBRI and Université de Bordeaux I [^1] #### Abstract : We construct an example of proof within the main formal system from [@Jan10], which is intended to capture the bisimulation equivalence for non-deterministic first-order grammars, and show that its conclusion is semantically false. We then locate and analyze the flawed argument in the soundness (meta)-proof of [@Jan10].\ The grammar =========== We consider the alphabet of actions ${\cal A}$, an intermediate alphabet of labels ${\cal T}$ and a map ${{\rm LAB}_{\cal A}}: {\cal T} \rightarrow {\cal A}$ defined by: $${\cal T} := \{ x,y,z,\ell_1\},\;\;{\cal A} := \{a,b,\ell_1\},\;\;\mbox{ and }$$ $${{\rm LAB}_{\cal A}}: x \mapsto a,\;\;y \mapsto a,\;\; z \mapsto b,\;\; \ell_1 \mapsto \ell_1.$$ (these intermediate objects ${\cal T}$, ${{\rm LAB}_{\cal A}}$ will ease the definition of ${{\rm ACT}}$ below). We define a first-order grammar ${\cal G} = ({\cal N},{\cal A},{\cal R})$ by: $${\cal N} := \{A, A', A'', B, B', B'', C, D, E, L_1\}$$ and the set of rules ${\cal R}$ consists of the following: $$\begin{aligned} A(v) &{\stackrel{y}{\longrightarrow_{}}} & C(v)\\ A(v) &{\stackrel{x}{\longrightarrow_{}}} & A'(v)\\B(v) &{\stackrel{x}{\longrightarrow_{}}} & C(v)\\B(v) &{\stackrel{y}{\longrightarrow_{}}} & B'(v)\\C(v) &{\stackrel{x}{\longrightarrow_{}}} & D(v)\\C(v) &{\stackrel{y}{\longrightarrow_{}}} & E(v)\\A'(v) &{\stackrel{x}{\longrightarrow_{}}} & A''(v)\\B'(v) &{\stackrel{x}{\longrightarrow_{}}} & B''(v)\\A''(v) &{\stackrel{x}{\longrightarrow_{}}} & D(v)\\B''(v) &{\stackrel{x}{\longrightarrow_{}}} & E(v)\\D(v) &{\stackrel{x}{\longrightarrow_{}}} & v \label{ruleD}\\E(v) &{\stackrel{x}{\longrightarrow_{}}} & v \label{ruleE1}\\E(v) &{\stackrel{z}{\longrightarrow_{}}} & v \label{ruleE2}\\L_1 &{\stackrel{\ell_1}{\longrightarrow_{}}} & \bot \label{ruleL1}$$ Let us name rule $r_i$ (for $1 \leq i \leq 14$), the rule appearing in order $i$ in the above list. We define a map ${{\rm LAB}_{\cal T}}: {\cal R} \rightarrow {\cal T}$ by: ${{\rm LAB}_{\cal T}}(r_i)$ is the terminal letter used by the given rule $r_i$. Subsequently we define ${{\rm ACT}}(r_i):= {{\rm LAB}_{\cal A}}({{\rm LAB}_{\cal T}}(r_i))$. Namely, ${{\rm ACT}}$ maps all the rules $r_1, \ldots , r_{12}$ onto $a$, $r_{13}$ on $b$ and $r_{14}$ on $\ell_1$. The formal system ================= We consider the formal systems ${\cal J}(T_0,T'_0,S_0,{\cal B})$ defined in page 22 of [@Jan10], which are intended to be sound and complete for the bisimulation-problem for non-deterministic first-order grammars. Let us denote by $\TERMS$ the set of all terms over the ranked alphabet ${\cal N} \cup \{L_i\mid i \in \N\} \cup\{\bot\}$ (here the symbols $L_i$ have arity $0$). Prefixes of strategies ---------------------- The notion of [*finite prefix of a D-strategy*]{} is mentionned p. 23, line 11. We assume it has the following meaning Let $T,T' \in \TERMS$. A finite prefix of a D-strategy w.r.t. $(T,T')$ is a subset $S \subseteq ({\cal R}\times{\cal R})^*$ of the form $$S = S'\cap ({\cal R}\times{\cal R})^{\leq n}$$ for some $n \in \N$ and some D-strategy $S'$ w.r.t. $(T,T')$. \[def-PDstrategy\] In order to make clear that the above notion is effective, we consider the following notion of D-q-strategy (Defender’s quasi-strategy). Let $T,T' \in \TERMS$. A [*D-q-strategy*]{} w.r.t. $(T,T')$ is a subset $S \subseteq ({\cal R}\times{\cal R})^*$ such that:\ DQ1: $(\varepsilon,\varepsilon) \in S$\ DQ2: $S$ is prefix-closed\ DQ3: $S\subseteq {{\rm PLAYS}}(T,T')$\ DQ4: $\forall \alpha \in S$,\ either $\alpha \backslash S=\{(\varepsilon,\varepsilon)\}$\ or ${{\rm NEXT}}((T,T'),\alpha) \notin \sim_1$\ or \[${{\rm NEXT}}((T,T'),\alpha) \in \sim_1$ and the set $\{ (\pi,\pi') \in {\cal R}\times{\cal R} \mid \alpha\cdot (\pi,\pi')\in S\}$ is full for ${{\rm NEXT}}((T,T'),\alpha)$\]. \[def-Dqstrategy\] Note that a D[*-strategy*]{} is a D-q-strategy where, condition DQ4 is replaced by:\ DQ’4: $\forall \alpha \in S$,\ ${{\rm NEXT}}((T,T'),\alpha) \notin \sim_1$\ or \[${{\rm NEXT}}((T,T'),\alpha) \in \sim_1$ and the set $\{ (\pi,\pi') \in {\cal R}\times{\cal R} \mid \alpha\cdot (\pi,\pi')\in S\}$ is full for ${{\rm NEXT}}((T,T'),\alpha)$\].\ A [*winning*]{} D-strategy, is a D-q-strategy where condition DQ4 is replaced by:\ DQ”4: $\forall \alpha \in S$,\ ${{\rm NEXT}}((T,T'),\alpha) \in \sim_1$ and the set $\{ (\pi,\pi') \in {\cal R}\times{\cal R} \mid \alpha\cdot (\pi,\pi')\in S\}$ is full for ${{\rm NEXT}}((T,T'),\alpha)$.\ Every finite prefix of a strategy is a D-q-strategy. \[L-PD\_implies\_DQ\] Let $S'$ be a D-strategy w.r.t. $(T,T')$ and $$S= S'\cap ({\cal R}\times{\cal R})^{\leq n}$$ for some $n \in \N$, $S'$ D-strategy w.r.t. $(T,T')$.\ DQ1: Since $S'$ is non-empty and prefix-closed $(\varepsilon,\varepsilon) \in S'$, hence $(\varepsilon,\varepsilon) \in S'\cap S({\cal R}\times{\cal R})^{\leq n}$.\ DQ2: $S'$ and $({\cal R}\times{\cal R})^{\leq n}$ are both prefix-closed, hence their intersection is also prefix-closed.\ DQ3: $S'\subseteq {{\rm PLAYS}}(T,T')$ and $S \subseteq S'$, hence $S\subseteq {{\rm PLAYS}}(T,T')$\ DQ4: $\forall \alpha \in S$,\ ${{\rm NEXT}}((T,T'),\alpha) \notin \sim_1$\ or \[${{\rm NEXT}}((T,T'),\alpha) \in \sim_1$ and the set $\{ (\pi,\pi') \in {\cal R}\times{\cal R} \mid \alpha\cdot (\pi,\pi')\in S'\}$ is full for ${{\rm NEXT}}((T,T'),\alpha)$\]. If $|\alpha| < n$, the above property holds in $S$.\ If $|\alpha| = n$, the property $\alpha \backslash S=\{(\varepsilon,\varepsilon)\}$ holds. In all cases DQ4 is fulfilled.\ We define the [*extension*]{} ordering over ${\cal P}(({\cal R}\times{\cal R})^*)$ as follows: for every $S_1,S_2 \in {\cal P}(({\cal R}\times{\cal R})^*)$, $S_1 \sqsubseteq S_2$ iff the two conditions below hold:\ E1- $S_1 \subseteq S_2$\ E2- $\forall \alpha \in S_2-S_1, \exists \beta \in S_1, \mbox{ which is maximal in } S_1 \mbox{ for the prefix ordering and such that },\\ \beta \preceq \alpha.$ \[def-extension\] Let $T,T'\in \TERMS$. The extension ordering over the set of all D-q-strategies w.r.t. $(T,T')$, is inductive. \[L\_inclusion\_is\_inductive\] We recall that a partial order $\leq$ over a set $E$ is [*inductive*]{} iff, every totally ordered subset of $E$ has some upper-bound.\ One can check that, if $P$ is a set of D-q-strategies w.r.t. $(T,T')$, which is totally ordered by $\sqsubseteq$, then the set $$S := \bigcup_{s \in P} s$$ is still a D-q-strategy and fulfills: $$\forall s \in P, s \sqsubseteq S.$$ Hence the extension ordering over the set of D-q-strategies w.r.t. $(T,T')$ is inductive. Let $S \subseteq ({\cal R}\times{\cal R})^*$ be finite and let $n:= \max\{ |\alpha| \mid \alpha \in S\}$.\ $S$ is a finite prefix of a D-strategy w.r.t. $(T,T')$ iff\ (1) $S$ is a D-q-strategy w.r.t. $(T,T')$\ (2) $\forall \beta \in S, [ \beta \backslash S = \{(\varepsilon,\varepsilon)\} \Rightarrow (|\beta| = n \mbox{ or } {{\rm NEXT}}((T,T'),\beta) \notin \sim_1] $). \[L-characterisation\_PDstrategies\] [**Direct implication**]{}:\ Let $S'$ be a D-strategy w.r.t. $(T,T')$ and $$S= S'\cap ({\cal R}\times{\cal R})^{\leq n}$$ for some $n \in \N$ and some $S'$ which is a D-strategy w.r.t. $(T,T')$.\ 1- By Lemma \[L-PD\_implies\_DQ\] $S$ is a D-q-strategy w.r.t. $(T,T')$.\ 2- Suppose that $\beta \in S, \beta \backslash S = \{(\varepsilon,\varepsilon)\}$ and $|\beta| < n$. Then $\beta \backslash S' = \{(\varepsilon,\varepsilon)\}$ too. Since $S'$ is a D-strategy w.r.t. $(T,T')$, this implies that ${{\rm NEXT}}((T,T').\beta) \notin \sim_1$.\ [**Converse**]{}:\ Suppose that $S$ fulfills conditions (1)(2). By Lemma \[L\_inclusion\_is\_inductive\], Zorn’s lemma applies on the set of D-q-strategies w.r.t. $(T,T')$: there exists a maximal D-q-strategy $S'$ (for the extension ordering) such that $S \sqsubseteq S'$. Since $S'$ is maximal, if $\alpha \in S'$ and $\alpha \backslash S=\{(\varepsilon,\varepsilon)\}$, ${{\rm NEXT}}((T,T'),\alpha) \notin \sim_1$. Thus, instead of the weak property DQ4, $S'$ fulfills the property: $$\forall \alpha \in S', {{\rm NEXT}}((T,T'),\alpha) \notin \sim_1\;\; \mbox{ or }$$ $$[{{\rm NEXT}}((T,T'),\alpha) \in \sim_1 \mbox{ and }\{ (\pi,\pi') \in {\cal R}\times{\cal R} \mid \alpha\cdot (\pi,\pi')\in S\} \mbox{ is full for } {{\rm NEXT}}((T,T'),\alpha)].$$ Hence $S'$ is a strategy w.r.t. $(T,T')$.\ Clearly $$S \subseteq S' \cap ({\cal R}\times{\cal R})^{\leq n}.$$ Let us prove the reverse inclusion.\ Let $\alpha \in S' \cap ({\cal R}\times{\cal R})^{\leq n}$. Let $\beta$ be the longuest word in ${{\rm PREF}}(\alpha) \cap S$.\ If $\beta = \alpha$, then $\alpha \in S$, as required.\ Otherwise $\alpha \in S'-S$. By condition E2 of definition \[def-extension\], there exists some $\beta \in S$, which is maximal in $S$ for the prefix ordering and such that $$\beta \prec \alpha.$$ Maximality of $\beta$ implies, by condition (2) of the lemma, that, $$|\beta| = n \mbox{ or } {{\rm NEXT}}((T,T').\beta) \notin \sim_1.$$ Since $\beta \prec \alpha$ we are sure that $|\beta| < n$ so that $${{\rm NEXT}}((T,T').\beta) \notin \sim_1.$$ This last statement contradicts the fact that $\beta \backslash S'$ is a D-strategy, w.r.t ${{\rm NEXT}}((T,T').\beta)$ which is non-reduced to $\{(\varepsilon,\varepsilon)\}$ (since it posesses $\beta^{-1}\alpha$).\ We can conclude that $\alpha \in S$. Finally: $$S = S' \cap ({\cal R}\times{\cal R})^{\leq n}.$$ Let $T,T'\in \TERMS$ and let $S \subseteq ({\cal R}\times{\cal R})^*$ be finite. One can check whether $S$ is a finite prefix of a D-strategy w.r.t. $(T,T')$ \[L-decidability\_PDstrategies\] This follows immediately from the characterisation given by Lemma \[L-characterisation\_PDstrategies\]. Formal systems {#subsec_formal_systems} -------------- For every $T_0,T'_0 \in \TERMS$, $S_0$ finite prefix of strategy w.r.t $(T_0,T_0)$ and finite ${\cal B} \subseteq \TERMS \times \TERMS,$ is defined a formal system $${\cal J}(T_0,T'_0,S_0,{\cal B})$$ The set of judgments of all the systems are the same. But the axiom and one rule (namely R7), is depending on the parameters $(T_0,T'_0,S_0,{\cal B})$. Judgments --------- A [*judgment*]{} has one of the three forms:\ [**FORM 1**]{}:\ $$m {\:|\!\!\!=\!\!\!\!=\:}(T,T',S)$$ where $m \in \N$, and $T,T'\in \TERMS$ are regular terms and $S$ is a finite prefix of a strategy. w.r.t. $(T,T')$ (D-strategies are defined p.20, lines 27-30; finite prefixes are mentionned, though in a fuzzy way. at p. 23, line 11; we shall apply here Definition \[def-PDstrategy\]).\ [**FORM 2**]{}:\ $$m {\:|\!\!\!=\!\!\!\!=\:}(T,T',S) \leadsto \alpha {\:|\!\!\!=\!\!\!\!=\:}(T_1,T'_1,S_1)$$ where $m \in \N$, $(T,T',S), (T_1,T'_1,S_1)$ fulfilling the above conditions, $\alpha \in S$ and $\alpha \backslash S = S_1$.\ [**FORM 3**]{}:\ $$m {\:|\!\!\!=\!\!\!\!=\:}(T,T',S) \leadsto \alpha {\:|\!\!\!=\!\!\!\!=\:}SUCC$$ where $m \in \N$, $(T,T',S)$ fulfill the above conditions and $\alpha \in S$.\ For all systems ${\cal J}(T_0,T'_0,S_0,{\cal B})$ the set of judgments is the same and consists of all the items of one of the three above forms. Basis ----- We call [*basis*]{} every finite set $${\cal B} \subseteq \TERMS \times \TERMS.$$ Axioms ------ ${\cal J}(T_0,T'_0,S_0,{\cal B})$ has a single axiom: $$0 {\:|\!\!\!=\!\!\!\!=\:}(T_0,T'_0,S_0)$$ Deduction rules --------------- All the systems ${\cal J}(T_0,T'_0,S_0,{\cal B})$ have the set of rules described page 22 of [@Jan10]. We name them $R1,R2, \ldots, R10$, the number corresponding to the one in the text. Note that R7 depends on the basis ${\cal B}$. Proofs ------ Let $T_0,T'_0 \in \TERMS$. A [*proof*]{} of $T_0 \sim T'_0$ within the family of formal systems defined above is a finite basis ${\cal B}$, together with, for each $(T,T') \in {\cal B} \cup \{(T_0,T'_0)\}$ a finite prefix of D-strategy $S$ w.r.t. $(T,T')$ and a proof, within system ${\cal J}(T,T',S,{\cal B})$ of the judgment $$0 {\:|\!\!\!=\!\!\!\!=\:}(T,T',S) \leadsto (\varepsilon,\varepsilon) {\:|\!\!\!=\!\!\!\!=\:}{\rm SUCC}.$$ The Equivalence proof {#sec-equivalence_proof} ===================== We exhibit here a proof of $$A(\bot) \sim B(\bot).$$ According to the above notion of proof, it consists of the following items.\ Basis: $${\cal B} := \{(C(L_1),C(L_1)), (D(L_1),D(L_1)), (E(L_1),E(L_1))\}.$$ Proofs:\ - a proof of the judgment $0 {\:|\!\!\!=\!\!\!\!=\:}A(\bot),B(\bot),S\leadsto (\varepsilon,\varepsilon) {\:|\!\!\!=\!\!\!\!=\:}{\rm SUCC}$ in the formal system ${\cal J}(A(\bot),B(\bot),S,{\cal B})$ (see $\pi_3$). - a proof of the judgment $0 {\:|\!\!\!=\!\!\!\!=\:}C(L_1),C(L_1),{\rm Id}_{C,1}\leadsto (\varepsilon,\varepsilon) {\:|\!\!\!=\!\!\!\!=\:}{\rm SUCC}$ in the formal system ${\cal J}(C(L_1),C(L_1),{\rm Id}_{C,1},{\cal B})$ (see $\pi_4$). - a proof of the judgment $0 {\:|\!\!\!=\!\!\!\!=\:}D(L_1),D(L_1),{\rm Id}_{D,2}\leadsto (\varepsilon,\varepsilon) {\:|\!\!\!=\!\!\!\!=\:}{\rm SUCC}$ in the formal system ${\cal J}(D(L_1),D(L_1),{\rm Id}_{D,2},{\cal B})$ (see $\pi_5$). - a proof of the judgment $0 {\:|\!\!\!=\!\!\!\!=\:}E(L_1),E(L_1),{\rm Id}_{E,2}\leadsto (\varepsilon,\varepsilon) {\:|\!\!\!=\!\!\!\!=\:}{\rm SUCC}$ in the formal system ${\cal J}(E(L_1),E(L_1),{\rm Id}_{D,2},{\cal B})$ (see $\pi_6$). {width="14cm"} where $H(A,B)$ stands for $0 {\:|\!\!\!=\!\!\!\!=\:}A(\bot),B(\bot), S$.\ Proof $\pi_2$: {width="10cm"} {width="10cm"} {width="16cm"} where $H(C,C)$ stands for $0 {\:|\!\!\!=\!\!\!\!=\:}C(L1),C(L1),{\rm Id}_{C,1}$.\ {width="10cm"} {width="14cm"} In the above proofs the following defender strategies (or prefix of strategies) were used (in fact, they can be deduced from the proofs):\ Let $${\cal S} := \{(yx,xy),(yy,xx),(xxx,yxx)\}.$$ For every subset $Z$ of $({\cal A} \times {\cal A})^* $, by ${{\rm PREF}}(Z)$ we denote its set of prefixes.\ We define $${\cal P} := {{\rm PREF}}({\cal S})$$ namely: $${\cal P} = \{(\varepsilon,\varepsilon),(y,x),(yx,xy),(x,y),(xx,yx),(xxx,yxx)\}$$ Finally, we define $S$ as the subset of $({\cal R} \times {\cal R})^* $ obtained by replacing, in ${\cal P}$, every 2-tuple $(u,v) \in ({\cal A} \times {\cal A})^* $ by the unique 2-tuple $(r_u,r_v) \in ({\cal R} \times {\cal R})^*$, such that $r_u$ (resp. $r_v$) is applicable on $A$ (resp. on $B$), ${{\rm LAB}_{\cal T}}(r_u) = u$ and ${{\rm LAB}_{\cal T}}(r_v) = v$. Namely: $$S = \{ (\varepsilon,\varepsilon),(r_1,r_2),(r_1r_5,r_2r_6),(r_1r_6,r_2r_5),(r_2,r_1), (r_2r_7,r_1r_8),(r_2r_7r_9,r_1r_8r_{10})\}.$$ (See figures \[figure\_stratST\]-\[figure\_stratSR\]).\ Subsequently: $$\begin{aligned} S_1 & := &\{(\varepsilon,\varepsilon),(r_5,r_6),(r_6,r_5)\}\\ S_2 & := &\{(\varepsilon,\varepsilon)\}\\ S_3 & := &\{ (\varepsilon,\varepsilon),(r_7,r_8),(r_7r_9,r_8r_{10})\}\\ S_4 & := &\{ (\varepsilon,\varepsilon),(r_9,r_{10})\}\\ S_5 & := &\{ (\varepsilon,\varepsilon)\}\\ S_6 & := & {{\rm INDSTR}}(S_2,S_5)= S_2^{-1} \circ S_5= \{ (\varepsilon,\varepsilon)\}\end{aligned}$$ (90,90) (DE)(0,30)[$(D,E)$]{} (ED)(60,30)[$(E,D)$]{} (CC)(30,60)[$(C,C)$]{} (AB)(60,90)[$(A,B)$]{} (APBP)(90,60)[$(A',B')$]{} (ASBS)(90,30)[$(A'',B'')$]{} (DEBOT)(90,0)[$(D,E)$]{} (AB,CC)[$(y,x)$]{} (CC,DE)[$(x,y)$]{} (CC,ED)[$(y,x)$]{} (AB,APBP)[$(x,y)$]{} (APBP,ASBS)[$(x,x)$]{} (ASBS,DEBOT)[$(x,x)$]{} (90,90) (DE)(0,30)[$(D,E)$]{} (ED)(60,30)[$(E,D)$]{} (CC)(30,60)[$(C,C)$]{} (AB)(60,90)[$(A,B)$]{} (APBP)(90,60)[$(A',B')$]{} (ASBS)(90,30)[$(A'',B'')$]{} (DEBOT)(90,0)[$(D,E)$]{} (AB,CC)[$(r_1,r_3)$]{} (CC,DE)[$(r_5,r_6)$]{} (CC,ED)[$(r_6,r_5)$]{} (AB,APBP)[$(r_2,r_4)$]{} (APBP,ASBS)[$(r_7,r_8)$]{} (ASBS,DEBOT)[$(r_9,r_{10})$]{} $S$ is a prefix of D-strategy w.r.t. $(A(\bot),B(\bot))$. \[S\_is\_strategy\] Let us check that $S$ fulfills the critetium given by Lemma \[L-characterisation\_PDstrategies\]. Here $n=3$. Point (1) is easily checked.\ Let $\beta \in ({\cal R}\times{\cal R})^{*}$ such that $\beta \backslash S = \{(\varepsilon,\varepsilon)\}$. Either (${{\rm NEXT}}((A,B),\beta)\in \{(E,D),(D,E)\}$, while $D \not{\!\!\sim_1} E$) or $|\beta| = 3$. Hence Point (2) holds. For proving the equivalences of the members of the basis we shall use the “trivial” prefixes of strategies, consisting of 2-tuples of identical rules on both sides: $$\begin{aligned} {\rm Id_{C,1}} & := & \{(\varepsilon,\varepsilon), (r_5,r_5),(r_6,r_6)\})\\ {\rm Id_{D,2}} & := & \{(\varepsilon,\varepsilon), (r_{11},r_{11}), (r_{11}r_{14},r_{11}r_{14})\})\\ {\rm Id_{E,2}} & := & \{(\varepsilon,\varepsilon), (r_{12},r_{12}), (r_{13},r_{13}),(r_{12}r_{14},r_{12}r_{14}), (r_{13}r_{14},r_{13}r_{14})\}).\end{aligned}$$ One can check that ${\rm Id_{C,1}}$ is a prefix of the strategy, for the game with initial position $(C,C)$, $${\rm Id}_{C,\infty}:=\{(u,u) \mid u \in {\cal R}^*, C(L_1) {\stackrel{u}{\longrightarrow_{}}}\}.$$ The set ${\rm Id_{D,2}}$ (resp. ${\rm Id_{E,2}}$) is really a strategy for the game with initial position $(D,D)$ (resp. $(E,E)$) since no rule $r_i$ is applicable on $\bot$. For every $N \in \{C,D,E\}$, the symbol ${\rm Id_{N,i}}$ will denote a residual of length $i$ of the strategy ${\rm Id_{N,n}}$: $$\begin{aligned} {\rm Id_{C,0}} & = & {\rm Id_{D,0}} = {\rm Id_{E,0}} = \{(\varepsilon,\varepsilon)\},\\ {\rm Id_{D,1}} & = & {\rm Id_{E,1}} = \{(\varepsilon,\varepsilon), (r_{14},r_{14})\}\end{aligned}$$ The Non-equivalence (meta-) proof ================================= $A(\bot) \not{\!\!\sim} B(\bot)$ \[lem-nonequivalence\_proof\] $$\forall u \in {\cal R}^*, ACT(u) = aaab \Rightarrow A(\bot) \not{\!\!{\stackrel{u}{\longrightarrow_{}}}}$$ while $$\exists u \in {\cal R}^*, ACT(u) = aaab \mbox { and } B(\bot) {\stackrel{u}{\longrightarrow_{}}}$$ hence $A(\bot) \not{\!\!\sim} B(\bot)$. From section \[sec-equivalence\_proof\] and Lemma \[lem-nonequivalence\_proof\] we conclude The family of formal systems $({\cal J}(T_0,T'_0,S_0,{\cal B}))$ is not sound. Variations ========== Let us describe variations around this example.\ #### Description of the proofs $\;\;$\ We chosed to write the proofs with judgments of the form $m {\:|\!\!\!=\!\!\!\!=\:}(T,T',S)$ or $m {\:|\!\!\!=\!\!\!\!=\:}(T,T',S) \leadsto \alpha {\:|\!\!\!=\!\!\!\!=\:}(T_1,T'_1,S_1)$ or $m {\:|\!\!\!=\!\!\!\!=\:}(T,T',S) \leadsto \alpha {\:|\!\!\!=\!\!\!\!=\:}{{\rm SUCC}}$, where, in the case of forms 2,3, the prefix $\alpha$ is given by its image under the map ${{\rm LAB}_{\cal T}}$ (its image is enough to determine $\alpha \in ({\cal R} \times {\cal R})^*$ just because the grammar is deterministic). Of course the proofs can be rewritten with prefixes $\alpha \in ({\cal R} \times {\cal R})^*$. #### Strategies $\;\;$\ The formal systems ${\cal J}(T_0,T'_0,S_0,{\cal B})$ described in subsection \[subsec\_formal\_systems\] were devised so that their set of judgments is recursive. Let us consider now the formal systems $\hat{{\cal J}}(T_0,T'_0,S_0,{\cal B})$ really considered in pages 21-24. Their judgments are also of the forms $$m {\:|\!\!\!=\!\!\!\!=\:}(T,T',S),\;\;m {\:|\!\!\!=\!\!\!\!=\:}(T,T',S) \leadsto \alpha {\:|\!\!\!=\!\!\!\!=\:}(T_1,T'_1,S_1),\;\; m {\:|\!\!\!=\!\!\!\!=\:}(T,T',S) \leadsto \alpha {\:|\!\!\!=\!\!\!\!=\:}{{\rm SUCC}}$$ but where $S,S_1$ are D-strategies (instead of finite prefixes of strategies), “except when a judgment is obtained by rule R2”: see the fuzzy remark on page 23, line 11, followed by the enigmatic remark that “we could complete the definition anyhow for such cases”. Since $S,S_1,S_2,S_3,S_4,S_5,{\rm Id_{D,2}},{\rm Id_{E,2}}$ are really D-strategies and $S_6$ is obtained by an application of rule R2, it seems that our proofs $\pi_3,\pi_5,\pi_6$ are also proofs in the systems $\hat{{\cal J}}(T_0,T'_0,S_0,{\cal B})$. As well, replacing ${\rm Id_{C,1}}$ by ${\rm Id_{C,\infty}}$ in $\pi_4$, we obtain a proof of judgment $0 {\:|\!\!\!=\!\!\!\!=\:}(C(L_1),C(L_1),{\rm Id_{C,\infty}}) \leadsto (\varepsilon,\varepsilon) {\:|\!\!\!=\!\!\!\!=\:}{{\rm SUCC}}$ in the system $(\hat{{\cal J}}(C(L_1),C(L_1),{\rm Id_{C,\infty}},{\cal B}))$. #### Depth of the examples $\;\;$\ One can devise such proofs of non-bisimilar pairs, with an arbitrary long initial strategy: it suffices to add non-terminals $D_1,D_2,\ldots, D_k,E_1,E_2,\ldots ,E_k$ and to replace rules (\[ruleD\],\[ruleE1\],\[ruleE2\],\[ruleL1\]) by the sequence of rules: $$\begin{aligned} D(v) &{\stackrel{x}{\longrightarrow_{}}} & D_1(v) \label{nruleD}\\E(v) &{\stackrel{x}{\longrightarrow_{}}} & E_1(v) \label{nruleE}\\\vdots&&\vdots \nonumber\\ D_1(v) &{\stackrel{x}{\longrightarrow_{}}} & D_2(v) \label{nruleD1}\\E_1(v) &{\stackrel{x}{\longrightarrow_{}}} & E_2(v) \label{nruleE1}\\\vdots&&\vdots \nonumber\\ D_k(v) &{\stackrel{x}{\longrightarrow_{}}} & v \label{nruleDk}\\E_k(v) &{\stackrel{x}{\longrightarrow_{}}} & v \label{nruleEk}\\E_k(v) &{\stackrel{z}{\longrightarrow_{}}} & v \label{nruleEkz}\\L_1 &{\stackrel{\ell_1}{\longrightarrow_{}}} & \bot $$ A proof of $0 {\:|\!\!\!=\!\!\!\!=\:}A(\bot),B(\bot),\hat{S} \leadsto (\varepsilon,\varepsilon) {\:|\!\!\!=\!\!\!\!=\:}{\rm SUCC}$ can still be written, but with a longer initial strategy $\hat{S}$ where the maximal length of words is $3+k$, and a prefix of strategy $\hat{S}_6$ of length $k$. Note that the sizes of the proofs $\pi_3,\pi_4,\pi_5,\pi_6$ still remain the same. The flawed argument =================== Let us locate precisely, in [@Jan10], the crucial [*flawed*]{} argument in favor of soundness of the systems.\ Page 24, line \$-4, the following assertion (FA) is written:\ “The final rule in deriving $m {\:|\!\!\!=\!\!\!\!=\:}(U,U',S') \leadsto (\varepsilon,\varepsilon) {\:|\!\!\!=\!\!\!\!=\:}SUCC$ could not be the Basis rule, due to the least eq-level assumption for $T,T'$ (recall Prop. 17)”.\ In our example: $$(T,T') = (A(\bot),B(\bot)),\;\; EqLv((A(\bot),B(\bot))=3$$ Let us take $$(U,U',S')= (E(\bot),E(\bot),S_6)$$ We have: $$EqLv(U,U',S') =0 = EqLv(T,T',S)-3$$ And the judgment $$3 {\:|\!\!\!=\!\!\!\!=\:}E(\bot),E(\bot),S_6 \leadsto (\varepsilon,\varepsilon) {\:|\!\!\!=\!\!\!\!=\:}SUCC$$ can be derived by the proof $\pi_7$ below. {width="14cm"} Hence $(T,T')$ has the [*least*]{} equivalence level, among the EqLevels of the elements of $\{ (T,T') \} \cup {\cal B}$ while $m,U,U'$ fulfills the [*maximality*]{} hypothesis of the text (line \$-7).\ But the final rule used in this proof is the basis rule (R7), contradicting the assertion (FA).\ The bug seems to be the following: by Proposition 17 $$EqLv(E(L_1),E(L_1)) \leq EqLv(E(\bot),E(\bot)) \label{prop17}$$ BUT $$EqLv(E(L_1),E(L_1)) > EqLv(E(\bot),E(\bot),S_6)\;\; ! \label{true_inequality}$$ A superficial look at the instance (\[prop17\]) of Proposition 17 can induce the idea that, for [*every*]{} D-strategy ${\cal S}$ (in particular for $S_6$), the inequality $$EqLv(E(L_1),E(L_1)) \leq EqLv(E(\bot),E(\bot),{\cal S}) \label{false_inequation}$$ holds. In fact, what shows Proposition 17, is that inequality (\[false\_inequation\]) does hold but, only for strategies ${\cal S}$ which are [*optimal*]{} for the defender, hence realizing exactly the equivalence level of $(E(\bot),E(\bot))$. [Jan10]{} P. Jancar. Short decidability proofs for dpda language equivalence and 1st order grammar bisimilarity. , pages 1–35, 2010. [^1]: mailing adress:LaBRI and UFR Math-info, Université Bordeaux1\ 351 Cours de la libération -33405- Talence Cedex.\ email:ges@labri.u-bordeaux.fr; fax: 05-40-00-66-69;\ URL:http://dept-info.labri.u-bordeaux.fr/$\sim$ges/

--- abstract: 'The transport and magnetic properties of correlated La$_{0.53}$Sr$_{0.47}$MnO$_{3}$ ultrathin films, grown epitaxially on SrTiO$_{3}$, show a sharp cusp at the structural transition temperature of the substrate. Using a combination of experiment and theory we show that the cusp is a result of resonant coupling between the charge carriers in the film and a soft phonon mode in the SrTiO$_{3}$, mediated through oxygen octahedra in the film. The amplitude of the mode diverges towards the transition temperature, and phonons are launched into the first few atomic layers of the film affecting its electronic state.' author: - 'Y. Segal' - 'K. F. Garrity' - 'C. A. F. Vaz' - 'J. D. Hoffman' - 'F. J. Walker' - 'S. Ismail-Beigi' - 'C. H. Ahn' bibliography: - 'cusp.bib' title: 'Resonant phonon coupling across the La$_{1-x}$Sr$_{x}$MnO$_{3}$/SrTiO$_{3}$ interface' --- The coupling of phonons to charge carriers is a process of key importance for a broad set of phenomena, ranging from carrier mobility in semiconductors to Cooper pairing. In recent times, phonon effects at interfaces emerged as a topic of great importance in the understanding and design of nano-structured materials [@interfacialphonons]. Coupling between charge, structure and magnetic ordering is particularly strong in the Mn oxides [@RefWorks:462], which are used as a component in heterostructure multiferroics [@CarlosPRLPaper]. In these materials, localized spins and mobile carriers reside on the Mn sites, each surrounded by an oxygen octahedra. Intersite hopping occurs through orbital overlap of the Mn with neighbouring oxygens, making it highly sensitive to the static orientation of the octahedra and to phonons that alter the octahedra’s orientation [@RefWorks:463]. This interplay between structure and properties has been exploited to control the electronic phase of CMR films via strain, and also via coherent photoexcitation of a specific octahedra vibration mode [@RefWorks:454].\ In this Letter, we use a specially designed thin film device to isolate and characterize phonon-carrier coupling within a few atomic layers of an interface between the perovskite SrTiO$_{3}$ (STO) and the CMR oxide La$_{0.53}$Sr$_{0.47}$MnO$_{3}$ (LSMO). A soft octahedral rotation phonon with a divergent amplitude in the STO couples to the corresponding mode of the film. This coupling results in a marked change in the electronic and magnetic properties, including a sharp cusp in the resistivity and a dip in the magnetic moment. The sensitivity of LSMO to octahedra orientation allows us to experimentally probe the microscopic character of this interfacial phonon coupling, and compare it to theory. The thin film devices consist of La$_{0.53}$Sr$_{0.47}$MnO$_{3}$ films grown by molecular beam epitaxy on TiO$_{2}$-terminated STO (001) substrates and overlaid by Pb(Zr$_{0.2}$Ti$_{0.8}$)O$_{3}$ (PZT), which is used to provide ferroelectric field effect modulation of the number and distribution of carriers in the film. Details concerning fabrication and structural characterization are described elsewhere [@CarlosGrowthPaper]. In the bulk LSMO phase diagram, the $x=0.5$ composition separates the ferromagnetic metallic phase from an insulating antiferromagnetic phase [@RefWorks:476]. When grown commensurate to the STO, the substrate induces tensile strain in the film, which is known to stabilize an A-type antiferromagnetic metallic phase (AF-M) [@RefWorks:414]. Using X-ray diffraction, we verified that our films are under tensile strain, with $c/a=0.975$, in agreement with previous studies [@RefWorks:414]. Transport measurements of an 11unit cell (uc) LSMO film are shown in Fig.\[fig:Transport\]a. The broad peak in resistivity at 250K corresponds to a metal-insulator transition, typical of this material. In addition, a unique feature is observed in our films: a large and sharp resistance peak centered around 108K, which corresponds to the temperature of the STO soft phonon peak. We observe further that the magnitude of the resistivity cusp decreases when the thickness of the film increases by a few unit cells. Indeed, in previous studies of films $\approx$80uc thick, only a trace of this feature was observed [@RefWorks:477]. This film thickness dependence implies that the strength of the mechanism creating the cusp decays quickly away from the STO/LSMO interface. We can verify this by switching the polarization state of the PZT. When the PZT is switched to the “depletion” state, holes are removed from the top layer of the LSMO, pushing the conducting region closer to the substrate. The opposite occurs in the “accumulation” state [@CarlosPRLPaper]. We find that the PZT has a pronounced effect on the cusp (Fig.\[fig:Transport\]b), making it much larger in the depletion state, in agreement with the notion of a rapid decay into the film. We note, however, that presence of PZT is not required to observe the effect: the same features are observed on uncapped LSMO films. We also observe a striking dip in the magnetic moment centered around the STO transition temperature (Fig.\[fig:Transport\]a). While the majority of the LSMO is in an antiferromagnetic-metallic state, a small ferromagnetic component remains [@RefWorks:477]. The dip in magnetic moment corresponds to a decrease in magnetic order within the ferromagnetic phase.\ ![\[fig:Transport\]Enhanced carrier-phonon scattering. (a) Left axes: Resistivity of an 11uc La$_{0.53}$Sr$_{0.47}$MnO$_{3}$ film showing a strong cusp at 108K. The PZT overlayer is in the depletion state. Right axis: Magnetic moment of a 15uc La$_{0.55}$Sr$_{0.45}$MnO$_{3}$ film. The moment is measured along the [\[]{}100[\]]{} direction under an applied magnetic field of 1kOe. A dip in the moment is observed, overlapping the temperature range of the resistivity cusp (emphasized by grey box). The dashed line is a linear interpolation between the edges of the dip region. b) The resistivity of the 11uc film for the two polarization states of the PZT. c) Energy of the $\Gamma_{25}$ phonon mode in STO, showing the softening around the STO transition temperature (after Ref.[@RefWorks:458]). Lines are a guide to the eye. Below the structural phase transition the mode splits due to the breaking of cubic symmetry.](fig1){width="8.5cm"} We attribute the transport and magnetism anomaly to a coupling between the LSMO and the phonon softening that occurs in STO around the 108K structural transition. The $\Gamma_{25}$ $(111)$ zone edge phonon [@RefWorks:460; @RefWorks:458] becomes lower in energy as the transition is approached from both temperature directions. Fig.\[fig:Transport\]c, reproduced from Ref.[@RefWorks:458], shows the $\Gamma_{25}$ phonon energy as a function of temperature. The softening leads to a divergent increase in mode occupation or amplitude. The motion associated with this mode is a rotation of the TiO$_{6}$ octahedra. Below the transition temperature, the octahedra stabilize into a rotated antiferrodistortive (AFD) configuration accompanied by a tetragonal distortion of the unit cell. Since the film is mechanically constrained to the substrate at the atomic level, motions of the TiO$_{6}$ octahedra couple to the MnO$_{6}$ ones, inducing both static and dynamic changes in their configuration. ![Side view of STO-LSMO interface geometry. The plot shows calculated ground-state atomic positions. Away from from the interface, the STO is fixed to have bulk-like octahedral rotations around the $x$ axis (into the page). The LSMO geometry at the interface is modified by the STO; however, the LSMO relaxes to its bulk-like octahedral rotations around both in-plane axes within 2-3 unit cells. \[fig:interface\]](fig2) We examine two mechanisms whereby the resistance of the LSMO layer might increase: $(i)$ static changes of the LSMO structure causing a change of electronic band parameters; $(ii)$ decreased carrier relaxation times due to enhanced phonon scattering, i.e. a dynamic effect. The static and dynamic contributions are reflected in the expression for the conductivity in the relaxation time approximation $\sigma_{ij}\propto\tau m_{ij}^{-1}$, where $\tau$ is the relaxation time and $m_{ij}^{-1}$ is the reciprocal effective mass tensor [@ashcroft]. To treat the temperature-dependent character of the coupling phenomena, we perform finite temperature simulations by building a classical model of the energetics of the system as a function of oxygen displacements. Our model includes harmonic coupling between oxygens, 4$^{th}$ order on-site anharmonic terms to stabilize the symmetry breaking, and lowest order coupling between oxygen displacements and stress, thus capturing the STO phase transition [@sto1]. Model parameters are obtained via density functional theory calculations using the spin-polarized PBE GGA functional [@GGA] and ultrasoft pseudopotentials [@ultrasoft]. Ground states for both bulk strained LSMO (using the virtual crystal approximation [@vca_vand]) and the LSMO/STO system are calculated, reproducing the experimental A-type ordering. In addition, the ground state calculation shows how the octahedral orientation is continuous going from substrate to film (see Fig.\[fig:interface\]), as was recently shown in a similar theoretical study [@rondinellioctahedra]. The harmonic interatomic force constants are calculated with DFT perturbation theory (DFPT) [@ModelH2], and the remaining parameters are fit to strained bulk calculations. We then perform classical Monte Carlo sampling on this model in a periodic box. The box contains $10\times10\times100$ perovskite unit cells composed of 60 STO and 40 LSMO unit cells in the $z$ direction. To evaluate the role of static structural changes, we compute the conductivity tensor of bulk strained LSMO for the static octahedra rotation angles obtained from the Monte-Carlo model. The conductivity is calculated from direct first principles evaluation of the reciprocal effective mass tensor, by summing over all bands at the Fermi energy [@ashcroft]. The upper bound of conductivity change is estimated by using the angles at the LSMO/STO interface, which change the most due to the substrate-film coupling. We find that the static coupling effect appears only below the phase transition temperature. The phase transition causes the octahedra angles in the LSMO to increase somewhat; however, the magnitude of resulting change in conductivity is too small to account the experimental findings. The computed Mn-O-Mn hopping elements change only by 1-2% due to static structural changes, while the experimental conductivity changed by more than 10%. Because the static structural change manifests only below 108K and does not yield a large enough change in conductivity, we examine whether the $\Gamma_{25}$ phonon might extend into the LSMO and cause dynamic carrier scattering. Hence, we compute the oxygen-oxygen correlation matrix $c_{ij}=\langle x_{i}x_{j}\rangle-\langle x_{i}\rangle\langle x_{j}\rangle$ where $x_{i}$ are the oxygen displacements from their equilibrium pseudocubic positions. We find that near the STO phase transition, the correlation length in STO diverges, as expected. Furthermore, oxygen motions in the interfacial LSMO layers become correlated with those deep in the STO (Fig.\[fig:correlation\]a) demonstrating that STO soft phonons extend into the LSMO. We quantify this relation more precisely by extracting the dominant eigenvectors of the correlation matrix, which are the softest phonon modes in the finite-temperature harmonic system. Fig. \[fig:correlation\]b shows the lowest frequency eigenvector: it decays exponentially into the LSMO with a decay length of 2.3 unit cells. ![\[fig:correlation\]DFT/Monte Carlo results of phonon transfer from STO to LSMO. a) Absolute magnitude of the correlation function of oxygen displacements with those deep in STO versus layer number at $T=T_{C}$. $x$ and $z$ indicate displacements stemming from octahedra rotation around the $x$ and $z$ axis respectively. (The $y$ component is equal to $x$ by symmetry). b) $x$ and $z$ components of the lowest frequency eigenvector at $T=T_{C}$. Layer-to-layer sign changes reflect the AFD nature of the oxygen displacments. ](fig3) Building upon our theoretical results, we use the following simple model to fit the experimental resistivity data: the scattering due to the soft mode is $cne^{-2z/\lambda}$, where $n$ is the $\Gamma_{25}$ occupation number in STO (given by the Bose-Einstein function using the energies in Fig.\[fig:Transport\]c). $c$ is a conversion factor from $n$ to resistivity (linearly related in phonon scattering theory [@Ziman]); and $\lambda$ is the decay length of the induced octahedra motion amplitude ($\lambda/2$ is the decay length for the phonon number). The total conductivity is obtained by summing over the film layers: $\sigma=\sum\limits _{layers}(\rho_{\mathrm{base}}+cne^{-2z/\lambda})^{-1}$. $\rho_{\mathrm{base}}$ is the unperturbed LSMO resistivity, which we find by passing a smooth line under the cusp, following the method used in an electron paramagnetic resonance study of the softening-induced disorder in STO [@PhysRevB.7.1052]. This model describes the experimental data well, as shown in Fig.\[fig:fits\], and yields a decay length of 1.8uc, in good agreement with theory. This verifies our attribution of the resistivity cusp to resonant coupling of the divergent $\Gamma_{25}$ mode into the LSMO, where it strongly scatters the carriers by disturbing the Mn-O-Mn hopping path. The phonon coupling picture also explains the dip in magnetic moment, when phonon-magnon interactions are taken into account. For the manganites, it has been shown theoretically [@RefWorks:484] that when phonons involving Mn or O distortions are added to the Heisenberg Hamiltonian, the magnon spectrum is softened. The phonons injected from the substrate cause a softening of the magnon spectrum in the LSMO, with maximal softening occuring at the STO transition. Magnon occupation increases with spectrum softening, leading to the reduction in the magnetic moment. The overlap between the temperature range of the transport cusp and the moment dip is striking (Fig.\[fig:Transport\]a), confirming that they are both driven by $\Gamma_{25}$ phonon softening.\ ![\[fig:fits\]Model fitting: resistivity of an 11uc film of La$_{0.53}$Sr$_{0.47}$MnO$_{3}$ as a function of temperature, for several out-of-plane magnetic field values. The PZT overlayer is in the depletion state. Black dashed lines are interpolations excluding the cusp and red dashed lines are fits to $\Gamma_{25}$ phonon coupling.](fig4) Previous work on the influence of the STO transition on manganite films dealt with effects attributed to the appearance of $a$ and $c$ domains in the STO and the resulting change in the strain state of the film below the transition temperature [@RefWorks:432; @RefWorks:434; @RefWorks:431]. In the current work, we find that effects appear both above and below the transition temperature and correspond to the temperature range of phonon softening. In addition, the short decay length that we find does not agree with a strain-mediated phenomenon. The LSMO remains strained to the substrate up to a thickness of at least 80 uc [@RefWorks:414], so that changes related to strain should be evident at these thicknesses as well. These two facts preclude the strain configuration of the STO below the transition from being the source of the cusp. To further verify that the cusp feature is independent of the $a/c$ domain structure formed in the STO, we applied an electric field of $2\times10^{5}$V/m using a back gate on the substrate. This field should break the symmetry between $a$ and $c$ domains and lead to a different domain structure compared to a zero field case. No difference in resistivity was observed with and without the electric field. Our observation of resonant phonon-carrier coupling illuminates a key feature of conduction in LSMO. The coupling manifests strongly near the $x=0.5$ composition, while films of similar thickness at the $x=0.2$ composition [@CarlosPRLPaper] showed a resistivity cusp much smaller in magnitude. We relate this effect to an increased carrier coupling to the $\Gamma_{25}$ phonon in the AF-M phase of the LSMO. In this phase, the Mn $d_{x^{2}-y^{2}}$ $e_{g}$ orbital is occupied while the $d_{3z^{2}-r^{2}}$ orbital is depopulated [@RefWorks:489]. This causes the carriers’ wavefunctions to be concentrated on the $xy$ MnO$_{2}$ planes, which underpins the 2D character of metallicity and ferromagnetism in this phase, in contrast to the 3D character of the $x=0.2$ to $0.4$ composition range. The transport measurements of our thin films probe carrier hopping in the $x$ and $y$ directions. Perturbation of the bridging oxygen positions due to the $\Gamma_{25}$ phonon will have a larger scattering effect on carriers in the AF-M phase compared to the 3D FM phase. This is because in the AF-M case, the electron density case is concentrated closer to the perturbed oxygens in the $xy$ plane through which the conduction occurs. This configuration also explains why an out-of-plane magnetic field reduces the effect of the phonon coupling, as can be seen in Fig. \[fig:fits\]. The magnetic field causes the Mn spins to cant so that they are partially aligned out-of-plane. This allows for some inter-plane hopping and reduces the confinement of carriers to the $xy$ MnO$_{2}$ planes, similarly to the “spin valve” effect in A-type Nd$_{0.45}$Sr$_{0.55}$MnO$_{3}$ [@spinvalve]. In conclusion, we show how a single phonon mode originating in the substrate extends resonantly across an epitaxial interface and into the film. The effects of this coupling are amplified by the properties of both materials: phonon softening in the substrate causes the phonon amplitude to diverge, while the LSMO’s electronic phase and charge distribution are tuned using strain and a ferroelectric gate. This work was supported by the National Science Foundation under Contract MRSEC No. DMR-0520495, DMR-1006265, and FENA. Computational resources were provided by Yale High Performance Computing, partially funded by grant CNS 08-21132 and by TeraGrid/NCSA under grant number TG-MCA08X007.

--- abstract: 'We introduce the positive intersection product in Arakelov geometry and prove that the arithmetic volume function is continuously differentiable. As applications, we compute the distribution function of the asymptotic measure of a Hermitian line bundle and several other arithmetic invariants.' address: 'Université Paris Diderot — Paris 7, Institut de mathématiques de Jussieu, case 247, 4 place Jussieu, 75252 Paris Cedex' author: - Huayi Chen bibliography: - 'chen.bib' title: Differentiability of the arithmetic volume function --- On introduit le produit d’intersection positive en géométrie d’Arakelov et on démontre que la fonction volume arithmétique est continuement dérivable. Comme applications, on calcule la fonction de répartition de la mesure de probabilité asymptotique d’un fibré inversible hermitien ainsi que quelques d’autres invariants arithmétiques. Introduction ============ Let $K$ be a number field, $\mathcal O_K$ be its integer ring and $\pi:X\rightarrow\operatorname{Spec}\mathcal O_K$ be an arithmetic variety of relative dimension $d$. Recall that the [*arithmetic volume*]{} of a continuous Hermitian line bundle $\overline L$ on $X$ is by definition $$\label{Equ:arithmetic volume}\widehat{\mathrm{vol}}(\overline L):=\limsup_{n\rightarrow\infty} \frac{\widehat{h}^0(X,\overline L^{\otimes n })}{n^{d+1}/(d+1)!},$$ where $$\widehat{h}^0(X,\overline L^{\otimes n}) =\log\#\{s\in\pi_*(L^{\otimes n})\mid \forall\,\sigma:K\rightarrow\mathbb C,\; \|s\|_{\sigma,\sup}\leqslant 1\}.$$ The properties of the arithmetic volume $\widehat{\mathrm{vol}}$ (see [@Moriwaki07; @Moriwaki08; @Yuan07; @Yuan08; @Chen_bigness; @Chen_Fujita]) are quite similar to the corresponding properties of the classical volume function in algebraic geometry. Recall that if $Y$ is a projective variety defined over a field $k$ and if $L$ is a line bundle on $Y$, then the volume of $L$ is defined as $$\mathrm{vol}(L):=\limsup_{n\rightarrow\infty} \frac{\operatorname{rk}_k H^0(Y,L^{\otimes n})}{n^{\dim Y}/(\dim Y )!}.$$ In [@Bou_Fav_Mat06], Boucksom, Favre and Jonsson have been interested in the regularity of the geometric volume function. They have actually proved that the function $\mathrm{vol}(L)$ is continuously differentiable on the big cone. The same result has also been independently obtained by Lazarsfeld and Musţatǎ [@Lazarsfeld_Mustata08], by using Okounkov bodies. Note that the geometric volume function is not second order derivable in general, as shown by the blow up of $\mathbb P^2$ at a closed point, see [@LazarsfeldI 2.2.46] for details. In the differential of $\mathrm{vol}$ appears the positive intersection product, initially defined in [@Boucksom_Demailly_Paun_Peternell] in the analytic-geometrical framework, and redefined algebraically in [@Bou_Fav_Mat06]. Inspired by [@Bou_Fav_Mat06], we introduce an analogue of the positive intersection product in Arakelov geometry and prove that the arithmetic volume function $\widehat{\mathrm{vol}}$ is continuously differentiable on $\widehat{\mathrm{Pic}}(X)$. We shall establish the following theorem: \[Thm:main theorem\] Let $\overline L$ and $\overline M$ be two continuous Hermitian line bundles on $X$. Assume that $\overline L$ is big. Then $$D_{\overline L}\widehat{\mathrm{vol}}(\overline M):=\lim_{n\rightarrow+\infty}\frac{\widehat{\mathrm{vol}} (\overline L^{\otimes n}\otimes\overline M)-\widehat{\mathrm{vol}}(\overline L^{\otimes n})}{n^d}$$ exists in $\mathbb R$, and the function $D_{\overline L}\widehat{\mathrm{vol}}$ is additive on $\widehat{\mathrm{Pic}}(X)$. Furthermore, one has $$D_{\overline L}\widehat{\mathrm{vol}}(\overline M)= (d+1)\big\langle\widehat{c}_1(\overline L )^d\big\rangle\cdot\widehat{c}_1(\overline M).$$ Here the [*positive intersection product*]{} $\big\langle\widehat{c}_1(\overline L )^d\big\rangle$ is defined as the least upper bound of self intersections of ample Hermitian line bundles dominated by $\overline L$ (see §\[SubSec:Posit\] [*infra*]{}). In particular, one has $\big\langle\widehat{c}_1(\overline L )^d\big\rangle\cdot\widehat{c}_1(\overline L)=\big\langle\widehat{c}_1(\overline L )^{d+1}\big\rangle=\widehat{\mathrm{vol}}(\overline L)$, which shows that the arithmetic Fujita approximation is asymptotically orthogonal. As an application, we calculate explicitly the distribution function of the asymptotic measure (see [@Chen08; @Chen_bigness]) of a generically big Hermitian line bundle in terms of positive intersection numbers. Let $\overline L$ be a Hermitian line bundle on $X$ such that $L_K$ is big. The asymptotic measure $\nu_{\overline L}$ is the vague limit (when $n$ goes to infinity) of Borel probability measures whose distribution functions are determined by the filtration of $H^0(X_K,L_K^{\otimes n})$ by successive minima (see ). Several asymptotic invariants can be obtained by integration with respect to $\nu_{\overline L}$. Therefore, it is interesting to determine completely the distribution of $\nu_{\overline L}$, which will be given in Proposition \[Pro:distribution function\] by using the positive intersection product. The article is organized as follows. In the second section, we recall some positivity conditions for Hermitian line bundles and discuss their properties. In the third section, we define the positive intersection product in Arakelov geometry. It is in the fourth section that we establish the differentiability of the arithmetic volume function. Finally in the fifth section, we present applications on the asymptotic measure and we compare our result to some known results on the differentiability of arithmetic invariants. [**Acknowledgement:**]{} I would like to thank R. Berman, D. Bertrand, J.-B. Bost, S. Boucksom, C. Favre and V. Maillot for interesting and helpful discussions. I am also grateful to M. Jonsson for remarks. Notation and preliminaries ========================== In this article, we fix a number field $K$ and denote by $\mathcal O_K$ its integer ring. Let $\overline K$ be an algebraic closure of $K$. Let $\pi:X\rightarrow\operatorname{Spec}\mathcal O_K$ be a projective and flat morphism and $d$ be the relative dimension of $\pi$. Denote by $\widehat{\mathrm{Pic}}(X)$ the group of isomorphism classes of (continuous) Hermitian line bundles on $X$. If $\overline L$ is a Hermitian line bundle on $X$, we denote by $\pi_*(\overline L)$ the $\mathcal O_K$-module $\pi_*(L)$ equipped with sup norms. In the following, we recall several notions about Hermitian line bundles. The references are [@Gillet-Soule; @Zhang95; @BGS94; @Moriwaki00]. Assume that $x\in X(\overline K)$ is an algebraic point of $X$. Denote by $K_x$ the field of definition of $x$ and by $\mathcal O_x$ its integer ring. The morphism $x:\operatorname{Spec}\overline K\rightarrow X$ gives rise to a point $P_x$ of $X$ valued in $\mathcal O_x$. The pull-back of $\overline L$ by $P_x$ is a Hermitian line bundle on $\operatorname{Spec}\mathcal O_x$. We denote by $h_{\overline L}(x)$ its normalized Arakelov degree, called the [*height*]{} of $x$. Note that the height function is additive with respect to $\overline L$. Let $\overline L$ be a Hermitian line bundle on $X$. We say that a section $s\in\pi_*(L)$ is [*effective*]{} (resp. [*strictly effective*]{}) if for any $\sigma:K\rightarrow\mathbb C$, one has $\|s\|_{\sigma,\sup}\leqslant 1$ (resp. $\|s\|_{\sigma,\sup}< 1$). We say that the Hermitian line bundle $\overline L$ is [*effective*]{} if it admits a non-zero effective section. Let $\overline L_1$ and $\overline L_2$ be two Hermitian line bundles on $X$. We say that $\overline L_1$ is [*smaller*]{} than $\overline L_2$ and we denote by $\overline L_1\leqslant\overline L_2$ if the Hermitian line bundle $\overline L_1^\vee\otimes\overline L_2$ is effective. We say that a Hermitian line bundle $\overline A$ is ample if $A$ is [*ample*]{}, $c_1(\overline A)$ is semi-positive in the sense of current on $X(\mathbb C)$ and $\widehat{c}_1(\overline L|_Y)^{\dim Y}>0$ for any integral sub-scheme $Y$ of $X$ which is flat over $\operatorname{Spec}\mathcal O_K$. Here the intersection number $\widehat{c}_1(\overline L|_Y)^{\dim Y}$ is defined in the sense of [@Zhang95] (see Lemma 6.5 [*loc. cit.*]{}, see also [@Zhang95b]). Note that there always exists an ample Hermitian line bundle on $X$. In fact, since $X$ is projective, it can be embedded in a projective space $\mathbb P^N$. Then the restriction of $\mathcal O_{\mathbb P^N}(1)$ with Fubini-Study metrics on $X$ is ample. Note that the Hermitian line bundle $\overline L$ thus constructed has strictly positive smooth metrics. Thus, if $\overline M$ is an arbitrary Hermitian line bundle with smooth metrics on $X$, then for sufficiently large $n$, $\overline M\otimes\overline L^{\otimes n} $ is still ample. We say that a Hermitian line bundle $\overline N$ is [*vertically nef*]{} if the restriction of $N$ on each fiber of $\pi$ is nef and $c_1(\overline N)$ is semi-positive in the sense of current on $X(\mathbb C)$. We say that $\overline N$ is [*nef*]{} if it is vertically nef and $\widehat{c}_1(\overline N|_Y)^{\dim Y}\geqslant 0$ for any integral sub-scheme $Y$ of $X$ which is flat over $\operatorname{Spec}\mathcal O_K$. By definition, an ample Hermitian line bundle is always nef. Furthermore, if $\overline A$ is an ample Hermitian line bundle and if $\overline N$ is a Hermitian line bundle such that $\overline N^{\otimes n}\otimes\overline A$ is ample for any integer $n\geqslant 1$, then $\overline N$ is nef. We denote by $\widehat{\mathrm{Nef}}(X)$ the subgroup of $\widehat{\mathrm{Pic}}(X)$ consisting of nef Hermitian line bundles. If $f:X(\mathbb C)\rightarrow\mathbb R$ is a continuous function, we denote by $\overline{\mathcal O}(f)$ the Hermitian line bundle on $X$ whose underlying line bundle is trivial, and such that the norm of the unit section $\mathbf{1}$ at $x\in X(\mathbb C)$ is $e^{-f(x)}$. Note that, if $f$ is positive, then $\overline{\mathcal O}(f)$ is effective. If $f$ is positive and plurisubharmonic, then $\overline{\mathcal O}(f)$ is nef. In particular, for any $a\in\mathbb R$, $\overline{\mathcal O}(a)$ is nef if and only if $a\geqslant 0$. If $\overline L$ is a Hermitian line bundle on $X$, we shall use the notation $\overline L(f)$ to denote $\overline L\otimes\overline{\mathcal O}(f)$. We say that a Hermitian line bundle $\overline L$ is [*big*]{} if its arithmetic volume $\widehat{\mathrm{vol}}(\overline L)$ is strictly positive. By [@Moriwaki07; @Yuan07], $\overline L$ is big if and only if a positive tensor power of $\overline L$ can be written as the tensor product of an ample Hermitian line bundle with an effective one. Furthermore, the analogue of Fujita’s approximation holds for big Hermitian line bundles, cf. [@Chen_Fujita; @Yuan08]. The arithmetic volume function $\widehat{\mathrm{vol}}$ is actually a limit (cf. [@Chen_bigness]): one has $$\widehat{\mathrm{vol}}(L)=\lim_{n\rightarrow\infty} \frac{\widehat{h}^0(X,\overline L^{\otimes n })}{n^{d+1}/(d+1)!}.$$ Moreover, it is a birational invariant which is continuous on $\widehat{\mathrm{Pic}}(X)_{\mathbb Q}$, and can be continuously extended to $\widehat{\mathrm{Pic}}(X)_{\mathbb R}$, cf. [@Moriwaki07; @Moriwaki08]. The analogue of Siu’s inequality and the log-concavity hold for $\widehat{\mathrm{vol}}$, cf. [@Yuan07; @Yuan08]. 1) In [@Zhang95] and [@Moriwaki00], the notions of ample or nef line bundles were reserved for line bundles with smooth metrics, which is not the case here. 2) Note that there exists another (non-equivalent) definition of arithmetic volume function in the literature. See [@Ber_Bou08 §10.1] and [@Chambert-Loir_Thuillier §5] where the “arithmetic volume” of a Hermitian line bundle $\overline L$ was defined as the following number: $$\label{Equ:sectional capa}S(\overline L):=\lim_{n\rightarrow+\infty}\frac{\chi(\pi_*(\overline L^{\otimes n}))}{ n^{d+1}/(d+1)!}\in[-\infty,+\infty[,$$ which is also called [*sectional capacity*]{} in the terminology of [@Rumely_Lau_Varley]. However, in the analogy between Arakelov geometry and relative algebraic geometry over a regular curve, it is that corresponds to the geometric volume function. Note that one always has $$\widehat{\mathrm{vol}}(\overline L)\geqslant \lim_{n\rightarrow+\infty}\frac{\chi(\pi_*(\overline L^{\otimes n}))}{ n^{d+1}/(d+1)!},$$ and the equality holds when $\overline L$ is nef. Under this assumption, both quantities are equal to the intersection number $\widehat{c}_1(\overline L)^{d+1}$. This is a consequence of the Hilbert-Samuel formula. See [@Gillet-Soule; @Abbes-Bouche; @Zhang95; @Autissier01; @Randriam06; @Moriwaki07] for details. In the following, we present some properties of nef line bundles. Note that Propositions \[Pro:critere de nef\] and \[Pro:positivity of nef intersection produc\] have been proved in [@Moriwaki00 §2] for Hermitian line bundles with smooth metrics. Here we adapt these results to continuous metric case by using the continuity of intersection numbers. \[Pro:critere de nef\] Let $\overline N$ be a Hermitian line bundle on $X$ which is vertically nef. Assume that for any $x\in X(\overline K)$, one has $h_{\overline N}(x)\geqslant 0$, then the Hermitian line bundle $\overline N$ is nef. Choose an ample Hermitian line bundle $\overline A$ on $X$ such that $h_{\overline A}$ has strictly positive lower bound. For any integer $n\geqslant 1$, let $\overline L_n:=(L_n,(\|\cdot\|_\sigma)_{\sigma:K\rightarrow\mathbb C })$ be the tensor product $\overline N^{\otimes n}\otimes\overline A$. The height function $h_{\overline L_n}$ is bounded from below by a strictly positive number $\varepsilon_n$. Note that the metrics of $\overline L_n$ are semi-positive. By [@Maillot00 Theorem 4.6.1] (see also [@Randriam06 §3.9]), there exists a sequence of smooth positive metric families $(\alpha_m)_{m\geqslant 1}$ with $\alpha_m=(\|\cdot\|_{\sigma,m})_{\sigma:K\rightarrow\mathbb C }$, such that $\|\cdot\|_{\sigma,m}$ converges uniformly to $\|\cdot\|_{\sigma}$ when $m$ tends to the infinity. Denote by $\overline L_{n,m}=(L_n,\alpha_m)$. For sufficiently large $m$, $h_{\overline L_{n,m}}$ is bounded from below by $\varepsilon_n/2$. Thus [@Zhang95 Corollary 5.7] implies that, for any integer subscheme $Y$ of $X$ which is flat over $\operatorname{Spec}\mathcal O_K$, one has $n^{-\dim Y }\widehat{c}_1(\overline L_{n,m}|_Y )^{\dim Y}\geqslant 0$. By passing successively $m$ and $n$ to the infinity, one obtains $\widehat{c}_1(\overline N|_Y)^{\dim Y}\geqslant 0$. Therefore $\overline N$ is nef. We say that a Hermitian line bundle $\overline L$ on $X$ is [*integrable*]{} if there exist two ample Hermitian line bundles $\overline A_1$ and $\overline A_2$ such that $\overline L=\overline A_1\otimes\overline A_2^\vee$. Denote by $\widehat{\mathrm{Int}}(X)$ the subgroup of $\widehat{\mathrm{Pic}}(X)$ formed by all integrable Hermitian line bundles. If $(\overline L_i)_{i=0}^d$ is a family of integrable Hermitian line bundles on $X$, then the intersection number $$\widehat{c}_1(\overline L_0)\cdots\widehat{c}_1(\overline L_d)$$ is defined (see [@Zhang95 Lemma 6.5], [@Zhang95b §1] and [@Maillot00] §5). Furthermore, it is a symmetric multi-linear form which is continuous in each $\overline L_i $. Namely, for any family $(\overline M_i)_{i=0}^d$ of integrable Hermitian line bundles, one has $$\lim_{n\rightarrow+\infty}n^{-d-1}\widehat{c}_1( \overline L_0^{\otimes n}\otimes\overline M_0)\cdots\widehat{c}_1(\overline L_d^{\otimes n}\otimes\overline M_d)=\widehat{c}_1(\overline L_0)\cdots\widehat{c}_1(\overline L_d).$$ \[Pro:positivity of nef intersection produc\] Let $(\overline L_i)_{i=0}^{d-1}$ be a family of nef Hermitian line bundles on $X$ and $\overline M$ be an integrable Hermitian line bundle on $X$ which is effective. Then $$\label{Equ:positivity de intese}\widehat{c}_1(\overline L_0)\cdots\widehat{c}_1(\overline L_{d-1}) \widehat{c}_1(\overline M)\geqslant 0$$ Choose an ample Hermitian line bundle $\overline A$ on $X$ such that $h_{\overline A}$ is bounded from below by some strictly positive number. By virtue of the proof of Proposition \[Pro:critere de nef\], for any $i\in\{0,\cdots,d-1\}$ and any integer $n\geqslant 1$, there exists a sequence of nef Hermitian line bundles with smooth metrics $(\overline L_{i,n}^{(m)})_{m\geqslant 1}$ whose underlying line bundle is $L_i^{\otimes n}\otimes A$ and whose metrics converge uniformly to that of $\overline L_i^{\otimes n }\otimes\overline A$. By [@Moriwaki00 Proposition 2.3], one has $$\widehat{c}_1(\overline L_0^{\otimes n}\otimes\overline A)\cdots\widehat{c}_1(\overline L_{d-1}^{\otimes n}\otimes \overline A)\widehat{c}_1(\overline M)\geqslant 0.$$ By passing to limit, one obtains . Using the same method, we can prove that, if $(\overline L_i)_{i=0}^d$ is a family of nef Hermitian line bundles on $X$, then $$\widehat{c}_1(\overline L_0)\cdots\widehat{c}_1(\overline L_d)\geqslant 0.$$ \[Pro:critere de nefness\] Let $\overline{L}$ be a Hermitian line bundle on $X$ such that $c_1(\overline L)$ is semi-positive in the sense of current on $X(\mathbb C)$. Assume that there exists an integer $n> 0$ such that $L^{\otimes n}$ is generated by its effective sections. Then the Hermitian line bundle $\overline{L}$ is nef. Since $L^{\otimes n}$ is generated by its sections, the line bundle $L$ is nef relatively to $\pi$. After Proposition \[Pro:critere de nef\], it suffices to verify that, for any $x\in X(\overline K)$, one has $h_{\overline{L}}(x)\geqslant 0$. For any integer $m\geqslant 1$, let $B_m=\pi_*(L^{\otimes m})$ and let $B_m^{[0]}$ be the saturated sub-$\mathcal O_K$-module of $B_m$ generated by effective sections. Since $L^{\otimes n}$ is generated by its effective sections, also is $L^{\otimes np}$ for any integer $p\geqslant 1$. In particular, one has surjective homomorphisms $x^*\pi^* B_{pn,K_x}^{[0]}\rightarrow x^* L_{K_x}^{\otimes np}$. By slope inequality (see [@BostBour96 Appendix A]), one has $nph_{\overline{ L}}(x)\geqslant\widehat{\mu}_{\min}(\overline{ B}_{np} ^{[0]}) $. By passing to limit, one obtains $h_{\overline{L}}(x)\geqslant 0$. We say that a Hermitian line bundle $\overline L$ on $X$ is [*free*]{} if $c_1(\overline L)$ is semi-positive in the sense of current on $X(\mathbb C)$ and if some positive tensor power of $L$ is generated by effective global sections. We denote by $\widehat{\mathrm{Fr}}(X)$ the subgroup of $\widehat{\mathrm{Pic}}(X)$ consisting of free Hermitian line bundles. After Proposition \[Pro:critere de nefness\], one has $\widehat{\mathrm{Fr}}(X)\subset\widehat{\mathrm{Nef}}(X)$. Unlike the ampleness, the properties of being big, nef, or free are all invariant by birational modifications. That is, if $\nu:X'\rightarrow X$ is a birational projective morphism, and if $\overline L$ is a Hermitian line bundle on $X$ which is big (resp. nef, free), then also is $\nu^*(\overline L)$. Positive intersection product ============================= In this section, we shall define the positive intersection product for big (non-necessarily integral) Hermitian line bundles. When all Hermitian line bundles are nef, the positive intersection product coincides with the usual intersection product. Furthermore, the highest positive auto-intersection number is just the arithmetic volume of the Hermitian line bundle. We shall use the positive intersection product to interpret the differential of the arithmetic volume function. Admissible decompositions ------------------------- Let $\overline L$ be a big Hermitian line bundle on $X$. We call [*admissible decomposition*]{} of $\overline L$ any triplet $(\nu,\overline N,p)$, where 1) $\nu:X'\rightarrow X$ is a birational projective morphism, 2) $\overline N$ is a free Hermitian line bundle on $X'$, 3) $p\geqslant 1$ is an integer such that $\nu^*(\overline L^{\otimes p}) \otimes\overline N^\vee$ is effective. Denote by $\Theta(\overline L)$ the set of all admissible decompositions of $\overline L$. We introduce an order relation on the set $\Theta(\overline L)$. Let $D_i=(\nu_i:X_i\rightarrow X,\overline N_i,p_i )$ ($i=1,2$) be two admissible decompositions of $\overline L$. We say that $D_1$ is [*superior*]{} to $D_2$ and we denote by $D_1\succ D_2$ if $p_2$ divides $p_1$ and if there exists a projective birational morphism $\eta:X_1\rightarrow X_2$ such that $\nu_2\eta=\nu_1$ and that $\overline N_1\otimes(\eta^*\overline N_2)^{\vee\otimes(p_1/p_2)}$ is effective. \[Rem:pull back of admissible decomp\] (1) Assume that $D=(\nu:X'\rightarrow X, \overline{N},p)$ is an admissible decomposition of $\overline{L}$. Then for any birational projective morphism $\eta: X''\rightarrow X'$, the triplet $\eta^*D:=(\nu\eta,\eta^*\overline{N},p)$ is also an admissible decomposition of $\overline L$, and one has $\eta^*D \succ D$. (2) Assume that $D=(\nu, \overline{N},p)$ is an admissible decomposition of $\overline{L}$. Then for any integer $n\geqslant 1$, $D_n=(\nu,\overline{N}^{\otimes n}, np)$ is also an admissible decomposition of $\overline{L}$. Furthermore, one has $D_n\succ D$. (3) Assume that $D_1=(\nu,\overline{N}_1,p)$ and $D_2=(\nu,\overline{N}_2,q)$ are two admissible decompositions of $\overline{L}$ whose underlying birational projective morphisms are the same. Then $D_1\otimes D_2:=(\nu,\overline{N}_1\otimes\overline{N}_2,p+q)$ is an admissible decomposition of $\overline{L}$. (4) Assume that $\overline M$ is an effective Hermitian line bundle on $X$. By definition, any admissible decomposition of $\overline L$ is also an admissible decomposition of $\overline L\otimes\overline M $. In the following proposition, we show that the set $\Theta(\overline L)$ is filtered with respect to the order $\succ$. \[Pro:filtered\] if $D_1$ and $D_2$ are two admissible decompositions of $\overline L$, then there exists an admissible decomposition $D$ of $\overline{L}$ such that $D\succ D_1$ and $D\succ D_2$. After Remark \[Rem:pull back of admissible decomp\] (1)(2), we may assume that the first and the third components of $D_1$ and $D_2$ are the same. Assume that $D_1=(\nu,\overline{N}_1,p)$ and $D_2=(\nu,\overline{N}_2,p)$, where $\nu: X'\rightarrow X$ is a birational projective morphism. Let $\overline{M}_i=\nu^*\overline{L}^{\otimes p}\otimes\overline{N}_i^\vee$ ($i=1,2$). Since $\overline M_1$ and $\overline M_2$ are effective, there exist homomorphisms $u_i: M_i^\vee\rightarrow\mathcal O_{X'}$ corresponding to effective sections $s_i:\mathcal O_{ X'}\rightarrow M_i$ ($i=1,2$). Let $\eta:X''\rightarrow X'$ be the blow up of the ideal sheaf $\mathrm{Im}(u_1\oplus u_2 )$. Let $ M$ be the exceptional line bundle and $s:\mathcal O_{X''}\rightarrow M$ be the section which trivializes $M$ outside the exceptional divisor. The canonical surjective homomorphism $\eta^*( M_1^\vee\oplus M_2^\vee )\rightarrow M^\vee$ induces by duality an injective homomorphism $\varphi: M\rightarrow M_1\oplus M_2$. We equip $ M_1\oplus M_2$ with metrics $(\|\cdot\|_{\sigma})_{\sigma:K\rightarrow\mathbb C}$ such that, for any $x\in X''_\sigma(\mathbb C)$ and any section $(u,v)$ of $ M_{1,\sigma}\oplus M_{2,\sigma} $ over a neighbourhood of $x$, one has $\|(u,v)\|_{\sigma}(x)=\max\{\|u\|_{\sigma,1}(x),\|v\|_{\sigma,2}(x)\}$. As $\varphi s=(\eta^*s_1,\eta^*s_2)$, and the sections $s_1$ and $s_2$ are effective, one obtains that the section $s$ is also effective. Let $\overline{N}=(\nu\eta)^*\overline{L}^{\otimes p }\otimes\overline{M}^\vee$. One has a natural surjective homomorphism $$\psi:\eta^*{N}_1\oplus \eta^*{N}_2\longrightarrow N.$$ Furthermore, if we equip $\eta^*{N}_1\oplus \eta^*{N}_2$ with metrics $(\|\cdot\|_{\sigma})_{\sigma:K\rightarrow\mathbb C}$ such that, for any $x\in X''_\sigma(x)$, $\|(u,v)\|_\sigma(x)=\|u\|_\sigma(x)+\|v\|_{\sigma}(x)$, then the metrics on $N$ are just the quotient metrics by the surjective homomorphism $\psi$, which are semi-positive since the metrics of $\eta^*\overline N_1$ and of $\eta^*\overline N_2$ are. As both Hermitian line bundles $N_1$ and $N_2$ are generated by effective global sections, also is $\overline{N}$. Therefore, $(\nu\eta,\overline{N},p)$ is an admissible decomposition of $\overline{L}$, which is superior to both $D_1$ and $D_2$. Intersection of admissible decompositions ----------------------------------------- Let $(\overline L_i)_{i=0}^d$ be a family of Hermitian line bundles on $X$. Let $m\in\{0,\cdots d\}$. Assume that $\overline L_i$ is big for any $i\in\{0,\cdots,m\}$ and is integrable for any $i\in\{m+1,\cdots, d\}$. For any $i\in\{0,\cdots,m\}$, let $D_i=(\nu_i:X_i\rightarrow X,\overline N_i,p_i)$ be an admissible decomposition of $\overline L_i$. Choose a birational projective morphism $\nu:X'\rightarrow X$ which factorizes through $\nu_i$ for each $i\in\{0,\cdots,m\}$. Denote by $\eta_i:X_i\rightarrow X$ the projective birational morphism such that $\nu=\nu_i\eta_i$ ($0\leqslant i\leqslant m$). Define $(D_0\cdots D_{m})\cdot\widehat{c}_1(\overline L_{m+1})\cdots\widehat{c}_1(\overline L_d)$ as the normalized intersection product $$\widehat{c}_1(\eta_0^{*}\overline N_{0})\cdots\widehat{c}_1( \eta_{m}^*\overline N_{m})\widehat{c}_1(\nu^*\overline L_{m+1})\cdots\widehat{c}_1(\nu^*\overline L_d)\prod_{i=0}^mp_i^{-1}.$$ This definition does not depend on the choice of $\nu$. \[Pro:decomposed intersection product\] Let $(\overline{L}_i)_{0\leqslant i\leqslant d}$ be a family of Hermitian line bundles on $X$. Let $m\in\{0,\cdots,d\}$. Assume that $\overline{L}_i$ is big for $i\in\{0,\cdots,m\}$, and is nef for $i\in\{m+1,\cdots,d\}$. For any $i\in\{0,\cdots,m\}$, let $D_i$ and $D_i'$ be two admissible decompositions of $\overline{L}_i$ such that $D_i\succ D_i'$. Then $$\label{Equ:comparison of product}(D_0\cdots D_{m})\cdot\widehat{c}_1(\overline{ L}_{m+1})\cdots \widehat{c}_1(\overline{ L}_{d})\geqslant (D_0'\cdots D_{m}')\cdot\widehat{c}_1(\overline{ L}_{m+1})\cdots \widehat{c}_1(\overline{ L}_{d})$$ By substituting progressively $D_i$ by $D_i'$, it suffices to prove that $$(D_0\cdot D_1\cdots D_{m})\cdot \widehat{c}_1(\overline{L}_{m+1})\cdots \widehat{c}_1(\overline{L}_{d})\geqslant (D_0'\cdot D_1\cdots D_{m})\cdot\widehat{c}_1(\overline{ L}_{m+1})\cdots \widehat{c}_1(\overline{ L}_{d}),$$ which is a consequence of Proposition \[Pro:positivity of nef intersection produc\]. With the notation and the assumptions of Proposition \[Pro:decomposed intersection product\], the supremum $$\label{Equ:intersection product D}\sup\Big\{(D_0\cdots D_{m}) \cdot\widehat{c}_1(\overline{L}_{m+1})\cdots \widehat{c}_1(\overline{L}_{d})\;\Big|\; 0\leqslant i\leqslant m,\, D_i\in\Theta(\overline{L}_i)\Big\}$$ exists in $\mathbb R_{\geqslant 0}$. For any $i\in\{0,\cdots,m\}$, let $\overline{A}_i$ be an arithmetically ample Hermitian line bundle on $X$ such that $\overline{A}_i\otimes\overline{L}_i^{\vee}$ is effective. Then all numbers of the set is bounded from above by $\widehat{c}_1(\overline{A}_0)\cdots \widehat{c}_1(\overline{A}_{m})\cdot\widehat{c}_1( \overline{L}_{m+1})\cdots\widehat{c}_1(\overline{ L}_{d})$. Positive intersection product {#SubSec:Posit} ----------------------------- Let $(\overline L_i)_{i=0}^{m}$ be a family of big Hermitian line bundles on $X$, where $0\leqslant m\leqslant d$. Denote by $\big\langle\widehat{c}_1(\overline L_0)\cdots\widehat{c}_1(\overline L_{m})\big\rangle$ the function on $\widehat{\mathrm{Nef}}(X)^{d-m}$ which sends a family of nef Hermitian line bundles $(\overline L_j)_{j=m+1}^d$ to the supremum $$\sup\Big\{(D_0\cdots D_{m})\cdot \widehat{c}_1(\overline L_{m+1})\cdots\widehat{c}_1(\overline L_d)\;\Big|\;0\leqslant i\leqslant m,\, D_i\in\Theta(\overline L_i) \Big\}.$$ Since all $\Theta(\overline L_i)$ are filtered, this function is additive in each $\overline L_j$ ($m+1\leqslant j\leqslant d$). Thus it extends naturally to a multi-linear function on $\widehat{\mathrm{Int}}(X)$ which we still denote by $\big\langle\widehat{c}_1(\overline L_0)\cdots\widehat{c}_1(\overline L_{m})\big\rangle$, called the [*positive intersection product*]{} of $(\overline L_i)_{i=0}^{m}$. \[Rem:property of admi decom\] (1) If all Hermitian vector bundles $(\overline{L}_i)_{i=0}^{m}$ are nef, then the positive intersection product coincides with the usual intersection product. (2) The positive intersection product is homogeneous in each $\overline{L}_i$ ($0\leqslant i\leqslant m$). However, in general it is not additive in each variable. If we consider it as a function on $\widehat{\mathrm{Nef}}(X)$, then it is super-additive in each variable. (3) Assume that all Hermitian line bundles $(\overline{L}_i)_{i=0}^{m}$ are the same. That is, $\overline{L}_0=\cdots=\overline{L}_{m}= \overline{ L }$. We use the expression $\big\langle\widehat{c}_1(\overline{L})^{m+1}\big\rangle$ to denote the positive intersection product $$\big\langle \underbrace{\widehat{c}_1(\overline{ L}) \cdots\widehat{c}_1(\overline{ L})}_{m+1\,\text{copies}}\big\rangle.$$ With this notation, for any $(\overline{ L}_j)_{j=m+1}^{d}\in\widehat{\mathrm{Nef}}( X)^{d-m}$, one has $$\begin{split}&\quad\;\big\langle\widehat{c}_1(\overline{L} )^m\big\rangle\cdot\widehat{c}_1(\overline{ L}_{m+1})\cdots \widehat{c}_1(\overline{ L}_{d})=\sup_{D\in\Theta(\overline{L })}(D\cdots D)\cdot\widehat{c}_1(\overline{L}_{m+1})\cdots \widehat{c}_1(\overline{L}_{d}). \end{split}$$ This equality comes from the fact that the ordered set $\Theta(\overline{L})$ is filtered (Proposition \[Pro:filtered\]) and from the comparison . In particular, the Fujita’s approximation theorem (see [@Chen_Fujita] and [@Yuan08]) implies that $\big\langle\widehat{c}_1(\overline{ L })^{d+1}\big\rangle=\widehat{\mathrm{vol}}(\overline{ L })$. \[Lem:comparaison des produVit positive\] Let $(\overline L_i)_{i=0}^m$ be a family of big Hermitian line bundles on $X$, where $m\in\{0,\cdots,d\}$. For any $i\in\{0,\cdots,m\}$, let $\overline M_i$ be an effective Hermitian line bundle on $X$ and let $\overline N_i=\overline L_i\otimes \overline M_i$. Then one has $$\label{Equ:comparaison de produit positif} \big\langle\widehat{c}_1(\overline L_0)\cdots\widehat{c}_1(\overline L_m)\big\rangle\geqslant\big\langle\widehat{c}_1(\overline N_0 )\cdots\widehat{c}_1(\overline N_m)\big\rangle,$$ where we have considered the positive intersection products as functions on $\widehat{\mathrm{Nef}}(X)^{d-m}$. By Remark \[Rem:property of admi decom\] (4), if $D_i$ is an admissible decomposition of $\overline{L}_i$, then it is also an admissible decomposition of $\overline L_i$. Hence by the definition of the positive intersection product, the inequality is true. The following proposition shows that the positive intersection product is continuous in each variable. \[Pro:continuite\] Let $(\overline{L}_i)_{0\leqslant i\leqslant m }$ be a family of big Hermitian line bundles on $X$, where $m\in\{0,\cdots,d\}$. Let $(\overline{ M}_i)_{0\leqslant i\leqslant m}$ be a family of Hermitian line bundles on $\mathscr X$. Then $$\label{Equ:continuity}\lim_{n\rightarrow\infty}n^{-m} \big\langle \widehat{c}_1(\overline{L}_0^{\otimes n}\otimes\overline{ M}_0)\cdots\widehat{c}_1(\overline{L}_m^{\otimes n}\otimes\overline{M}_m)\big\rangle= \big\langle\widehat{c}_1(\overline{L}_0 )\cdots\widehat{c}_1(\overline{L}_m )\big\rangle$$ We consider firstly both positive intersection products as functions on $\widehat{\mathrm{Nef}}(X)$. Let $\alpha_n=\big\langle\widehat{c}_1(\overline L_0^{\otimes n} \otimes\overline M_0)\cdots\widehat{c}_1(\overline L_m^{\otimes n} \otimes\overline M_m)\big\rangle$. Since $\overline L_i$ is big, there exists an integer $q\geqslant 1$ such that the Hermitian line bundles $\overline L_i^{\otimes q}\otimes\overline M_i$ and $\overline L_i^{\otimes q}\otimes\overline M_i^\vee$ are both effective. Thus the Lemma \[Lem:comparaison des produVit positive\] implies that $$\begin{gathered} \alpha_n\geqslant \big\langle\widehat{c}_1(\overline L_0^{\otimes(n-q)})\cdots\widehat{c}_1(\overline{ L}_m^{\otimes(n-q)})\big\rangle=(n-q)^m\big\langle\widehat{c}_1 (\overline L_0)\cdots\widehat{c}_1(\overline L_m) \big\rangle,\\ \alpha_n\leqslant \big\langle\widehat{c}_1(\overline L_0^{\otimes(n+q)})\cdots\widehat{c}_1(\overline{L}_m^{ \otimes(n+q)})\big\rangle=(n+q)^m\big\langle \widehat{c}_1(\overline L_0)\cdots\widehat{c}_1(\overline L_m) \big\rangle.\end{gathered}$$ By passing to limit, we obtain as an equality of functions on $\widehat{\mathrm{Nef}}(X)^{d-m}$. The general case follows from the multi-linearity. \[Rem:continuite\] Proposition \[Pro:continuite\] implies in particular that, if $(f_n^{(i)})_{n\geqslant 1}$ ($i=0,1,\cdots,m$) are families of continuous functions on $X(\mathbb C)$ which converge uniformly to zero. Then one has $$\lim_{n\rightarrow+\infty}\big\langle\widehat{c}_1(\overline L_0(f_0))\cdots\widehat{c}_m(\overline L_m(f_m))\big\rangle=\big\langle\widehat{c}_1(\overline L_0)\cdots\widehat{c}_m(\overline L_m)\big\rangle.$$ In particular, the mapping $$t\mapsto\big\langle \widehat{c}_1(\overline L_0(t))\cdots\widehat{c}_1(\overline L_m(t))\big\rangle$$ is continuous on the (open) interval that it it well defined. \[Pro:positivite de positive intersection product\] Let $(\overline L_i)_{i=0}^{d-1}$ be a family of big Hermitian line bundles on $X$. If $\overline M$ is an effective integrable Hermitian line bundle on $X$, then $$\big\langle\widehat{c}_1(\overline L_0)\cdots\widehat{c}_1(\overline L_{d-1})\big\rangle \cdot\widehat{c}_1(\overline M)\geqslant 0.$$ This is a direct consequence of Proposition \[Pro:positivity of nef intersection produc\]. \[Rem:generations of positive product\] Proposition \[Pro:positivite de positive intersection product\] permits us to extend the function $\big\langle\widehat{c}_1(\overline L_0)\cdots\widehat{c}_1(\overline L_{d-1})\big\rangle$ on $\widehat{\mathrm{Pic}}(X)$. Let $\overline M$ be an arbitrary Hermitian line bundle on $X$. By Weierstrass-Stone theorem, there exists a sequence $(f_n)_{n\geqslant 1}$ of continuous functions on $X(\mathbb C)$ which converges uniformly to $0$, and such that $\overline M(f_n)$ is of smooth metrics for any $n$. Thus $\overline M(f_n)$ is integrable and $a_n=\big\langle\widehat{c}_1(\overline L_0)\cdots\widehat{c}_1(\overline L_{d-1})\big\rangle\cdot\widehat{c}_1(\overline M(f_n))$ is well defined. Let $\varepsilon_{n,m}=\|f_n-f_m\|_{\sup}$. Choose an ample Hermitian line bundle $\overline A$ such that $\overline A\otimes\overline L_{i}^\vee$ is effective for any $i\in\{0,\cdots,d-1\}$. Note that $$\label{Equ:difference a m et an}\begin{split}a_n-a_m&=\big\langle\widehat{c}_1(\overline L_0)\cdots\widehat{c}_1(\overline L_{d-1})\big\rangle\cdot\widehat{c}_1(\overline{\mathcal O }(f_n-f_m))\\ &\leqslant \langle\widehat{c}_1(\overline L_0)\cdots\widehat{c}_1(\overline L_{d-1})\big\rangle\cdot\widehat{c}_1(\overline{\mathcal O }(\varepsilon_{n,m}))\\ &\leqslant\big\langle\widehat{c}_1(\overline A)^d\big\rangle\cdot\widehat{c}_1(\overline{\mathcal O}(\varepsilon_{n,m}))=\varepsilon_{n,m}c_1(A_K)^d \end{split}$$ By interchanging the roles of $n$ and $m$ in and then combining the two inequalities, one obtains $|a_n-a_m|\leqslant\varepsilon_{n,m}c_1(A_K)^d$. Therefore, $(a_n)_{n\geqslant 1}$ is a Cauchy sequence which converges to a real number which we denote by $\big\langle\widehat{c}_1(\overline L_0)\cdots\widehat{c}_1(\overline L_{d-1})\big\rangle\cdot\widehat{c}_1(\overline M)$. By an argument similar to the inequality , this definition does not depend on the choice of the sequence $(f_n)_{n\geqslant 1}$. The extended function is additive on $\widehat{\mathrm{Pic}}(X)$, which is positive on the subgroup of effective Hermitian line bundles, and satisfies the conclusion of Proposition \[Pro:continuite\]. Differentiability of the arithmetic volume function =================================================== In this section, we establish the differentiability of the arithmetic volume function. We begin by presenting the following lemma, which is analogous to [@Bou_Fav_Mat06 Corollary 3.4]. \[Lem:Siu-Yuan\] Let $\overline L$ and $\overline N$ be two nef Hermitian line bundles on $X$. Let $\overline M $ be an integrable Hermitian line bundle on $X$. Assume that $\overline M\otimes\overline N $ and $\overline M^\vee\otimes\overline N$ are nef and that $\overline L^\vee\otimes\overline N$ is effective. Then there exists a constant $C>0$ only depending on $d$ such that $$\widehat{\mathrm{vol}}(\overline L^{\otimes n} \otimes\overline M)\geqslant n^{d+1}\widehat{\mathrm{vol}}(\overline L) +(d+1)n^d\widehat{c}_1(\overline L)^{d}\widehat{c}_1(\overline M)-C\widehat{\mathrm{vol}}(\overline N)n^{d-1}.$$ Recall that (see [@Yuan07 Theorem 2.2], see also [@Moriwaki07 Theorem 5.6]) if $\overline A$ and $\overline B$ are two nef Hermitian line bundles on $X$, then $$\label{Equ:Yuan's inequality}\widehat{\mathrm{vol}}(\overline B\otimes\overline A^\vee) \geqslant \widehat{c}_1(\overline B)^{d+1}-\widehat{c}_1(\overline B)^d\widehat{c}_1(\overline A).$$ Let $\overline B=\overline L^{\otimes n}\otimes\overline M\otimes\overline N$. It is a nef Hermitian line bundle on $X$. If one applies on $\overline B$ and on $\overline A=\overline N$, one obtains $$\begin{split}\widehat{\mathrm{vol}}(\overline L^{\otimes n} \otimes\overline M)&=\widehat{\mathrm{vol}} (\overline B\otimes\overline N^\vee)\geqslant \widehat{c}_1(\overline B)^{d+1}-(d+1)\widehat{c}_1(\overline B)^d\widehat{c}_1(\overline N)\\ &=n^{d+1}\widehat{c}_1(\overline L)^{d+1}+ (d+1)n^d\widehat{c}_1(\overline L)^d\widehat{c}_1(\overline M)+O(n^{d-1}), \end{split}$$ where the implicit constant is a linear combination of intersection numbers of Hermitian line bundles of the form $\overline L $ or $\overline M\otimes\overline N$, and hence can be bounded from above by a multiple of $\widehat{c}_1(\overline N)^{d+1}=\widehat{\mathrm{vol}}(\overline N)$, according to Proposition \[Pro:positivity of nef intersection produc\]. In [@Yuan07], Yuan has actually proved a stronger inequality by replacing the $\widehat{\mathrm{vol}}(\overline B\otimes\overline A^\vee)$ in by $$\lim_{n\rightarrow+\infty}\frac{\chi(\pi_*(\overline B^{\otimes n}\otimes\overline A^{\vee\otimes n}))}{n^{d+1}/(d+1)!}.$$ In fact, this quantity is always bounded from above by $\widehat{\mathrm{vol}}(\overline B\otimes\overline A^\vee)$. We first assume that the metrics of $\overline M$ are smooth. Choose an ample Hermitian line bundle $\overline N$ such that $\overline N\otimes\overline M$ and $\overline N\otimes\overline M^\vee$ are ample, and that $\overline N\otimes\overline L^\vee $ is effective. Let $D=(\nu:X'\rightarrow X,\overline A,p)$ be an admissible decomposition of $\overline L$. One has $$\widehat{\mathrm{vol}}(\overline L^{\otimes n} \otimes\overline M)=p^{-d-1} \widehat{\mathrm{vol}}(\nu^*\overline L^{\otimes np}\otimes\nu^*\overline M^{\otimes p})\geqslant p^{-d-1}\widehat{\mathrm{vol}}(\overline A^{\otimes n}\otimes\overline{M}^{\otimes p }).$$ Note that $\overline N^{\otimes p}\otimes\overline M^{\otimes p }$ and $\overline N^{\otimes p}\otimes\overline M^{\vee\otimes p}$ are nef and $$\overline N^{\otimes p }\otimes\overline A^{\vee}=(\overline N\otimes \overline L^{\vee})^{\otimes p}\otimes(\overline L^{\otimes p }\otimes\overline A^\vee)$$ is effective. After Lemma \[Lem:Siu-Yuan\], one obtains $$\widehat{\mathrm{vol}}(\overline A^{\otimes n} \otimes\overline M^{\otimes p}) \geqslant n^{d+1}\widehat{c}_1(\overline A)^{d+1}+ (d+1)n^dp\widehat{c}_1(\overline A)^d \widehat{c}_1(\overline M)-Cp^{d+1}\widehat{\mathrm{vol}}(\overline N )n^{d-1}.$$ Therefore, $$\widehat{\mathrm{vol}}(\overline L^{\otimes n} \otimes\overline M)\geqslant n^{d+1}(D^{d+1})+(d+1)n^d(D^d)\cdot\widehat{c}_1(\overline M)-C\widehat{\mathrm{vol}}(\overline N )n^{d-1}.$$ Since $D $ is arbitrary, one has $$\label{Equ:estimation de siu} \widehat{\mathrm{vol}}(\overline L^{\otimes n} \otimes\overline M)\geqslant n^{d+1}\widehat{\mathrm{vol}}(\overline L )+(d+1)n^d\big\langle\widehat{c}_1(\overline L )^d\big\rangle\cdot\widehat{c}_1(\overline M)-C\widehat{\mathrm{vol}}(\overline N)n^{d-1}.$$ By passing to limit, one obtains $$\liminf_{n\rightarrow+\infty}\frac{\widehat{\mathrm{vol}} (\overline L^{\otimes n}\otimes\overline M)-\widehat{\mathrm{vol}}(\overline L^{\otimes n})}{n^d}\geqslant (d+1)\big\langle\widehat{c}_1(\overline L )^d\rangle\cdot\widehat{c}_1(\overline M).$$ If we apply in replacing $\overline L$ by $\overline L^{\otimes n}\otimes\overline M$, $\overline M$ by $\overline M^{\vee\otimes n}$ and $\overline N$ by $\overline N^{\otimes 2n}$, we obtain $$\widehat{\mathrm{vol}}(\overline L^{\otimes n^2}) \geqslant n^{d+1}\widehat{\mathrm{vol}}(\overline L^{\otimes n}\otimes \overline M)-(d+1)n^{d+1}\big\langle\widehat{c}_1(\overline L^{\otimes n}\otimes\overline M)^d\big\rangle\cdot\widehat{c}_1(\overline M)-C(2n)^{d+1}\widehat{\mathrm{vol}}(\overline N) n^{d-1},$$ or equivalently $$\widehat{\mathrm{vol}}(\overline L^{\otimes n}) \geqslant\widehat{\mathrm{vol}}(\overline L^{\otimes n}\otimes\overline M)- (d+1)\big\langle\widehat{c}_1(\overline L^{\otimes n}\otimes\overline M)^d\big\rangle\cdot\widehat{c}_1(\overline M)-2^{d+1}C\widehat{\mathrm{vol}}(\overline N)n^{d-1}.$$ Thus $$\begin{split}\limsup_{n\rightarrow\infty}\frac{\widehat{\mathrm{vol}} (\overline L^{\otimes n}\otimes\overline M)-\widehat{\mathrm{vol}}(\overline L^{\otimes n})}{n^d}&\leqslant\lim_{n\rightarrow\infty}n^{-d}(d+1) \big\langle\widehat{c}_1(\overline L^{\otimes n}\otimes\overline M)^d\big\rangle\cdot\widehat{c}_1(\overline M)\\ &=(d+1)\big\langle\widehat{c}_1(\overline L)^d\big\rangle\cdot\widehat{c}_1(\overline M). \end{split}$$ Therefore, one has $$\lim_{n\rightarrow+\infty}\frac{\widehat{\mathrm{vol}} (\overline L^{\otimes n}\otimes\overline M)-\widehat{\mathrm{vol}}(\overline L^{\otimes n})}{n^d}=(d+1)\big\langle\widehat{c}_1(\overline L )^d\rangle\cdot\widehat{c}_1(\overline M).$$ For the general case, by Weierstrass-Stone theorem, for any $\varepsilon>0$, there exist two Hermitian line bundles with smooth metrics $\overline M_{\varepsilon,1}=(M,(\|\cdot\|_{\sigma,\varepsilon}')_{\sigma:K\rightarrow\mathbb C })$ and $\overline M_{\varepsilon,2}=(M,(\|\cdot\|_{\sigma,\varepsilon}'')_{\sigma:K\rightarrow\mathbb C })$ such that $$\|\cdot\|_{\sigma,\varepsilon}'\leqslant\|\cdot\|_\sigma \leqslant\|\cdot\|_{\sigma,\varepsilon}'',\quad\text{and}\quad \max_{\sigma}\sup_{x\in X_\sigma(\mathbb C)} \big| \log\|\cdot\|_{\sigma,\varepsilon}'(x)- \log\|\cdot\|_{\sigma,\varepsilon}''(x)\big| \leqslant\varepsilon,$$ where $\|\cdot\|_\sigma$ is the norm of index $\sigma$ of $\overline M$. Note that $\overline M_{\varepsilon,2}\leqslant\overline M\leqslant\overline M_{\varepsilon,1} $. By the special case that we have proved, one has $$\begin{split}&\quad\;(d+1)\big\langle\widehat{c}_1(\overline L)^d\big\rangle\cdot \widehat{c}_1(\overline M_{\varepsilon,2})\leqslant\liminf_{n\rightarrow+\infty}\frac{\widehat{\mathrm{vol}} (\overline L^{\otimes n}\otimes\overline M)-\widehat{\mathrm{vol}}(\overline L^{\otimes n})}{n^d}\\&\leqslant\limsup_{n\rightarrow+\infty}\frac{\widehat{\mathrm{vol}} (\overline L^{\otimes n}\otimes\overline M)-\widehat{\mathrm{vol}}(\overline L^{\otimes n})}{n^d}\leqslant (d+1)\big\langle\widehat{c}_1(\overline L )^d\big\rangle\cdot\widehat{c}_1(\overline M_{\varepsilon,1} )\end{split}$$ Let $f_\varepsilon:X(\mathbb C)\rightarrow\mathbb R$ be the function such that $\log\|\cdot\|_{\sigma,\varepsilon}''(x)= \log\|\cdot\|_{\sigma,\varepsilon}'(x)+f_\varepsilon(x)$. Denote by $\overline{\mathcal O}(f_\varepsilon)$ the Hermitian line bundle on $X$ whose underlying Hermitian line bundle is trivial, and such that $\|\mathbf{1}\|(x)=e^{-f_\varepsilon(x)}$. It is an effective Hermitian line bundle since $f_{\varepsilon}\geqslant 0$. Furthermore, one has $\overline M_{\varepsilon,1}\otimes\overline M_{\varepsilon, 2}^\vee\cong\overline{\mathcal O}(f_\varepsilon)$. Let $\overline F$ be an ample Hermitian line bundle on $X$ such that $\overline F\otimes\overline L^\vee$ is effective. One has $$\begin{split}&\quad\;\big\langle\widehat{c}_1(\overline L)^d\big\rangle \cdot(\widehat{c}_{1}(\overline M_{\varepsilon,2})-\widehat{c}_1(\overline M_{\varepsilon,1}))=\big\langle\widehat{c}_1(\overline L)^d\big\rangle\cdot\widehat{c}_1(\overline {\mathcal O}(f_\varepsilon) )\\ &\leqslant\big\langle\widehat{c}_1(\overline L)^d\big\rangle\cdot\widehat{c}_1(\overline {\mathcal O}(\varepsilon) )\leqslant \widehat{c}_1(\overline F)^d\widehat{c}_1(\overline {\mathcal O}(\varepsilon) )= \varepsilon\int_{X(\mathbb C)}{c}_1(\overline F)^d,\end{split}$$ where in the first inequality, we have applied Proposition \[Pro:positivite de positive intersection product\] (see also Remark \[Rem:generations of positive product\]), and in the second inequality, we have used the fact that $\overline{\mathcal O}(\varepsilon)$ is nef and then applied Lemme \[Lem:comparaison des produVit positive\]. Since $\varepsilon$ is arbitrary, we obtain that $$D_{\overline L }\widehat{\mathrm{vol}}(\overline M ):=\lim_{n\rightarrow+\infty}\frac{\widehat{\mathrm{vol}} (\overline L^{\otimes n}\otimes\overline M)-\widehat{\mathrm{vol}}(\overline L^{\otimes n})}{n^d}$$ exists in $\mathbb R$. Since $$(d+1)\big\langle\widehat{c}_1(\overline L)^d\big\rangle\cdot \widehat{c}_1(\overline M_{\varepsilon,2})\leqslant D_{\overline L }\widehat{\mathrm{vol}}(\overline M )\leqslant (d+1)\big\langle\widehat{c}_1(\overline L )^d\big\rangle\cdot\widehat{c}_1(\overline M_{\varepsilon,1} ),$$ by virtue of Remark \[Rem:generations of positive product\], we obtain $ D_{\overline L }\widehat{\mathrm{vol}}(\overline M )=(d+1)\big\langle\widehat{c}_1(\overline L )^d\big\rangle\cdot\widehat{c}_1(\overline M)$, which is additive with respect to $\overline M$. A direct consequence of Theorem \[Thm:main theorem\] is the asymptotic orthogonality of arithmetic Fujita approximation. Assume that $\overline L$ is a big Hermitian line bundle on $X$. One has $$\label{Equ:vol comm intersection}\big\langle \widehat{c}_1(\overline L)^d\big\rangle\cdot\widehat{c}_1(\overline L) =\widehat{\mathrm{vol}}(\overline L).$$ By definition, $$\begin{split} &\quad\;D_{\overline L}\widehat{\mathrm{vol}}(\overline L)=\lim_{n\rightarrow+\infty}\frac{\widehat{\mathrm{vol}}(\overline L^{n+1} )-\widehat{\mathrm{vol}}(\overline L^{\otimes n })}{n^d}\\&=\widehat{\mathrm{vol}}(\overline L) \lim_{n\rightarrow+\infty}\frac{(n+1)^{d+1}-n^{d+1}}{n^d} =(d+1)\widehat{\mathrm{vol}}(\overline L). \end{split}$$ So follows from Theorem \[Thm:main theorem\]. As mentioned in Introduction, the differentiability of the geometrical volume function can be obtained by using the method of Okounkov bodies developed in [@Okounkov96]. See [@Lazarsfeld_Mustata08] for a proof of this result and other interesting results concerning geometric volume functions. Recently, Yuan [@Yuan08] has proposed a partial analogue of the construction of Lazarsfeld and Musţatǎ in Arakelov geometry. In fact, he has defined the Okounkov bodies of a Hermitian line bundle with respect to so-called “vertical flags”. This permits him to obtain the log-concavity of the arithmetic volume function and the analogue of Fujita’s approximation theorem in Arakelov goemetry, where the latter has also been independently obtained by the present author [@Chen_Fujita], using asymptotic measures. Quite possibly, an analogue of Lazarsfeld and Musţatǎ’s construction with respect to “horizontal flags” could also imply the differentiability of the arithmetic volume function. Applications and comparisons ============================ In this section, we shall apply our differentiability result to study several arithmetic invariants of Hermitian line bundles. Asymptotic measure ------------------ Let $\overline L$ be a Hermitian line bundle on $X$ such that $L_K$ is big. The [*asymptotic measure*]{} of $\overline L$ is the vague limit in the space of Borel probability measures $$\label{Equ:nu}\nu_{\overline L}:=-\lim_{n\rightarrow+\infty}\frac{\mathrm d}{\mathrm{d}t} \frac{\operatorname{rk}\Big(\mathrm{Vect}_K\big(\{s\in\pi_*L^{\otimes n }\mid\forall\sigma,\,\|s\|_{\sigma,\sup}\leqslant e^{-\lambda n} \}\big)\Big)}{\operatorname{rk}(\pi_*L^{\otimes n })},$$ where the derivative is taken in the sense of distribution. It is also the limit of normalized Harder-Narasimhan measures (cf. [@Chen08; @Chen_bigness; @Chen_Fujita]). Note that the support of the probability measure $\nu_{\overline L}$ is contained in $]-\infty,\widehat{\mu}_{\max}^\pi(\overline L)]$, where $\widehat{\mu}_{\max}^\pi(\overline L)$ is the limit of maximal slopes (see [@Chen08 Theorem 4.1.8]): $$\widehat{\mu}_{\max}^{\pi}(\overline L):=\lim_{n\rightarrow+\infty} \frac{\widehat{\mu}_{\max}(\pi_*\overline L^{\otimes n})}{n}.$$ Recall that in [@Chen_bigness Theorem 5.5], the present author has proved that $\widehat{\mu}_{\max}^{\pi}(\overline L)>0$ if and only if $L_K$ is big. Furthermore, by definition, one has $\widehat{\mu}_{\max}^{\pi}(\overline L(a))=\widehat{\mu}_{\max}^{\pi}(\overline L)+a$ for any $a\in\mathbb R$. Therefore, $\widehat{\mu}_{\max}^{\pi}(\overline L)$ is also the infimum of all real numbers $\varepsilon$ such that $\overline L(-\varepsilon)$ is big. The asymptotic measure is a very general arithmetic invariant. Many arithmetic invariants of $\overline L$ can be represented as integrals with respect to $\nu_{\overline L}$. In the following, we discuss some examples. The [*asymptotic positive slope*]{} of $\overline L$ is defined as $$\widehat{\mu}_+^\pi(\overline L):=\frac{1}{[K:\mathbb Q]} \frac{\widehat{\mathrm{vol}}(\overline L)}{(d+1)\mathrm{vol}(L_K)}.$$ In [@Chen_bigness], the author has proved that $\widehat{\mu}_+^\pi(\overline L)$ is also the maximal value of the asymptotic Harder-Narasimhan polygon of $\overline L$ and that the asymptotic positive slope has the following integral form: $$\widehat{\mu}_+^\pi(\overline L)=\int_{\mathbb R}\max(x,0)\,\nu_{\overline L}(\mathrm{d}x).$$ More generally, for any $a\in\mathbb R$, one has $$\label{Equ:call et mu plus} \int_{\mathbb R}\max(x-a,0)\,\nu_{\overline L}(\mathrm{d}x)=\widehat{\mu}_+^\pi(\overline L(-a)).$$ Another important example is the [*asymptotic slope*]{} of $\overline L$, which is $$\widehat{\mu}^\pi(\overline L):=\frac{1}{[K:\mathbb Q]} \frac{S(\overline L)}{(d+1)\mathrm{vol}(L_K)}\in[-\infty,+\infty[,$$ where $S(\overline L)$ is the sectional capacity of $\overline L$ as in . The asymptotic slope has the following integral form $$\widehat{\mu}^\pi(\overline L)=\int_{\mathbb R}x\,\nu_{\overline L} (\mathrm{d}x).$$ Observe that we have $$\widehat{\mu}_{\max}^\pi(\overline L)\geqslant \widehat{\mu}_+^\pi(\overline L)\geqslant\widehat{\mu}^\pi(\overline L).$$ Using Theorem \[Thm:main theorem\] and the differentiability of geometric volume function in [@Bou_Fav_Mat06], we prove that the asymptotic positive slope $\widehat{\mu}_+^\pi$ is differentiable and calculate its differential. \[Pro:derivabilite de mu plus\] Assume that $\overline L$ is a big Hermitian line bundle on $X$. For any Hermitian line bundle $\overline M$, one has $$D_{\overline L}\widehat{\mu}_+^\pi(\overline M):=\lim_{n\rightarrow+\infty} \big({ \widehat{\mu}_+^\pi(\overline L^{\otimes n}\otimes\overline M )-\widehat{\mu}_+^\pi(\overline L^{\otimes n})}\big)$$ exists in $\mathbb R$. Furthermore, one has $$D_{\overline L}\widehat{\mu}_+^\pi(\overline M):= \frac{\big\langle\widehat{c}_1(\overline L)^d\big\rangle\cdot\widehat{c}_1(\overline M )}{[K:\mathbb Q]\mathrm{vol}(L_K)}-\frac{d\big\langle c_1(L_K)^{d-1}\big\rangle\cdot c_1(M_K) }{\mathrm{vol}(L_K)}\widehat{\mu}_+^\pi(\overline L),$$ where $\big\langle c_1(L_K)^{d-1}\big\rangle\cdot c_1(M_K)$ is the geometric positive intersection product ([@Bou_Fav_Mat06 §2]). This is a direct consequence of Theorem \[Thm:main theorem\] and [@Bou_Fav_Mat06 Theorem A], where the latter asserts that $$\lim_{n\rightarrow+\infty}\frac{\mathrm{vol}(L_K^{\otimes n} \otimes M_K)-\mathrm{vol}(L_K^{\otimes n})}{n^{d-1}}=d\big\langle c_1(L_K)^{d-1}\big\rangle\cdot c_1(M_K).$$ We then deduce from Proposition \[Pro:derivabilite de mu plus\] the expression of the distribution function of the measure $\nu_{\overline L}$. \[Pro:distribution function\] The distribution function $F_{\overline L}$ of $\nu_{\overline L}$ satisfies the equality $$F_{\overline L}(a):=\nu_{\overline L}(]-\infty,a])=1-\frac{\big\langle\widehat{c}_1(\overline L(-a))^d\big\rangle\cdot\widehat{c}_1(\overline{\mathcal O}(1))}{[K:\mathbb Q]\mathrm{vol}(L_K)}, \quad a<\widehat{\mu}_{\max}^\pi(\overline L).$$ One has $$F_{\overline L}(a)=1+\frac{\mathrm{d}}{\mathrm{d}a}\int_{\mathbb R}\max(x-a,0)\,{\nu}(\mathrm{d}x)=1+\frac{\mathrm{d}}{\mathrm{d}a} \widehat{\mu}_+^\pi(\overline L(-a)).$$ By Proposition \[Pro:derivabilite de mu plus\], one obtains $$\frac{\mathrm{d}}{\mathrm{d}a}\widehat{\mu}_+^\pi(\overline L(-a))=-\frac{\big\langle\widehat{c}_1(\overline L(-a))^d\big\rangle\cdot\widehat{c}_1(\overline{\mathcal O}(1))}{[K:\mathbb Q]\mathrm{vol}(L_K)}.$$ 1) Since the support of $\nu_{\overline L}$ is bounded from above by $\widehat{\mu}_{\max}(\overline L )$, one has $F_{\overline L}(a)=1$ for $a\geqslant\widehat{\mu}_{\max}(\overline L )$. 2) As a consequence of Proposition \[Pro:distribution function\], one obtains that the function $$\frac{\big\langle\widehat{c}_1(\overline L(-a))^d\big\rangle\cdot\widehat{c}_1(\overline{\mathcal O}(1))}{[K:\mathbb Q]\mathrm{vol}(L_K)}$$ is decreasing with respect to $a$ on $]-\infty,\widehat{\mu}_{\max}^\pi(\overline L)[$, which is also implied by Lemma \[Lem:comparaison des produVit positive\]. Furthermore, this function takes values in $]0,1]$, and converges to $1$ when $a\rightarrow-\infty$. 3) Let $a\in]-\infty,\widehat{\mu}_{\max}^\pi(\overline L)[$. The restriction of $$\frac{1}{[K:\mathbb Q]\mathrm{vol}(L_K)}\big\langle \widehat{c}_1(\overline L(-a))^d\big\rangle$$ on $C^0(X(\mathbb C))$ (considered as a subgroup of $\widehat{\operatorname{Pic}}(X)$ via the mapping $f\mapsto\mathcal O(f)$) is a positive linear functional, thus corresponds to a Radon measure on $X(\mathbb C)$. Furthermore, by 1), its total mass is bounded from above by $1$, and converges to $1$ when $a\rightarrow-\infty$. 4) After Remark \[Rem:continuite\], we observe from Proposition \[Pro:distribution function\] that the only possible discontinuous point of the distribution function $F_{\overline L}(a)$ is $a=\widehat{\mu}_{\max}^\pi(\overline L)$. As an application, we calculate the sectional capacity in terms of positive intersection product. Let $\overline L$ be a Hermitian line bundle on $X$ such that $L_K$ is big. Let $A=\widehat{\mu}_{\max}^\pi(\overline L)$. One has $$S(\overline L)=(d+1)A\lim_{x\rightarrow A-}\big\langle\widehat{c}_1(\overline L(-x))^d\big\rangle\cdot\widehat{c}_1(\overline {\mathcal O}(1))-\int_{-\infty}^{A} (d+1)x\,\mathrm{d}\big\langle\widehat{c}_1(\overline L(-x))^d\big\rangle\cdot\widehat{c}_1(\overline{\mathcal O}(1)) .$$ Lower bound of the positive intersection product ------------------------------------------------ Our differentiability result permits to obtain a lower bound for positive intersection products of the form $\big\langle\widehat{c}_1(\overline L)^d\big\rangle\cdot\widehat{c}_1(\overline M)$, where $\overline L$ is a big Hermitian line bundle on $X$ and $\overline M$ is an effective Hermitian lien bundle on $X$, by using the log-concavity of the arithmetic volume function proved in [@Yuan08]. Let $\overline L$ and $\overline M$ be two Hermitian line bundles on $X$. Assume that $\overline L$ is big and $\overline M$ is effective. Then $$\label{Equ:isoperimetric}\big\langle\widehat{c}_1(\overline L)^d\big\rangle\cdot\widehat{c}_1(\overline M)\geqslant \mathrm{vol}(\overline L)^{\frac{d}{d+1}}\mathrm{vol}(\overline M)^{\frac 1{d+1}}.$$ Theorem \[Thm:main theorem\] shows $$\lim_{n\rightarrow+\infty}\frac{\mathrm{vol}(\overline L^{\otimes n}\otimes\overline M)-\mathrm{vol}(\overline M)}{n^d}=(d+1)\big\langle\widehat{c}_1(\overline L)^d\big\rangle\cdot\widehat{c}_1(\overline M).$$ By [@Yuan08 Theorem B], one has $$\widehat{\mathrm{vol}}(\overline L^{\otimes n}\otimes\overline M)\geqslant\Big(\widehat{\mathrm{vol}}(\overline L^{\otimes n})^{\frac{1}{d+1}}+\widehat{\mathrm{vol}}(\overline M)^{\frac{1}{d+1}}\Big)^{d+1}.$$ By passing to limit, we obtain the required inequality. The inequality could be considered as an analogue in Arakelov geometry (suggested by Bertrand [@Bertrand95]) of the [*isoperimetric inequality*]{} proved by Federer [@Federer69 3.2.43]. See [@Fulton93 §5.4] for an interpretation in terms of intersection theory, and [@Bertrand95 §1.2] for an analogue in geometry of numbers. Comparison to other differentiability results --------------------------------------------- We finally compare our results to several differentiability results on arithmetic invariants. ### Intersection number {#intersection-number .unnumbered} Recall that the self-intersection number $\widehat{c}_1(\overline L)^{d+1}$ is well defined for integrable Hermitian line bundles $\overline L$. See [@Gillet-Soule90; @Zhang95; @Zhang95b]. Furthermore, it is a polynomial function. Therefore, for any integrable Hermitian line bundles $\overline L$ and $\overline M$, one has $$\lim_{n\rightarrow+\infty}\frac{\widehat{c}_1(\overline L^{\otimes n}\otimes\overline M)^{d+1}-\widehat{c}_1(\overline L^{\otimes n})}{n^d} =(d+1)\widehat{c}_1(\overline L)^{d}\widehat{c}_1(\overline M).$$ This formula shows that the intersection number is differentiable at $\overline L$ along all directions in $\widehat{\mathrm{Int}}(X)$. ### Sectional capacity {#sectional-capacity .unnumbered} By using the analogue of Siu’s inequality in Arakelov geometry, Yuan [@Yuan07] has actually proved that the sectional capacity $S$ is differentiable along integrable directions at any Hermitian line bundle $\overline L$ such that $L$ is ample and that the metrics of $\overline L$ are semi-positive. Furthermore, for such $\overline L$, one has $$D_{\overline L}S(\overline M):=\lim_{n\rightarrow+\infty} \frac{S(\overline L^{\otimes n}\otimes\overline M)-S(\overline L^{\otimes n})}{n^{d}}=(d+1)\widehat{c}_1(\overline L)^d\widehat{c}_1(\overline M),$$ where $\overline M$ is an arbitrary integrable Hermitian line bundles. This result has been established by Autissier [@Autissier01] in the case where $d=1$. Recently Berman and Boucksom [@Ber_Bou08] have proved a general differentiability result for the sectional capacity. They have proved that the function $S$ is differentiable along the directions defined by continuous functions on $X(\mathbb C)$ on the cone of generically big Hermitian lien bundles. Namely, for any continuous function $f$ on $X(\mathbb C)$ and any Hermitian line bundle $\overline L$ on $X$ such that $L_K$ is big and that $S(\overline L)$ is finite, the limit $$\lim_{n\rightarrow+\infty}\frac{S(\overline L^{\otimes n}(f))-S(\overline L^{\otimes n})}{n^d}$$ exists. They have also computed explicitly the differential in terms of the Monge-Ampère measure of $\overline L$ (see Theorem 5.7 and Remark 5.8 [*loc. cit.*]{}). Our differentiability result for arithmetic volume function (Theorem \[Thm:main theorem\]), combined with , implies the differentiability of arithmetic invariants which can be written as the integration of a (fixed) smooth function of compact support with respect to the asymptotic measure of the Hermitian line bundle, by using integration by part. It would be interesting to know if a similar idea permits to deduce the differentiability of the sectional capacity, which can be written as the integral of the function $f(x)=x$ with respect to the asymptotic measure, along any direction at any Hermitian line bundle $\overline L$ such that $L_K$ is big and that $S(\overline L)$ is finite.

--- author: - Shauna Revay - Matthew Teschke bibliography: - 'sample.bib' title: Multiclass Language Identification using Deep Learning on Spectral Images of Audio Signals --- Introduction ============ Recently, voice assistants have become a staple in the flagship products of many big technology companies such as Google, Apple, Amazon, and Microsoft. One challenge for voice assistant products is that the language that a speaker is using needs to be preset. To improve user experience on this and similar tasks such as automated speech detection or speech to text transcription, automatic language detection is a necessary first step. The technique described in this paper, language identification for audio spectrograms (LIFAS), uses spectrograms of raw audio signals as input to a convolutional neural network (CNN) to be used for language identification. One benefit of this process is that it requires minimal pre-processing. In fact, only the raw audio signals are input into the neural network, with the spectrograms generated as each batch is input to the network during training. Another benefit is that the technique can utilize short audio segments (approximately 4 seconds) for effective classification, necessary for voice assistants that need to identify language as soon as a speaker begins to talk. LIFAS binary language classification had an accuracy of 97%, and multi-class classification with six languages had an accuracy of 89%. Background ========== Finding a dataset of audio clips in various languages sufficiently large for training a network was an initial challenge for this task. Many datasets of this type are not open sourced [@mozilla]. VoxForge [@voxforge], an open-source corpus that consists of user-submitted audio clips in various languages, is the source of data used in this paper. Previous work in this area used deep networks as feature extractors, but did not use the networks themselves to classify the languages [@conference; @unified]. LIFAS removes any feature extraction performed outside of the network. The network is fed a raw audio signal, and the spectrogram of the data is passed to the neural network during training. The last layer of the network outputs a vector of probabilities with one prediction per language. Thus, the whole process from raw audio signal to prediction of language is performed automatically by the neural network. In [@lstmpaper], a CNN was combined with a long short-term memory (LSTM) network to classify language using spectrograms generated from audio. The network presented in [@lstmpaper] classified 4 languages using 10-second audio clips for training [@blog], while LIFAS achieves similar performance for 6 languages using 4-second audio clips. This demonstrates the robustness of the architecture and its improvement upon earlier techniques. Residual and Convolutional Neural Networks ------------------------------------------ CNNs have been shown to give state of the art results for image classification and a variety of other tasks. As neural networks using back propagation were constructed to be deeper, with more layers, they ran into the problem of vanishing gradient [@gradient]. A network updates its weights based on the partial derivatives of the error function from the previous layers. Many times, the derivatives can become very small and the weight updates become insignificant. This can lead to a degradation in performance. One way to mitigate this problem is the use of Residual Neural Networks (ResNets [@resnet]). ResNets utilize skip connections in layers which connects two non-adjacent layers. ResNets have shown state-of-the-art performance on image recognition tasks, which makes them a natural choice for a network architecture for this task [@imageresidual]. Spectrogram Generation ---------------------- A spectrogram is an image representation of the frequencies present in a signal over time. The frequency spectrum of a signal can be generated from a time series signal using a Fourier Transform. In practice, the Fast Fourier Transform (FFT) can be applied to a section of the time series data to calculate the magnitude of the frequency spectrum for a fixed moment in time. This will correspond to a line in the spectrogram. The time series data is then windowed, usually in overlapping chunks, and the FFT data is strung together to form the spectrogram image which allows us to see how the frequencies change over time. Since we were generating spectrograms on audio data, the data was converted to the mel scale, generating “melspectrograms”. These images will be referred to as simply “spectrograms” for the duration of this paper. The conversion from $f$ hertz to $m$ mels that we use is given by, $$m = 2595 \log_{10} \left( 1 + \frac{f}{700} \right).$$ An example of a spectrogram generated by an English data transmission is shown in figure \[spec\]. ![Spectrogram generated from an English audio file.[]{data-label="spec"}](spec.png){width="\textwidth"} Data Preparation ================ Audio data was collected from VoxForge [@voxforge]. Each audio signal was sampled at a rate of 16kHz and cut down to be 60,000 samples long. In this context, a sample refers to the number of data points in the audio clip. This equates to 3.75 seconds of audio. The audio files were saved as WAV files and loaded into Python using the librosa library and a sample rate of 16kHz. Each audio file of 60,000 samples was saved separately and is referred to as a clip. The training set consisted of 5,000 clips per language, and the validation set consisted of 2,000 clips per language. Audio clips were gathered in English, Spanish, French, German, Russian, and Italian. Speakers had various accents and were of different genders. The same speakers may be speaking in more than one clip, but there was no cross contamination in the training and validation sets. Spectrograms were generated using parameters similar to the process discussed in [@audioblog] which used a frequency spectrum of 20Hz to 8,000Hz and 40 frequency bins. Each FFT was computed on a window of 1024 samples. No other pre-processing was done on the audio files. Spectrograms were generated on-the-fly on a per-batch basis while the network was running (i.e. spectrograms were not saved to disk). Network ======= We utilized the fast.ai [@fastai] deep learning library built on PyTorch [@pytorch]. The network used was a pretrained Resnet50. The spectrograms were generated on a per-batch basis, with a batch size of 64 images. Each image was $432 \times 288$ pixels in size. During training, the 1-cycle-policy described in [@leslie] was used. In this process, the learning rate is gradually increased and then decreased in a linear fashion during one cycle [@onecycleblog]. The learning rate finder within the fast.ai library was first used to determine the maximum learning rate to be used in the 1-cycle training of the network. The maximum learning rate was then set to be $1 \times 10^{-2}$. The learning rate increases until it hits the maximum learning rate, and then it gradually decreases again. The length of the cycle was set to be 8 epochs, meaning that throughout the cycle 8 epochs are evaluated. Experiments =========== Binary Classification with Varying Number of Samples ---------------------------------------------------- Binary classification was performed on two languages using clips of 60,000 samples. English and Russian were chosen to use for training and validation. To test the impact of the number of samples on classification while keeping the sample rate constant, binary classification was also performed on clips of 100,000 samples. Multiple Language Classification -------------------------------- For each language (English, Spanish, German, French, Russian, and Italian), 5,000 clips were placed in the training set. Each clip was 60,000 samples in length. 2,000 clips per language were placed in the validation set, and no speakers or clips appeared in both the training and validation sets. Results ======= Accuracy was calculated for both binary classification and multiclass classification as: $$Accuracy = \frac{Number \; of \; Correct \; Predictions}{Total \;Number \;of \; Predictions}.$$ LIFAS binary classification accuracy for Russian and English clips of length 60,000 samples was 94%. In comparison, LIFAS binary classification accuracy on the clips of 100,000 samples was 97 %. The accuracy totals given in the experiments above are calculated on the total number of clips in the validation set. The accuracy can also be broken up into accuracy for English clips, or accuracy for Russian clips, where there was essentially no difference in the accuracy for English clips and the accuracy for Russian clips. To confirm that the network performance was not dependent on English and Russian language data, binary classification was tested on other languages with little to no impact on validation accuracy. LIFAS accuracy for the multi-class network with six languages was 89 %. These results were based on clips of 60,000 samples since a sufficient number of longer clips were unavailable. Results from the 100,000 sample clips in the binary classification model suggest performance could be improved in the multi-class setting with longer clips. The confusion matrix for the multi-class classification is shown in figure \[confusion\]. ![The confusion matrix for the multiclass language identification problem.[]{data-label="confusion"}](confusion.png){width="80.00000%"} Discussion and Limitations ========================== Notably, the highest rate of false negative classifications came when Spanish clips were classified as Russian, and when Russian clips were classified as Spanish. Additionally, almost no other language is misclassified as Russian or Spanish. One hypothesis for this observation is the fact that Russian is the only Slavic language in the training set. Therefore, the network may be performing some thresholding at one layer that separates Russian from other languages, and by chance Spanish clips are near the threshold. One limitation in our findings is that all of the data came from the same dataset. Since audio formats can have a wide variety of parameters such as bit rate, sampling rate, and bits per sample, we would expect clips from other datasets collected in different formats to potentially confuse the network. There is potential for this drawback to be overcome assuming appropriate pre-processing steps were taken for the audio signals so that the spectrograms did not contain artifacts from the dataset itself. This is a problem that should be explored as more data becomes available. Conclusion ========== This work shows the viability of using deep network architectures commonly used for image classification in identifying languages from images generated from audio data. Robust performance can be accomplished using relatively short files with minimal pre-processing. We believe that this model can be extended to classify more languages so long as sufficient, representative training and validation data is available. A next step in testing the robustness of this model would be to include test data from additional (e.g. non-VoxForge) datasets. Additionally, we would want the network to be performant on environments with varying levels of noise. VoxForge data is all user submitted audio clips, so the noise profiles of the clips vary, but more regimented tests should be done to see how robust the network is to different measured levels of noise. Simulated additive white Gaussian noise could be added to the training data to simulate low quality audio, but still might not fully mimic the effect of background noise such as car horns, clanging pots, or multiple speakers in a real life environment. Another way to potentially increase the robustness of the model would be to implement SpecAugment [@specaugment] which is a method that distorts spectrogram images in order to help overfitting and increase performance of networks by feeding in deliberately corrupted images. This may help to add scalability and robustness to the network, assuming the spectral distortions generated in SpecAugment accurately represent distortions in audio signals observed in the real world.

--- abstract: 'We have assembled a sample of high spatial resolution far-UV (Hubble Space Telescope Advanced Camera for Surveys Solar Blind Channel) and H$\alpha$ (Maryland-Magellan Tunable Filter) imaging for 15 cool core galaxy clusters. These data provide a detailed view of the thin, extended filaments in the cores of these clusters. Based on the ratio of the far-UV to H$\alpha$ luminosity, the UV spectral energy distribution, and the far-UV and H$\alpha$ morphology, we conclude that the warm, ionized gas in the cluster cores is photoionized by massive, young stars in all but a few (Abell 1991, Abell 2052, Abell 2580) systems. We show that the extended filaments, when considered separately, appear to be star-forming in the majority of cases, while the nuclei tend to have slightly lower far-UV luminosity for a given H$\alpha$ luminosity, suggesting a harder ionization source or higher extinction. We observe a slight offset in the UV/H$\alpha$ ratio from the expected value for continuous star formation which can be modeled by assuming intrinsic extinction by modest amounts of dust (E(B-V) $\sim$ 0.2), or a top-heavy IMF in the extended filaments. The measured star formation rates vary from $\sim$ 0.05 M$_{\odot}$ yr$^{-1}$ in the nuclei of non-cooling systems, consistent with passive, red ellipticals, to $\sim$ 5 M$_{\odot}$ yr$^{-1}$ in systems with complex, extended, optical filaments. Comparing the estimates of the star formation rate based on UV, H$\alpha$ and infrared luminosities to the spectroscopically-determined X-ray cooling rate suggests a star formation efficiency of 14$^{+18}_{-8}$%. This value represents the time-averaged fraction, by mass, of gas cooling out of the intracluster medium which turns into stars, and agrees well with the global fraction of baryons in stars required by simulations to reproduce the stellar mass function for galaxies. This result provides a new constraint on the efficiency of star formation in accreting systems.' author: - 'Michael McDonald, Sylvain Veilleux, David S. N. Rupke, Richard Mushotzky, and Christopher Reynolds' title: Star Formation Efficiency in the Cool Cores of Galaxy Clusters --- Introduction ============ The high densities and low temperatures of the intracluster medium (hereafter ICM) in the cores of some galaxy clusters suggests that massive amounts (100–1000 M$_{\odot}$ yr$^{-1}$) of cool gas should be deposited onto the central galaxy. The fact that this gas reservoir is not observed has been used as prime evidence for feedback-regulated cooling (see review by Fabian 1994). By invoking feedback, either by active galactic nuclei (hereafter AGN) (e.g., Guo [et al. ]{}2008; Rafferty [et al. ]{}2008; Conroy [et al. ]{}2008), mergers (e.g., Gómez [et al. ]{}2002; ZuHone 2010), conduction (e.g., Fabian [et al. ]{}2002; Voigt [et al. ]{}2004), or some other mechanism, theoretical models can greatly reduce the efficiency of ICM cooling, producing a better match with what is observed in high resolution X-ray grating spectra of cool cores (0–100 M$_{\odot}$ yr$^{-1}$, Peterson [et al. ]{}2003). However, these modest cooling flows had remained unaccounted for at low temperatures until only recently. The presence of warm, ionized gas in the form of H$\alpha$ emitting filaments has been observed in the cores of several cooling flow clusters to date (e.g., Hu [et al. ]{}1985, Heckman [et al. ]{}1989, Crawford [et al. ]{}1999, Jaffe [et al. ]{}2005, Hatch [et al. ]{}2007). More recently, it has been shown by McDonald [et al. ]{}(2010, 2011; herafter M+10 and M+11, respectively) that this emission is intimately linked to the cooling ICM and may be the result of cooling instabilities. However, while it is possible that the warm gas may be a byproduct of ICM cooling, the source of ionization in this gas remains a mystery. A wide variety of ionization mechanisms are viable in the cores of clusters (see Crawford [et al. ]{}2005 for a review), the least exotic of which may be photoionization by massive, young stars. ------------ ------------- -------------- -------- -------- ------ ----------- -------------- Name RA Dec z E(B-V) M F$_{1.4}$ Proposal No. (1) (2) (3) (4) (5) (6) (7) (8) Abell 0970 10h17m25.7s -10d41m20.3s 0.0587 0.055 – $<$ 2.5 11980 Abell 1644 12h57m11.6s -17d24m33.9s 0.0475 0.069 3.2 98.4 11980 Abell 1650 12h58m41.5s -01d45m41.1s 0.0846 0.017 0.0 $<$ 2.5 11980 Abell 1795 13h48m52.5s +26d35m33.9s 0.0625 0.013 7.8 924.5 11980, 11681 Abell 1837 14h01m36.4s -11d07m43.2s 0.0691 0.058 0.0 4.8 11980 Abell 1991 14h54m31.5s +18d38m32.4s 0.0587 0.025 14.6 39.0 11980 Abell 2029 15h10m56.1s +05d44m41.8s 0.0773 0.040 3.4 527.8 11980 Abell 2052 15h16m44.5s +07d01m18.2s 0.0345 0.037 2.6 5499.3 11980 Abell 2142 15h58m20.0s +27d14m00.4s 0.0904 0.044 1.2 $<$ 2.5 11980 Abell 2151 16h04m35.8s +17d43m17.8s 0.0352 0.043 8.4 2.4 11980 Abell 2580 23h21m26.3s -23d12m27.8s 0.0890 0.024 – 46.4 11980 Abell 2597 23h25m19.7s -12d07m27.1s 0.0830 0.030 9.5 1874.6 11131 Abell 4059 23h57m00.7s -34d45m32.7s 0.0475 0.015 0.7 1284.7 11980 Ophiuchus 17h12m27.7s -23d22m10.4s 0.0285 0.588 0.0 28.8 11980 WBL 360-03 11h49m35.4s -03d29m17.0s 0.0274 0.028 – $<$ 2.5 11980 ------------ ------------- -------------- -------- -------- ------ ----------- -------------- *-0.2 in (1): Cluster name, (2–4): NED RA, Dec, redshift of BCG (<http://nedwww.ipac.caltech.edu>), (5): Reddening due to Galactic extinction from Schlegel [et al. ]{}(1998), (6): Spectroscopically-determined X-ray cooling rates (M$_{\odot}$ yr$^{-1}$) from McDonald [et al. ]{}(2010), (7): 1.4 GHz radio flux (mJy) from NVSS (<http://www.cv.nrao.edu/nvss/>) (8) HST proposal number for FUV data. Proposal PIs are W. Jaffe (\#11131), W. Sparks (\#11681), S. Veilleux (\#11980).\ $^a$: No available *Chandra* data. \[sample\]* The identification of star-forming regions in cool core clusters has a rich history in the literature. Early on, it was noted by several groups that brightest cluster galaxies (hereafter BCGs) in cool core clusters have higher star formation rates than non-cool core BCGs (Johnstone [et al. ]{}1987; Romanishin [et al. ]{}1987; McNamara and O’Connell 1989; Allen 1995; Cardiel [et al. ]{}1995). These studies all found evidence for significant amounts of star formation in cool cores, but the measured star formation rates were orders of magnitudes smaller than the X-ray cooling rates (e.g. McNamara and O’Connell 1989). In recent history, two separate advances have brought these measurements closer together. First, as mentioned earlier in this section, the X-ray spectroscopically-determined cooling rates are roughly an order of magnitude lower than the classically-determined values based on the soft X-ray luminosity. Secondly, large surveys in the UV (e.g., Rafferty [et al. ]{}2006; Hicks [et al. ]{}2010), optical (e.g., Crawford [et al. ]{}1999; Edwards [et al. ]{}2007; Bildfell [et al. ]{}2008; McDonald [et al. ]{}2010), mid-IR (e.g., Hansen [et al. ]{}2000; Egami [et al. ]{}2006; Quillen [et al. ]{}2008; O’Dea [et al. ]{}2008; hereafter MIR), and sub-mm (e.g., Edge 2001; Salomé and Combes 2003) have allowed a much more detailed picture of star formation in BCGs. The typical star formation rates of $\sim$ 1–10 M$_{\odot}$ yr$^{-1}$ (O’Dea [et al. ]{}2008) imply that gas at temperatures of $\sim10^{6-7}$ K is being continuously converted into stars with an efficiency on the order of $\sim$ 10%. The fact that most of these studies consider the *integrated* SF rates makes it difficult to determine the exact role of young stars in ionizing the extended warm gas observed at H$\alpha$, since the two may not be spatially coincident or the measurements may be contaminated by the inclusion of a central AGN. In order to understand both the role of star formation in ionizing the warm gas and the efficiency with which the cooling ICM is converted into stars, we have conducted a high spatial resolution far-UV survey of BCGs in cooling and non-cooling clusters. We describe the collection and analysis of the data from this survey in §2. In §3 we desribe the results of this survey, while in §4 we discuss the implications of these results in the context of our previous work (M+10,M+11). Finally, in §5 we summarize our findings and discuss any outstanding questions. Throughout this paper, we assume the following cosmological values: H$_0$ = 73 km s$^{-1}$ Mpc$^{-1}$, $\Omega_{matter}$ = 0.27, $\Omega_{vacuum}$ = 0.73. Data Collection and Analysis ============================ To study the far-UV (hereafter FUV) emission in cluster cores, we selected 15 galaxy clusters from the larger samples of M+10 and M+11, which have deep, high spatial resolution (FWHM $\sim$ 0.6$^{\prime\prime}$) H$\alpha$ imaging from the Maryland-Magellan Tunable Filter (hereafter MMTF; Veilleux [et al. ]{}2010) on the Baade 6.5-m telescope at Las Campanas Observatory. Additionally, most of these systems have deep Chandra X-ray Observatory (hereafter CXO) spectroscopic imaging, as well as Two-Micron All Sky Survey (hereafter 2MASS; Skrutskie [et al. ]{}2006) and NRAO VLA Sky Survey (hereafter NVSS; Condon [et al. ]{}1998) fluxes. This broad energy coverage provides an excellent complement to a FUV survey, allowing for the source of emission to be carefully identified. A summary of these 15 clusters can be found in Table 1. For further information about the reduction and analysis of the H$\alpha$ and X-ray data, see M+10. FUV imaging was acquired using the Advanced Camera for Surveys Solar Blind Channel (hereafter ACS/SBC) on the Hubble Space Telescope (hereafter HST) in both the F140LP and F150LP bandpasses whenever possible, with a total exposure time of $\sim$ 1200s each (PID \#11980, PI Veilleux). The pointings were chosen, based on the results of our MMTF survey, to include all of the H$\alpha$ emission in the field of view. Exposures with multiple filters are required to properly remove the known ACS/SBC red leak, which has a non-negligible contribution due to the fact that the underlying BCG is very luminous and red. Since the aforementioned filters are long-pass filters, they have nearly identical throughputs at longer wavelengths. Thus, by subtracting the F150LP exposure from the F140LP exposure we can effectively remove the red leak and consider only a relatively small range in wavelength, from 1400Å–1500Å. Due to the small bandpass, it is possible for line emission to dominate the observed flux – we investigate this possibility in §4. We have carried out this subtraction for 13/15 of the BCGs in our sample which have both F140LP and F150LP imaging. For Abell 1795 and Abell 2597, we are unable to remove the red leak contribution due to the lack of paired exposures, but we point out that, conveniently, these two systems have the brightest FUV flux in our sample and, thus, are largely unaffected by the inclusion of a small amount of non-FUV flux. Following the red-leak subtraction, we also bin the images $8\times8$ and smooth the images with a 1.5 pixel smoothing radius, yielding matching spatial resolution at FUV and H$\alpha$. This process is also necessary in order to increase the signal-to-noise of the FUV image, allowing us to identify interesting morphological features. The final pixel scale for both the H$\alpha$ and FUV images is 0.2$^{\prime\prime}$/pixel. All FUV and H$\alpha$ fluxes were corrected for Galactic extinction following Cardelli [et al. ]{}(1989) using reddening estimates from Schlegel [et al. ]{}(1998). Results ======= {width="90.00000%"} In the Appendix, we show the stellar continuum, H$\alpha$ and FUV images for each of the 15 BCGs in our sample. At a glance there does not appear to be consistent agreement between the H$\alpha$ and FUV morphologies. We observe systems having H$\alpha$ filaments without accompanying FUV emission (Abell 1991, Abell 2052), systems with complex FUV emission without accompanying H$\alpha$ (Abell 1837, Abell 2029), and systems with coincident H$\alpha$ and FUV extended emission (Abell 1644, Abell 1795). Thus, it is obvious that a single explanation (e.g., star formation) is unable to account for the variety of FUV and H$\alpha$ emission that we observe. As we did with the H$\alpha$ emission in M+10, the FUV morphology can be classified as either nuclear or extended. We find, in the FUV, 7/15 systems have extended emission, 5/15 have nuclear emission, while 3/15 have no emission at all. In order to quantitatively examine both the nuclear and extended emission, we extract FUV and H$\alpha$ fluxes in several regions, as shown in the Appendix. ![Distribution of the FUV to H$\alpha$ luminosity ratios for the 10 clusters in our sample with high S/N detections in either the FUV or H$\alpha$. Each panel considers a different region, from top to bottom: the nucleus, the central 3$^{\prime\prime}$, excluding the nucleus, the filaments, and the entire system. The red vertical band represents the region consistent with shocks (Dopita & Sutherland 1996), while the blue band represents the region consistent with continuous star formation (Leitherer [et al. ]{}1999). In the nucleus, the majority of the observed H$\alpha$ emission is consistent with having been shock-heated, while in the outer regions, including filaments, the FUV/H$\alpha$ ratio lies between the two highlighted regions. In the bottom panel, the dotted histogram shows how the distribution would be altered if we corrected the fluxes for an i intrinsic extinction of E(B-V)=0.2 (dust screen model), while the blue dashed line represents the expected value for O8V stars (symbolic of a top-heavy IMF).[]{data-label="uvha_hists"}](uvha_hists.pdf){width="48.00000%"} In Figure \[uvha\]a, we show the correlation between the FUV and H$\alpha$ luminosity for the regions identified in the Appendix. We find a significant amount of FUV emission in all 5 of the systems for which we do not detect any H$\alpha$ emission. Additionally, we see that at least 3 systems are consistent with being shock-heated (Dopita & Sutherland 1996) – a point we will return to later in this section. As discussed by Hicks [et al. ]{}(2010), a significant fraction of the FUV emission may be due to old stellar populations (e.g., horizontal branch stars) in the BCG. To proceed, we must isolate the FUV excess due to young, star-forming regions. In order to remove the contribution from old stars, we consider the inner 3$^{\prime\prime}$ and plot the K-band (from 2MASS; Skrutskie [et al. ]{}2006) versus the FUV luminosity (Figure \[uvha\]b). Hicks [et al. ]{}(2010) show that the FUV luminosity from old stars is highly concentrated in the central region, thus removing this contribution in the inner region will act as a suitable first-order correction. In order to calibrate this correction for our sample, which lacks a control sample of confirmed non-star-forming galaxies, we opt to fit a line which is chosen to pass through the four points with the lowest $L_{FUV}$/$L_{K^{\prime}}$ ratio. We make the assumption that these four galaxies with the lowest $L_{FUV}/L_{K^{\prime}}$ ratio are non-star-forming, which is supported by non-detections at H$\alpha$. The equation for this relation is: log$_{10}(L_{FUV,3^{\prime\prime}}) = 2.35$log$_{10}(L_{K^{\prime},3^{\prime\prime}}) - 42.98$. The fact that four points with non-detections at H$\alpha$ lie neatly along the same line suggests that this correction is meaningful. Figure \[uvha\]c shows the FUV excess due to young stars versus the H$\alpha$ luminosity for the total, nuclear and extended regions in our complete sample of BCGs. With the contribution from old stellar populations removed, we find a tight correlation between $L_{FUV}$ and $L_{H\alpha}$ over four orders of magnitude. The majority of systems in our sample are consistent with the continuous star formation scenario (Kennicutt 1998, Leitherer [et al. ]{}1999), suggesting that much of the warm gas found in cluster cores may be photoionized by young stars. Two systems, Abell 0970 and Abell 2029 have anomolously high FUV/H$\alpha$ ratios, suggesting that star formation may be proceeding in bursts. As a starburst ages, the UV/H$\alpha$ ratio will climb quickly due to the massive stars dying first. This means that, by 10 Myr after the burst, the UV/H$\alpha$ ratio can already be an order of magnitude higher than the expected value for continuous star formation (see M+10 for further discussion). We find that the filaments in Abell 1991 and Abell 2052 are consistent with being heated by fast shocks, along with the nuclei of Abell 2052 and Abell 2580. In the case of Abell 2052, there exist high quality radio and X-ray maps which show that the observed H$\alpha$ emission is coincident with the inner edge of a radio-blow bubble. In Abell 1991, the H$\alpha$ morphology is reminiscent of a bow shock, and is spatially coincident with a soft X-ray blob which is offset from the cluster core. Much of the FUV and H$\alpha$ data is clustered between the regions depicting continuous star formation and shock heating, as shown in the zoomed-in portion of Figure \[uvha\]. These regions may indeed be heated by a combination of processes, or they may simply be reddened due to intrinsic extinction. Based on their FUV/H$\alpha$ ratios, H$\alpha$ morphology, disrupted X-ray morphology, and high radio luminosity, we propose that the optical emission in Abell 1991, Abell 2052, and Abell 2580 is the product of shock heating, while the remaining 12 systems are experiencing continuous or burst-like star formation. We will return to this classification in the discussion. The SF rates that we measure in the systems with nuclear emission only range from 0.01–0.1 M$_{\odot}$ yr$^{-1}$, which are typical of normal red-sequence ellipticals (Kennicutt 1998). For the systems with extended emission, excluding those that are obviously shock-heated, the measured star formation rates range from 0.1–5 M$_{\odot}$ yr$^{-1}$ which is similar to the rates of 0.008–3.6 M$_{\odot}$  yr$^{-1}$ observed in “blue early-type galaxies” (Wei [et al. ]{}2010). -------------------------------------- ------------------------------------------ {width="49.00000%"} {width="47.00000%"} -------------------------------------- ------------------------------------------ In Figure \[uvha\_hists\] we present the distribution of the FUV/H$\alpha$ ratio in various regions for the 10 systems with detections ($>$1-$\sigma$) at either H$\alpha$ or FUV. In the innermost region ($r<0.8^{\prime\prime}$), the warm gas appears to be shock-heated in 60% of systems – these shocked nuclei may be associated with AGN-driven outflows. Due to the small radial extent of this bin, the 2MASS data is of insufficient spatial resolution to remove any contribution from old stars. Thus, these FUV/H$\alpha$ ratios are upper limits. Of the remaining 4 systems, 2 are consistent with continuous star formation or a young starburst, while the remaining two are consistent with an aged starburst. At larger radius ($0.8^{\prime\prime}<r<3.0^{\prime\prime}$) the FUV/H$\alpha$ ratio is slightly larger, with the distribution peaking in between the regions describing shocks and star formation (see inset of Figure \[sfr\]). These data have had the contribution from old stellar populations removed, as described earlier in this section. The FUV/H$\alpha$ ratio at this radius is similar to what we observe in the filaments, as is seen in the third panel of Figure \[uvha\_hists\]. The fact that the distribution of FUV/H$\alpha$ peaks between the values for shocks and star formation supports a number of scenarios, including a mixture of the two processes, dusty star formation and star formation with an IMF skewed towards high-mass stars (see right-most panel of Figure \[uvha\_hists\]). We will investigate these scenarios in §4. In M+10 and M+11, we showed that the H$\alpha$ emission observed in the cool cores of galaxy clusters is intimately linked to the cooling ICM. In general, the thin, extended filaments observed in many of these clusters are found in regions where the ICM is cooling most rapidly, suggesting that this warm gas may be a byproduct of the ongoing cooling. If this is the case, it is relevant to ask what fraction of the cooling ICM is turning into stars. We address this question in Figure \[sfr\] by comparing the star formation rate with the X-ray cooling rate (dM/dt) for 32 galaxy clusters. In order to compute the star formation rate, we use the prescriptions in Kennicutt (1998). For the systems observed with HST, we use the average of the FUV- and H$\alpha$-determined star formation rates (filled blue circles). For an additional 10 clusters from M+10 and M+11 we make use of archival GALEX data for 5 clusters (open blue circles) and assume that both the UV and H$\alpha$ emisison trace star forming regions. These data have also had the contribution from old stellar populations removed, as described in M+10. In the remaining 5 cases where there is no accompanying UV data, we assume that the H$\alpha$ emission is the result of photoionization by young stars, and convert the H$\alpha$ luminosity into a continuous star formation rate (green triangles). For the three shock-heated systems (red crosses) mentioned above, we determine the SF rate based on the FUV luminosity alone – this represents an upper limit on the amount of continuous star formation. Finally, we also include 10 clusters from the sample of O’Dea [et al. ]{}(2008), who compute SF rates based on [*Spitzer*]{} data, and dM/dt in a similar manner to us (open purple circles). We note that the SFRs derived from FUV data are systematically lower than those derived from 24$\mu m$ [*Spitzer*]{} data. This is most likely due to our lack of an intrinsic extinction correction for these FUV data. We find that the “efficiency” of star formation, defined as the current ratio of stars formed to gas cooling out of the ICM, can range from 1% to 50% over the full sample. This spread is independent of whether we only consider systems classified as star-forming based on their FUV/H$\alpha$ ratios in Figure \[uvha\]. For the majority of the 32 clusters in Figure \[sfr\], there is a tight correlation between SF rate and dM/dt with a typical efficiency between 10–50%. In the right panel of Figure \[sfr\], we show that the efficiency is nearly independent of the cluster temperature, with only a weak dependence which is primarily driven by highly extincted systems. This suggests that the temperature of the surrounding ICM does not hamper the BCG’s ability to form stars. While not shown here, we also investigated the distribution of star formation efficiencies with the central entropy, $K_0$, from Cavagnolo [et al. ]{}(2009) and, similarly to $T_X$, find no correlation. In the following section we will discuss possible interpretations of this efficiency measure. Discussion ========== Star Formation as an Ionization Source -------------------------------------- ------------------------------------------- ------------------------------------------- {width="45.00000%"} {width="44.50000%"} ------------------------------------------- ------------------------------------------- In Crawford [et al. ]{}(2005), a variety of ionization sources are discussed which could produce the observed H$\alpha$ emission in the cool cores of galaxy clusters. The purpose of this HST survey was to investigate one of the most plausible ionization sources: photoionization by young stars. In Figures \[uvha\] and \[uvha\_hists\] we showed that, once the contribution to the FUV emission from old stellar populations is removed, the majority of the H$\alpha$ and FUV emission that we observe in cluster cores is roughly consistent with the star formation scenario. Based on the FUV/H$\alpha$ ratios, we identify three different types of system: 1. FUV/H$\alpha$ $\gtrsim$ $10^{-12}$ Hz$^{-1}$. Suggests a starburst that has aged by at least 10 Myr. Two systems, Abell 0970 and Abell 2029, fulfill this criteria, while several others may fall into this category if their H$\alpha$ luminosity is significantly less than the measured upper limits. 2. FUV/H$\alpha$ $\sim$ $10^{-13}$ Hz$^{-1}$. The FUV/H$\alpha$ ratios of these systems are consistent with continuous star formation or a recent (0–5 Gyr ago) burst of star formation. The filaments in Abell 1644, Abell 1795, Abell 2597, and part of Abell 2052, along with the nuclei of Abell 1795, Abell 1991, Abell 2151, and Abell 2597 appear to be star-forming. 3. FUV/H$\alpha$ $\lesssim$ $10^{-14}$ Hz$^{-1}$. Suggests heating by fast shocks or some other source of hard ionization (e.g., cosmic rays, AGN). The filaments of Abell 2052 and Abell 1991, and the nuclei of Abell 2052 and Abell 2580 have FUV/H$\alpha$ ratios which are consistent with this picture, in the absence of internal reddening. Figures \[uvha\] (inset) and \[uvha\_hists\] show that a large fraction of the systems which we observe fall between the regions describing shock heating and star formation. However, these data have not been corrected for intrinsic reddening due to dust. Correcting for a very modest reddening (E(B-V) $\sim$ 0.2) would boost the FUV luminosity of these systems such that the FUV/H$\alpha$ ratio is consistent with star formation (see Figure \[uvha\]c and the lower panel of Figure \[uvha\_hists\]). Unfortunately, the amount of reddening in the filaments and nuclei of these systems is currently not well constrained for very many systems, but typical values of E(B-V) can range from 0–0.4 in the cores of galaxy clusters (Crawford [et al. ]{}1999). In the case of Abell 2052, for which we measure FUV/H$\alpha$ ratios indicative of shock-heating *and* have an estimate of the amount of intrinsic reddening from Crawford ($E(B-V)=0.22^{+0.36}_{-0.21}$), we can investigate whether correcting for this extinction would provide FUV/H$\alpha$ ratios consistent with star-forming regions. Assuming a simple dust-screen model, correcting for a reddening of $E(B-V)=0.22$ would transform a FUV/H$\alpha$ ratio of $4.4\times10^{-15} Hz^{-1}$ in the filaments of Abell 2052 to $1.2\times10^{-14} Hz^{-1}$, which is consistent with the upper limit for shock-heated systems (see Figure \[uvha\_hists\]). However if, contrary to expectations, the filaments have a slightly higher reddening than the nucleus, the FUV/H$\alpha$ ratio may be even higher. Thus, it is certainly possible that the systems which we classify as shock-heated, or those which have ambiguous FUV/H$\alpha$ ratios, may in fact be highly-obscurred star-forming systems. We will address this possibility in significantly more detail in an upcoming paper which will include long-slit spectroscopy of the H$\alpha$ filaments providing, for the first time, reddening estimates away from the nucleus in these systems. An alternative explanation for the intermediate FUV/H$\alpha$ ratios is that the IMF in the filaments is top-heavy ($\alpha \ll 2.35$). Again, the lower panel of Figure \[uvha\_hists\] shows that the peak of the FUV/H$\alpha$ distribution is consistent with the value expected for O8V stars. There is a substantial amount of literature providing evidence for a top-heavy IMF in various environments including the Galactic center (Maness [et al. ]{}2007) and disturbed galaxies (Habergham [et al. ]{}2010). Thus, regardless of whether there is a small amount of dust or a slightly altered IMF, we suspect that the majority of the systems with intermediate FUV/H$\alpha$ ratios are in fact star-forming, with the exception of Abell 1991, Abell 2052, and Abell 2580, which have low FUV/H$\alpha$ ratios *and* morphologies which resemble bow-shocks and/or jets. Due to our use of the F150LP filter to remove red leak contamination, we are considering only a very small wavelength range from 1400–1500Å. In this region, there may be line emission from \[OIV\] and various ionization states of sulfur due to gas cooling at $\sim 10^5$ K. In order to establish that we are indeed observing continuum emission from young stars, we have computed UV spectral energy distributions (SEDs) for 6 BCGs which have deep GALEX, XMM-OM, and HST UV data. These data are presented in the left panel of Figure \[sed\]. We see that, in general, the UV SED follows a powerlaw over the range of 1500–3000Å. The new HST data, depicted as colored stars in this plot, agree well with the extrapolation of the continuum to shorter wavelengths, suggesting that there is very little contamination from line emission. This also suggests that there is little contribution from a diffuse UV component. This is further emphasized in the right panel of Figure \[sed\] where we show the residuals from the continuum fit for each of the 6 BCGs. Our measured FUV fluxes from these new HST data are consistent with the measurement of a UV continuum from archival GALEX and XMM-OM data. The idea that massive, young stars may be responsible for heating the majority of the warm, ionized filaments observed in cool core clusters is certainly not a new one (see e.g., Hu [et al. ]{}1985; Heckman [et al. ]{}1989; McNamara and O’Connel 1989). Most recently, O’Dea [et al. ]{}(2008), Hicks [et al. ]{}(2010) and McDonald [et al. ]{}(2010a) conducted MIR, UV and H$\alpha$ surveys, respectively, of cool core clusters and found a strong correlation between the SF rate and the cooling properties of cluster cores. However, this work extends these findings to include spatially-resolved SF rates, which the previous studies have been unable to provide. This allows us to say conclusively that the young stars and the warm, ionized gas are in close proximity ($\lesssim 1^{\prime\prime}$) in the vast majority of systems, offering a straightforward explanation for the heating of these filaments. Star Formation Efficiencies in Cooling Flows -------------------------------------------- In §3, we provide estimates of the efficiency with which the cooling ICM is converted into stars, assuming that this is indeed the source of star formation. This assumption is based on the results of M+10 and M+11, which provided several strong links between the X-ray cooling properties and the warm, ionized gas. These estimates of star formation efficiency represent a constraint on the so-called “accreting box model” of star formation. The simplified model that we propose is that the ICM is allowed to cool rapidly in regions where cooling *locally* dominates over feedback (Sharma [et al. ]{}2010). Our estimates of the ICM cooling rate, based on medium-resolution CXO spectra, are consistent with estimates based on high-resolution XMM grating spectroscopy by Peterson [et al. ]{}(2003) for the five overlapping systems. Once the gas reaches temperatures of $\sim$10$^{5.5}$K, it can continue to cool rapidly via UV/optical/IR line emission without producing fluxes that are inconsistent with what are observed. In the standard way, star formation will proceed once the gas reaches low enough temperature and high enough density. Observations by Edge (2001) and Salomé & Combes (2003) show evidence for molecular gas in the cool cores of several galaxy clusters, consistent with this picture. ![Distribution of measured star formation efficiencies from Figure \[sfr\] for 26 systems. Systems which are likely shock-heated (Abell 1991, Abell 2052, Abell 2580) have not been included. The additive contribution to the total histogram (solid black line) from UV+H$\alpha$, H$\alpha$ and MIR data are shown in different colors following the color scheme in Figure \[sfr\]. The dotted line shows a Gaussian fit to this histogram, which peaks at $14^{+18}_{-8}$% efficiency.[]{data-label="sfre"}](sfre_hist.pdf){width="49.00000%"} If the above scenario is correct, our estimate of the star formation efficiency provides a constraint on the fraction of hot gas that will be converted to stars, assuming a steady inflow of gas. In Figure \[sfre\] we provide a histogram of SF efficiencies for all of the systems in Figure \[sfr\] with non-zero X-ray cooling rates (dM/dt). The peak of this distribution is well defined at an efficiency of $14^{+18}_{-8}$%, regardless of which SF indicator (UV, H$\alpha$, MIR) is used. We note that, while the distribution for UV- and MIR-determined SF rates both peak at roughly the same value, the UV-determined SF histogram extends to much lower values. The low-efficiency tail of this distribution may be an artifact produced by intrinsic extinction due to dust, to which the UV will be most sensitive, or may be indicative of a selection bias in the MIR sample. If we measure the peak efficiency based on the subsamples excluding the MIR and MIR+H$\alpha$ (with no accompanying UV) data, we get 10$^{+25}_{-7}$% and 14$^{+20}_{-8}$%, respectively. Thus, the peak value of 14% is not solely driven by the inclusion of MIR data. The average efficiency of 14$^{+18}_{-8}$%, based on MIR, H$\alpha$ and UV data, is consistent with the estimates of star formation efficiency over the lifetime of a typical molecular cloud (20–50%; Kroupa [et al. ]{}2001, Lada & Lada 2003). This large variance in star formation efficiency may be due to differences in the ICM cooling and star formation timescales. Naturally, one would expect that there is some delay between the ICM cooling and the formation of stars, so that a reservoir of cold gas can accumulate and the formation of stars can be triggered. If this is indeed the case, one would expect to observe cooling-dominated periods (low SFE) followed by periods of strong star formation (high SFE) once the cold gas reservoir has reached some critical mass. Over an ensemble of systems, the average SFE is then an estimate of the time-averaged efficiency of an accreting system in converting a steady stream of cooling gas into stars. An alternative explanation for the spread of observed efficiencies is that the source of feedback is episodic (e.g., AGN). In this scenario, an episode of strong feedback from the AGN would re-heat the reservoir of cool gas, severely reducing the potential for star formation. This may indeed be the case, since two of the three systems with the highest 1.4 GHz luminosity (Abell 2052, Abell 2597, and Perseus A) have SFE $\lesssim$ 0.1. The fact that Figure \[sfre\] shows a well-defined peak suggests that the fraction of stars formed in an accreting system is constant over long enough timescales. Our estimate of an average efficiency indicates that, for a steady-state system accreting hot gas which is then allowed to cool, roughly 4 M$_{\odot}$ of gas will either be re-heated or expelled via winds for every 1 M$_{\odot}$ of stars formed. This fraction of baryons in stars is consistent with the global fraction of $\sim$ 20–30% required by simulations to reproduce the observed stellar mass function of galaxies (Somerville [et al. ]{}2008). Unlike measurements of SF efficiency for giant molecular clouds, this estimate does not require the use of a specific timescale, since we are assuming that stars are forming out of the inflow of hot gas and that the reservoir for this hot gas is inexhaustible. Summary and Future Prospects ============================ We have assembled a unique set of high spatial resolution far-UV and H$\alpha$ images for 15 cool core galaxy clusters. These data provide an unprecedented view of the thin, extended filaments in the cores of galaxy clusters. Based on the ratio of the far-UV to H$\alpha$ luminosity, the UV SED, and the far-UV and H$\alpha$ morphology, we conclude that the warm, ionized gas in the cluster cores is photoionized by massive, young stars in all but a few (Abell 1991, Abell 2052, Abell 2580) systems. We show that the extended filaments, when considered separately, appear to be forming stars in the majority of cases, while the nuclei tend to have slightly lower FUV/H$\alpha$ ratios, suggesting either a harder ionization source or higher extinction. The slight deviation from expected FUV/H$\alpha$ ratios for continuous star formation (Leitherer [et al. ]{}1999) may be due to the fact that we have made no attempt to correct for intrinsic extinction due to dust or due to a top-heavy ($\alpha \ll 2.35$) IMF. We note that modest amounts of dust (E(B-V) $\sim$ 0.2) in the most dense regions of the ICM can account for this deviation. Ideally, one would like spatially-resolved optical spectra of the filaments in order to constrain the heat source and intrinsic reddening of the filaments. We plan on addressing this issue in upcoming studies. Comparing the estimates of the star formation rates based on FUV, H$\alpha$ and MIR luminosities to the spectroscopically-determined X-ray cooling rate suggests a star formation efficiency of 14$^{+18}_{-8}$%. This value represents the time-averaged fraction, by mass, of gas cooling out of the ICM which turns into stars and agrees well with the stars-to-gas fraction of $\sim$20–30% required by simulations to reproduce the observed stellar mass function. This result provides a new constraint for studies of star formation in accreting systems. Many aspects of this simplified scenario are still not well understood, including whether the star formation is similar to that seen in nearby spirals or vastly different. We intend to investigate such differences via an assortment of star formation indicators from the UV to radio in future work. Acknowledgements {#acknowledgements .unnumbered} ================ Support for this work was provided to M.M. and S.V. by NSF through contracts AST 0606932 and 1009583, and by NASA through contract HST GO-1198001A. We thank E. Ostriker and A. Bolatto for useful discussions. We also thank the technical staff at Las Campanas for their support during the ground-based observations, particularly David Osip who helped in the commissioning of MMTF. 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--- abstract: 'The aim of this work is to show how Einstein’s quantum hypothesis leads immediately and necessarily to a departure from classical mechanics. First we note that the classical description and predictions are in terms of idealized measurements that are exact, instantaneous, non-perturbative, independent of each other and process agnostic. If we assume we cannot arbitrarily reduce the strength of a signal, measurements are ultimately perturbative to some degree. We show how a physical description in which the best measurement conceivable, i.e. the ideal measurement, perturbs the system leads to all the concepts present in quantum mechanics including conjugate variables, probabilistic predictions and measurements connected to symmetries.' author: - Gabriele Carcassi date: 'February 15, 2009' title: 'How Einstein’s quantum hypothesis requires a departure from classical mechanics' --- Introduction ============ It is unfortunate that, after more than half a century that quantum mechanics has been a core part of our scientific understanding, it is still surrounded by a cloud of mystery and perceived as strange and nonintuitive. It is true that it does predict behavior that is odd and counter to our intuition, but does that have to imply we are bound to feel like something escapes us? Quantum mechanics is not the only 20th century theory that has strange consequences: the concept of spacetime, time dilation, length contraction, equivalence of mass and energy, curved space and black holes are some of the landmarks of special and general relativity, yet both are usually presented as natural, in fact necessary. We believe that the main difference is that they are presented as coming from a simple physical idea, the invariance of the speed of light in the first case and the equivalence principle in the second, which help us make sense of all the other physical consequences. Quantum mechanics, with its uncertainty principle, interference and probabilistic predictions, is usually presented as a set of mathematical postulates[@sudbery], usually prefaced by a historical perspective[@liboff] or by a heuristic account that gives some sort of justification for them[@feynman; @griffiths; @shankar; @sakurai]. We are not told why a Hilbert space must be used as the phase space, or why observables are associated with operators: that is the starting point. The mathematical results derived from the postulates need to be subsequently interpreted physically, with nothing else to connect them together but the mathematical framework. Should the physics not come first? Should the math not be derived from the physics? We are left to wonder whether we are missing something: what is the “big physical idea” that requires us to abandon the classical description? Maybe if we were to present quantum mechanics derived from it, it would increase our sense of understanding: what is understanding if not being able to identify, in the midst of all that is confusing and misleading, that simple truth from which all others descend? We are convinced that this idea is something that is already present in all quantum textbooks: Einstein’s quantum hypothesis. This states that light consists of and propagates in discrete packets of energy. The aim of this paper is to convince the reader that this assumption, by itself, is sufficient to require departure from the classical description. The language and the level of math used are appropriate for an introductory class, where typically more importance is given to thought experiments and concepts. In fact, the arguments are designed so that they could be used “as is” during the first lecture of an introductory class in quantum mechanics. In section II we will show how the classical description is in terms of idealized measurements that are exact, instantaneous, non-perturbative, independent of each other and process agnostic. In section III we will show such an idealized measurement is in principle possible using the classical electromagnetic field. In section IV we will show how the introduction of the quantum hypothesis requires all measurements to be perturbative and that this leads to many of the features present in quantum mechanics. In section V we show that the concepts we developed fit extremely well in the mathematical framework. In section VI we extend the arguments to show how the introduction of the equivalence principle and gravitation necessarily leads to a departure from quantum mechanics, as measurements are no longer independent of each other and cannot be regarded as exact or instantaneous. In section VII we go through some common reactions to these arguments, comparing to other types of works. Ideal measurements in classical mechanics ========================================= Predictions and measurements are fundamental in physics: devising experiments, performing them and comparing their results to our predictions are in essence the activities of a physicist. In this section we are going to review some aspects of these basic concepts in the context of classical mechanics. In classical mechanics we describe a system by a set of quantities that vary in time. For example, we write $x=x(t)$ for position or $p=p(t)$ for momentum. At each moment in time, we have a prediction for each quantity: if we were to measure, and our description were correct, we would obtain that value. There are a few details, though, that we have to keep in mind. First we note that while an actual experiment will only measure a quantity within a certain accuracy, the prediction is at least in principle exact. If we increase the accuracy of our measurement, the result will have to be closer to our prediction for the prediction to be correct. The prediction is really for an idealized measurement: one for which the uncertainty is so small that it can be neglected. [^1] Secondly, an actual experiment will measure a quantity within a finite interval of time while the prediction is given for an instant. We can imagine that we improve our measurement so that the interval is smaller and smaller. The prediction is again in terms of an idealized measurement, one in which the time interval is so small that it can be neglected. The third thing that we notice is that the classical description does not require us to say what quantity we are measuring and when: the evolution does not depend on it. In general, an actual experiment will modify the evolution of the system. We can, once again, imagine that we improve our technique so that it affects less and less the future evolution. The prediction is really in terms of an idealized measurement, one in which the perturbation caused by the experiment is so small that it can be neglected. The fourth point: if we had two or more instances of the same experiment conducted at the same time, or within an interval so small that it could be neglected, the prediction would be the same. While in practice this may be difficult to achieve, ideal measurements do not interfere with each other, so our description does not depend on how many idealized measurements are performed at a particular time: we can have several observers or just one and we obtain the same result. Fifth and last point: the prediction does not depend on which particular physical process or measurement technique we use for our measurement. So, when we write $x(t)$ there are quite a few assumptions that go with it. It assumes that, at least in principle, we can imagine an ideal measurement that is exact, instantaneous, non-destructive, independent of other measurements (or “inter-independent”) and process agnostic. *This is the best measurement we can conceive.* Our description is given by the outcomes of such idealized measurements at every moment in time. Even when we do describe a “real” measurement, we do so by describing the entire process in terms of these idealized quantities. In other words: we describe the world using the best measurement we can conceive, and from this determine predictions for our real, less perfect, measurement. It should be noted that if any of those conditions cannot be satisfied conceptually, our description of position as $x=x(t)$ would not make much sense. Also note that since the ideal measurement is non-perturbative, inter-independent and process agnostic, our description does not depend on in what way, how many times, what or even if we measure. *This is what allows us to imagine the outcomes as properties of the system we are studying*: they are going to be the same no matter what we do. But we always have to keep in the back of our mind that even these quantities are the outcome of a process, however idealized it may be. An ideal process for an ideal measurement ========================================= We have seen how the classical description is in terms of ideal measurements. In this section we identify a physical process that we could conceptually use in classical mechanics to perform such an ideal measurement. Given that we assume that the results do not depend on what process we use, we can choose the process we prefer. For reasons that will become obvious in the next section, we will focus on an ideal position measurement through the use of the electromagnetic field. In the simplest case, we can think of sending an electromagnetic signal toward our target. The electromagnetic signal will interact with the target, leaving it affected in general. The signal itself will also be changed and when it is received by the detector it will give us some information about the target. In measuring position, we can imagine the signal bounces off the target, changing its momentum a bit, and by knowing the initial and final position and angles of the electromagnetic signal, we can infer the position of our target at the time of impact. Can this process satisfy, under ideal conditions, the requirements of an ideal measurements? To measure the position with greater accuracy, we can reduce the width of the electromagnetic signal and, ideally, we would make the packet so small that its length can be disregarded. We can also make the duration of the signal as short as we desire, and ideally it will be so short that it can be disregarded. We can make the measurement non-perturbative by lowering the intensity of the signal enough so that the effect on our target can be neglected. And regarding inter-independency, we note that the equations describing the electromagnetic field are linear: different electromagnetic signals of different ideal experiments are not going to interact with each other. This idealized process can conceptually be used to perform our idealized measurement. This does not in any way prove that the underlying assumptions are right, or that we can actually make such a measurement in practice: it just shows that classical mechanics is consistent. Also note that this example is specifically chosen to remind the reader of thought experiments used for special relativity such as the ideal clock. The quantum hypothesis and the ideal measurement ================================================ In this section we show how the introduction of the quantum hypothesis no longer allows us to work under the previous idealized conditions. Even an idealized measurement must be perturbative, and we show that this perturbation has to be such that it randomizes another physical quantity, making the future evolution independent of it. The classical description, then, needs to be abandoned for one that takes into account such perturbation and that can only be in terms of probabilistic predictions. In the previous section we discussed how we can make the ideal measurement non-perturbative by decreasing the intensity of the field. If we introduce the quantum hypothesis, that the electromagnetic field consists of and propagates in discrete packets of energy, we are no longer able to do that arbitrarily: at some point we will reach those units of energy and we cannot go lower than that; a photon is a photon and we cannot divide it. If the target is massive enough so that the effect of one photon can be disregarded, then we can still assume that our ideal measurement is non-perturbative. If that is not the case, then our assumptions regarding our ideal measurement have to change. *The best measurement we can hope to achieve will still affect the system.* Given that the classical description and predictions are in terms of the non-pertubartive assumption, this means that the quantum hypothesis has more profound effects than one would at first expect. This assumption was the one that allowed us to give a description of the evolution without having to specify what we were measuring and, most importantly, whether we were measuring anything at all: if ideal measurements do not affect the system, we can ignore them. But now even ideal measurements will affect the system, so we will at least need to specify what is it that we are measuring at what moment in time to properly describe the evolution of the system. Now the question becomes whether we can hold onto the other ideal assumptions, and what do we need to sacrifice to keep them. Ideally we can imagine producing a photon that is extremely well localized both in space and in time, so we can still assume that our ideal measurement is instantaneous and exact. The measurement, as we said, is now perturbative: *measuring one quantity will change at least one other quantity*. So measuring position will need to change some other quantity. Given that ideal measurements are process independent, if measuring position using the electromagnetic field will change momentum, it will have to do so for any other process. Therefore the link between these two quantities is something deep: whenever we describe one we need to describe the other and the effects caused by our idealized measurement. We still want to assume that our ideal measurement is inter-independent: while the evolution may depend on when and what we measure, it shall not depend on how many times. Let us imagine a time interval $dt$ where we will perform $N$ idealized measurements. Let $x(t)$ and $p(t)$ be the initial position and momentum, and $x(t+dt)$ and $p(t+dt)$ be position and momentum after all measurements. We will need $x(t+dt)$ and $p(t+dt)$ to give us the same description regardless of the value of $N$. It is clear what needs to happen for the position, the quantity we are measuring: all ideal measurements will have to give us the same precise value, a real number, so $x(t+dt)$ will be simply a real number. It is trickier to consider what needs to happen to the quantity that changes: *each measurement will need to change the momentum, yet the future evolution of the system must not depend on the number of changes*. We can write: $$\label{measurementEffect} p(t+dt)=p(t) + \displaystyle\sum_N \Delta p$$ where $\Delta p$ is a stochastic variable that describes the change. We assume all changes obey the same distribution, are independent of each other and of the measured position. What we want to understand is the $\Delta p$ distribution: does it give the same value every time? Is it a Gaussian distribution? What is the expectation? What is the variance? Note that for the expectation of $p(t+dt)$ to be independent of $N$ we need the expectation of one perturbation to be the same as many perturbations: this can only happen if the expectation for $\Delta p$ is zero. $$<\Delta p>=N<\Delta p>$$ $$<\Delta p>=0$$ Since the change cannot be identical to zero (or the ideal measurement would not be perturbative), the variance needs to be non-zero. The change is then a random change, and does not privilege any direction. This means that both $\Delta p$ and $p(t+dt)$ will necessarily be given in terms of probability distributions. As $N$ increases, $p(t+dt)$ will spread out, given that $$\sigma^2_{p(t+dt)}=\sigma^2_{p(t)}+N \sigma^2_{\Delta p}.$$ The contribution given by $p(t)$ will get smaller, and in the limit of large $N$ $p(t+dt)$ will not depend on $p(t)$ at all. But if this is true for large $N$ it has to be true for any value of $N$, since $p(t+dt)$ can not depend on $N$: *the value of momentum after an idealized measurement of position does not depend on the value of momentum previous to such measurement*! Let us now imagine we start from a slightly different initial condition: $$p'(t)=p(t) + \delta p.$$ Given that $p(t+dt)$ is independent of $p(t)$ is, $p'(t+dt)$ will not change: $$p'(t+dt) = p(t+dt).$$ The final distribution is symmetric under boosts: *after a measurement we will have a symmetry in the conjugate variable, the quantity that changes.* Each value of momentum is therefore equally likely: $p(t+dt)$, as well as $\Delta p$, will be uniform distributions from minus infinity to infinity. *Our ideal position measurement, then, is going to change momentum so that each value is equally likely.* The way that we can describe processes if we assume that ideal measurements are interactions is quite different: some of our predictions will only be in statistical terms. But we have to realize that this is not just a problem in our description or limited to our measurements. Not only can we not measure position and momentum precisely at the same time: no electromagnetic process can depend deterministically on both position and momentum or can prepare a state in which both are well defined. If we had such a process, we could use it to make a more precise measurement. But the ideal process we described before represents the best measurement we could make using the electromagnetic force, so such a process cannot exist. As future evolution cannot depend on the value of the momentum at the time we measured position, no process and no measurement will be able to distinguish between those configurations. As what can be distinguished is radically different, we will need to redefine our concept of physical state. This is still the output of a set of idealized measurements, as in classical mechanics, but the output of an idealized measurement is radically different: for each measurement quantity we have a symmetry. The uncertainty that comes out of assuming ideal measurements are perturbative is not a simple matter of *our* inability to measure precisely (as it is in classical mechanics): it is an intrinsic part of the physical description because electromagnetic processes will depend on that uncertainty, and in fact they do. The stability of the atomic orbits, the tunneling effect, and all the other quantum effects are based, one way or another, on this. And if we assume that ideal measurements and our description do not depend on what process we use, this reasoning extends not only to electromagnetic processes, but to all processes. The scale of this perturbation has to be the same for all forces: it is universal and so is the constant $\hbar$ that defines it. We have reached the main point of this article: assuming the quantum hypothesis leads necessarily to a physical description that is radically different form the one of classical mechanics. One could go further: we should note that Rovelli[@rovelli] derives the whole mathematical framework of quantum mechanics starting from the following postulates 1. There exists a maximum amount of relevant information that can be extracted from a system. 2. It is always possible to obtain new information about the system. If we define the state as the output of a set of ideal measurements, we imply the first. Noting that one measurement creates a symmetry in another quantity implies the second. Given that these two postulates lead to quantum mechanics, and the quantum hypothesis leads to these two postulates, one could show that the quantum hypothesis, by itself, not only necessarily requires a departure from classical mechanics, but leads to quantum mechanics itself. We are not going to do so, as it is outside the scope of this particular work, but we believe it is worth pointing it out. In this section we saw how the quantum hypothesis requires us to abandon the classical description. Even ideal measurement, the best measurement we can possibly conceive, must be perturbative: the measurement of one quantity will change another. Such change makes the value after the measurement independent of the value before the measurement: future evolution cannot depend on the previous value, which also mean that it can no longer be measured. This independence of the initial value means that the state has a symmetry in the quantity: we can ideally change it yet future evolution is the same. A symmetry corresponds to a uniform distribution for the description of such quantity: if we measured it we would need to have an equal chance to get any value. Realizing how these elements are so tightly connected is the key, in our opinion, to understanding quantum mechanics: one cannot have one without the others, they are all different sides of the same coin. This way of presenting the subject does a better job, in our opinion, of bringing these concepts closer in the mind of the reader than what is found in standard textbooks. A dictionary ============ Even if our aim is not to derive quantum mechanics we want to show that the concepts that we developed above find their place very naturally in the mathematical framework of quantum mechanics. We do this so as to not leave ambiguities, but also to show how disparate characteristics of the theory can find significance using the few concepts we have outlined. We will do this briefly in this section touching only the important points, as the full treatment of each particular item would essentially result in writing a textbook, which is not the purpose of this work either. The notion of the ideal measurement translates, as the reader might expect, to the projection postulate: a measurement changes the state of the system to one of the eigenstates of the hermitian operator corresponding to the measurement, with the probability of doing so given by the projection of one onto the other. The eigenvalue is the outcome of the measurement and it is a real value, meaning that the result of the ideal measurement is exact. The projection is done *at a particular time t*, meaning the ideal measurement is instantaneous. The state is changed during a measurement, so it is perturbative, but if the measurement is performed by more than one observer the final state will be the same, as further projections would yield the same result. It also does not depend on what process is used to measure. It satisfies all the properties we discussed. But it has to be understood in terms of an *ideal* measurement: a simplification, a limit, something that, in practice, we cannot do. What it is important is not really whether we can realize it in practice, but that *each state can be seen as the outcome of this idealized process*. In mathematical terms, this means that for any state there exists a set of Hermitian operators that have that state as an eigenstate. This can be easily demonstrated to be true: let ${|\Psi_a\rangle}$ and ${|\Psi_b\rangle}$ be two normalized vectors in a Hilbert space $\mathcal{H}$, and let the first be an eigenket of Hermitian operator $A$. Given that they are normalized, we can always define a unitary operator $U$ such that ${|\Psi_b\rangle}=U{|\Psi_a\rangle}$: we do so by rotating on the plane defined by the two vectors and leaving the other directions unchanged. If we now consider the operator $B=UAU^{\dagger}$, it is easy to show that ${|\Psi_b\rangle}$ is an eigenket of $B$ and that $B$ is Hermitian. In this light, *the concept of state in both classical mechanics and quantum mechanics is the same*: the state is the outcome of a set of idealized measurements that must be specified to reproduce a system. The only thing that changes is that ideal measurements are perturbative in quantum mechanics. This creates the strongest possible connection between classical and quantum mechanics: it says how they are different and how they are the same. The idea that one quantity is changed by measuring another is represented by conjugate quantities, such as position/momentum and spin/angle. The symmetry after a measurement corresponds to the unitary transformation generated by the Hermitian operator. An eigenstate of Hermitian operator $A$ will also be an eigenstate of the transformation $1 + A d\beta / i \hbar$, which means it is invariant under that transformation, and the distribution for an observable $B(\beta)$ that depends on parameter $\beta$ will be uniform, flat: *uniform distributions and symmetries are the same thing*. With this in mind, it makes much more sense to say that *final states of a measurement are eigenstates of the generated transformation*, since this actually has physical meaning, which is once again: for every quantity measured, or well defined, we have a symmetry. This also works in reverse: a state with a symmetry will have an associated observable well defined. This provides physical intuition for the Noether theorem: if a Lagrangian is invariant for a given transformation, the equations of motion preserve that symmetry and the associated observable. Other important concepts can be understood in terms of just the ideas presented. Briefly: the commutator describes the change of one observable under a unitary transformation generated by a second, with the Heisenberg equation of motion being a specific example; a state with defined momentum is symmetric under translation, the evolution does not depend on position, hence non-locality; if the measured quantity spans two systems, such as the sum of their momentum, the symmetry also spans both systems, such as the sum of their position: we cannot describe the systems independently, the systems are entangled. This should give the reader a glimpse of how deeply these concepts are rooted in the theory. As we said, we cannot go through an entire curriculum in this article. But we believe that we have at least shown how not only the concepts we used in the previous sections fit perfectly within the mathematical framework, but many more elements acquire a very straightforward significance if related to such ideas. This ability to explain the same number of concepts with fewer ideas can lead to a more elegant way to present the theory. Limitations of quantum mechanics ================================ We want to stress that the assumptions underneath ideal measurements in quantum mechanics are, well, ideal: at some point they will break down and need to be abandoned. To that end, we show in this section how the equivalence principle directly requires such a departure, which implies a limit of applicability for quantum mechanics. We do not intend this treatment to be exhaustive and it will be only in qualitative terms: our aim is just to show that problems exist and not to give a solution. In our previous description of the ideal position measurement through the electromagnetic field we said that it is inter-independent because the laws that describe that field are linear. This is true, but we have not considered that each photon, having a finite energy, is going to interact with others through the gravitational force. This means that, if we introduce the equivalence principle, measurements affect each other and it will matter how many observers perform the same measurement at the same time: measurements are inter-dependent. If we want to describe the evolution of a system, we not only have to say what is measured at each time, but also, for example, what is the energy used as this will somehow affect the result of other measurements and future evolution in general: that information will have to be part of the state. It is also evident that the projection postulate of quantum mechanics is not at all suitable to describe an ideal measurement in which we consider the effect of gravity: it describes inter-independent measurements and is agnostic regarding the energy involved in the measurement. Given that the gravitational force is orders of magnitudes smaller than the electromagnetic force, we do have cases in which the energy is such that the gravitational pull can be neglected (inter-independence), while the effect on the system being measured cannot (perturbation). This is the region where quantum mechanics will be valid. If we require higher energies, then we reach its limit. When is this the case? From quantum mechanics itself we know that to probe shorter time scales we need to use higher energies: if the time scale required is “fast enough” the energy will be “large”. Since instantaneous measurements would ideally require an infinite amount of energy, and infinite qualifies as large, we most likely need to abandon this assumption as well: we will need to specify what the duration of our idealized measurement is, which will tell us how much energy is needed. But we also need to note that, due to special relativity, the duration in time is connected to the precision in space. So our exact measurement assumption must also be abandoned. The progression should be now evident: in classical mechanics we consider quantities as outcomes of completely ideal measurements; in quantum mechanics we start to consider the perturbation caused by measurements; here we see that to go further we also need to consider the strength, which also affects the precision in time and space. In other words, we are making our ideal measurement less and less ideal.[^2] It is no surprise that quantum mechanics and general relativity are difficult to integrate. What is interesting is that the above argument is not *technical* (as in whether the equations are linear or not) but it is *conceptual* and pretty straightforward. Developing these ideas further is, naturally, not in the scope of this work, which is just to show how the quantum hypothesis requires us to abandon classical mechanics. What we did want to show is that the assumptions underneath quantum mechanics are, in a way, already known to be “wrong”. As there is no “right” theory in physics, only ones that are accurate within the constraints of their validity, this section is meant to underline such constraints for quantum mechanics and to show how thinking in terms of idealized measurement allows one to do that very concisely. Discussion ========== We feel that this work could end here. However, given that a lot has been written on quantum mechanics and many issues are still deemed as controversial, it may be important to better qualify our work so that it is at least clear what we do not expect it to be. Before one forms a definite opinion, we will discuss common reactions to these ideas and typical comparisons that have been made. Some people, typically ones that only give only a superficial reading, argue that this work does not provide anything new: there is no new math developed, no new physical concepts, no new prediction. This shows a misunderstanding on our aim, which is to better understand a theory that is already now established for almost a century: it is unlikely that some important piece is missing. It is more likely, instead, that we do have all the pieces but we have not assembled them in a way that allows the big picture to emerge. Introducing new math or radical new physical concepts would be, in our opinion, either admitting our own failure or implying the failure of the people that came before us, and we are not holding our breath for the latter. *The overall sense of familiarity that one may experience should be seen as an actual sign of strength of the argument*, as it lacks the implication that everyone has to review what he knows. And despite the familiarity, the arguments as presented are not in any work of which we are aware. The differences are more in shifting importance between different elements to provide a better, more cohesive narrative. Such changes are subtle and we cannot expect each and every reader, having read this work once, to jump back in disbelief at how much their understanding has increased. The main reason being that *the reader already knows quantum mechanics*. But if someone, as we did, uses this set of arguments to give a half an hour introduction on the subject to someone who is scientifically literate, you will discover that he can follow the thought experiments, he does not find the reasoning hard, is often convinced of the necessity of the arguments and of the striking consequences (like non locality and entanglement). This has not been our experience when we first learned quantum mechanics. This work can provide a mental scaffolding to students, who can get the big picture more quickly, and who will have a placeholder in which to put the rest of the content. If someone has already built the structure, it is clear the scaffolding is of less use. Some readers will still be intrigued by the simplicity and the elegance of the arguments, which to us has huge value in itself, but not everybody is bound to because, in the end, this judgement is subjective. Some readers will try to put this work in the context of quantum mechanics reconstructions, which we can briefly describe as a set of different works that aim to derive the theory from a different set of postulates than the standard more mathematical ones.[@grinbaum] While we do believe that we could derive quantum mechanics from Einstein’s quantum hypothesis, we are happy to simply note Rovelli’s work and leave it at that: we feel that why we have to leave classical mechanics is as important, or possibly more, than where we have to go. When in dark nights we wonder about the mysteries of quantum mechanics, knowing that it is the consequence of “no bit commitment” does not comfort us (which may be due to our minimal background in quantum information theory). We want to know why we are forced to abandon classical mechanics. We also note that the choice of Einstein’s quantum hypothesis as the foundational idea is not arbitrary: it makes the arguments work at multiple levels. It works because it is a clear physical statement of the world around us that is firmly experimentally established and is (at least now) non-controversial. It works because many feel that historically it is what started quantum theory in the first place. It works because it has been so much in front of our noses that it was difficult to notice its importance. It works because it allows to introduce quantum in a similar fashion to special and general relativity: through ideas and thought experiments. It works because it is what Einstein got his Nobel prize for. It *feels* right. Even if we could not derive all of quantum mechanics from it, we personally do not feel the arguments would diminish in value. The fact that we could derive the mathematical framework as a whole is just the cherry on top. Other readers will try to put this work in the context of interpretations. We can see how providing a dictionary from math to physics might make one think that this work belongs in that category. Here there are even more important distinctions, and we believe it instructive to compare and contrast with a couple of questions interpretations try to answer. The first question we address is whether the wave function represents a real physical object or not. There is no claim in our work of what the wave function represents: in fact, there is no mention of the wave function at all. The state as a whole, though, is fully determined by a set of observables and their outcomes. This is also true during the evolution, so at each moment in time there is a set of observables that are well defined, possibly a different set every time. These observables are as real as real can get: they are defined whether we measure or not, their values are the same for all observers and their knowledge is enough to determine the state. This is the same as classical mechanics. What is different is that there are other observables, the conjugate ones, which do not determine the future evolution of the system and therefore cannot be measured. These, one might say, are not real at all: they are not determined a priori. The problem here, we believe, is simply the focus on valued measurement, and we hope the reader forgives this obvious statement: *the symmetry in the conjugate variable is as measurable as the value of the original one*. The symmetry as a whole is as real as the valued quantities: it is there regardless of whether we measure it or not and it is the same for all observers. We always have a set of defined quantities and a set of symmetries no matter what, and it is obvious that the two are mutually exclusive for the same quantity. Time evolution or the act of measurement simply changes which quantity and symmetry. We let the reader decide whether this is enough to satisfy his thirst for realism, but it is for us. Second question: can we have hidden variables? Most definitely: we can simply pick any value from the quantities in which we have a symmetry. There is nothing that says we should or we should not do so. The caveat is that, no matter what value we pick, it will never come into play in determining the future evolution of the quantities that fully determine the quantum state. If it did, then we would have a process that we can use to measure better than the ideal measurement, which is forbidden. That is why these variables are, in the end, hidden.[^3] The third question is the so-called measurement problem, which we briefly sum up. To put it simply, we have two ways of evolving a system: the Schroedinger equation for physical processes, which evolves the state continuously and is deterministic (given an initial state we can determine exactly the final state), and the projection postulate for measurements, which evolves the state discontinuously and is non-deterministic. The problem is that each measurement should be a process, but how can measurements be non-deterministic if all processes are deterministic? To address this problem let us construct this thought experiment. We start at time $t$ in the eigenket ${|p_0\rangle}$ of $P$, and we perform an ideal measurement of some other quantity $A$. At time $t+dt$ we will be in an eigenket ${|a_i\rangle}$ of $A$, and we will perform an ideal measurement of $P$ so that at time $t+2dt$ we will be back in an eigenket of $P$. At the end will have a distribution in $P$, and by studying this distribution depending on what observable $A$ is we can understand a couple of things. First: what is the chance we are going back to the same eigenket? The probability to go from ${|p_0\rangle}$ to ${|a_i\rangle}$ is the same as for going from ${|a_i\rangle}$ to ${|p_0\rangle}$, and for the probability of going back we need to take into account all possible paths: $$\begin{aligned} Prob(p_0, a_i) &= {\langle p_0|a_i\rangle} {\langle a_i|p_0\rangle} \\ Prob_{total} &= \displaystyle\sum_{a_i} Prob^2(p_0, a_i)\end{aligned}$$ If the $A$ corresponds to $P$, it is clear that we return to ${|p_0\rangle}$ with complete certainty: we never leave it. Let us consider the case where $A$ is not $P$, but one of its eigenstates ${|\hat{a}\rangle}$ is infinitesimally close to ${|p_0\rangle}$. In that case, we will change state but we are almost certain of our final state: determinism implies continuity and vice versa. We also require this change to be reversible, that is if we switch the two states, we get the same change but opposite in sign. Mathematically the two assumptions are: $$\begin{aligned} {\langle \hat{a}|p_0\rangle} &= 1 - \epsilon \\ {\langle p_0|\hat{a}\rangle} &= 1 + \epsilon\end{aligned}$$ Under these condition, the probability to return is $1-\epsilon^2$: we return to the same state with certainty as we can disregard the second order infinitesimal term. But these are the same conditions under which the Schroedinger equation works: determinism, continuity and reversibility. Assuming reversible determinism implies continuous and unitary evolution, and from these two conditions we can actually *derive* the Schroedinger equation![^4] So, in this special case, there is no actual problem. The more the eigenkets of $A$ differ, though, the more the jump will be non-deterministic and the final distribution in $P$ will be spread. The “worst” case is when $A$ corresponds to $X$, and we will have a uniform distribution in $P$. But here we also have a bigger problem: if the momentum changes, the energy changes. Where is our system getting the energy from (or losing it to)? The only answer possible is: from the probe of our idealized measurement. Now think about it: if our intermediate measurement is $X$, the ideal measurement has to change the momentum to values that are possibly very very large. This means that the smallest amount of energy required by our ideal measurement is potentially infinite. But in the previous section we argued that if we *require* energies that are large (and infinity still counts as large) we cannot use quantum mechanics. Whether and how these jumps do occur we still cannot say, but it seems clear that they cannot be described within quantum mechanics itself. Therefore it actually makes a lot of sense we do not have an equation describing such process of measurement: it would be nonsensical. It would describe a process based on the assumption that such process does not exist. This does not make quantum mechanics consistent, it simply shows at a conceptual level why such an inconsistency occurs. So, it is true that this work may say something about the problems that interpretations try to address, but it is also true that it does not really provide answers: if anything it tells us the reason why we do not have them. Depending on the inclination of the reader, though, this may be far more interesting than the answers themselves, which is definitely the case for us. We leave up to the reader to decide for himself what this work is closer to; we simply suggest that he keep these ideas in consideration. No matter what one may conclude, our claim is simply that this is a better way to introduce quantum mechanics. Conclusion ========== We have presented a heuristic that can be used to conceptually justify the departure from classical mechanics, which we believe useful in the context of an undergraduate or graduate course of quantum mechanics. This material is complementary to the more standard ways of introducing quantum mechanics, because while it says from where we need to depart from and justifies some of the more puzzling aspects of quantum mechanics, it does not say where we need to end up. The advantage of introducing the subject in this manner is that it brings it closer to the way that special and general relativity are usually introduced, using thought experiments that stimulate intuition, which we chose on purpose to be as similar as possible to those used for those theories. The other side effect of this introduction is the realization of how much is implied in classical mechanics: it is beneficial to remind students that the use of exact quantities, among other things, is an idealization. Though quantum mechanics is the theory that uses probability, it is classical mechanics that is in fact less precise and we believe this way of presenting reinforces that. We also hope that this work is of interest to those who, like us, spent a significant amount of time reading a great number of textbooks in their youth trying to find a satisfying conceptual justification of quantum mechanics. May this work provide them at least part of it. [5]{} A. Sudbery, Quantum Mechanics and the Particles of Nature (1986). Mathematical postulates are introduced only after a very brief review of some key experiments. R. L. Liboff, Introductory Quantum Mechanics, third edition (1997). After a historical review, the mathematical postulates are introduced. R. Shankar, Principles of Quantum Mechanics, second edition (1994). Detailed analysis of the double-slit experiment, followed by the introduction of the mathematical postulates. D. T. Griffiths, Introduction to Quantum Mechanics (1994). The postulates are never formally introduced. The necessary math is presented throughout the book with heuristic introductions. R. P. Feynman, R. B. Leighton, M. L. Sands, The Feynman lectures on Physics, Third printing (1966). The postulates are never formally introduced. The necessary math is presented throughout the book with heuristic introductions. J. J. Sakurai, Modern Quantum Mechanics, revised edition (Addison-Wesley, Reading, MA, 1994). Detailed analysis of the Stern–Gerlach experiment, after which the mathematical framework is introduced. C. Rovelli, Relational quantum mechanics, Int. J. of Theor. Phys., 35: pp.1637, 1996. For a nice introduction see Alexei Grinbaum, On the notion of reconstruction of quantum theory, arXiv:0509104v2 \[quant-ph\]. [^1]: This notion of idealized measurements is not new. For example, J. V. José, E. J. Saletan, Classical dynamics: a contemporary approach (Cambridge University Press, 1998), p. 6: the principles of classical mechanics “must be understood as statements about idealized experiments”. We expand on this notion to identify all the underlying assumptions. [^2]: Note that the invariance of the speed of light, the quantum hypothesis and the equivalence principle all say something about ideal measurements: that is why their introduction always requires a new way of describing physical phenomenona. [^3]: A hidden variable may only determine the future evolution of another hidden variable. [^4]: See for example Sakurai[@sakurai], pp 68-72. The derivation starts from probability conservation and continuity requirements.

--- abstract: | Space-based microlens parallax measurements are a powerful tool for understanding planet populations, especially their distribution throughout the Galaxy. However, if space-based observations of the microlensing events must be specifically targeted, it is crucial that microlensing events enter the parallax sample without reference to the known presence or absence of planets. Hence, it is vital to define objective criteria for selecting events where possible and to carefully consider and minimize the selection biases where not possible so that the final sample represents a controlled experiment. We present objective criteria for initiating observations and determining their cadence for a subset of events, and we define procedures for isolating subjective decision making from information about detected planets for the remainder of events. We also define procedures to resolve conflicts between subjective and objective selections. These procedures maximize planet sensitivity of the sample as a whole by allowing for planet detections even if they occur before satellite observations for objectively-selected events and by helping to trigger fruitful follow-up observations for subjectively-chosen events. This paper represents our public commitment to these procedures, which is a necessary component of enforcing objectivity on the experimental protocol. author: - 'Jennifer C. Yee, Andrew Gould, Charles Beichman, Sebastiano Calchi Novati, Sean Carey, B. Scott Gaudi, Calen Henderson, David Nataf, Matthew Penny, Yossi Shvartzvald, Wei Zhu' title: 'Criteria for Sample Selection to Maximize Planet Sensitivity and Yield from Space-Based Microlens Parallax Surveys' --- [Introduction]{} \[sec:intro\] ============================== Measuring the Distances to Microlensing Planets ----------------------------------------------- While more than 6000 planets (and strong planetary candidates) have been found within about 1 kpc of the Sun (the great majority discovered via the transit and radial velocity techniques), there are only a handful of confirmed planets with known distances that are greater than 4 kpc and only one confirmed planet in the Galactic bulge [@mb11293B]. All of these distant planets were found using gravitational microlensing, and in most cases the distances were determined using the “microlens parallax” technique [@gould92]. Microlensing would therefore appear to be the most natural method to measure the Galactic distribution of planets, i.e., to determine planet frequency as a function of Galactic environment. Such a measurement would provide important constraints on planet formation theories. For example, @thompson13 has suggested that gas-giant formation may have been inhibited in the Galactic bulge due to the high intensity of ambient radiation during the main epoch of star formation. However, while roughly half of the $\sim30$ published microlensing planets have measured distances, this sample is heavily biased toward nearby systems. The reasons for this are well understood and are closely related to the general biases in astronomy toward nearby objects. First, nearby lenses have larger lens-source trigonometric parallaxes, $\pi_\rel = \au(D_L^{-1}-D_S^{-1})$, which gives rise to larger microlens parallaxes $$\bpi_\e \equiv {\pi_\rel\over\theta_\e}\,{\bmu\over\mu}; \qquad \theta_\e^2 = \kappa M\pi_\rel, \qquad \kappa \equiv {4 GM\over c^2\au}\simeq 8.1\,{{\rm mas}\over M_\odot}, \label{eqn:pie}$$ where $\bmu$ is the lens-source relative proper motion (in either the heliocentric or geocentric frame), $\theta_\e$ is the angular Einstein radius, and $M$ is the lens mass. As explained in some detail by @gouldhorne, the magnitude of $\bpi_\e$ quantifies the amplitude of the parallax distortion on the microlens light curve, so that all other things being equal, larger $\pi_\e$ implies easier detection. The most common method for measuring microlens parallax has been to observe the effect of Earth’s acceleration on the light curve (so-called orbital parallax). However, for typical Einstein timescales $t_\e\sim 20\,$day, this effect is quite modest. This means that in addition to nearby lenses and low mass lenses, one is biased toward abnormally long duration events. It is difficult (though probably not impossible) to quantify these biases, but the main problem is that due to these biases, there are simply no microlens planets in the Galactic bulge with measured microlens parallaxes. Indeed, the one confirmed Bulge planet had its distance measured by other means. This brings us to the other method of measuring lens distances: direct detection of the lens. The main difficulty is that the lens is superposed on a (usually) substantially brighter source, and remains so for typically a decade or more after the event. If the lens is sufficiently bright, then it is possible to directly detect it by measuring the combined source and lens light using high-resolution imaging (adaptive optics or [*Hubble Space Telescope (HST)*]{}) and subtracting out the source contribution, which is known from the light curve model. This, in fact, is how the distance to the only planet known to be in the Galactic bulge was measured [MOA-2011-BLG-293Lb; @mb11293B]. At the present time, this method is primarily limited to lenses that are at least 15% as bright as the source: otherwise the excess light due to the lens cannot be reliably detected. Hence, it is biased toward luminous (i.e., massive) and nearby lenses. The alternative is to wait until the source and lens separate due to their relative proper motions (typically a few mas yr$^{-1}$) and can be individually resolved. Again, this method is more easily applied to brighter lenses and with current facilities, one must wait $\sim10$ yr for the source and lens to separate sufficiently. When the next generation of 30 m telescopes are available, it will be applicable to much fainter lenses because these will separate sufficiently from the sources to be resolved within a few years due to their relative proper motions [@alcock01; @gould14; @ob05169a; @ob05169b; @henderson15]. Therefore, the only path at present to routinely measure the distances to lenses (especially faint lenses), and hence to measure the Galactic distribution of planets, is via space-based microlens parallaxes. In this approach, one observes a microlensing event simultaneously from Earth and from a satellite in solar orbit, and derives $\bpi_\e$ from the difference in the two light curves [@refsdal66]. There are some challenges to this method (over and above the problem of gaining routine access to such a satellite). First, the results are subject to a four-fold degeneracy in $\bpi_\e$, including a two-fold degeneracy in $\pi_\e$. However, @21event showed that it is possible in practice to break this degeneracy in the great majority of cases. Second, $\pi_\e$ does not by itself yield distances and masses. Rather this requires knowledge of $\theta_\e$, $$\pi_\rel = \theta_\e \pi_\e, \qquad M = {\theta_\e\over \kappa \pi_\e}, \label{eqn:massidst}$$ and of the source parallax $\pi_S$ ($\pi_L = \pi_\rel + \pi_S$), although the latter is usually known quite adequately. Fortunately, $\theta_\e$ is usually measured for planetary events because the normalized source size $\rho\equiv\theta_*/\theta_\e$ can usually be measured from the source crossing of the planetary caustic, while the angular source size $\theta_*$ is almost always known from its color and magnitude. Moreover, even for non-planetary (and non-binary) events, which generally lack such crossings, the lens distance (and so mass) can usually be estimated quite well from the measured $\bpi_\e$ and kinematic arguments [@21event]. Finally, for the case that the source proper motion can be measured, this estimate becomes even more accurate [@ob140939]. Hence, as shown by @21event, one can obtain an accurate estimate of the cumulative distribution of lens distances from a given sample, and can in principle compare this to the cumulative distance distribution of detected planets. [[*Spitzer*]{}]{} and the Galactic Distribution of Planets ---------------------------------------------------------- To determine the Galactic distribution of planets, however, the detected planets must be compared to the underlying distribution of planet sensitivities, not simply of events. @21event did not attempt to do this because there was only one planet in their sample [@ob140124], making a meaningful comparison impossible. The small number of planet detections was rooted in the nature of the observing campaign, which was a 100-hr “pilot project” to determine the feasibility of making such microlens parallax measurements using [*Spitzer*]{}. Thus, the [*Spitzer*]{} observations were limited to the subset of events judged most likely to yield $\bpi_\e$, and no special effort was made to find planets within these events via, for example, intensive follow-up observations. @21event argued, nevertheless, that it would be possible to estimate the cumulative distribution of sensitivities, simply by measuring the sensitivity of each event in the standard fashion [@rhie00; @gaudisackett; @gaudi02] and multiplying these sensitivities by the distance distributions in their Figure 3, even though the selection function of the events was unknown (and probably unknowable). This argument rested critically on the fact that the events were monitored from the ground and chosen for [*Spitzer*]{} observations without regard to the presence of absence of planets. This is a very similar argument to the one made by @gould10 in the first study to derive planet frequencies from microlensing planet detections. @21event further argued that their sample could be concatenated with future space-based samples, regardless of whether these were carried out using [*Spitzer*]{} or other satellites such as [*Kepler*]{}, and regardless of whether the selection function was the same or different, known or unknown. The only proviso was that, as with the @gould10 and @21event samples, the events were monitored without regard to the presence or absence of planets. This Paper ---------- The goal of the [[*Spitzer*]{}]{} microlensing parallax program (and indeed any space-based parallax program) is to create a sample of events with well-measured parallaxes. If these events are observed by the satellite without regard to whether or not they have planets, the final sample can be used to determine the Galactic distribution of planets, e.g., by comparing the frequency of planets in the Galacitic bulge with the frequency of planets in the disk. Hence, achieving this scientific goal has three primary considerations. First and foremost, the decision to select an event for [[*Spitzer*]{}]{} observations must be independent of any knowledge of the presence or absence of a planet[^1]. Second, these observations must lead to a measurable parallax. Finally, maximizing the constraints on the Galactic distribution of planets requires maximizing not only the number of planets detected but also the range of planets that [*could*]{} be detected (i.e. the planet sensitivity), since the detection efficiency is a crucial component to any measurement of the planet occurrence rate. The primary goal of the present paper is to determine a strategy to monitor events with [[*Spitzer*]{}]{} to ensure the final sample of events with parallaxes is monitored without reference to the presence or absence of planets, because this is the property of the sample that is most difficult to control. At the same time, this strategy is driven by the additional goals of maximizing both the planet sensitivity of the monitored events and the likelihood of measuring parallaxes. By defining this strategy in advance of the observations, we can create a sample of events with measured parallaxes with maximal leverage for measuring the occurrence rate of planets as a function of Galactic distance. We begin, in Section \[sec:objsub\], with a general discussion of how events may be selected, either objectively or subjectively, and how that selection affects the resulting planet sensitivities of those events. Since much of that discussion is guided by planet sensitivity and the practical considerations of the [[*Spitzer*]{}]{} campaign, the reader may also wish to refer to Sections \[sec:planet-sens\] and \[sec:spitzobs\]. Sections \[sec:ingredients\]–\[sec:parallaxprob\] cover the various ingredients necessary to define criteria for selecting events, namely the planet sensitivity and the probability of measuring parallax. Then in Section \[sec:objcrit\], we formally define objective criteria for the [[*Spitzer*]{}]{} campaign to select events and also determine their observing cadences. Section \[sec:subchoice\] then discusses specific guidelines for subjectively choosing events for this campaign, and Section \[sec:realloc\] specifies how the available observations will be distributed amongst the targets. Finally, we give a brief summary in Section \[sec:conclude\]. We have provided a glossary of terms in Table \[tab:glossary\] to clarify some qualitative statements we may use and also colloquialisms that have arisen in microlensing. Objective vs. Subjective Selection Criteria {#sec:objsub} =========================================== There are many choices that must be made with respect to [*Spitzer*]{} observations of any individual event. One must decide when to begin making such observations, when to commit to the target[^2], with what initial cadence, whether and when to change this cadence, and whether and when to halt the observations. This entire chain must be carefully established in order to ensure the fundamental requirement that the observational sequence be indifferent to the presence or absence of planets. Table \[tab:definitions\] gives a brief overview of the relevant decision points, and Section \[sec:spitzobs\] discusses how the specifics of [[*Spitzer*]{}]{} operations set the quantitative definitions. The starting point is the choice to begin monitoring an individual event, i.e., “triggering” observations. This choice can be made either because the event meets some objective criteria (in which case the “choice” is automatic) or according to some subjective criteria of the team organizing the observations. Table \[tab:selection\] summarizes the various channels through which observations may be triggered. However, all other decisions about the monitoring are heavily influenced by the first dichotomy (objective vs. subjective), so we divide the discussion according to it. As we will describe, optimal event selection requires a combination of objective and subjective selection. Because this is so, one must also decide what to do in advance if the objective and subjective selection procedures collide. That is, what should be done if an event is selected subjectively, but later meets the objective criteria for selection. We discuss the architecture of the selection procedure before discussing the criteria themselves because the architecture is both non-trivial and logically independent of the criteria. Within the framework of this discussion, one must keep in mind that the overall goal is to maximize the sensitivity of the experiment to planets and that planet sensitivity rests primarily on ground-based observations (see Section \[sec:planet-sens\]). At the same time, after an event is selected, its entry into the final sample to measure the Galactic distribution of planets requires that its parallax is measured, which depends primarily on [*Spitzer*]{} observations. In the following sections, we occasionally give examples to illustrate the points under discussion. For these examples, it may be helpful to keep in mind that some of observables that affect planet sensitivity and parallax measurements include the time of the event peak $t_0$, the magnification of the event (larger is better), and the magnitude of the event as seen from the ground or from [[*Spitzer*]{}]{} (brighter is better). These observables and their relationship to planet sensitivity and parallax are discussed in detail in Sections \[sec:planet-sens\] and \[sec:parallaxprob\]. The final criteria are given in Section \[sec:objcrit\]. Objectively Chosen Events {#sec:obj} ------------------------- ### Objective Selection The great advantage of choosing events by objective criteria is that any planet that is discovered during an event that is so chosen can be included in the sample, and similarly, the planet sensitivity of the event over its entire duration can be included in the analysis as well. For example, suppose an event is announced by a survey group on May 1 but [[*Spitzer*]{}]{} observations cannot begin until June 8. The event undergoes a planetary deviation on May 15, peaks on May 28, and on June 3 is scheduled for [[*Spitzer*]{}]{} observations beginning June 8 because it is found to meet previously chosen objective criteria. Then the planet can be included in the sample, even though it was discovered before the [*Spitzer*]{} observations began, and even before it was known that it would eventually satisfy the objective conditions that triggered observations. By contrast, in the absence of such criteria, the event could have been selected for [*Spitzer*]{} observations subjectively. In that case, neither the previously discovered planet, nor the planet sensitivity from the the entire pre-decision period, could be included in the analysis. Otherwise, the presence or absence of the planet could influence the event’s “selection” (i.e., inclusion in the final sample with measured parallaxes; Section \[sec:measurement\] discusses what is meant by a “measured parallax”). ### Objective Cadences A large fraction of objectively chosen events will be similar to the hypothetical one described above in that most of their planet sensitivity will be past at the time that the [*Spitzer*]{} observations begin. Therefore, it is absolutely essential that the cadence be chosen objectively as well. In order to enter the sample, the event must have a measured parallax. If the cadence is not chosen objectively, events with planets could receive extra observations to help ensure they have measured parallaxes. We will discuss specific algorithms to make this choice in Section \[sec:objcrit\]. Finally, we will just mention that to avoid wasting observations, there must also be a mechanism for halting this objectively-determined [*Spitzer*]{} observation schedule when these observations are no longer useful. However, the only permitted reasons for doing so are that the microlens parallax has been measured or that the event (as seen from [*Spitzer*]{}) has already returned to baseline (i.e., is now essentially unmagnified), so that no further improvement is possible. Subjectively Chosen Events {#sec:sub} -------------------------- ### The Need for Subjective Selection At first glance, the advantage conferred by the objective approach appears to be so great that one might wonder why one would consider the subjective approach at all. The main problem is it is impossible to define objective criteria loosely enough to capture all events of interest without at the same time introducing a large number of events either with poor planet sensitivity or a low probability of yielding a parallax measurement (“bad” events). Hence, the objective criteria must be strictly defined so that all of the selected events are both highly sensitive to planets and have a high likelihood of yielding a parallax (“good” events). Otherwise, a large amount of observing time will be wasted on events of little value. A second issue is whether or not an event chosen objectively will yield a parallax measurement. The central difficulty is that the event’s objectively-chosen [*Spitzer*]{} observational sequence must yield a parallax measurement. It can be quite difficult to choose such observational sequences based solely on objective criteria, or to determine which events might be worth the observational effort to obtain parallaxes, or even to determine which might yield parallax measurements with any sort of effort. These difficulties can all be more effectively addressed by subjectively choosing the event, in which case one can also choose a cadence (or cadence algorithm) that is individually tailored to that event. For example, suppose that it is known that an event will meet objective criteria in 2 weeks, but this will allow for only 1 week of [[*Spitzer*]{}]{} observations. If we wait to start observations until this date, we risk the possibility that 1 week of data will be insufficient to measure a parallax, in which case the entire event and its sensitivity is lost. In contrast, we could select the event subjectively now to get 3 weeks of [[*Spitzer*]{}]{} observations and vastly improve the probability that those observations will yield a parallax. Hence, there are two reasons that events might be chosen subjectively. First, because the objective criteria cannot capture all events of interest. Secondly, because an earlier subjective trigger may make the difference between measuring a parallax or not. As we discuss in the next section, not much planet sensitivity is lost by subjectively choosing events. We will also discuss in Sections \[sec:collide\] and \[sec:reversions\] the resolution of conflicts for events that may be selected both subjectively and objectively. ### Subjective Selection If, for whatever reason, an event fails to meet the objective selection criteria, the team may decide to observe it anyway. The reasons one might want to do this are discussed in some detail below, but first we focus on the consequences of this decision, which leads to three types of subjective selection as defined in Table \[tab:selection\]. Actually, the key decision is not whether to begin [*Spitzer*]{} observations but when to (publicly) commit to a schedule of [*Spitzer*]{} observations. Once such a decision is made, it must be accompanied by a public commitment of an observational sequence (or to an objective algorithm for determining that sequence). Otherwise, one could choose events without any knowledge of whether they would later show planets, discover that they do indeed host planets, and then be biased to observe them more frequently in order to preferentially increase the probability that their parallax will be measured, thus placing them in the sample for measuring the Galactic distribution of planets. In the case of subjective decisions, all planets that show up after the public commitment date ($t_{\rm com}$), as well as all sensitivity to planets after this date, would be kept in the analysis, while all planets and planet sensitivity from before this date must be excluded. We illustrate the need to separate the [*decision*]{} to observe from the [*commitment*]{} to observe with two examples. First, suppose that at a given [*Spitzer*]{} decision time (typically $t_{\rm sel}=$Monday, see Section \[sec:spitzobs\] for details of the logistical constraints on [*Spitzer*]{} observations) an event that has not yet peaked has an ambiguous future, with some chance that it will rise in magnification sufficiently to have good sensitivity to planets and to enable a viable parallax measurement (e.g., panel 2a of Figure \[fig:lcs\]). However, as of Monday, this cannot be established with any confidence, although ground-based data are likely to resolve this ambiguity with a few days. Based on this assessment, the team decides to observe this event once per day during the week beginning at the next upload three days hence, i.e., $t_{j,\rm next}=$Thursday. Even though the future of the event is uncertain, preemptive observations (rather than waiting for the next upload cycle, one week hence) could make the difference between a good parallax measurement and a meaningless upper limit. This is especially true since the [*Spitzer*]{} observations can only be updated once per week. Then on Sunday, ground-based data show that this event has risen sufficiently that its future behavior can be predicted well enough to determine that it is an interesting target (e.g. panel 2c of Figure \[fig:lcs\] as compared to panel 2a). The team then announces that it is committed to monitoring this event and also announces its chosen cadence (or objective procedure for determining such cadence). All planets and planet sensitivity from $t_{\rm com}=$Sunday forward can then be included in the analysis. Similarly, only [*Spitzer*]{} data beginning on Sunday can be used in the initial parallax measurement measurement that determines whether the event enters the sample. Even though the Thursday–Saturday observations cannot be used, this preemptive decision to observe has resulted in extra observations (Sun–Wed) that can be used to improve the parallax measurement as compared to waiting until the following Thursday to begin [*Spitzer*]{} observations. With regard to the specific role of [*Spitzer*]{} data in planet sensitivity and discovery: if the team has seen the first few days of [*Spitzer*]{} data prior to the Sunday commitment, then these cannot be included, but if they have not yet seen them, these can be (since in this case they would not have influenced the decision). For a second example, consider the same case as above but with the event peaking at low magnification, and hence having both low sensitivity to planets and low likelihood of a measurable parallax (e.g., the data instead follow the solid black line in panel 2a of Figure \[fig:lcs\]). The team then decides not to continue monitoring this event. Because the team never committed to the event, they have no obligation to continue monitoring it, and so it is entirely dropped from the sample. By the same token, any planets discovered from this event cannot be included in the analysis. In contrast, if the decision and commitment were the same process, in order to avoid bias, the team would be required to continue monitoring an event that it recognized as worthless. In the first example given above, the public commitment to observe the event was made after the [*Spitzer*]{} observation sequence was uploaded to the spacecraft and indeed after those observations began ($t_{\rm com}>t_{j,\rm next}>t_{\rm sel}$). This is the situation described as “Subjective, secret” in Table \[tab:selection\]. However, the same principles apply to, for example, an event that is newly recognized a few days before the upload decision and is already recognized to be promising (e.g. panels 2c or 2d of Figure \[fig:lcs\]). The team could publicly commit to observing this event immediately with specified cadence (i.e. “Subjective, immediate” in Table \[tab:selection\]), and then all planets and planet sensitivity from that date forward would be included (provided the parallax was measured well enough to put the event in the sample). Finally, in the first example, the team might publicly commit immediately to observe the ambiguous event, but with explicitly stated criteria for halting such observations (Table \[tab:selection\], “Subjective, conditional”). For example, it might specify that observations would be discontinued if the event failed to reach some specified magnification before the time of the next upload. In this case, whether or not the observations were continued, the event might have sufficient [*Spitzer*]{} data to measure the parallax and enter the sample. If so, all planets and planet sensitivity from after $t_{\rm sel}=t_{\rm com}$ could enter the sample. At first sight, “Subjective, conditional” seems clearly superior to “Subjective, secret” because it enables inclusion of more planets and more planet sensitivity. However, this is not always the case. In fact, it often happens that the uncertainty in predicting rising events includes not only to their time of peak and peak magnification, but extends even to when this knowledge will be reliably available. For example, consider the evolution of the fits to the data in column 1 of Figure \[fig:lcs\], which illustrates how the fits may change dramatically due to small fluctuations in the data and may not capture the true, underlying behavior of the event. If the team cannot reliably predict the future course of the event, it may not be possible to correctly pre-define criteria for halting observations. This creates the risk of being committed to observing many bad events or being forced to halt observations of an event that turns out to be good, but with different parameters than initially supposed. Thus, as summarized in Table \[tab:selection\], subjective decisions can take a considerable variety of forms. The only constraint is that they must be constructed to avoid the possibility that the presence or absence of planets detected after $t_{\rm com}$ will influence the cadence of observations. ### Subjective Cadences In contrast to objectively chosen events, the cadences for subjectively chosen events can be chosen by the team. However, they must be fully specified at the time of the commitment to observations ($t_{\rm com}$). As with objectively chosen events, after $t_{\rm com}$ the only permitted reasons for halting the scheduled [*Spitzer*]{} observations are that the parallax has already been measured or the event has returned to baseline (as seen from [*Spitzer*]{}). Collisions: Subjectively Chosen Events that Meet Objective Criteria {#sec:collide} ------------------------------------------------------------------- An event that has been subjectively chosen may, at a later date, meet the objective criteria. In this case, the objective selection of the event [*must*]{} take precedence. Otherwise, there is no point in having objective criteria since they could always be subjectively overridden. The objective criteria will specify an objectively-determined cadence of observations. The more frequent of the subjective or objective cadences will take precedence for future observations. However, only the observations taken after the objective selection and at the objective cadence can be used to determine whether the parallax measurement is adequate to enter the sample. If it does enter, then all planets that are detected and all planet sensitivity (including from before the commitment to the event, $t_{\rm com}$) enter the analysis. The next section will clarify the reason for maintaining a higher, but subjective, cadence even if only a fraction of the data can be used for the objectively-measured parallax. Note that for these cases all planets discovered in [*Spitzer*]{} data (as well as all planet sensitivity from these data) should be included in the analysis, regardless of whether these data were taken before or after or in response to the commitment to the subjective selection of the event. The only exception would be if the planet were detected purely in the [[*Spitzer*]{}]{} data [*and*]{} subsequent [[*Spitzer*]{}]{} observations were increased because of it (see Section \[sec:characterization\]). Reversions from Objective to Subjective {#sec:reversions} --------------------------------------- If an event has 1) been chosen subjectively, 2) subsequently satisfied the objective criteria and thus triggered a conversion to being objectively-chosen, 3) fails to yield a parallax based on the objective portion of the [*Spitzer*]{} light curve, and 4) does yield a parallax based on all, post-commitment [*Spitzer*]{} data, then it automatically reverts to subjective status. In this case, only planets and planet sensitivity from the post-commitment part of the light curve can be included in the analysis. The basic reason is that none of the decisions made after this commitment depended in any way on the presence or absence of a planet (other than possible planets before this date, which must be excluded from the calculation of planet occurrence). Planet-Characterization Observations {#sec:characterization} ------------------------------------ As discussed above, whether the decision to observe an event with [*Spitzer*]{} is objective or subjective, the cadence must be chosen without reference to the presence or absence of a planet. For the objectively-chosen events, this cadence must be determined objectively by pre-determined criteria. For subjectively chosen events, they must be announced at the times of the decision, i.e., prior to the discovery of any planets. However, once a planet is discovered, it can be important to increase the pace of observations from the ground and/or [*Spitzer*]{} in order to improve the [*characterization*]{} of the planet. In this case, one can increase the observational cadence, but only the observational data that would have been taken under the pre-determined schedule can be used to assess the [*detectability*]{} of planets and the measurability of the parallax. See, for example, @mb11293. Application of this rule is straightforward for [*Spitzer*]{} observations, which is the main focus in this section, because, as we have specified, the observational cadence is in fact pre-determined. The situation is more complicated for ground-based observations of the same event, to which we now turn. Ground-based Follow-up Observations {#sec:gbfollow} ----------------------------------- The majority of microlensing planets published to date were discovered by a combination of microlensing surveys that find large numbers of events, mostly with a low-to-moderate cadence of observations, and follow-up surveys that target individual events for more intensive monitoring in order to enhance the discovery and characterization of planets. In fact, survey groups sometimes go into “follow-up mode” by increasing the cadence in the survey field that contains a particularly interesting event [@mb11293] or even re-centering an existing field to incorporate a particular event [@mb13220]. For present purposes, observations by specialized follow-up groups and survey teams in “follow-up mode” are equally considered as “follow up”. The only exception would be follow-up observations that are determined by purely objective criteria. Follow-up observations must be evaluated with respect to two questions: first, how do they impact planet sensitivity and planet detection, and second how do they impact the measurement of microlens parallax (and so entry into the sample). ### Follow-Up and Planet Sensitivity/Detection \[sec:followupsens\] As with survey observations, follow-up observations contribute to planet sensitivity and detection for the entire duration of objectively chosen events and for all post-commitment observations of subjectively chosen events (which may be a time-span much longer than the window for [[*Spitzer*]{}]{} observations). Indeed, while the fundamental point of announcing subjective choices of events for [*Spitzer*]{} observations (i.e., the commitment to observe an event) is to establish a record of what planet sensitivity can be included, a major secondary goal is to encourage observations of these events, particularly those that are not well covered by the surveys, (see Section \[sec:ground\] for a general outline of standard survey strategies). Such subjective announcements automatically have the effect of encouraging follow-up observations because prior to such announcement, the planet detections can enter only if the event has already satisfied, or ultimately proves to satisfy, objective criteria. The only question is whether changes in the adopted cadence of follow-up observations due to the perceived presence of planets influences the detectability of the planets. This can happen in principle if the planet generates an observed perturbation (in either survey or follow-up data) that is strong enough to trigger interest but not, by itself, sufficient to confirm the presence of the planet. In this case, follow-up observations aimed at characterizing the planet can make the difference between it being undetectable and being detectable. This issue is not particular to parallax experiments: it pertains to any experiment that aims to make a statistical statement about planets using microlensing follow-up data. For example, @gould10 noted that two of their six detected planets occurred in an event that showed an early (and in itself, not comprehensible) perturbation that ultimately proved to be due to a planet. This early perturbation did trigger additional observations, but these subsided over the next few days. Observations only intensified again when the event approached high magnification, which was the standard trigger for high-cadence follow-up observations. In brief, these issues arise in a minority of planetary events, and usually can be resolved based on records of the decision making. While it is possible that there may be unresolved cases in the future, the importance of characterizing planets is too great to allow this possibility to interfere with aggressive follow-up response to the tentative detection of planets. ### Follow-Up and Parallax The other aspect that must be considered is the role of follow-up observations in measuring microlens parallaxes and so in putting individual events into the final sample used to measure the Galactic distribution of planets. There are two relatively distinct ways that this can happen. First, the planetary-induced features in the light curve may substantially increase the precision of the parallax measurement from [*ground-based observations alone*]{} and so make the difference between whether it is included in or excluded from the sample. Second, the follow-up observations may improve the precision of the non-planetary light curve parameters $(t_0,u_0,t_\e)_\oplus$ (i.e., the time of maximum, the impact parameter, and the Einstein timescale) and thus improve the precision of the parallax measurement that comes from the comparison with [*Spitzer*]{} data. We discuss these in turn. @angould01 argued that events with three peaks (features due to caustics induced by a companion) would gain significantly improved parallax measurements relative to otherwise similar point-lens events. However, in the intervening years, almost nothing has been done to investigate the role of the perturbations in the parallax measurements for detected planets. For example, early modeling showed that the immediate post-peak light curve of MOA-2009-BLG-266 yielded a surprisingly good parallax measurement, despite the fact that it is extremely rare for orbital parallax to be measurable before an event substantially returns to baseline. This was attributed to the sharp deviations in the light curve caused by a Neptune mass-ratio planet. Yet @mb09266 say that the dominant source of the $\Delta\chi^2=2789$ parallax signal derives from the MOA data in the wings and that very little parallax signal comes from the perturbed region. However, we find from fitting the MOA data alone (and excluding the perturbed region) that the parallax signal from these data is only $\Delta\chi^2=205$, implying that the perturbation could in fact play a major role in the strength of the parallax signal, as @angould01 anticipated. It is very likely that planetary perturbations play a significant role in the strength of the parallax signal in many other events as well. This is likely to be a partial explanation for the fact that roughly half of all microlens planets published to date have measured parallaxes. Although historically, there has been a lack of interest in where the parallax signal comes from, with respect to [*Spitzer*]{} observations, this question is of cardinal importance. If planetary events have more easily measured parallaxes than non-planetary events, then the sample of objects with measured parallaxes is biased. Hence, in determining whether the event enters the sample, it is essential to ask whether it would have a sufficiently well-measured parallax even if there had been no planet. This means both eliminating follow-up observations that were triggered by the presence of a planet and also (for this purpose) replacing the actual light curve with a fake, point lens light curve based on the event’s parameters $(t_0,u_0,t_\e)$. This fake light curve could then be fit to determine the strength of the parallax signal from the point lens event. This procedure must also be applied in cases in which the planetary perturbation is seen from [[*Spitzer*]{}]{}. A closely related issue is that an incipient planetary anomaly might be misinterpreted as evidence for an approach to high magnification, and hence trigger an “honest” (i.e., seemingly non-planet-related) decision to observe the event, either ground-based follow-up observations or observations by [*Spitzer*]{}. This occurred, for example, for ground-based observations of the second microlensing planet, OGLE-2005-BLG-071 [@ob05071]. Such observations by [*Spitzer*]{} might enable a parallax measurement that would not have been possible for a point-lens event. This must be checked in all cases, which again can be done through the use of fake light curves. By contrast, it will be relatively rare for follow-up data to play a major role in the determination of the event’s point-lens parameters simply because the main features of the event that enter the parallax measurement usually derive from long-term observations and so are well measured from survey data. However, high-magnification events can be an exception, primarily because dense coverage of the peak is often required to determine the ground-based impact parameter, particularly if the surveys cover the field at low cadence. This was exactly the case for OGLE-2014-BLG-1049, one of the 21 events analyzed by @21event. However, in the great majority of cases (including this one), an improved parallax measurement is simply one of the benefits of conducting follow-up observations of parallax candidates. The primary motivation is generally the increased probability of detecting planets [@griest98]. The only exception would be if these critical near-peak observations were triggered by the known presence of a planet (rather than the hope of finding one). We expect such planet-triggered parallax-assisting measurements will be extremely rare and mention them primarily for completeness. Principal Ingredients for Developing Selection Criteria {#sec:ingredients} ======================================================= At a fundamental level, there are only two considerations in deciding whether to choose one event over another for [*Spitzer*]{} observations. First, how sensitive to planets is that event? Second, how likely is it that if [*Spitzer*]{} observations are undertaken, a microlens parallax will actually be measured? Thus we may schematically define a “quality factor” $Q$ for the experiment $$Q = \sum_i S_i P_i , \label{eqn:qual}$$ where $S_i$ is the planet sensitivity of the $i$th event chosen, and $P_i$ is the probability that it will yield a parallax measurement. Then the goal in developing selection criteria should be to maximize $Q$. One issue posed by Equation (\[eqn:qual\]) is that planet sensitivity is actually a function of planet properties. However, this is easily resolved by adding two additional indices to $S_i$ to specify these properties. We return to this complication in Section \[sec:quantitative\]. A more fundamental challenge is that both $S_i$ and $P_i$ may be poorly known at the time that the decision must be made to initiate observations (c.f. Figure \[fig:lcs\]). The first step toward figuring out how to proceed in the face of these uncertainties is to review what makes an event sensitive to planets and how much about this can be known at any given stage in its evolution, and to then address the corresponding questions about the measurability of its parallax. Planet Sensitivities {#sec:planet-sens} ==================== After an event is over, its “sensitivity” to planets[^3] can be rigorously defined as a function of two variables, the planet-star mass ratio $q$ and the planet-star separation $s$ (in units of $\theta_\e$). See, for example, @gaudisackett, @gaudi02, @gould10, and @cassan12. However, when choosing an event for additional observations while it is evolving, one must be guided by a more qualitative understanding of what properties make the event sensitive to planets and judge how likely it is that these will appear. Ground-based Microlensing Observations {#sec:ground} -------------------------------------- Because microlensing events depend on the chance alignment of two stars at radically different distances, for the most part, these events cannot be predicted in advance. Hence, microlensing surveys monitor millions of stars toward the Galactic bulge, where the stellar density, and therefore the microlensing event rate, is highest. If the survey team sees a star brightening in a manner consistent with microlensing, they issue an alert announcing a new microlensing event. Microlensing surveys monitor these fields at a variety of cadences\ (e.g. [http://www.astrouw.edu.pl/\$\\sim\$jskowron/ogle4-BLG/](http://www.astrouw.edu.pl/$\sim$jskowron/ogle4-BLG/)). “High-cadence” fields are monitored one to several times per hour (e.g. column 2 of Figure \[fig:lcs\]), which is sufficient to characterize small planetary signals from terrestrial-mass planets. “Moderate-cadence” fields are observed several times per night, which can capture planetary signals from ice giants. “Low-cadence” fields are monitored once per night or less than once per night (e.g. column 1 of Figure \[fig:lcs\]) and are generally focused on producing alerts of ongoing microlensing events that can then be monitored more intensively by follow-up groups, although these survey observations themselves are occasionally sufficient to characterize large (gas-giant) planets. In the context of this paper, we will focus on survey data from two sources. The primary data will come from the OGLE-IV survey [@ogleiv], whose sky coverage is given in the above URL. In addition, we will consider data from the new Korea Microlensing Telescope Network (KMTNet) fields, which will be conducting its first year of routine microlensing survey observations. Under the assumption that these observations are carried out, we will include in our evaluations data from the four core KMTNet fields, which will be observed many times per hour. Planet Sensitivity: Qualitative Features {#sec:qualitative} ---------------------------------------- Microlensing detects planets around the lens stars as perturbations to a standard (point-lens) microlensing event. That is, the microlensing event is overall dominated by the gravitational potential of the lens star (host), which splits the light into two images whose position and size change as a function of time. If a planet interacts with one of those images, it creates a perturbation that distorts the shape and total magnification of that image, which can lead to a detectable signal. See @gaudi12, in particular Figures 4 and 5. The total “planet sensitivity” of a given event depends on two factors. The first is the intrinsic sensitivity of the event to planets. Larger planets, and planets that are closer to the “Einstein ring” (circle with angular radius $\theta_\e$) are easiest to detect. The larger the images are, the more sensitive they are to planetary perturbations. Hence, more highly magnified events are more sensitive to planets, i.e., to smaller planets and to planets that are farther inside or outside the Einstein ring [@gouldloeb]. The most extreme example would be the high magnification events (peak magnification of the underlying point lens event $A(t_0)=A_\max\ga 100$) in which the images form an almost perfect ring that probes a wide range of separations and is easily perturbed [@griest98]. The second factor affecting planet sensitivity is how well the data cover potential perturbations. The quality of this coverage is defined by two principal characteristics: cadence and photometric precision. Regarding the first, planetary perturbations typically last between a few hours and a few days, so a cadence that is a factor of $\sim 10$ more frequent than that is necessary to characterize those perturbations. Since such perturbations may occur at any time during the event, the sensitivity will be greatest if the data are continuous during this period. However, if observing resources are limited, then restricting continuous observations to the most highly magnified (hence most sensitive) parts of the light curve may be the most productive approach. Nevertheless, planets can appear in all parts of the light curve, even after the main event is over or before it began (e.g., @ob08092), so that observations are never “wasted”. Photometric precision is mainly governed by source brightness. This factor therefore favors intrinsically bright sources, but highly magnified sources can also be bright (just at the moment that they are most sensitive to planets). Even though we have not yet examined the other key determinant of $Q$ (i.e., the probability of measuring $\bpi_\e$, Section \[sec:parallaxprob\]), we can already draw a few general lessons from the above analysis of planet sensitivity. First, higher peak magnification is the best single indicator for choosing events (provided that the peak region can be intensively monitored from the ground). Second, for objectively chosen events, those that are in high-cadence survey fields are substantially more valuable than those that are not. This is because high-cadence events can yield planets long before the onset of [*Spitzer*]{} observations, or even before the event was recognized as microlensing, whereas low-cadence events generally cannot. Third, for events in low-cadence fields that achieve relatively high-magnification, it is important to mobilize follow-up observations prior to peak. If these are chosen subjectively, then the desirability of choosing this target will depend critically on the expectation that such follow-up observations will be carried out. Planet Sensitivity: Quantitative Determination {#sec:quantitative} ---------------------------------------------- The question of quantifying the planet sensitivity $S_i$ of each event is mainly outside the scope of the present paper because this can only be done after the event is largely over and hence after all of the observing decisions that are the subject of this paper have already been made. That is, these decisions must be made on the basis of the qualitative indicators discussed above. Nevertheless, for completeness we present here a broad overview of the relevant issues. Planet sensitivity is measured as a function of two parameters $(s,q)$ and so can be formally written $S^{s,q}_i$. These are two of the seven basic parameters that are the minimum needed to describe a planetary event. Three of the others are $(t_0,u_0,t_\e)$, i.e., the parameters of the underlying point lens event. The remaining two are the angle between the source-lens trajectory and planet-star axis, $\alpha$, and the ratio of the source radius to the Einstein radius, $\rho=\theta_*/\theta_\e$. Historically there have been two approaches to determining planet sensitivity. In the first approach [@rhie00], one constructs an ensemble of planetary light curves that vary in $\alpha$ but are fixed in the remaining six parameters. The values of $(t_0,u_0,t_\e)$ are adopted from the best fit of the single-lens event. We address the choice of $\rho$ further below. The remaining two parameters are just those being tested $(s,q)$. For each light curve, one creates fake data points at the times of each of the real measurements, with values equal to those predicted by the model and error bars equal to the those of the real data points. With the adopted parameters, the fit to a fake curve without a planet would be “perfect”, i.e., $\chi^2=0$, so any $\chi^2$ in excess of this value must be due to the planet. One then fits these fake data to a point-lens model. Therefore, if the $\chi^2$ is above some threshold (perhaps $\Delta\chi^2>200$ for events with moderate magnification, $A_\max<100$, @mb11293) then the planet is said to be detectable. The fraction of all the $\alpha$ at fixed $(s,q)$ for which the planet is detectable is then said to be the sensitivity $S_i^{s,q}$. In the other approach [@gaudisackett], one fits the actual data with planetary models with the same sampling of parameters, and measures the $\Delta\chi^2$ improvement between the planetary model and the best-fit point lens model. This method is more time-consuming but has the advantage of simultaneously searching for all planets that may be lurking in the data down to the adopted threshold [@gaudi02]. The choice of $\rho$ is a subtle one. For planetary events, one often measures $\rho$ from the smearing out of the light curve as the source passes the sharp edge of a caustic. Hence, when constructing fake planetary light curves, one must insert some value of $\rho$ even though this quantity is very rarely measured in point-lens events. The problem is that while $\theta_*$ is usually well-determined from the color and magnitude of the source (e.g., @ob03262), $\theta_\e$ is not known. We do not review the various methods used to estimate $\rho$ in the past but simply note that for [*Spitzer*]{} events, $\theta_\e = \pi_\rel/\pi_\e$ is usually known quite well because $\pi_\e$ is measured and $\pi_\rel$ is well constrained [@21event], allowing a well-constrained estimate of $\rho=\theta_*/\theta_\e$. Current microlensing experiments have far too few detections to constrain the full two-dimensional distribution of planets as a function of $(s,q)$. However, for comparing to data, one can marginalize over one or both indices, e.g., $$S_i^q = \int_{-\infty}^\infty d\log s S_i^{s,q}; \qquad S_i^{<\rm jup} = \int_{-\infty}^{-3} d\log q S_i^q. \label{eqn:sisq}$$ There is one further issue that has not been previously considered in the literature but is quite relevant here. In the above-described procedures, it is implicitly assumed that the observations were carried out without reference to the presence or absence of a planet. This has usually been the case, and in the one notable example that it was not, the authors took the trouble to remove the extra observations that were triggered by the presence of the planet [@mb11293]. See also @ob120406. However, for [*Spitzer*]{} events that are chosen subjectively, a large fraction of the nominal planet sensitivity may be due to observations before the public announcement. However, as discussed in Section \[sec:followupsens\], there is the possibility that a planetary perturbation in the early stages could trigger additional observations or selection before the perturbation is well-understood or even recognized, e.g. if the perturbation was mistaken for a rise toward high-magnification. To determine which hypothetical planets should be excluded from the planet sensitivity, we suggest that the following additional test be conducted for each hypothetical planet that is regarded as “detected” based on the full light curve: truncate the fake-data light curve at the decision date and fit only the points prior to this date to a point-lens model. We suggest that if $\Delta\chi^2<10$, then the signal from the hypothetical planet would be too small to trigger either follow-up or [*Spitzer*]{} observations. Hence, the hypothetical-planet detection should be accepted, but otherwise it should be rejected. This limit is chosen because in our experience such $\Delta\chi^2<10$ deviations are extremely common and so cannot possibly trigger resource-expending actions. However, substantially higher $\Delta\chi^2$ might well trigger an unconscious [*Spitzer*]{} decision announcement. Probability of Measuring Parallax {#sec:parallaxprob} ================================= While parallax measurements derive from a combination of ground-based and space-based data, the limiting factor will be [*Spitzer*]{} data in almost all cases. The main reason is that the [*Spitzer*]{} observations of any given event are restricted to 38 days by Sun-angle constraints[^4], and these 38 days fall in an arbitrary part of the light curve. Second, both the cadence and quality of the data are very likely to be lower than the ground-based data. Finally, if the parallax is large enough, the event as seen from [*Spitzer*]{} may pass entirely outside the Einstein radius during the [*Spitzer*]{} observations and so be effectively unmagnified. Hence, the probability that $\bpi_\e$ will be measurable reduces in essence to probability that adequate [*Spitzer*]{} observations can be obtained. Hence, most of the discussion about “measuring parallax” is rooted in the specific nature and procedures for [*Spitzer*]{} observations. However, we begin by briefly discussing what it means to “measure parallax”. Meaning of “Parallax Measurement” {#sec:measurement} --------------------------------- Quantities are usually said to be “measured” if a numerical value can be assigned to them with some error bar and if this value is determined to be inconsistent with zero with some specified level of confidence, e.g., $3\,\sigma$. For space-based microlensing parallaxes, there are circumstances in which upper and lower limits are sufficiently constraining, and therefore the definition of a “measurement” requires explicit discussion. For example, suppose that an event is observed from the ground with $(u_0,t_\e)_\oplus = (0.2,10\,{\rm day})$ and with $t_{0,\oplus}$ within the window of [*Spitzer*]{} observations, but the [*Spitzer*]{} light curve is completely flat. Also suppose that the source is bright enough that 10% variations in its flux would have been detected. From these (lack of) measurements, together with the fact that [*Spitzer*]{} was $\sim 1 {\rm AU}$ from Earth at the time of observations, one could conclude that $\pi_\e > 1.5$, but no specific value of $\pi_\e$ could be assigned. Although not a “measurement” by traditional standards, this lower limit would be highly constraining. That is, it would imply that the projected velocity would be $\tilde v \equiv {\rm AU}/(\pi_\e t_\e)> 115\,{\rm km\,s^{-1}}$, implying that the lens was very likely in the near disk, $\pi_{\rm rel}>0.1\,{\rm mas}$. Hence, if there were no planet discovered, the distance would be statistically well enough constrained to enter the cumulative distance distribution function. If the event proved to have a planet, then it is likely that $\rho$ (and so $\theta_\e$ would be measured), which would permit a strict lower limit on $\pi_{\rm rel}=\theta_\e\pi_\e$ and a strict upper limit on $M=\theta_\e/\kappa\pi_\e$. At the opposite extreme, if for the same $(t_0,u_0,t_\e)_\oplus$, the [*Spitzer*]{} and Earth-based lightcurves appeared identical, this would be consisent with $\pi_\e=0$, which in traditional terms might be considered as “no measurement”. However, if this consistency were quite tight, say $\pi_\e<0.01$, then the projected velocity would be constrained $\tilde v > 17,300\,{\rm km\,s^{-1}}$, implying that the lens certainly lies in the bulge. Hence, the final sample must be defined as events that yield true measurements or [*either*]{} upper or lower limits on $\pi_\e$ (or both). The exact limits cannot be defined in advance because there is not yet enough experience with [*Spitzer*]{} parallax measurements to determine what are reasonable limits. [*Spitzer*]{} Procedure {#sec:spitzobs} ----------------------- [*Spitzer*]{} observation sequences can only be uploaded to the spacecraft once per week, and hence targets can only be changed on this timescale, and furthermore, the entire week of observations must be planned in advance. In addition, it takes several days to prepare the observation sequence for upload to the spacecraft even after the targets and observation sequence have been set. The net result is that the targets and sequence are set 3–10 days before the observations are actually carried out. See Figure 1 of @ob140124. In light of these considerations, and to facilitate the discussion, we define several variables summarized in Table \[tab:definitions\]. First, we define $t_{j, \rm dec}$ to be 6 hours prior to the time that observing choices must be forwarded to [ *Spitzer*]{} operations for a given observing “week” $j$, i.e., $t_{j, \rm dec}=$ Monday UT 15 - 6 hrs. Experience shows that this is the latest time that new information can be reliably incorporated into the observing request without risking the introduction of serious errors. Given the day-of-the-week constraints, the $\sim 40$ day campaign, and a start date of June 3rd, $j$ takes on values from 1 to 7. The time of the first possible [[*Spitzer*]{}]{} observations of a given event is defined as $t_{\rm first}$. For simplicity, we will let $t_{\rm first}$ be when those coordinates could first be observed by [[*Spitzer*]{}]{}, even if the event is not discovered until afterward this date. Finally, we define $t_{j, \rm next}$ as the time of the first possible observation that can be requested at $t_{j, \rm dec}$, and we define $t_{\rm fin}$ as the final possible observation of an event before the [*Spitzer*]{} observing season ends (because of Sun-angle restriction and/or the end of the allocated observations). Note that $t_{j, \rm dec}$ and $t_{j, \rm next}$ change each week, while $t_{\rm first}$ and $t_{\rm fin}$ are defined for each particular event (set by the 38-day Sun-angle constraint). In general, for events selected for the first week of observations, $t_{1, \rm next}=t_{\rm first}$. [*Spitzer*]{} Observation Cycles {#sec:spitzcyc} -------------------------------- Generally speaking, we expect most or all of the available [[*Spitzer*]{}]{} time to be devoted to this program during the $\sim 40$ day observing window. Hence, given continuous observing time, [*Spitzer*]{} observations in a given week can be carried out most efficiently if the targets are organized in concatenated “cycles” moving West-to-East through the Southern bulge and then East-to-West through the Northern bulge. Each event can then be given a priority $n$, which designates that it will be observed each $1/n$ cycles. That is, if $n=1$ it will be observed every cycle through the Bulge, and if $n=8$ it will be observed every eighth cycle. We expect approximately eight cycles per day, each lasting $\sim 2.4$ hours, with the exact number determined by the total observation time allotted and the total number of targets per cycle. This is discussed in more practical detail in Section \[sec:realloc\]. [*Spitzer*]{}’s Role in Parallax Measurements {#sec:spitzrole} --------------------------------------------- As originally conceived, the standard way to measure satellite parallax was to observe the peak of the light curve from the satellite and the full light curve from the ground. This requires only partial, but very specific, light curve coverage from space. Therefore, one of the goals of [[*Spitzer*]{}]{} observations is to try to capture this peak in as many cases as possible. Although this measurement is nominally still subject to the four-fold degeneracy in $\bpi_\e$ [@refsdal66; @gould94], @gould95 showed that these degeneracies could be partially or fully broken by measuring the very small difference in $t_\e$ as seen from the two vantage points, and this idea was then investigated in extensive simulations [@boutreux96; @gaudi97]. Hence, this goal of observing the peak of the event guided the 2014 [[*Spitzer*]{}]{} campaign. For many years it was believed that because of the four-fold degeneracy, parallax measurements would not be possible if the satellite observed only the rising or falling side of the event, but did not capture the peak. Coverage of the peak would in fact be required if one needed to derive independent point-lens parameters $(t_0,u_0,t_\e)$ from ground-based and space-based observations (as is necessary in @gould95 to break the degeneracy). However, @21event showed that the four-fold degeneracy can usually be broken by a combination of the so-called “Rich argument” and kinematic priors derived from a Galactic model. Once the problem of the four-fold degeneracy is removed, the requirements on the satellite data are drastically reduced. First, $t_{\e,\rm sat}$ can be regarded as “essentially known,” so that it is only necessary to determine two satellite parameters ($t_{0,\rm sat},u_{0,\rm sat}$) to measure the parallax. Of course, $t_\e$ is actually slightly different as seen from Earth and the satellite because they have a relative motion of $\sim 30\,\kms$ in the East direction. However, the resulting difference in $t_\e$ is directly determined by $\bpi_\e$, so while it is not strictly the case that $t_\e$ is irrelevant, it remains true that only two independent light curve parameters must be derived from the satellite light curve. Second, the source flux parameter $f_{s,\rm sat}$ for the satellite can be determined independently of the satellite light curve using a color-color relation derived from field stars combined with the measured color ($I-H$ or $V-I$) derived from the ground-based light curve. @21event obtained typical precisions for $f_{s,\rm sat}$ of 5% in the cases for which they had good $H$ or $V$ data. However, it remains necessary to determine $f_{b,{\rm sat}}$ from the light curve, which constitutes a third parameter that must be derived from the [*Spitzer*]{} data. Then, from simple parameter counting, it is in principle enough to measure three non-colinear points on the light curve to measure the parallax [@dong07], e.g., one point that is “known” to be at baseline and two others at different magnifications. In practice, more points are usually needed to have confidence in the measurement and to have checks against discrete degeneracies. However, it would be enough, for example, to track the falling part of the light curve from the time that the source exited the Einstein ring until it had dropped by 30% in magnification, i.e., approached baseline. For events that are well before peak as the [*Spitzer*]{} window ends, the situation is less straightforward because there would probably not be any baseline and the short duration of the observations might not yield any measurable change of slope. However, such events could be recovered by post-event baseline observations, either six months later (when the Bulge is not visible from Earth, so ordinary satellite parallax observations are not feasible) or the following year. Hence, $t_{\rm fin}$ could be considered as a date in the distant future rather than the end of the current 38-day observing window. Therefore, there are two different channels through which parallaxes can be measured with [[*Spitzer*]{}]{} for point lens events. First, [[*Spitzer*]{}]{} can observe just the peak of the light curve. Second, [[*Spitzer*]{}]{} can observe either the rising or the falling side of the event plus some measurement of the baseline. Finally, if the event has a binary lens, features from the binary may be used to measure the parallax. However, this situation is more complex since binary perturbations last long enough that they may not be fully captured by the [[*Spitzer*]{}]{} data and so the four-fold degeneracy may persist [@ob141050]. [*Spitzer*]{} Photometric Pipeline Issues {#sec:pipeline} ----------------------------------------- The feasibility of measuring $\bpi_\e$ from a given set of [*Spitzer*]{} observations obviously depends on the quality of the photometry that can be extracted from these observations. Remarkably, none of the wide range of publicly available [*Spitzer*]{}-specific photometry packages is well matched to the problem of time series of variable stars in crowded fields. As a result, the limits of what can be achieved from such photometry are not well understood. All the main elements required to solve this problem are at hand, but they have not so far been combined. First, the [*Spitzer*]{} pixel response function (PRF) is extremely well understood. That is, if a point source has a known flux and known position relative to the optical axis, then the response of all pixels can be predicted to much higher precision than is relevant for the relatively faint sources that are studied in microlensing experiments. The positions and fluxes of the great majority of sources in the microlensing fields are known to be constant on the timescales of the 38-day [*Spitzer*]{} observing window. Moreover, the locations of all field sources that are bright enough to be relevant are known from ground-based optical astrometry at much higher precision than is needed for [*Spitzer*]{} photometry, while the location of the microlensed source is typically known with even higher precision in this optically based frame. Even the approximate $3.6\,\mu$m fluxes (other than the microlensed source) are known from optical $V/I$ photometry and fairly robust local-field $V/I/3.6\mu$m color-color diagrams. Hence, a conceptually straightforward procedure would be to forward model the ensemble of $n$ images with one flux parameter for each non-microlensed source and $n$ flux parameters for the microlensed source. Intrinsically variable stars could be recognized as poor fits in this process and either ignored (if they were sufficiently far from the lensed source) or modeled with $n$ parameters instead of just one. We are working on such a pipeline, but since criteria for 2015 observations (beginning in June) are required several months in advance, we must assess likely [*Spitzer*]{} performance based on applying existing pipelines to 2014 [*Spitzer*]{} microlens data. These each contain some (but not all) of the advantages of the ideal pipeline outlined above. For example, the MOPEX pipeline fully incorporates the PRF but does not hold stellar positions constant, nor does it hold the flux of field stars constant. The well-known DoPhot [@dophot] pipeline can be applied to images formed by combining the six 30s dithered images at each epoch. It can hold stellar positions constant but does not incorporate any information about the PRF. We also applied a variant of the ISIS pipeline, which uses image subtraction to the same combined images. Although this pipeline normally outperforms DoPhot for ground-based microlensing data (with some exceptions), we find that the lack of PRF information generally affects ISIS more adversely than DoPhot. We conduct a purely empirical investigation, using 47 events from the 2014 [*Spitzer*]{} microlensing “pilot program” that have enough points to potentially construct a coherent light curve. We consider the photometry from MOPEX, DoPhot, and a preliminary version of our own pipeline. We create an [*optically-based*]{} effective [ *Spitzer*]{} (“$L$-band”) magnitude (since prior to obtaining [ *Spitzer*]{} data we have no independent knowledge of the true [ *Spitzer*]{} flux). This is defined $$L_\eff \equiv I - 0.93 A_I - 1.3 + 0.5\Theta(I - A_I - 17.2), \label{eqn:optspitz}$$ where $A_I$ is the extinction in $I$-band [@nataf13] and $\Theta$ is the Heaviside step function. We stress that no precise physical meaning should be attached to $L_\eff$. It is simply an approximate predictor of the [*Spitzer*]{} flux based on optical data. The $\Theta$ function divides all stars into two types: turnoff stars ($\Theta=1$) and low-luminosity giants ($\Theta=0$). The justification for this approximation is that significantly fainter (and redder) dwarfs generally will not enter our sample and significantly brighter (and redder) giants are very rare. Of course, by limiting ourselves to two classes of stars we are still ignoring evolution over the sub-giant branch. However, in the general case, it is not possible to make a finer distinction, particularly before a detailed investigation of an individual event has been made, as is almost always the case when one must make the decision about whether to monitor a particular event. Using this proxy, we find that it is usually not possible to obtain good photometry with existing software unless there is at least one point with $L_\eff<15.5$. We therefore use this criterion as our principal guideline for deciding whether parallaxes can be measured for particular events. This may appear too conservative in that there will almost certainly be photometry improvements by the time that the data are analyzed. On the other hand, when making decisions about [*Spitzer*]{} observations, one must use the simplified “assumption” that [*Spitzer*]{} will see the same brightness source star as it would if it were observing from Earth because the true magnification as seen from [*Spitzer*]{} is unknown. That is, the whole point of the experiment is that the [*Spitzer*]{} and Earth-based light curves will differ by an intrinsically unpredictable amount. In particular, the source could be less magnified as seen from [*Spitzer*]{} than from the ground. Thus, we adopt $L_\eff<15.5$ as a good balance between these two considerations. An additional consideration, given that the IRAC pixels are 1.2$^{\prime\prime}$, is that the target may be blended with other stars in the crowded field, which can affect the quality of the photometry. The severity of this blending depends both on the separation of the blend from the target and on their relative fluxes at $3.6\mu$m. While the separation of potential blends can generally be determined from existing, higher-resolution, ground-based data, the relative fluxes cannot. Furthermore, in part because of the problems with the photometric pipelines, we have not been able to determine exactly what criteria can be used to assess whether or not a given blend star will cause a problem for the photometry. Hence, while we are aware of this issue, it is not possible to account for it at the present time. Objective Criteria {#sec:objcrit} ================== ![Flowchart illustrating the process for objectively selected events. An event may be objectively selected either before or after the peak, but must meet all of the selection criteria for that category. \[fig:obj\]](flowchart_obj.pdf){width="90.00000%"} As discussed in Section \[sec:obj\], there is an extremely strong reason for choosing as many events as possible based on purely objective criteria: all planets (and planet sensitivity) from the entire event can be included in the sample. However, there is also a huge potential for wasted [*Spitzer*]{} observations if these criteria are not sufficiently restrictive. Hence, we have opted for a conservative approach. An important point to keep in mind is that for events in low-cadence fields, there is no major advantage to selecting the event objectively because such events have very little sensitivity to planets in the absence of follow-up data (Section \[sec:planet-sens\]). Their sensitivity will only be substantial if higher-cadence (usually ground-based) observations are triggered. If this recognition also triggers [*Spitzer*]{} observations (or rather, commitment to such observations) at the same time, then essentially no planet sensitivity is lost. Column 1 of Figure \[fig:lcs\] demonstrates that these low-cadence events are also extremely hard to predict. Another point to keep in mind is that it is substantially easier to predict the future course of events that have already peaked than those that are still rising (compare panels 1e and 2e to earlier panels in Figure \[fig:lcs\]), and hence to estimate accurately whether a successful parallax measurement can be made. This fact is especially important for events that have peaked before the [*Spitzer*]{} campaign has begun. For events that peak during the campaign, the probability of measuring $\bpi_\e$ can be substantially enhanced if [*Spitzer*]{} observations are made over peak, i.e., before such secure information about the event parameters is available. Guided by these considerations (and others related to subjective selection that are discussed below), J.C.Y. and A.G. developed some preliminary objective criteria, and then (independent of these criteria) each individually analyzed 242 events based on OGLE and MOA data obtained up through June 3, 2013, which is the analogous time to the first decision time in 2015. These events had been pre-selected based on very loose criteria from about 1000 events that had been found by these collaborations by this date. For each event, they decided whether it should be chosen for hypothetical [*Spitzer*]{} observations to begin three days later, and if so at what cadence. All disagreements were discussed and the final joint decisions were subsequently evaluated based on comparison to the full 2013 light curves. The agreed-upon sample contained all nine events that were selected by the preliminary objective criteria, and also many that were not for a total of 44 events. Based on this detailed analysis J.C.Y. and A.G. refined the objective criteria for selection and the objective cadence choice, both of which are listed immediately below, and also developed general guidelines to subjectively choose events, which are discussed in Section \[sec:subchoice\]. Figure \[fig:obj\] summarizes the process for objectively selecting events. In this scheme, events may be divided into two categories: events that have already peaked ($t=t_0+2$ days) and events “before” the peak ($t<t_0+2$ days). In order to be objectively selected, the event must meet all criteria for the appropriate category. We begin by discussing events that have peaked, because they are generally better understood (i.e. the model fits have converged and their future behavior is well-constrained). Events that have Already Peaked {#sec:pastpeak} ------------------------------- 1. [A1) $t_0 + 2\,{\rm days} <t_{j, \rm dec}$]{} 2. [A2) Either]{} 1. [a) in an OGLE field w/ cadence $\geq 10\,{\rm day}^{-1}$, or]{} 2. [b) in a “core” KMTNet field, or]{} 3. [c) $A_{\rm max}>3$ and in an OGLE field w/ cadence $= 3 \,{\rm day}^{-1}$]{} 3. [A3) $I_{\rm S}-2.5\log(A_{\rm max})<17$]{} 4. [A4) $A(t_{j, \rm next})-A(t_{\rm fin})>0.3$]{} 5. [A5) $L_{\rm S,eff} - 2.5\log[A(t_{j, \rm next})]<15.5$]{} 6. [A6) $L_{\rm S,eff} - 2.5\log[A(t_{j, \rm next})-A(t_{\rm fin})]<17$]{} Criterion (A1) is simply a practical definition of “post-peak”. Criterion (A2) selects for events that have significant planet sensitivity. For events in high-cadence fields this is essentially any event (provided it meets the other criteria) because planets can be discovered in these events far out into the wings and even at baseline. For other, somewhat lower-cadence events, the criterion demands $A_\max>3$ as a minimum indicator that the [*Spitzer*]{} parallax observations will be worthwhile. However, for events in low-cadence survey fields, it is essential to carry out ground-based follow-up observations to gain substantial planet sensitivity. Hence, for events in low-cadence fields, there is no triggering of [*Spitzer*]{} observations via objective criteria, and it will therefore be necessary to subjectively select these events prior to peak (which may be before the [*Spitzer*]{} observing window). Criterion (A3) demands that the peak flux from the magnified source (not including any blended light) be $I<17$. It increases the likelihood that the ground-based photometric precision will enable good planet sensitivity. To date, the overwhelming majority of planetary microlensing events have peaked $I<17$. In addition, this criterion is important not only to secure an accurate fit to the light curve but to permit application of color-color relations to determine the [*Spitzer*]{} source flux, both of which affect the final parallax measurement. Criterion (A4) is driven by the fact that parallax measurements require a well-measured change in magnification as seen from [*Spitzer*]{}. In practice, this means both measuring a flux change and independently determining the [*Spitzer*]{} source flux. For observations that begin past peak, it is impossible to reliably fit the [*Spitzer*]{} light curve for the source flux, so it must be determined from color-color relations, which can be done reliably to about 5%. Hence, we require at least a 0.3 change in magnification (i.e., 6-fold larger than 5%) based on our estimate of what will be required for reliable parallax measurements (Section \[sec:spitzrole\]). This criterion is equivalent to demanding that the source is still in the Einstein ring (at the time of the next possible observation) for the case that the final possible observation is well outside the Einstein ring. Criterion (A5) derives directly from the difficulty of extracting [*Spitzer*]{} photometry unless at least one point is brighter than $L_\eff<15.5$ (Section \[sec:pipeline\]). Criterion (A6) demands a minimal flux change in [*Spitzer*]{} flux. It will be automatically satisfied for the great majority of stars that satisfy criterion (A5) and is included to guard against including (at least automatically) events that are not predicted to change much over the remainder of the observations. As noted in the justification for criterion (A3), measurable flux changes are crucial for parallax measurement. Criterion (A6) ensures that not only the magnification changes, but the flux itself changes by a significant amount. Rising Events {#sec:rising} ------------- 1. [B1) $t_0 > t_{\rm dec}-2\,$days]{} 2. [B2) in OGLE field w/ cadence $\geq 3\,{\rm day}^{-1}$ or in a “core” KMTNet field]{} 3. [B3) $I_{\rm now} < 17.5$]{} 4. [B4) $I_{\rm base}-I_{\rm now} > 0.3$]{} 5. [B5) $L_{\rm eff, dwarf, now} = I_{\rm now}-0.93A_I-0.8 < 15.5$]{} Criterion (B1) is just the practical definition of a rising event[^5] (i.e., the complement of criterion (A1). Criterion (B2) restricts this entire class of objective selection to fields with moderate-to-high cadence, $\geq 3\,{\rm day}^{-1}$, to ensure that the selected events have good sensitivity to planets over the entire light curve. The primary concern is that once an event has met the full set of rather restrictive criteria, it will be close to the peak and a large fraction of the event will be over. As discussed in Section \[sec:planet-sens\], without substantial follow-up data, low-cadence events have very little practical planet sensitivity. Hence, it is not worthwhile to observe them with [[*Spitzer*]{}]{} to try to measure a parallax because they will add very little to the final analysis of the planet occurrence rate as a function of Galactic distance. We note that most high-quality events that fail this criterion can be selected subjectively (Section \[sec:subchoice\]) well before peak with the specific goal of triggering additional follow-up observations to improve the planet sensitivity. The remaining three criteria make no reference to a light curve model and instead rely on purely empirical observables. Again, this is because experience shows that such models are not reliable for pre-peak light curves. All three make reference to “$I_{\rm now}$” which is the last measured OGLE point. Criterion (B3) assures that the event will be bright enough for accurate measurements (necessary for both planet sensitivity and parallax). Criterion (B4) assures that even if the source is not blended, the event has risen at least 32% above baseline, i.e., the source must be (nearly) inside the Einstein ring. The combination of (B3) and (B4) ensure that even if the source turns out to be heavily blended, then at least the magnified flux will change significantly compared to its present value relative to a future baseline measurement, which will enable a measurement of the parallax. Finally, criterion (B5) attempts to assure that there will be at least one point above the photometric threshold for measuring a [*Spitzer*]{} light curve (Section \[sec:pipeline\]). This is only “attempted” (rather than guaranteed) since the event may not be as magnified from [*Spitzer*]{}’s vantage as from Earth. Because the source magnitude is most likely not known at the time of this algorithmic selection, we conservatively assume it is a dwarf (i.e., relatively blue and so fainter as seen by [*Spitzer*]{} for fixed $I$-band brightness). Objectively Determined Cadences ------------------------------- As we have discussed, events that are selected objectively must have objectively determined cadences. In practice, cadences are actually defined by “priorities”, where priority $n$ means that the event is observed during $1/n$ of the cycles through the microlensing fields (Section \[sec:spitzcyc\]). However, we state these here in terms of cadences, since there is a clear-cut conversion from one to the other once the target sample is selected. We designate the following algorithm for setting the observation cadence for Week $j$ 1. [C1) Default cadence: $1\,{\rm day}^{-1}$]{} 2. [C2) $2\,{\rm day}^{-1}$ provided that all of the following are true]{} 1. [This is first [*Spitzer*]{} observation period of the event, and]{} 2. [$t_{j, \rm next}>t_0$, and]{} 3. [$A(t_{j, \rm next})<1.35$]{} 3. [C3) $2\,{\rm day}^{-1}$ for events beginning the when less than two full weeks remain]{} 4. [C4) Stop observing the event, provided that all of the following are true]{} 1. [At least 2 weeks of objectively-determined observations are complete, and]{} 2. [$t_{j, \rm next} > t_0+t_\e$]{} 3. [$t_{j+1, \rm next} > t_0 + 2 t_\e$]{} 5. [C5) Stop observing the event, if either]{} 1. [The parallax of an event has been measured from [*Spitzer*]{} data already collected, or]{} 2. [The [*Spitzer*]{} light curve has already reached baseline (so no more parallax information could be extracted from additional observations)]{} Criterion (C1) has been shown by @21event to be generally adequate to make parallax measurements. However, for events that are leaving the Einstein ring as the [*Spitzer*]{} observations begin (C2) or for which there is only a short rising observational sequence at the end of the [*Spitzer*]{} window (C3), the cadence is doubled. These events have significantly more restricted light curve coverage than the typical events analyzed by @21event and therefore require higher cadence to obtain more points (so higher signal-to-noise ratio) while the event is still significantly magnified. (C4) imposes a reasonably conservative criterion for halting observations. In principle, this may cause [*Spitzer*]{} to miss a key portion of the light curve because it can in principle peak either earlier or later as seen from [*Spitzer*]{} than the Earth-based light curve would predict. However, because bulge lenses have small microlens parallaxes, the light curve peaks as seen from Earth and [*Spitzer*]{} are very close in time. On the other hand, for disk lenses, the [*Spitzer*]{} peak is usually earlier (or not much later) than from the ground because these disk lenses tend to move in the direction of Galactic rotation, i.e., about $30^\circ$ East of North, whereas [*Spitzer*]{} is roughly due West of Earth. See Figure 2 of @21event. Finally, (C5) provides more specific conditions for halting observations if the [*Spitzer*]{} data can be reduced and analyzed in real-time. Binary Events {#sec:binaries} ------------- For completeness, we also specify the objective selection criteria for binary events. Unlike planetary events, binary events show prominent anomalies that modify the single-lens light curve significantly. Therefore, most binary events can be recognized in advance, and the inability to model them with single-lens light curves makes all selection criteria based on single-lens modeling (i.e., Sections \[sec:pastpeak\] and \[sec:rising\]) fail in most cases. As a consequence, in the 2014 pilot program we subjectively selected binary events, such as OGLE-2014-BLG-1050 [@ob141050], for observations because their nature as binary events had been confirmed. Therefore, in order to enable statistical studies of stellar binaries one has to have objective selection criteria. After reviewing those binary events from the 2014 season, we decide to use the following criteria and cadence: 1. [Begin [[*Spitzer*]{}]{} observations if]{} 1. [The ground-based light curve is in a U-shaped trough,]{} and 2. [$L_{\rm eff, trough} < 16$.]{} 2. [End [[*Spitzer*]{}]{} observations]{} 1. [Either:]{} 1. [One full week after the [*Spitzer*]{} light curve exits the caustic,]{} or 2. [Both:]{} 1. [One full week has passed since the ground-based light curve exits the caustic,]{} and 2. [The [*Spitzer*]{} light curve is shown never to have entered the caustic.]{} 3. [Default cadence of 1 day$^{-1}$.]{} Guidelines for Subjectively Chosen Events {#sec:subchoice} ========================================= ![Flowchart illustrating the process for subjectively selecting an event. By definition, this is less quantitative than the process for objectively selecting an event (see Figure \[fig:obj\]), but the underlying considerations are the same in both cases, namely: “Does the event have good planet sensitivity?” and “Is this event likely to yield a [[*Spitzer*]{}]{} parallax?”.\[fig:sub\]](flowchart_sub.pdf){width="90.00000%"} We have already discussed in Section \[sec:sub\] the various reasons one might want to subjectively select an event and how that might affect the type of subjective selection (Table \[tab:selection\]). Figure \[fig:sub\] summarizes the decision process that can lead to subjective selection of an event. The most straightforward type of subjective selection is “Subjective, immediate” in which an event is immediately selected for observations, committed to, and has its cadence specified. Such a decision may be made at any time, including before the start of [[*Spitzer*]{}]{} observations. There are two primary types of events that might be selected this way. First, events that are discovered and well understood before $t_{j, \rm dec}$ but might not meet the objective criteria. Committing to observations immediately allows more of the planet sensitivity of that event to be captured. The second type of event is a well-understood, low-cadence event that requires follow-up observations in order to capture significant planet sensitivity. However, we expect that the great majority of events will be selected under the categories “Subjective, conditional” or “Subjective, secret” because at time $t_{j, \rm dec}$ there is no way to distinguish between events that will reach moderate or higher magnification ($A_\max>3$) sometime before the start of the next cycle of [*Spitzer*]{} observations, $t_{j+1, \rm next}$ (i.e., $t_{j, \rm dec} + 10\,{\rm day}$), and events that will turn over at low magnification and low flux levels (e.g. compare the two fits in panel 1b of Figure \[fig:lcs\]). Moreover, even among those that are likely to achieve satisfactory magnification, it cannot be decided automatically whether suitable follow-up resources can be allocated to a specific event, given a wide variety of operational constraints. And finally, it may be impossible to determine which of these events will get to high or very high magnification based on routine survey data. However, additional investigation, including additional follow-up data and/or color information from survey and/or follow-up observations may resolve this question. As we have already discussed above, subjective decisions may be made before or after $t_{\rm first}$, but they must specify the cadence (or cadence algorithm) at the time of commitment. To be robust, any algorithm must be based on readily available data, such as the $I$-band light curve and the $I$-band field extinction. Given the above factors, together with the fact that planet sensitivity is heavily skewed toward higher-magnification events, it is inevitable that [*Spitzer*]{} observations will be triggered for a large number of events of uncertain prospects, and therefore that the majority of these must be terminated promptly after the event fails to rise to the level that permits significant planet sensitivity and also permits its parallax to be measured. Otherwise, a substantial fraction of observing time will be wasted on useless events. Hence, the principal issue will be deciding how to frame this failure in either of the two cases (“Subjective, conditional” or “Subjective, secret”). This will have to be done on a case-by-case basis at the time the events are subjectively chosen, since the uncertain nature of such events makes it impossible to develop strict guidelines in advance. In the first case, this framework must provide a guide to terminating observations (e.g., “stop observations next week if event does not reach $I<16.2$ by the next decision point, $t_{j+1,\rm dec}$”). In the second, it must provide a guide as to whether or not the team should commit to observations of the event. To the extent possible, the criteria governing such terminations (or continued observations) should be framed in terms of post-peak observables because these are more robust. They should then be formulated as proxies for the criteria (A1)–(A6), but with $t_{j, \rm next}\rightarrow t_0$. However, this implicitly assumes that $t_{j, \rm dec}<t_0<t_{j+1,\rm dec}$ (essentially criterion (A1)). It will in general be necessary to specify what should be done if the event does not ultimately satisfy that criterion. For example, it could be stated that in this case, the observations continue at the same cadence and the decision is made during the next week. Objective Allocation of Remaining Observing Time {#sec:realloc} ================================================ General Considerations ---------------------- An important aim of the [*Spitzer*]{} observations, as outlined in the proposal [@gouldcareyyee14] is to detect and characterize planets from [*Spitzer*]{} observations themselves. Because planet sensitivity scales with magnification, this can best be done by monitoring higher-magnification events more intensively from [*Spitzer*]{}, in particular events that are at higher magnification as seen by [*Spitzer*]{}. Hence, after the fundamental goal is met by allocating enough observation time to reliably measure parallaxes (as outlined in Section \[sec:objcrit\] and \[sec:subchoice\]) and also to obtaining parallaxes of microlens binaries (which was also part of the proposal but is not the subject of the present study), the remaining time should be allocated to this purpose. Because these additional planet-finding observations involve allocation of additional time to the [*same*]{} set of events that are the object of microlens parallax measurements, it is important to isolate the decisions about this allocation from the prospect of improving parallax measurements. Otherwise, events with known planets could receive additional measurements aimed at measuring parallax, making them more likely to have good parallax measurements relative to those that failed to show any planets. And since only events with good parallax measurements can be included in the sample to measure the Galactic distribution of planets, we must eliminate the potential for bias. At the same time, it is difficult to develop completely objective criteria for these allocations because of the wide range of the possible quantities of available time and wide range of the possible quality of events to which they might be applied. In addition, the amount of additional time will depend on the precise number of events being monitored. Finally, the targets for a given week will have a wide range of magnifications, whose distribution cannot be predicted in advance. Practical Execution ------------------- We propose the following algorithm to effectively separate these decisions while leaving adequate freedom to respond to potential planet sensitivity by obtaining additional [*Spitzer*]{} observations. Recall from Section \[sec:spitzcyc\] that the most efficient way to observe these events is to cycle through the Bulge West-to-East and East-to-West. The priority of an event sets how often an event will be observed. Specifically, priority $n=1$ is observed every cycle whereas priority $n=8$ is observed every eighth cycle. The exact number of cycles depends on the the number of events, their priorities, and the amount of time available. All events that have been chosen for [*Spitzer*]{} observations, [*except newly selected subjective events*]{} are rank ordered according to the $2\,\sigma$ lower limit of their highest magnification ($A_{j,\max}$) in the observing window \[$t_{j,\rm next},t_{j+1,\rm next}$\]. These are then assigned priorities that map monotonically to this ranking. The break points in this mapping are decided manually. For example, $A_{j,\max}>20\Rightarrow n=1$, $10<A_{j,\max}\leq 20\Rightarrow n=2$, $5<A_{j,\max}\leq 10\Rightarrow n=3$, $3<A_{j,\max}\leq 5\Rightarrow n=4$. The only other rule is that the priorities and the number of events in each category result in a total number of observations that equals the number of observations available. Hence, there are likely many choices of break points and priorities that fulfill these two criteria (monotonic mapping and total observations) for any given rank-ordered set of events and amount of observing time. The final choice, which will set the number of cycles and their duration, is at the discretion of the [[*Spitzer*]{}]{} team. While the manual decision on break points might seem to allow skewing (conscious or unconscious) of the strength of different parallax measurements, in fact this is virtually impossible. The events whose parallax is poorly measured with existing data, and therefore might require “saving” with additional observations, will be at low magnification and hence cannot be helped by any manual decision that is constrained by the monotonic mapping described above. Here, the default meaning of “highest magnification” is highest as predicted for observations from the ground. However, it may be possible to download and process [*Spitzer*]{} data sufficiently quickly to make predictions about the course of the event as seen from [*Spitzer*]{}. In this case, the [*Spitzer*]{} magnifications should take precedence. Why do we exclude the subjectively chosen events whose observations are just starting? There are two reasons. First, the future course of these events is often very poorly understood, so the $2\,\sigma$ lower limit on the magnification is likely to be very low and hence unlikely to trigger the additional observations being considered here. More fundamentally, the cadence of these first-week observations is subjectively decided, so no objective procedure is required to allocate additional observations to planet hunting in these events if the team decides that is necessary. The cadence (or algorithm for determining the cadence) of [ *Spitzer*]{} observations must be specified at the time of the commitment to observe a subjectively selected event. However, in practice these cadences will almost always be set at $t_{\rm sel}$, since even for “subjective, secret” events $t_{\rm com}$ will generally happen before the next decision point when the cadence could be changed ($t_{j+1,\rm dec}$). In subsequent weeks, any change in the cadence of these events must be through the process described above. Conclusions {#sec:conclude} =========== We have outlined an approach for maximizing the planet sensitivity of space-based microlens parallax surveys, for the particular case that the satellite targets are chosen based on ground-based identification of events. This applies to current observations by [*Spitzer*]{} and any narrow field-of-view, targeted observations from space[^6]. The basic principles are: First, objective criteria are quite easy to establish for events that have already peaked because their fits are well constrained. Because the criteria are objective, the entire time span of those events, with respect to both planets and planet sensitivity, can be included in the analysis. This includes, in particular, all of the time before peak, which lies before the onset of [*Spitzer*]{} observations and also before the event was even recognized as microlensing. Second, it is also possible to establish objective criteria for a subset of pre-peak events. However, because these events are pre-peak, their model fits may not be reliable, so it is necessary to define these criteria in terms of observables. Third, for objectively chosen events, cadences must also be determined by objective criteria. Fourth, for those events in low-cadence survey fields, it is less important to define objective criteria, because the events have low sensitivity to planets unless additional follow-up observations are obtained. Fifth, the remaining events, including those in low-cadence fields, can be chosen subjectively, but the full cadence (or prescription for determining the cadence) must be specified at the time that they are selected. Subjective selection can take several forms, but the most important aspect is when a commitment is made (and announced) to observe the event. It is this date that determines what planet sensitivity and planet detections are included in the analysis. Sixth, in the case of events that were previously chosen on subjective grounds and that subsequently meet the objective criteria, their objective status must take precedence in evaluating the event as part of the sample. This assumes that a parallax measurement proved possible based only on the (more restricted subset of) objectively required [*Spitzer*]{} observations. If not, and if the full set of [*Spitzer*]{} observations yields a parallax measurement, then they revert to subjective status. This paper constitutes a public announcement of our objective criteria and procedures. If there are any updates to these, they will be posted on arXiv as a revision to or update of this paper. Work by JCY, AG, and SC was supported by JPL grant 1500811. Work by JCY was performed under contract with the California Institute of Technology (Caltech)/Jet Propulsion Laboratory (JPL) funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. 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For example, if an event does not show evidence of a planet, it could be excluded from selection for parallax measurements because it is “uninteresting.” Hence, the absence of planets has similar potential to create bias in the final sample but in the opposite direction as the presence of planets. [^2]: The distinction between deciding to observe a target and committing to a target is discussed in Section \[sec:sub\]. [^3]: In other planet-finding contexts, e.g. radial velocities, “planet sensitivity” is often referred to as “detection efficiency.” [^4]: In practice, the spread in targets over a few degrees in the Bulge allows us to stretch the time-frame of the campaign to 40 days. [^5]: Technically, this also includes events $<2\,$days past peak. The reason for this choice is that it can be difficult to be confident that an event has indeed peaked unless there are data after the peak to demonstrate this explicitly. [^6]: It does not apply to a microlens survey by [*Kepler*]{} (K2) because the field will be a large, pre-selected region rather than having the observations targeted at specific, known microlensing events.